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UNIVERSIDADE ESTADUAL DECAMPINAS

Instituto de Matemática, Estatística eComputação Científica

DOUGLAS FINAMORE

Entropy of Pseudogroups and Foliations

Entropia de Pseudogrupos e Folheações

Campinas2018

Douglas Finamore

Entropy of Pseudogroups and Foliations

Entropia de Pseudogrupos e Folheações

Tese apresentada ao Instituto de Matemática,Estatística e Computação Científica da Univer-sidade Estadual de Campinas como parte dosrequisitos exigidos para a obtenção do título deMestre em Matemática.

Thesis presented to the Institute of Mathemat-ics, Statistics and Scientific Computing of theUniversity of Campinas in partial fulfillment ofthe requirements for the degree of Master inMathematics.

Supervisor: Gabriel Ponce

Este exemplar corresponde à versão fi-nal da Tese defendida pelo aluno Dou-glas Finamore e orientada pelo Prof. Dr.Gabriel Ponce.

Campinas2018

A ficha catalográfica será fornecida pela biblioteca

A folha de aprovação será fornecida pela Secretaria de Pós-Graduação

To Elizete, for encouraging me on everyadventure, specially this one

“Deviner avant de démontrer! Ai-je besoin de rappeler quec’est ainsi que se sont faites toutes les découvertes importantes”

— Henri Poincaré, Le Valeur de la Science

Acknowledgements

Foremost, the completion of this thesis would not have been possible without thesupport and nurturing of my advisor, Dr. Gabriel Ponce, who were always available whenneeded, with his immense patience, understanding and knowledge. I’d like to thank you forall the insightful suggestions and feedback, ever steering me in the right direction, and mostimportantly, for giving me enough freedom to create a work of my own taste. I couldn’t havewished for a better advisor.

Besides, I extend my most sincere appreciation to Prof. Dr. Steven Hurder, whoseemails were a great source of references, clarification and insight, and also for providing me withso many articles and interesting reading material. Your work is an inspiration to me.

To my mother, for the advices and comforting words, and for always supporting myme, even though she never quite understood what mathematicians really do (it is my hope thepresent work will help clarify the matter a bit). Thank you, your role in getting me here cannot be overestimated.

Finally, to my ever moderately wise friends, who no matter how far might have been,always found a way to stand by me and make themselves present. Thank you, I miss you allterribly.

This work was funded by the Brazilian National Council of Scientific and Technological Devel-opment – CNPq.

AbstractEntropy plays a important role in the theory of Dynamical Systems by providing means tomeasure a given dynamics’ complexity. In this work we define concepts of entropy applicable topseudogroups and foliations, aiming to quantify the complexity of such systems. For foliations, ageometric characterisation of positive entropy is given by proving that the occurrence of positiveentropy is equivalent to the existence of resilient leaves. Finally, we provide sufficient conditionsunder which a foliation can be perturbed in order to acquire positive entropy.

Keywords: Dynamical systems, entropy, foliations, holonomy, pseudogroups.

ResumoA entropia é usada na teoria de Sistemas Dinâmicos como um indicativo da complexidade dadinâmica a que diz respeito. Neste trabalho definimos conceitos de entropias para pseudogrupose folheações, com o intuito de quantificar a complexidade desses sistemas. Uma caracterizaçãogeométrica da entropia de folheações é fornecida, demonstrando-se que a ocorrência de entropiapositiva é equivalente a existência de folhas resilientes. Por fim, estipulamos algumas condiçõessuficientes a fim de que uma folheação possa ser perturbada de modo a adquirir entropia positiva.

Palavras-chave: Entropia, folheações, holonomia, pseudogrupos, sistemas dinâmicos.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1 THE FOUNDATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1 Pseudogroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 Foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.1 Definition and first examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.2 Constructing foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2.2.1 Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.2.2.2 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.2.3 Turbulisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.3 The space of leaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2.4 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.2.5 Resiliency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.2.6 Foliated bundles and the suspension of representations . . . . . . . . . . . . . . 42

2 ENTROPY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.1 The growth and entropy of a pseudogroup . . . . . . . . . . . . . . . . . . 472.1.1 Orbit and expansion growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.1.2 The topological entropy of a pseudogroup . . . . . . . . . . . . . . . . . . . . . 572.1.2.1 The motivation from discrete dynamics . . . . . . . . . . . . . . . . . . . . . . . . 572.1.2.2 The pseudogroup case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.1.3 Pseudo-orbits and topological entropy . . . . . . . . . . . . . . . . . . . . . . . 662.1.4 A topological entropy for pseudogroups acting on noncompact sets . . . . . . . . 772.2 The geometric entropy of a foliation . . . . . . . . . . . . . . . . . . . . . 802.2.1 Pseudoleaves and geometric entropy . . . . . . . . . . . . . . . . . . . . . . . . 93

3 ENTROPY AND DYNAMICS OF C1 FOLIATIONS . . . . . . . . . . . . 993.1 The framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.2 The holonomy groupoid and foliation cocycles . . . . . . . . . . . . . . . . 1023.3 Ping-Pong games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.4 Foliated geodesic flow and transverse hyperbolicity . . . . . . . . . . . . . 1053.5 Flow invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193.5.1 Hyperbolic measures and the existence of resilient leaves . . . . . . . . . . . . . 1233.5.2 Positive entropy and the existence of ping-pong games . . . . . . . . . . . . . . 126

3.6 A continuous counterexample . . . . . . . . . . . . . . . . . . . . . . . . . 146

4 GENERATING ENTROPY . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.1 The Epstein topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.2 Recurrent leaves and families of jointly splitting charts . . . . . . . . . . . 1504.3 Perturbing foliations and generating entropy . . . . . . . . . . . . . . . . . 1524.3.1 Water slide functions and the subordinated partition . . . . . . . . . . . . . . . 1524.3.2 The perturbation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.4 On the density of foliations with positive geometric entropy . . . . . . . . 164

5 OPEN PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.1 Topology and geometry of foliations . . . . . . . . . . . . . . . . . . . . . . 1675.2 Entropy of pseudogroups and foliations . . . . . . . . . . . . . . . . . . . . 169

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

11

Introduction

“Only entropy comes easy”— Anton Chekhov

Entropy is one of the several hallmarks of chaos in classical dynamical systemstheory. The concept originated in Physics and made its way to Information Theory through theastonishing work of Claude Shannon [49] (where he also popularises the usage of the term “bit”for a unit of information). The name “entropy” comes from the fact the formulae for explicitcalculating this quantity is completely analogue to the formulae for the Gibbs-entropy of athermodynamical system, in Statistical Thermodynamics. Shannon’s entropy is a measure of astates impredictability, or in some contexts, of the noise involved in a process of transmittinginformation.

It was motivated by the work of Shannon that A. N. Kolmogorov and his doctoralstudent Y. G Sinai developed, in a series of articles [37, 38, 51, 50] (all of which can be foundtranslated at [34]), the concept of measure-entropy for measure-preserving dynamical systems.It is an invariant for measure-preserving dynamical systems, and it is a quantitative measure ofthe dynamics’s instability.

Later on, Adler, Konheim and McAndrew defined, by adapting the ideas of Kol-mogorov and Sinai, an entropy for topological spaces [1]. Their definition worked for compactspaces and used finite coverings instead of partitions. One of the most important results regardingentropy in dynamical systems is the famous variational principle, proved by E. Dinaburg in[17]. It states that, in a compact space, the topological entropy of a dynamical system is thesupremum of its measure-theoretic entropies over all the measures generating the topology. Thetopological entropy was then expanded by Bowen in [8] to general (noncompact) topologicalspaces, using a totally new approach inspired by the concept of Hausdorff dimension of a set.Dinaburg and Bowen’s work helped clarify the meaning of topological entropy, pointing to itsinterpretation as a measurement of the distinctiveness of orbit segments. It is an exponetialasymptotic rate of information necessary to distinguished distinct orbits: the greater the entropy,the more information is needed, that is, the harder is to make this distinction; and thus the

Introduction 12

more “complicated” is the system’s dynamics. In isometries, for instance, the orbits are alwaysat same distance from one another, thus the entropy vanishes.

The most well known notion of deterministic chaos is due to Devaney, who defined achaotic dynamical system as one which is sensitive to initial conditions, topologically transitiveand where the orbit of every periodic point is dense. While entropy does not measure chaositself, but rather complexity, its presence more often than not indicates that a system is chaotic,in a way or another. We know, for instance, that positive entropy implies chaos in the Li-Yorkesense [6]; as well as some counterexamples for the converse [52, 59]. There are also the so calleddistributional chaos: DC1, DC2 and DC3 [2]. While DC1 and DC2 imply chaos in the Li-Yorkesense, the same is not true for last one. We know DC2 is implied by positive topological entropy[19], while definitions of chaos have been suggested which are equivalent to positive entropy,and stronger than Devaney’s notion of chaos [41]. In any case, systems with positive entropyhave often intricate and complex dynamics, and form thus an extremely interesting object ofstudy within Dynamical Systems.

Here we investigate different types of entropy in objects that are, in some sense,generalised dynamical systems: foliations and pseudogroups of transformations. This might bea bit difficult to grasp at first, since dynamics is a rather extensive term, but we offer someinsight. In the pseudogroup case things are clearer, as pseudogroups are basically groupoidswith a couple of other properties. In a few words, a discrete-time classical dynamical system is apair consisting of a space X endowed with some structure and the action of the cyclic group(or semigroup) generated by a map f : X → X preserving the mentioned structure. As cyclicgroups are, in particular, groupoids, we see immediately how a pseudogroup acting on X givesrise to a generalised discrete dynamical system.

Foliations are certainly trickier. A foliation F of manifold M is a partition of thismanifold in submanifolds all of the same dimension. Intuitively, the study of the dynamics of Fis concerned with what happens as one follows paths on the leaves, which is what the manifoldsforming the partition F are called. Some authors, as Camacho and Neto [11], have suggested thatfoliation theory should be seen as a geometric theory for partial differential equations, much inthe same way the theory of flows can be thought of as a geometry theory for ordinary differentialequations. The leaves are, thus, generalised trajectories (in fact, nonsingular flows are dimensionone foliations). Besides, there are foliations which are not defined by PDE’s, and in this sensefoliations are more general dynamics systems. More than that, foliations and pseudogroups areclosely related, as every foliation comes with its own pseudogroup: the holonomy pseudogroup.It carries a lot of information about its foliation, and has its own generalised dynamics in theabove sense, which can also be thought of as the dynamics of F . Finally, when M comes withRiemannian metric, F induces a particular flow, called the foliated geodesic flow, and as we

Introduction 13

shall see in Chapter 3, with the right techniques much can be learned about the behaviour of Ffrom studying this flow. The investigation of all these different dynamical systems, generalisedor not, is usually referred to as “foliation dynamics”.

Chapter 1 gives the basics on pseudogroup and foliation theory, needed for thesubsequent chapters. In Chapter 2 we start with the main subject of the present work, bydefining several entropies on different systems. Results are given concerning their properties andhow they may be calculated. Chapters 3 and 4 focus on foliations and give some results relatingtheir geometry and general complexity, as well as some initial results concerning the question ofhow common is positive entropy among foliations. We finish by exhibiting some open problemsin Chapter 5. Throughout this text we assume the reader is familiar with some basic aspects ofErgodic Theory, as well as the theories of Dynamical Systems and Riemannian Geometry.

14

Chapter 1

The Foundations

We begin by defining the basic objects of our study, namely pseudogroups andfoliations, and in doing so we also set the notation to be used throughout the text.

1.1 PseudogroupsOne of the main objects of study in this work, as well as the most important tool in

understanding the dynamics of foliations, is a pseudogroup of transformations. Intuitively, onemay think of a pseudogroup as a groupoid of transformations with the additional property thattransformations can be glued together when they coincide in their domain’s intersection.

Definition 1.1.1 (Pseudogroup). Given a topological space X, denote by Homeo(X) the setof all homeomorphisms h : Dh → Rh where Dh and Rh are open subsets of X. A pseudogroupacting on X is a family G ⊂ Homeo(X) satisfying

(i) g h ∈ G for every g and h in G for which the composition is defined;

(ii) g−1 ∈ G whenever g ∈ G;

(iii) g|U ∈ G for every g ∈ G and U ⊂ Dg open;

(iv) if g ∈ Homeo(X) and its domain Dg has an open cover U such that g|U ∈ G for everyU ∈ U , then g ∈ G.

(v)⋃g∈G

Dg = X

In other words, the family G is closed under composition, inversion, restriction to open subdo-mains and under what we call an amalgamation, addressed in item (iv): we could rephrase it

Chapter 1. The Foundations 15

by asking that whenever two elements g, h of G coincide in the intersection Dg ∩Dh of theirdomains, then their amalgamation

g ∪ h : Dg ∪Dh −→ Rg ∪Rh

x 7−→

g(x) if x ∈ Dg,

h(x) if x ∈ Dh.

is also an element of G. In light of the first four conditions, the last condition on the domains isequivalent to asking that the identity idX belongs to G. When g = g1 g2 is a composition, itsdomain is always considered the largest one in which it is defined, namely Dg = g−1

2 (Dg1).

A subfamily G ′ of G which is a pseudogroup on its own is called a subpseudogroup ofG. Given a family Γ ∈ Homeo(X) satisfying

⋃g∈ΓDg ∪Rg = X, (1.1)

the pseudogroup generated by Γ is defined as the intersection of all the pseudogroups containingΓ, that is, the smallest pseudogroup containing Γ. It always exists as Homeo(X) is a pseudogroupitself, and it is denoted by G(Γ). An element h of Homeo(X) belongs to G(Γ) if and only if forevery x ∈ Dh there is an open neighbourhood U ⊂ Dh of x such that h|U is a finite composition

h|U = ge11 ge22 · · · genn |U (1.2)

where gi ∈ Γ and ei ∈ −1, 1. Remark that due to (i), (ii) and (iv) in 1.1.1, condition 1.1 isequivalent to asking that idX ∈ Γ.

When Γ = f1, ..., fn is finite we say G(Γ) is finitely generated and write G(f1, ..., fn)instead of G(Γ) for the generated pseudogroup. When Γ contains the identity and satisfiesΓ−1 = g−1; g ∈ Γ ⊂ Γ we say the generating set Γ is symmetric. When g = hi1 · · · himwe say g has length m and write |g| = m. Of course, the length of element depends on thegenerating set chosen.

Definition 1.1.2 (Regular Pseudogroup). Let Γ be a finite symmetric generating set for thepseudogroup G on X. If every element g 6= idx of Γ have a relatively compact domain containedin an open set U to which g is extensible, that is, if Dg ⊂ U is compact and g is extensible toan homeomorphism g : U → g(U) in G, then Γ is said to be a regular set of generators, and G aregular pseudogroup.

Note that every composition f = gi1 · · · gin of elements from a regular generating set wouldalso have a relatively compact domain and a continuous extension f to the compact closureof that domain. Basically, the pseudogroup only acts on the compact Ug∈ΓDg. This means we


can ignore everything but a compact subset of X when dealing with regular pseudogroups, thatis, we lose no generality in assuming X itself to be compact. Regular pseudogroups are alsoreferred to as good pseudogroups by some authors [56], while sometimes the pair (G,Γ) is saidto be a regular pair [12]. Remark that any subpseudogroup of a regular pseudogroup is regularas well. More than that, since any composition of elements in Γ retains the extension property,any finite generator for a subpseudogroup is a regular generator. Some examples are in order:

Example 1.1.3. (i) The sets G(idX) and Homeo(X) are the smallest and the greatest pseu-dogroup on X, respectively. The set of all global homeomorphisms of a space X onto itselfform a pseudogroup as well. If X is compact, since intX = X, then both G(idX) and thepseudogroup of all homeomorphisms are regular.

(ii) If fi : X → X are local homeomorphisms, i.e., continuous surjective functions such thatevery point x has a neighbourhood U for which fi|U : U → f(U) is a homeomorphism,then the set of all restrictions fi|U generate a pseudogroup. If X is compact and regular,this generating set can be chosen to be finite and symmetric and satisfying the regularitycondition.

(iii) If X is a Cr-manifold, (r = 1, 2, ...,∞, ω), and s ≤ r then the family Diffs(S) of allCs-diffeomorphisms between open sets of X is a pseudogroup. If X is orientable, the setDiffs+(S) of all orientation-preserving Cs-diffeomorphisms is a subpseudogroup of Diffs(S).

(iv) Given a metric space (X, d), the family of all the local isometries of X generate a pseu-dogroup Iso(X). We do not require an element f of Iso(X) to map its domain Df

isometrically onto its range Rf , the only condition is that every point x ∈ Df admits aneighbourhood U such that f |U : U → f(U) is an isometry. Iso(X) is a subpseudogroup ofanother pseudogroup on X: the pseudogroup Lip(X) generated by all the locally Lipschitzmaps f ∈ Homeo(X). If f ∈ Lip(X) then for every x ∈ Df there is a neighbourhood Uand a constant c ≥ 1 such that for every z, y ∈ U :

1c· d(f(y), f(z)) ≤ d(y, z) ≤ c · d(f(y), f(z)).

When X is a locally compact metric space then finitely generated pseudogroups of eitherlocal isometries or locally Lipschitz maps are regular.

A subset Y of X is G-invariant if g(x) ∈ Y for every g ∈ G and x ∈ Dg ∩ Y . For aG-invariant set Y we family of maps g|Dg ∩Y g∈G generate a pseudogroup denoted by G|Y . Thisconstruction can be extended for any subspace Y of X, even a non-invariant one, by definingG|Y as the pseudogroup generated by all the maps of the form g|Y ∩g−1(Y ), g ∈ G. Our most basicG-invariant subsets are the orbits of elements of X:


Definition 1.1.4 (G-orbit). Given x ∈ X, its orbit under G or G-orbit is the set

Gx := g(x); g ∈ G and x ∈ Dg.

All the elements of a pseudogroup are homeomorphisms from their domains onto their ranges,and this ensures that the closure of an invariant set is also invariant. In particular, closures oforbits are invariants. Closed invariant sets are partially ordered by inclusion: if Y and Y ′ aretwo closed G-invariant subsets of X, write Y Y ′ if and only if Y ′ ⊂ Y . When X is compactevery chain C of non-empty closed invariant sets admits an upper bound, for the intersection∩C is closed, non-empty, G-invariant and contained in all members of C. It follows from Zorn’sLemma that any non-empty closed invariant set contains minimal (with respect to the inclusion)non-empty closed invariant sets, which we call minimal here.

Definition 1.1.5 (Minimal Set). A non-empty closed G-invariant set Y is called minimal ifwhenever Y ′ ⊂ Y is a non-empty closed G-invariant set then Y ′ = Y .

Closed orbits are minimal sets: indeed, suppose Y ⊂ Gx minimal. Given y = g(x) ∈ Y andz = h(x) ∈ G, we have z = h(g−1(y)) ∈ Y , since h g−1 ∈ G and Y is invariant. ThereforeGx ∈ Y and Y = Gx. When all the G-orbits are dense in X the only minimal set is X itself. Inthis situation the pseudogroup G is called minimal.

Definition 1.1.6 (Exceptional Minimal Set). Minimal sets different from X and single orbitsare called exceptional.

It is worth noting the existence of a category of pseudogroups, its arrows beingdefined as follows:

Definition 1.1.7 (Pseudogroup morphisms). Let G ⊂ Homeo(X) and H ⊂ Homeo(Y ) bepseudogroups and Φ a family of homeomorphisms ϕ : Dϕ → Rϕ between open sets Dϕ ⊂ X

and Rϕ ⊂ Y . We say Φ is a morphism of G into H and write Φ : G → H if

(i)⋃ϕ∈Φ

Dϕ = X;

(ii) ϕ g ψ−1 ∈ H for every g ∈ G and ϕ, ψ ∈ Φ for which the composition is defined.

When a morphism Φ of G into H is such that Φ−1 = ϕ−1;ϕ ∈ Φ is again a morphism, thistime from H into G, we say Φ is an isomorphism of pseudogroups.

If Φ is an isomorphism then X = ∪Dϕ;ϕ ∈ Φ, Y = ∪Rϕ;ϕ ∈ Φ and

g ∈ G ⇔ ϕ g ψ−1 ∈ H, for every ϕ, ψ ∈ Φ.


In particular, the family Φ could consist of a single homeomorphism ϕ, in which case we recoverthe intuitive idea that the pseudogroups are conjugated. If X and Y are compact then theisomorphism Φ can be taken to be a finite family Φ = ϕ1, ..., ϕm. Indeed, for any isomorphismΦ0 between the two pseudogroups, just take ϕ1, ..., ϕr and ϕr+1, ..., ϕm subsets of Φ0 suchthat

r⋃i=1

Dϕi = X andm⋃

i=r+1Rϕi = Y.

Then both Φ and Φ−1 are morphisms because they cover their respective domains and Φ0 ⊃ Φ,Φ−1

0 ⊃ Φ−1 are both morphisms.This notion of isomorphism, though strange at first, is enough to guarantee a correspondencebetween minimal sets in both spaces and other such properties a pseudogroup might have, asregularity, for instance:

Example 1.1.8. Suppose G, H are isomorphic pseudogroups acting on X and Y , respectively.If both X and Y are compact and G is regular then H is also regular. To see this we constructa regular generating set for H from a regular generating set Γ for G. Let Φ = ϕ1, ..., ϕm be afinite isomorphism between the two pseudogroups, and consider

Γ′ := ϕ g ψ−1;ϕ, ψ ∈ Φ, g ∈ Γ ⊂ H.

The set Γ′ is clearly finite and symmetric. Given h = ψi g ϕ−1i ∈ Γ′, its domain Dh is

contained in the domain of ϕ−1i , and ϕ−1

i (Dh) ⊂ Dg. Since ϕ−1i is a homeomorphism and

ϕ−1i (Dh) = ϕ−1

i (Dh) ⊂ Dg we conclude Dh is compact. Furthermore, since g extends to g in∪iDϕi , the map h extends to h on ∪iRϕi . Any element f ′ ∈ H is of the form ϕi f ψ−1

j withϕi, ψj ∈ Φ and f ∈ G. Since Γ is a symmetric generator for G, for ψ−1(x) ∈ Df there is an openneighbourhood U ⊂ Df of ψ−1(x) such that f |U is a finite composition

f |U = g1 g2 · · · gn|U

of elements of Γ. Now, by choosing suitable elements of Φ we can write

f ′ = ϕi g1 · · · gn ψ−1j = (ϕi g1 ϕ−1

k1 ) (ϕk1 g2 ϕ−1k2 ) · · · (ϕkn−1 gn ψ−1).

This means Γ′ generates H and is therefore a regular generator.

When defining a pseudogroup acting on a manifold N in [12] (p. 58), Candel andConlon say it "is ‘trying’ to be a Cr transformation group on N . The problem is that thetransformations are not defined globally on N , so composition may not always be defined".In fact, the terminology pseudogroup is not arbitrary: complete abstract pseudogroups can bedefined in a strict algebraic setting as a special form of semigroup [39, 47], without the needfor an underlying topological space, and are therefore a generalisation of a concept of a group.The transformation pseudogroup defined here becomes then a special case of complete abstractpseudogroup, namely that of homeomorphisms between open sets of a topological space.


1.2 Foliations

1.2.1 Definition and first examples

In the simplest terms, a foliation is a partition of a manifold by immersed submanifoldsall of the same dimension: an equivalence relation whose classes are connected immersedsubmanifolds. They occur naturally in many contexts in Geometry and Dynamical Systems.

Definition 1.2.1 (Cr Foliated Atlas). For 0 ≤ s ≤ ∞, let M be a Cs n-dimensional manifold.A Cr foliated atlas on M is a Cr maximal atlas A with the following properties:

(i) the charts (ϕ,U) are such that ϕ(U) = U1 × U2 ⊂ Rp × Rn−p, where U1 and U2 are opensets in Rp and Rn−p, respectively;

(ii) if for two charts (ϕ,U) and (ψ, V ) there are constants c1, c2 ∈ Rn−p such that P :=ϕ−1(U1 × c1) and Q := ψ−1(V1 × c2) intersect, then the intersection P ∩Q is open inboth P and Q, equipped with the subspace topology.

The charts of A are called distinguished charts.

Figure 1 – A distinguished chart (U,ϕ) of a foliation of M by lines.

The foliated atlas A induces an equivalence relation F on M in the following way:for a distinguished chart ϕ(U) = U1 × U2 ⊂ Rp × Rn−p each connected component of the setϕ−1(U1 × c), c ∈ U2 is called a plaque of U (or for that matter, a plaque of F). Each of theseplaques is a connected open p-dimensional submanifold of M , since ϕ−1|U1×c : U1 × c → U

is a Cr-embedding. Note that if two plaques in a same chart U have a nonempty intersectionthey are equal, since they are both the same connected component of the preimage of thesame constant c ∈ U2. Define a relation ∼ on M by putting xFy whenever there exists a finitesequence P1, ..., Pk of plaques, Pi ∈ Ui, such that x ∈ P1, y ∈ Pk and Pi ∩ Pi+1 6= ∅ for everyi ∈ 1, ..., k − 1. Of course, since for different plaques in the same domain there cannot be


nonempty intersections, each Pi must be a plaque in a different chart. This relation is easilyseen to be reflective, symmetric and transitive, and since the distinguished charts of A cover M ,it is indeed an equivalence relation on all of M . The sequence P = (P1, ..., Pk) is called a chainof plaques.

Definition 1.2.2 (Foliation). For a Cr foliated atlas A of charts ϕ : U → U1×U2 ⊂ Rp×Rn−p,the equivalence relation F is called a Cr p-dimensional foliation on M , and the pair (M,F) issaid to be a foliated manifold. We write dim(F) = p and codim(F) = n− p for the dimensionand codimension of F , respectively, and call the distinguished chart in A the distinguishedcharts of F .

A leaf F(x) of F is this the set of all points of M which can be joined to x by a chain of plaques,that is, a equivalence class of M . The union P1 ∪ · · · ∪ Pk of all the plaques in a chain is pathconnected due to the condition Pi ∩ Pi+1 6= ∅, so the leaf itself is a path connected subset of M .If a single leaf is being considered and there is no need to emphasise that a certain point x is anelement of it, we may choose to call it L instead of F(x). When talking about foliations we areusually considering the partition of M by leaves rather than the equivalence relation itself.

Definition 1.2.3 (Leaves of a Foliation). The classes of MF := M∼ are called the leaves ofF , and we write [x] = F(x).

In light of our choice for A to be maximal, the set of all plaques of a leaf L becomesa basis for a topology in L. With respect to this topology, L ⊂M has a natural Cr-differentialstructure making it a p-dimensional immersed submanifold of M . Before proving this, however,we would like to show that the foliated atlas can actually be chosen to be very well behaved,and that this choice impose no loss of generality.

Definition 1.2.4 (Regular Atlas). A foliated atlas A = (ϕi, Ui)i∈I is said to be regular (as isthe covering of M by the domains of its charts) if:

(i) for each i ∈ I, ϕi(Ui) ⊂ Rn is an open cube;

(ii) the covering Uii∈I is locally finite;

(iii) if (ϕi, Ui) and (ϕj, Uj) are charts in F and the intersection of their domains is nonempty,then there is a chart (ψ,Wij) ∈ F such that Ui ∪ Uj ⊂ Wij and ϕi = ψ|Ui .

Note that due to condition (iii), for any given domains Ui and Uj of charts in F a plaque of Uiintercepts at most one plaque of Uj, which is equivalent to asking that for any two overlapping


distinguished charts (ϕ,U) and (ψ, V ) of F the transition map is of the form

ψ ϕ−1 : Rp × Rn−p → Rp × Rn−p

(x, y) 7→ (α(x, y), γ(y)). (1.3)

Remark 1.2.5. Some authors ([11], for instance) use the condition expressed in Equation 1.3as the characterizing condition of a foliated atlas. This is a equivalent condition to 1.2.1 (ii). Aproof of this assertion can be found in [12, Section 1.2].

Together with the locally finitude of the covering Uii∈I , Equation 1.3 gives us away to have some control over the chains of plaques in the leaves of M . What is more, we canalways assume a foliated atlas to be regular without any loss of generality, due to the followingresult:

Lemma 1.2.6. For any foliation F on any manifold M , regular coverings of M exist.

Proof. Our manifolds are Hausdorff and second countable, hence metrisable due to Urysohn’sMetrisation Theorem. Fix a metric for M and suppose for a moment M is a compact manifoldand F = (ψi, Vi)ki=1 is finite. Given a Lebesgue number ε > 0 for F , for any point x of M wecan find a foliated chart (ϕx, Ux) satisfying

x ∈ Ux ⊂ Ux ⊂ Vj,

ϕx = ψj|Ux ,

diam(Ux) <ε

2 .

for some chart (ψj, Vj) of A. We can recover from Uxx∈M a finite subcovering of domainsand hence a finite (and in particular, locally finite) subatlas A′ = (ϕi, Ui)li=1 such that forany Ui ∩ Uk 6= ∅ we have diam(Ui ∪ Uk) < ε, thus Ui ∪ Uk is contained in some Vj and satisfiesϕi = ψj|Ui . We can always compose ϕ with a diffeomorphism taking its range onto and opencube, so F ′ is indeed a regular refinement of A.

In the general case, we can use second countability and local compactness to constructa countable family of compacts Kii∈N such that Ki ⊂ int(Ki+1) and M = ∪iKi. AssumingA = (ψj, Vj) to be countable if necessary, we can find a strictly increasing sequence (nl)l∈Nsuch that Al = (ψj, Vj)nlj=0 covers Kl. Denoting by δl the distance between Kl and Kl+1 weconstruct a sequence (εl) such that εl is a Lebesgue number for both coverings Fl and Fl+1,satisfying

ε0 <δ0

2 ;

εl < minδl2 , εl−1

, l ≥ 1.


Finally, for each x ∈ Kl \Kl−1 we construct a chart (ϕx, Ux) as we did for a compact M , usingεl as the chosen Lebesgue number and requiring Ux to be a compact subset of some Vj of withj ≤ nl. As before, we pass to a finite subcover (ϕi, Ui)nlnl−1+1 (make n−1 = 0) of Kl \Kl−1. Theunion of these finite subcoverings gives us an atlas A′ = (ϕi, Ui)∞i=1 of M , which is a regularrefinement of A.

Given now a foliated manifold (M,F), we fix a regular covering and prove the following result

Proposition 1.2.7. The distinguished charts of F induce on each leaf L a differential structuremaking L a p dimensional Cr submanifold of M .

Proof. The Hausdorff property is inherited by L from M . To see the topology generated by theplaques is second countable, note that any two plaques of L must be the endpoints of some chainof plaques in L. On the other hand, because of conditions (ii) and (iii) of 1.2.4, given any plaquePi of L there are only finite many possibilities for the next plaque Pi+1 in the chain, so there areonly countably many possible chains of plaques in L, hence L has only countable many plaques.With L shown to be Hausdorff and second countable, the atlas is then defined in the followingway: given a plaque P of L, there is a distinguished chart ϕ = (ϕ1, ϕ2) : U → U1×U2 of F suchthat P = U ∩ L = ϕ−1(U1 × a) for some a ∈ U2. The restriction ϕ := ϕ1|P : P → U1 ⊂ Rp iscontinuous since ϕ is the restriction of a continuous function. It is bijective because if it were notit would contradict the fact ϕ is and homeomorphism. It is open because any open V ⊂ P is of theform V = V ′∩P where V ′ ⊂ U is open, thus ϕ(V ′) is open hence ϕ1(V ′) ≈ ϕ1(V ′)×a = ϕ(V )is open. The application ϕ is therefore a homeomorphism between P and Rp. We claim the set

B := (ϕ, P );P ⊂ L is a plaque of U where (ϕ,U) ∈ F

is a Cr atlas of dimension p. It suffices now to check the transition maps are well definedCr applications. By definition P ∩ Q is an open set of both P and Q. Since ϕ and ψ arehomeomorphisms the images ϕ(P ∩Q) and ψ(P ∩Q) are open sets of Rp. The transition mapis then ϕ ψ−1 : ψ(P ∩ Q) → ϕ(P ∩ Q), which is Cr because ϕ ψ−1(x) = α(x, b) wheneverx ∈ ψ(P ∩Q) and α is Cr in accordance to 1.3. An analogous argument can be used to show itsinverse ψ ϕ−1 is also Cr. Hence the transition maps are Cr diffeomorphisms, as we claimed.

The topology constructed in the last proposition’s proof is called the intrinsic topologyof the leaf L. It does not always coincide with the subspace topology of L, the latter beingstronger in general. The reason for this is that the leaf L can accumulate onto itself, causingany neighbourhood of the cluster points to contain an infinite number of plaques. It follows L is


not path connected in any neighbourhood of a cluster point in the subspace topology, whilewith the intrinsic topology L is a manifold and therefore path connected. The leaves for eachof the two topologies coincide are called proper leaves. Compact leaves are always proper, butnon-compact proper leaves occur as well. When considering distinguished charts, we have that aleaf L is proper if and only if any plaque P ⊂ U ∩ L has an open neighbourhood V ⊂ U suchthat V ∩ L = P .

Example 1.2.8 (Foliations defined via submersions). If f : M → N is a submersion thereis a natural way of foliate M via f by means of the Local Form of Submersions. Indeed, ifdimM = m and dimN = n, then for any point x of M there are charts (ϕ,U) around x and(ψ, V ) around f(x) such that f(U) ⊂ V , ϕ(U) = U1 × U2 ⊂ Rm−n × Rn, U2 ⊂ ψ(V ), and thelocal representation of f is ψ f ϕ−1 : (x, y) 7→ y. The collection F of all the charts (ϕ,U) onM is such that if ϕi(Ui)∩ϕj(Uj) 6= ∅ and ϕi f ψi is the projection π2 on the second coordinate,then f(ϕj(ϕi(Ui)∩ϕj(Uj))) ⊂ Dom(ψi) and one has π2 (ϕi ϕ−1

j )(x, y) = ψi f ϕ−1j (x, y) = y.

This implies all the overlapping charts in F satisfy 1.3 with γ = idRn−p , and the leaves of F areexactly the sets f−1(p), p ∈ N .

Example 1.2.9 (Foliations defined by fibrations). Let F → Ef−→ B be a locally trivial fibration

(also called a fibre bundle), that is, f : E → B is a submersion and for every point x ∈ B thereare an open neighbourhood U of x and a diffeomorphism ϕ : f−1(U)→ U × F such that

f−1(U) U × F

U

ϕ

f π1

is commutative, π1 being the projection on the first coordinate. Then the fibres of f , that is,the manifolds f−1(x), x ∈ B, form a foliation for E in accordance to Example 1.2.8 above.

Example 1.2.10 (The Frobenius Theorem). Given a foliation F on a manifold M , we denoteby TF the set of all elements in TM tangent to leaves of F . The Frobenius Theorem is aclassical result in Differential Topology (see, for instance, [43]) that guarantees that for anyinvolutive Cr-subbundle E ⊂ TM there is a foliation F of M for which E = TF . In terms ofvector fields, it says that a distribution by k-planes D is involutive if and only if it is completelyintegrable. The partition formed by all the integral submanifolds is the foliation F . In particular,since any 1-dimensional subbundle of TM is involutive, it determines a foliation. Vector fieldswithout singularities, for instance, determine foliations of dimension 1 for which each leaf is oneorbit of the vector field’s flow.


If the Frobenius Theorem is stated by means of differential forms, then it says thatany ideal I of the graded algebra Ω(M) of differential forms such that dI ⊂ I and

ker I(x) = v ∈ TxM ;ω(v) = 0 for all 1-forms ω ∈ I

has the same dimension q for all x ∈M , then the subbundle ker I = ∪x∈M ker I(x) determinesa foliation on M , whose codimension is q. In particular, any 1-form ω such that dω = η ∧ ωgenerates an ideal with the properties above, and thus define a foliation of codimension 1. Whenthis is the case we say the foliation is defined by the equation

ω = 0.

The cohomology class [η ∧ dη] is known as the Godbillon-Vey class of F .

1.2.2 Constructing foliations

In this paragraph, we address the exposition of some basic constructions in foliationtheory, namely pullbacks, tangential and transverse gluing, and turbulisation. All of these aredifferent ways of either modifying existing foliations in a space to create new ones, or use themto construct foliations in different spaces, usually via some diffeomorphism. Before we start withsuch constructions, however, we take a closer look at Definition 1.2.1 and generalise it in orderto admit foliations in manifolds with boundary and corners.

Suppose ∂M 6= ∅. Given x ∈ ∂M and a chart ϕ : U → Rp×Rq around x, then either

ϕ(U) ⊂ Hp × Rq and ϕ(U ∩ ∂M) ⊂ Rp0 × Rq

orϕ(U) ⊂ Rp ×Hq and ϕ(U ∩ ∂M) ⊂ Rp × Rq0,

where by Rn0 we denote the hyperplane (x1, ..., xn−1, 0). Let Fn ∈ Rn,Hn. A rectangularneighbourhood in Fn is a set of the form B = J1 × · · · × Jn, where each Ji is an interval (notnecessarily bounded) in F1, open in the subspace topology of F1 ⊂ R1. If Jn = [0, a) for somepositive number a, then B has a boundary ∂B := (x1, ..., xn−1, 0. With this in mind we makethe following definition:

Definition 1.2.11. In an n-dimensional manifold M (possibly with boundary and corners), aCr-distinguished chart of codimension q is a Cr-diffeomorphism

ϕ = (ϕ′, ϕ′′) : U → Bτ ×Bt ⊂ Fn−p × Fq.

For each point y ∈ Bt, the set ϕ−1(Bτ × y) is called a plaque of U , and for each x ∈ U theset Tx := ϕ−1(ϕ′(x) ×Bt) is a transverse section of U through x.


In these terms, a codimension q Cr foliation in M is the equivalence relation (or the partitionof M by its classes) induced by an Cr atlas A of distinguished charts such that the intersectionP ∩Q of any two plaques is open in both P and Q. This generalises the definition given before.

Let (M,F) be a foliated manifold and A and atlas of distinguished charts of F . For(ϕ,U) ∈ A, the set

∂τU := ϕ−1(Bτ × ∂Bt)

is called the tangential boundary of U , while

∂tU := ϕ−1(∂Bτ ×Bt)

is the transverse boundary of U . Accordingly, the sets

∂τM :=⋃U∈A

∂τU

and∂tM :=

⋃U∈A

∂tU

are the tangential and transverse boundaries of M with respect to F , respectively.

Figure 2 – The transverse (red) and tangential (blue) boundaries of a foliation of I2 by lines.

It should be noted that the transverse sections Tx of a distinguished chart are alllocally diffeomorphic. This is known as Transverse Uniformity:

Theorem 1.2.12 (Transverse Uniformity). Given any leaf L of a foliated manifold (M,F)and points x, y ∈ L, there are transverse sections T1 3 x and T2 3 y and a diffeomorphismf : T1 → T2, whose differentiability class is the same of F , such that f(L′ ∩ T1) = L′ ∩ T2 forany leaf L′ of F .


Proof. Fix a regular atlas for F . Consider a chain of plaques P1, ..., Pk joining x and y, with Pjbeing a plaque of the chart (ϕj, Uj), and write ϕj(Uj) = Bj

τ ×Bjt. Let x = x0, x1, ..., xk = y be

a sequence such that xj ∈ Pj ∩ Pj+1 for 1 ≤ j ≤ k − 1. For each j = 0, ..., k denote by Dj thetransverse section ϕ−1

j (ϕ′j(xj)×Bjt). Since our atlas is regular, each union Uj ∪Uj+1 is contained

in the domain of a distinguished chart of F , so there is a disc Bj of dimension codim(F) withxj ∈ Bj ⊂ Dj ∩ Uj+1 which intersects each plaque of Uj+1 at most once. There is therefore awell defined injection fj : Bj → Dj+1 mapping a point p of Bj to the unique point fj(p) in theintersection of Dj+1 with plaque through p. In particular, fj(xj) = xj+1, and fj : Bj → fj(Bj)has the same differentiability class then the map γ appearing in 1.3, that is, each fj has thesame differentiability class than F . We end the construction by taking

x ∈ T1 ⊂ B0 ∩ f−10 (B1) ∩ · · · ∩

(f−1

0 · · · fk−1(Dk))

andT2 := fk−1 · · · f0(T1),

and setting f := fk−1 · · · f0|T1 . Then f : T1 → T2 has the desired properties.

More generally, we say a submanifold T of M is transverse to F , which we denote byT t F , when at each point x ∈ T one has TxM = Tx T +TxF(x), i. e., if T when transverse toeach leaf it intersects. A transverse submanifold T whose dimension is equal to the codimensionof F is called a transverse section for F . In much the same way, we define a C1 mappingf : N → (M,F) to be transverse to F if f is transverse to each leaf. More specifically, f t Fwhenever

TxN = Tf(x)F(f(x)) + Im dxf

holds for every x ∈ N . We will often replace TxF(x) for the shorthand notation TxF , leavingimplicit the fact that the tangent space in question is that of the unique leaf containing x.TF ⊂ TM will be the subbundle of all vector tangent to the leaves of F .

1.2.2.1 Pullbacks

Suppose f : N → (M,F) is C1 and f t F . In this case we can define a foliation f ∗Fof N whose leaves are the connected components of preimages of leaves from F under f . Thisconstruction is called the pullback of F by f and generalises example 1.2.8, since any submersionf : N →M is automatically transverse to any possible foliations of M . The accurate statementof the theorem is the following:

Theorem 1.2.13. Let f be a surjective Cs map from a Cs manifold N to a manifold (M,F)equipped with a Cr foliation F . If s, r > 0 and f t F , then there is a unique Cminr,s foliation


f ∗F of N whose leaves are the connected components of f−1(L), with L ranging over all theleaves of F . Moreover, codim(f ∗F) = codim(F).

Proof. Let x be a point inN and (ϕ,U) a distinguished chart around f(x). The second coordinatefunction ϕ′′ : U → Fq is a submersion and therefore

ϕ′′ f : f−1(U)→ Fq

is also a submersion, since f t F by hypothesis. As in Example 1.2.8, we use the Local Form ofSubmersions to guarantee the existence of a chart ψ = (ψ′, ψ′′) defined on a neighbourhood Vof x such that

ϕ f ψ = ψ′′,

hence each preimage ψ′′−1(c), c ∈ Bt, is an embedded submanifold of N of codimension q, corre-sponding exactly to the preimage under f of the plaque ϕ′′−1(c) of U . Due to the surjectivenessof f , if we consider all the leaves of F then the family A of all the charts (ψ, V ) forms a foliatedatlas for N . This atlas is at least as smooth as the minimum between s and r. Given anotheratlas A′ satisfying the conditions above, then A ∪A′ is an foliated atlas again in it induces onN the same equivalence relation as A and A′, therefore the foliation f ∗F is unique.

In particular, note that for any covering (M, π) of a Cr-foliated manifold (M,F) there is aunique foliation F = π∗F of M .

Remark 1.2.14. Note that locally every foliation is a pullback. Indeed, if F is a codimensionp foliation on a dimension n manifold, the defining condition on a distinguished chart (U,ϕ)could be rephrased by saying that F|U is the foliation obtained by pulling back under ϕ thetrivial foliation of Rn by p-planes.

1.2.2.2 Gluing

We say M is the result of gluing the manifolds M1 and M2 when M is diffeomorphicto the identification of points in M1 with points in M2 via a diffeomorphism from M2 to M1. Weare specially interested in gluing manifolds along the boundary, that is, in the case that M1 andM2 are n-dimensional manifolds with boundary, Si ⊂ ∂Mi are unions of connected componentsof the boundary, f : S2 → S1 is a diffeomorphism and M = M1 ∪f M2 := (M1 tM2)∼, where∼ is the relation in M1 tM2 generated by x ≡ f(x).

We would like to assure that the topological space resulting from gluing manifoldsis again a manifold. In order to do so, we modify slightly the construction above by addingcollar neighbourhoods and a well chosen vector field to it: choose neighbourhoods of Si and


define vector fields Xi on these neighbourhoods such that Xi is transverse to Si and Xi|Si pointsinwards. If Si happens to have a boundary, then ∂Si will be a corner in M , separating Si fromanother connected component Wi ⊂ ∂Mi, and in this case the vector field Xi is also chosen insuch a way that it is tangent to Wi. Such Xi can be constructed locally and put together byusing a smooth partition of unity. Without loss of generality (by replacing Xi by αiXi for someappropriate function αi) we can assume the flow Φi of Xi is of the form

Φi : Si × [0, 2)→Mi.

We consider the neighbourhoods

Ni := Φi(Si × [0, 1)) ⊂Mi,

on which we shall use the coordinates imposed by Φi, that is, for any x ∈ Si the coordinates ofthe point Φi

t are (x, t), 0 ≤ t < 1. Now Mi is diffeomorphic to M1 \Ni, and we define the gluing

M := ((M1 \N1) t (M2 \N2))∼,

where ∼ is the equivalence relation generated by identifying (x, t) with (f(x), 1− t). Since weare dealing with an identification via a diffeomorphism between two open manifolds Ni \ Si it issimple to give M the desired differential structure.

Suppose now we have n-dimensional foliated manifolds (M1,F1) and (M2,F2), F1

and F2 being both of class Cr with dimF1 = dimF2. If the subsets Si belong either to ∂tM or∂τM , that is, if both foliations are simultaneously transverse or tangent to the boundaries beingidentified, then we can glue the foliations in order to provide a foliation for the glued manifoldM . For the case Si ⊂ ∂tM this is unsurprisingly called transverse gluing. In the same setting asabove, we consider a mapping

g : [0, 2)→ [0, 2)

such that g|[0,1] ≡ 0, g|[ 32 ,2) ≡ id and g′(t) > 0 for every t. For such an g the map (x, t) 7→ (x, g(t))

smoothly maps Si×[0, 2) onto itself and cab be therefore extended to a smooth map fi : Mi →Mi

which coincides with the identity when outside the collar Si× [0, 2). Moreover, fi is a submersionof Si × [0, 1] onto S1 and maps Si × (1, 2) diffeomorphically onto its image. Since Fi t Si, themap fi is transverse to Fi, hence the foliation Fi can be pulled back via fi to a foliation F ′iof same regularity class than F which coincides with F outside the collar Si × [0, 2). In thecollar neighbourhood Ci := Si × [0, 1) the leaves of F ′i |Ci take the form L× [0, 1), with L beinga leaf of Fi|Si . If the Cr diffeomorphism f : S2 → S1 matches the foliations, then the gluingidentification (x, t) ≡ (f(x), 1 − t) is of class Cr, matching the foliations F ′1|C1 and F2|C2 toform a new Cr foliation F = F1 ∪f F2 of M .

One can apply the same constructions as above to glue M to itself along separatedportions of this boundary. This is especially useful to induce foliations on quotient spaces:


Example 1.2.15 (Reeb Foliation). The Reeb foliation or the orientable Reeb component of asolid torus D2 × S1, where D2 is the closed unit disc in R2, is a classic and useful example inthe theory of foliations. We begin by considering the real function

λ : p 7→ exp(− exp

(1

1− |p|2

)),

whose domain is the interior of D2, and then defining the submersion f : int(D2)× R→ R byf(x, y, z) = λ(|(x, y)|)ez. Due to Theorem 1.2.13, the preimages of constants under f , which arethe graphs of functions

D2 3 p 7→ exp(

11− |p|2

)+ c,

are the leaves of a foliation on int(D2)× R. This foliation is invariant under translations in thez-axis, in the sense that the translation of a leaf along this axis is again a leaf. This foliationtogether with the cylinder S1×R form a foliation on the cylinder D2×R. We can now impose tothis cylinder the relation (x, y, z) ∼ (x′, y′, z′) if and only if x = x′, y = y′ and z = z′ + k, k ∈ Z.The quotient manifold (D2 × R)∼ is diffeomorphic to a solid torus D2 × S1, and due to itsinvariance under translations the foliation on D2 ×R induces a foliation on the torus, called theorientable Reeb foliation of D2×S1. The boundary S1×S1 is a leaf itself, the only compact one.All the other leaves are diffeomorphic to R2 and cluster on the compact leaf. The Reeb foliationis not given by any submersion since leaves of such foliations are all closed, being preimages ofsingletons, while the only closed leaf of the Reeb foliation is the boundary.

Figure 3 – The foliation of the cylinder by pre-images and a leaf of the Reeb foliation after theidentification.

We could repeat exactly the same procedure identifying (x, y, z) with (y, x, z+k), k ∈Z, instead. The resulting manifold is non-orientable and the compact leaf is diffeomorphic to a


Klein bottle. This is called the non-orientable Reeb foliation or a non-orientable Reeb component.Together they can be used to form the Reeb foliation of S3. (Example 1.2.32).

This construction is more general, as any smooth, symmetric and even functionλ : (−1, 1) → R can be used, as long as it satisfies lim

t→1λ = lim

t→−1λ = ∞. To obtain the

Reeb components one has to shift the graph grλ up and down in order to fill the rectangle[−1, 1]× [0, 1], rotate this rectangle around the segment 0 × [0, 1] and then glue together thediscs D2 × 0 and D2 × 1 doing the appropriate identifications. The foliation F inside theReeb component is given by the equation ω = 0 (Example 1.2.10) where

ω = cos(θ(r))dr + sen(θ(r))dt,

where θ(r) = arctgλ′(r), (r, ϑ) are the polar coordinates in D2 and t is a parameter alongS1. In this way it is clearer how the same equation can be written in Dp × S1 providing(p+ 1)-dimensional Reeb components.

We can do the same for n-dimensional foliated manifolds (M1,F1) and (M2,F2) whenf : S2 → S1 is a diffeomorphism between subsets of the tangential boundary, Si ⊂ ∂τMi. We saythen (M,F) is obtained by tangential gluing of (Mi,Fi) along f . A priori, the differentiabilityclass of F can be lower that the those of F1 and F2, but Theorem 1.2.31 gives us sufficientconditions on the holonomy maps that guarantee F to be of class Cr as well.

1.2.2.3 Turbulisation

Let D : M → 2M be a distribution by k-planes, that is, an application that assignsto each point x ∈M a k-dimensional subspace of TxM . A second distribution D : M → 2M issaid to be transverse to D if TxM = D(x) + D(x) for every x in M .

Definition 1.2.16. A foliation F of a manifold M is orientable if the distribution

D : x 7→ TxF

is orientable. In a similar way, we say F is transversely orientable if D admits an orientabletransverse distribution.

Recall that a manifold is said to be closed when it is compact and boundaryless. Aclosed transversal of (M,F) is an embedded closed manifold intersection all the leaves of F andtransverse to each one of them. In particular, for codimension 1 foliations, closed transversalsare diffeomorphic to S1. Let F is an orientable and transversely orientable codimension 1 Cr

foliation of a n-dimensional manifold M , with r ≥ 1. Given a closed transversal T , we canconstruct a tubular neighbourhood N(T ) ' Dn−1 × S1 built from discs in the leaves of F . We


Figure 4 – The closed transversal and the tubular neighbourhood N(T ). Inside it we put anorientable Reeb component.

consider in N(T ) cylindrical coordinates (r, ϑ, t), with t ∈ S1. Each level surface t = c0 is a leafof F|N(T ). Consider the one-form ω ∈ Ω1(N(T )) given by

ω(r, ϑ, t) = cosλ(r)dr + sen(λ(r))dt,

where λ : [0, 1]→[−π2 ,

π

2

]is any smooth function strictly increasing in the interval (0, 1− ε)

for some ε > 0 and such that

λ(0) = −π2 , λ|[1−ε,1] ≡π

2 and λ(k)(0) = 0 for every k = 1, 2, ... .

Since dω = λ′(r) cos(λ(r))dt ∧ dr = −λ′(r)dt ∧ ω, the equation ω = 0 defines a foliation F0 ofN(T ). As r ranges from 0 to the value r0 such that λ(r0) = 0 the form ω goes from −dt todr. As r goes forth until it reaches 1, ω goes from dr until it becomes constant equal to dt inthe closed interval [1− ε, 1]. Remark that the coordinate ϑ does not appear in the descriptionof ω, hence the foliation F0 has rotational symmetry. This foliation leaves F unchanged in aneighbourhood of ∂N(T ), thus it matches F along the boundary and gives us a new foliationF ′ of M which coincides with F0 in the interior of N(T ) and with F in the rest of the manifoldM . What we did here was to remove a piece of F along the closed transversal T and replace itby a Reeb component. We say F ′ is obtained from F by the process of turbulisation (from theFrench word tourbillonnement).


1.2.3 The space of leaves

Let us now take a closer look at the quotient space defined in 1.2.3: we call it thespace of leaves of the foliation F . First we show the natural projection π : M →M∼ mappinga point x to its class is an open mapping:

Proposition 1.2.17. The projection π is open.

Proof. Let U ∈ M be open. It is sufficient to show π−1(π(U)) is open in M . Remark thatπ−1(π(U)) is the family of all the leaves whose intersection with U is nonempty. Given anelement p of U , then F(p) ∩ U 6= ∅ and for any other q in the intersection there is a path ofplaques P1, ..., Pk joining p and q, where each Pi is a plaque of Ui, the domain of a distinguishedchart (ϕi, Ui) of F . Let ϕ = (ϕ1

i , ϕ2i ) and write ϕi(Ui) = U ′i ×U ′′i ⊂ Rp ×Rn−p. Suppose there is

i ∈ 1, ..., k such that some x ∈ Pi has a neighbourhood V ⊂ π−1(π(U)) ∩ Ui. Then ϕi(V ) isan open set of U ′i × U ′′i and for the projection π2 on the second coordinate the set π−1

2 (ϕ2i (V ))

is therefore an open of U ′i × U ′′i as well. Since the plaque Pi is the preimage of a constant itfollows that W = ϕ−1

i (π−12 (ϕ2

i (V ))) is a neighbourhood of the plaque Pi in π−1(π(U)), that is,Pi belongs to the interior of π−1(π(U)). Now since p ∈ P1, P1 is in the interior of π−1(π(U)) wecan repeat the above argument to conclude P2 is in the interior of π−1(π(U)) as well, and sothe entire chain of plaques belongs to the interior of π−1(π(U)), in particular the point q.

In accordance to our notation for a leaf through a point x, we fix the notationF(U) := π−1(π(U)) = q ∈M ; there is a p ∈ U with q ∼ p = ∪p∈UF(p). This set is called thesaturated of U by F . The last proposition can be rephrased as: the saturated of every open setU of M is again an open subset of M . We shall call U ∈M invariant under F , or F-invariant,if it is equal to its saturated, i.e., if F(U) = U .

It is clear that the complement of an invariant set is also invariant. If U is invariant, so is itsinterior, its closure and its boundary: indeed, int(U) ⊂ F(int(U)) ⊂ U because U is invariant,and since F(int(U)) is open the maximality of the interior implies int(U) = F(int(U)). Thecomplement M \ U is invariant, so int(M \ U) = M \ U is invariant, and thus U is invariant.Finally ∂U = U \ int(U) is invariant as well.

Transverse sections are our main tool in the study and classification of leaves. Our most importantresult in this sense is The Transverse Uniformity Theorem (Theorem 1.2.12), which gives us aclassification of the leaves based on how they intersect with their transverse sections

Proposition 1.2.18. Let (M,F) be a foliated manifold, L a leaf in M and T a transversesection of F whose intersection with L is nonempty. One of the possibilities occur


(i) T ∩ L is discrete and L is a proper leaf.

(ii) T ∩ L has nonempty interior. This happens if and only of L ⊂ int(L), in which case L issaid to be a locally dense leaf.

(iii) T ∩ L is a perfect set. In this case L is called a exceptional leaf.

Proof. Suppose x, y ∈ T ∩L are distinct, and let T1 3 x and T2 3 y be subsets of T as given bythe Transverse Uniformity theorem. There are three mutually exclusive possibilities:

(a) x is an isolated point of T ∩ L, and so are all the other points in this intersection, henceit is discrete.

(b) x is an interior point of T ∩ L and so are all the other points in T ∩ L.

(c) T ∩ L has an empty interior but is not discrete.

Consider first the case (a). If L is proper, then each of its plaques has a neighbourhood whoseintersection with L is again the plaque in question, so L ∩ T has to be discrete. Conversely, ifL ∩ T is discrete then for each y ∈ L we can consider a chart (ϕ,U) about y and the transversedisc D = ϕ−1(ϕ′(y)× U2). The intersection L∩ D must be discrete and it contains y, hence thereis an open U2 ⊂ U2 and a smaller disc D = ϕ−1(ϕ′(y)× U2) ⊂ D, transverse to F , such thatD∩L = y. This means ϕ−1|U1×U2 : U1×U2 →M is a Cr-embedding such that L∩ϕ−1(U1×U2)consists of only one plaque of U , hence L is an embedded submanifold of M . This guaranteesitem (i). If we are in the case (b), fix the notation as in the argument before. Since y is a pointin L ∩ D there is an open U2 ⊂ U2 and a smaller disc D = ϕ−1(ϕ′(y)× U2) ⊂ L ∩ D, and thenL contains the open set ϕ−1(U1×U2). Since the closure of a leaf is invariant L must contain thesaturated of this open set as well, which is also open. Furthermore, L ⊂ F(ϕ−1(U1 × U2)) ⊂ L

and F(ϕ−1(U1 × U2)) is exactly the interior of L, and thus (ii) holds. The alternative (c) iscomplementary to the other two, and it implies (iii).

Definition 1.2.19 (Minimal set of a foliation). A nonempty closed F -invariant subset U of Mis called minimal if whenever U ′ ⊂ U is a nonempty closed F -invariant set then U ′ = U .

Of course, every closed leaf is a minimal set, as is M itself. If every leaf of F is dense in M thenM is the only minimal set and we say the foliation F itself is minimal. Minimal sets that areneither single leaves nor the entire manifold are called exceptional. The existence of such setshas implications in the complexity of the geometry, topology and dynamics of the foliation F .As was the case for G-invariant sets of a pseudogroup, compact foliated manifolds always admitminimal sets. The demonstration is about the same:


Proposition 1.2.20. Every foliation on a compact manifold has a minimal set.

Proof. The collection C of all F -invariant subsets of M is nonempty because it contains M andthe closure of every leaf. If we consider the partial order given by the inclusion, the intersection∩Ui of every element of a chain U1 ⊃ U2 ⊃ ... is again an element of C, thus every chain admitsbounds and Zorn’s Lemma implies the existence of minimal sets.

One can use the same argument as above to show the closure of every leaf in a compact foliatedmanifold must contain a minimal set.

Theorem 1.2.21. For a minimal subset U of M the following holds

(i) Every leaf of F contained in U is dense in U .

(ii) If M is a connected manifold and int(U) 6= ∅ then U = M .

(iii) If U is not the closure of a single leaf and T is a transverse section to F whose intersectionwith U is nonempty, then U ∩ T is a perfect subset of T .

(iv) Moreover, if T is a transverse section with codim(F) = dim(T ) = 1, T ∩ U 6= ∅ and U isexceptional with an empty interior, then T ∩ U is a Cantor set.

Proof. (i) Since U is closed it also contains the closure of any leaf which is entirely in U .Since the closure of any leaf is invariant and U is minimal the closure must be U itself.

(ii) If int(U) is nonempty then it is intersected by all leaves in U , according to item (i). ThusF(int(U)) = F(U) = U , and U is therefore both open and closed.

(iii) We must show that the intersection is closed and has no isolated points. It is clearly closedin T since U is closed. Remark that due to (i) every leaf L contained in U must intersectT : indeed, given any point in T ∩ U we can find a neighbourhood V of this point suchthat T is transverse to every plaque of F intersecting V , and since L is dense by (i), theremust be a point x of L in V , and then the plaque Px of L through x intersects T . Forx ∈ T ∩ U , since F(x) 6= U , there must be at least one other leaf L in U . The previousargument implies that for any neighbourhood V of x there is a point of L in L ∩ T ∩ V ,then x is a cluster point of T ∩ U indeed.

(iv) Since U is not all of M we can find a neighbourhood V of T ∩ U such that T ∩ V isdiffeomorphic to an interval in R. Then (iii) together with the fact that int(U) = ∅


imply T ∩ U is a compact, perfect subset of R whose interior is empty, hence T ∩ U ishomeomorphic to Cantor’s Middle Thirds Set.

1.2.4 Holonomy

The similarity between definitions 1.2.19 and 1.1.5 and the terminology involvedis no coincide. It happens because associated to a foliation F there is a pseudogroup, calledthe holonomy pseudogroup of F , which acts on a manifold transverse to F and translates thenotions defined above to those of a pseudogroup. Much more than that though, the holonomypseudogroup is the main tool in studying the dynamics of a foliation.In order to construct the holonomy pseudogroup, let F be a p-dimensional Cr foliation on M ,dim(M) = n, and fix a regular covering U for (M,F), which we can do with no loss of generalitydue to lemma 1.2.6. For any U ∈ U , consider the quotient space TU := U(F|U), where by F|Uwe mean the foliation on U obtained by restricting all the leaves of F to U . In this particularcase where U is the domain of a distinguished chart, the leaves of F|U are just the plaques of U ,thus TU is the space of plaques. If ϕ = (ϕ′, ϕ′′) : U → Rp × Rn−p, then TU is Cr-diffeomorphicto the cube ϕ′′(U) of Rn−p via the mapping P ∈ TU 7→ ϕ′′(P ) ∈ Rn−p. Moreover, for any givenconstant c0 ∈ ϕ′(U), the Cr-submanifold T ′U := ϕ−1(c0×ϕ′′(U)) of M is homeomorphic to TUvia the homeomorphism h : P ∈ TU 7→ ϕ−1(c0×ϕ′′(P )) ∈M . Since c0×ϕ′′(U) is transverseto every plaque ϕ′(U)×c, c ∈ ϕ′′(U), the manifold T ′U is transverse to every plaque in U . Theunion of all these space of plaques for a regular covering of M is therefore homeomorphic to aCr submanifold of M whose dimension is n− p, which is intersects, and is transverse to, everyleaf of F .

Definition 1.2.22 (Complete Transversal). The disjoint union

T :=⊔U∈UTU

is called a complete transversal for F .

The complete transversal T is the topological space where the holonomy pseudogroupof F acts.

Definition 1.2.23 (Holonomy Maps). For any two intersecting open sets U and V of U , letDV U := π(U ∩ V ), where π : U → TU is the natural projection. Then DV U , which is the setof plaques in U intersecting a plaque in V is open by 1.2.17. We define the holonomy maphV U : DV U → TV by

hV U(P ) = Q⇔ P ∩Q 6= ∅ (1.4)


Due to (iii) in 1.2.4 each plaque of DV U intersects only one plaque of V , thus the holonomymap is well defined.

It follows from the format of the coordinates change in 1.3 that hV U maps DV U

homeomorphically onto the set DUV of TV (Cr-diffeomorphically when F is Cr). ClearlyhV U = h−1

UV . When U, V,W ∈ U and U ∩ V 6= ∅ and V ∩W 6= ∅ one has hWU = hWV hV U . Inother words, whenever they are defined, holonomy maps satisfy the cocycle conditions:

hWU = hWV hV U ,

hUU = idTU ,

hV U = h−1UV ,

and are therefore also called by some authors the cocycle maps of a regular covering. Since DUU

is just TU , the domains of the holonomy maps cover the complete transversal T , so that wecan use them to generate a pseudogroup acting on the transversal. Most often, the completetransversal T is identified with a subset Rq, and the holonomy pseudogroup is handled as ifacting on this space, in order to simplify things. In this case, the holonomy cocycles are just themaps γ from 1.3.

Definition 1.2.24 (Holonomy Pseudogroup of F). Given a Cr foliation F on M and anassociated regular covering U , let Γ be the family of all the holonomy maps hV U ∈ Homeo(T ).The holonomy pseudogroup of F is the pseudogroup of Cr diffeomorphisms H := G(Γ) generatedby all the holonomy maps on T .

A general element ofH is called a holonomy map as well. Given a chain of plaques P = (P0, ..., Pk),there is an associated chain of domains (U0, ..., Uk) of distinguished charts of U such that Pi ⊂ Ui.Of course, Ui−1 ∩ Ui 6= ∅ for every i ≥ 1, which means the holonomy map

hP := hU0U1 · · · hUk−1Uk

is well defined. We denote its domain by DP , and say it is the holonomy map of the chain P .When P0 = Pk the chain P is said to be a closed chain. We can define a composition operationin the set PF of all the plaque chains of a foliation F : if P = (P1, ..., Pk) and Q = (Q1, ..., Ql)are chains of plaques with Pk ∩Q1 6= ∅, then we say the pair (P,Q) is admissible. For admissiblepairs composition is well defined and we can set

Q P := (P1, ..., Pk, Q1, ..., Qk).

Any chain P has a inverse P−1 = (Pk, ..., P1), making PF into a groupoid. If a chain P isclosed then we can compose it with itself as many times as we wish, and we denote by P n such


compositions. The cocycle conditions of the holonomy maps showed above mean the mapping

PF −→ H

P 7−→ hP ,

satisfies hQP = hQ hP .

Figure 5 – Holonomy along the chain P = (P1, P2, P3, P4, P5).

The notation H used above for the holonomy pseudogroup of F makes no note onthe chosen regular covering. This was done so because two different regular coverings for F giverise to essentially the same holonomy pseudogroup, so it is indeed an object dependent only onF

Proposition 1.2.25. The holonomy pseudogroups H and H′ associated to two different regularcoverings U and U ′ of a foliated manifold (M,F) are isomorphic.

Proof. For the particular case when U ′ is subordinated to U , their union U ′′ = U ∪ U ′ is again aregular covering of M . Let H′′ be its holonomy pseudogroup and consider the family

Φ = h h′′UU ′ h′;h ∈ H, h′ ∈ H′, h′′UU ′ ∈ H′′, U ∈ U , U ′ ∈ U ′ and U ′ ⊂ U.

We claim Φ is a pseudogroup isomorphism Φ : H′ → H. Indeed, if P1 and Pk are plaques in Tbelonging to the same leaf, there is an element of H mapping P to P , namely the holonomymap hP associated to the chain of plaques P joining P1 to Pk. Thus the domains of the maps inΦ cover T ′, and for the same reason the images of maps in Φ cover T , so that Φ is indeed anisomorphism.


For the general case, we can always use the construction showed in Lemma 1.2.6 to create athird regular covering subordinated to both U and U ′, so that both H and H′ are isomorphic tothe holonomy pseudogroup of the subordinated covering, and therefore isomorphic one to eachother.

The holonomy pseudogroup of a pullback is also related to the original holonomy pseudogroupvia an isomorphism:

Proposition 1.2.26. Let f : N → (M,F) be transverse to the C1 foliation F . If f(N) meetsall the leaves of F then the holonomy pseudogroup H′ of the pullback F ′ = f ∗F is isomorphicto a subpseudogroup of H, the holonomy pseudogroup of F .

Proof. Let T ⊂ N and T ′ = f(T ) ⊂ M denote the complete transversal for F ′ and F ,respectively. The restriction

f = f |T ′ : T ′ → T

is a local diffeomorphism: indeed, we have

TxN = TxF ′ ⊕ Tx T′ and dfx(TxF ′) ⊂ Tf(x)F ,

and due to the transversality of f ,

Tf(x)M = Tf(x)F + Im dfx.

These two equations combined yield

Tf(x) M = Im dfx + Tf(x)F .

This implies dim Im dfx ≥ codimF , while on the other hand dim Im dfx = dim dfx(Tx T ′) ≤dim TxT ′ = codimF . Hence dim dfx(Tx T

′) = dim Tx T′ and dfx is an isomorphism.

We consider then the family Φ of all the restrictions f |W , where W ⊂ T ′ is anopen such that f |W : W → f(W ) is a diffeomorphism. Φ is an isomorphism of H′ onto thesubpseudogroup of H generated by every map h : f(W ′)→ f(W ) in H of the form

h = (f |W ) h′ (f |W ′)−1,

with h′ ∈ H′ and W,W ′ ⊂ T ′ sufficiently small open sets.

The holonomy pseudogroup could be defined in a analogous way for any submanifold T of Mwith the same codimension as F , intersecting all of its leaves and transverse to each one ofthem. Such T is also called a complete transversal, and the holonomy pseudogroup on T isagain isomorphic to those arising from regular coverings of M . For instance, in example 1.2.9the base space B of the fibration F → E

f−→ B is a complete transversal for F .


Remark 1.2.27. We shall sometimes abuse notation and apply the holonomy maps to points inM , specially for foliations whose dimension or codimension is one, since for those the intersectionof a transversal with a leaf is a single point. When this is the case, by h(x) = y we mean h mapsthe unique plaque of TU containing x to the unique plaque of TV containing y.

With H defined, we can now see that a subset A ⊂ M is saturated if and only ifA∩ T is invariant under H, and minimal whenever A∩ T is closed in T , invariant under H andhas no proper subset with the same properties. A is exceptional whenever A∩T is a exceptionalset for H in the sense of definition 1.1.6.

While the holonomy pseudogroup gives us global information on the foliation, wecan get local information around a point x ∈M by means of a holonomy group, constructedusing closed chains beginning at the plaque through x. This holonomy group is, not to muchsurprise, related to the fundamental group of the leaf F(x). In order to construct such group,we consider closed leaf curves γ : I → F(x), γ(0) = γ(1) = x. We let P = (P0, ..., Pk), P0 = P1

be a chain of plaques covering of γ, Ui ⊃ Pi, U0 = Uk opens of a regular covering U , and choosepoints 0 = t0 < t1 < · · · < tk = 1 such that γ([ti, ti+1]) ⊂ Pi. We call the holonomy map

hγ := hP = hU0U1 · · · hUk−1Uk

a holonomy along γ. Remark that the domain of this map depends on the choice of plaques (andconsequently of distinguished charts of U) covering γ, so it would be wise to make reference tothis choice in our notation. We will, however, be specially interested in the germs of such maps,which we denote by hγ, and such germs have the nice property of being constant on (relative)homotopy classes:

Proposition 1.2.28. hγ depends only on [γ]

Proof. Suppose both chains(P0, ..., Pk) and (P ′0, ..., P ′l )

cover γ and let

0 = t0 < t1 < · · · < tk = 1 and 0 = t′0 < t′1 < · · · < t′l = 1

be the corresponding partitions of I. We can, by making small perturbations if necessary, assumewithout loss of generality that these partitions have no values in common. This means they canbe used to construct a single partition refining them both, with an associated chain of plaques.If somewhere in these new partition and chain of plaques we find strings

tr < t′p < · · · < t′q < tr+1


Pr, P′p, ..., P

′q, Pr+1

where p could be equal q. Now by definition γ(t′j+1) ∈ P ′j ∩ P ′j+1 for all j and γ([tr, tr+1]) ⊂ Pr,hence

Pr ∩ P ′j ∩ P ′j+1, p ≤ j < q

andPr ∩ P ′q ∩ Pr+1.

The cocycle conditions satisfied by the holonomy maps gives us then

hUr+1Ur = hUr+1U ′q hU ′qU ′q−1 · · · hU ′pUr

on a neighbourhood of Pr in T . In particular, both sides have the same germ at Pr, and finiterepetition of this process shows both chains have the same germ at x, so hγ depends only onγ. To finish the proof, consider a homotopy Hs in F(x) relative to ∂I such that H0 = γ. Forsmall variations in s the homotopy does not change plaques in the chain, so the germ is locallyconstant in s. Since I is connected, this proves our assertion.

As a consequence the following homomorphism is well defined

ΦF(x) : π1(F(x), x) 3 [γ] 7→ hγ.

We call it the holonomy homomorphism of the leaf F(x).

Definition 1.2.29 (Germinal holonomy group). For a point x in M the image

HF(x) = Im(ΦF(x)

)is called the holonomy group of F(x). When this group is trivial one says the leaf F(x) hastrivial holonomy.

In [22] the authors prove that the generic leaf of a foliation has trivial holonomy, by which wemean the union of all the leaves with trivial holonomy group is a residual F -invariant subset ofM .

Germinal holonomy groups are the main tool in investigating sufficient conditionsfor the tangential gluing of two Cr foliations be also of class Cr:

Definition 1.2.30 (Infinitesimally Cr-trivial boundary leaves). Let F be a codimension 1 Cr

foliation of the manifold M . Let L be a component of ∂τM , x ∈ L and HF(x) be the holonomygroup of the boundary leaf L. Suitably parametrising the transversal using x as 0 we can realiseall the elements of HF(x) as germs at 0 ∈ [0,∞) of local diffeomorphisms hγ in R. If for everyhγ ∈ HF(x) the local diffeomorphism hγ satisfies h′γ(0) = 1 and h(k)

γ (0) = 0 for every k = 2, ..., rwe say the foliation F is infinitesimally Cr-trivial at the boundary leaf L.


Note that the same could be defined using germs at 0 ∈ (−ε, 0]. Using this concept we cancharacterise foliations for which tangential gluing does not lower the differentiability class:

Proposition 1.2.31. Let (Mi,Fi), i = 1, 2, be n-dimensional foliated manifolds, Fi of classCr and dim cF1 = dimF2. Let Si ⊂ ∂τMi and f : S2 → S1 be a diffeomorphism. If Fi isInfinitesimally Cr-trivial along each component Si then there is a Cr structure onM = M1∪fM2,agreeing omM1 andMs with the structures there, such that the resulting foliation F = F1∪f F2

of M is of class Cr.

Proof. This is Proposition 3.4.2 in [12].

Example 1.2.32 (The Reeb Foliation of S3). If we realise S3 as the subset (z, w) ∈ C2; |z|2 +|w|2 = 1 of unit vectors of C2, then it can be decomposed in two solid tori

T1 := (z, w) ∈ S3; |z| ≥ |w|

T2 := (z, w) ∈ S3; |z| ≤ |w|

whose common boundary is the torus T := T1 ∩ T2 = (z, w) ∈ S3; |z| = |wrvert =

√2−1. Circles on T that are meridians with respect to the solid torus T1, say, those

given by w = c0, are longitudes with respect to the second solid torus T2.

Now let λ be as in section 1.2.2.3, with the additional hypothesis that λ(r0)(m) = 0 forevery m ≥ 1, where r0 is the point where λ(r0) = 0. This makes the Reeb component 0 ≤ r ≤ r0

infinitesimally C∞-trivial along the border. We can then replace λ by the C∞-function

ρ : r 7→

λ(r), se 0 ≤ r ≤ r0,

0, otherwise.

We define a foliation of D1 × S1 ≈ Ti by the form

ω = cos ρ(r)dr + senρ(r)dt.

It consists of a Reeb component for [0, r0] and on [r0, 1] it is the product foliation whose leavesare parallel copies of S2−2 × S1 = 0, 1 × S1, that is, 0, 1 × S1 × rr∈[r0,1]. Since thefoliation is infinitesimally C∞-trivial along T , this can be glued on 0, 1 × S1 × r by theidentity map on 0, 1×S1 ≈ 0, 1×S1×0, obtaining a C∞ foliation. These two foliationscan be glued again by means of any C∞ diffeomorphism

f : 0, 1 × S1 → 0, 1 × S1

to obtain a C∞ foliation of S3, called the Reeb foliation.


The same could be done for any Reeb component ofDn−1×S1 and C∞ diffeomorphismf : Sn−2 × S1 → Sn−2 × S1 in order to obtain a C∞ foliation of the compact boundaryless C∞

manifold obtained from the gluing process. This is usually also referred to as the Reeb foliation.

1.2.5 Resiliency

Holonomy maps are also used to define a special type of leaf that will play aimportant role in the next chapters, being closely related to the entropy of a foliation. Let F bea codimension 1 foliation of M , H its holonomy pseudogroup. We consider T as a submanifoldof M . Suppose x ∈ T , and that P0 is the plaque containing x. Assume P = (P0, ..., Pk) is aclosed chain of plaques and note that hP (x) = x.

Definition 1.2.33 (Resilient Leaf ). A point y in the domain DP of the holonomy map hP issaid to be asymptotic to x if lim

n→∞hnP (y) = x. A point x ∈ T is said to be resilient for H if there

is a point y asymptotic to x and a chain of plaques Q = (Q0, ..., Ql) joining x to y. A leaf L ofF is a resilient leaf if it contains a resilient point, that is, if there exists a point x ∈ T ∩ L, aholonomy map h defined on an neighbourhood U of x and a point y 6= x in L∩ T ∩U such that

h(x) = x and hn(y)→ x as n→∞.

Equivalently, a leaf is resilient when it contains distinct points x, y and a loop γ with x ∈ γ(I)∩Tsuch that hnγ(y)→ x when n→∞. Two dimensional resilient leaves have the shape of a spring,which is the reason for the nomenclature. The term resilient is actually a somewhat literaltranslation (some authors call it “unfortunate” [12]) of the French word "ressort", which meansexactly that, "spring". Of course, leaves with trivial holonomy can not be resilient, and resilientleaves are never proper as they cluster on themselves. In this sense, they play a similar role incodimension one foliation theory as homoclinic orbits in the dynamics of flows. As we shall see inChapter 3, the existence of resilient leaves is what characterise foliations with positive geometricentropy, the same way homoclinic intersections characterise positive topological entropy fordiffeomorphisms.

Analogously, one can define resilient orbits for a pseudogroup acting on R.

Definition 1.2.34 (Resilient Orbit). A orbit Gx of a pseudogroup G in R is said to be a resilientorbit if there are elements g, h ∈ G with x ∈ Dg ∩ Dh such that g(x) = x, h(x) ∈ Dh andhn(g(x))→ x as n→∞.

Of course, the resilient leaves of F are exactly those corresponding to resilient orbits of theholonomy pseudogroup.


Figure 6 – A 2-dimensional resilient leaf. Note that the complete transversal is considered hereas submanifold, and the holonomy hγ acts on points of M .

Example 1.2.35. Consider X =[−1

3 ,43

]and the mappings h1 : x 7→ x

3 and h2 : x 7→ x+ 23 .

We let D1 = D2 = X andR1 =

[−19 ,

49

], R2 =

[59 ,

109

].

This makes hi : Di → Ri, i = 1, 2, into an element of Homeo(X). The orbit of 0 is resilient:h1(0) = 0, h2(0) = 2

3 and hn1(2

3

)= 2(3)−(n+1) → 0 as n→ 0.

1.2.6 Foliated bundles and the suspension of representations

An important class of foliations consists of foliations on fibre bundles having the fibresas complete transversals. They appear in a construction called the suspension of a representation,which is a very useful tools for constructing foliations with a given holonomy pseudogroup.

Definition 1.2.36 (Foliated Bundle). Let F → Mπ−→ B be a locally trivial fibration as in

example 1.2.9, where M,B and the fibres F are Cr-manifolds, 1 ≤ r ≤ ∞, and the base spaceB is connected. Given a codimension q foliation F on M , q = dimF , we say (M,F , π) is a Cr

foliated bundle if every point x in the base space B has a connected neighbourhood U and a


local trivialisation ϕ : π−1(U) → U × F of class Cr such that ϕ carries the foliation F|π−1(U)

onto the product foliation U × yy∈F ; in other words, for a leaf L of F , every connectedcomponent of L ∩ π−1(U) is mapped bijectively onto a set U × y for exactly one y ∈ F .

The first consequence of this definition is that the leaves of F are all transverse to the fibresof M : one says F is transverse to the fibration. Moreover, any leaf L has the same dimensionas the base space B. Since π : M → B is a Cr-submersion, the map π|L : L → B is a localdiffeomorphism, and for any leaf V of F|π−1(U), that is, any connected component of L∩π−1(U),we have π|L(V ) = U . This amounts to say each neighbourhood U of B realising the diagram

π−1(U) U × F

U

ϕ

π p,

where p is the projection in the first coordinate, is π|L-admissible, hence π|L : L → B is acovering transformation. In particular, this means every fibre of M meets every single leaf atleast once, so that the fibres are complete transversals to F . This is the opposite situationfrom what we had in example 1.2.9, where the fibres were the leaves and B was the completetransversal. The definition could be extended to encompass the case r = 0. If all the structuresinvolved are continuous then we ask for B to admit a C1 differential structure. Any maximal C1

atlas contains a smooth atlas, so we loose no generality by assuming B as regular as we want.

Example 1.2.37 (Mapping Cylinder and the Suspension of a Diffeomorphism). Let f : F → F

be a Cr-diffeomorphism of a compact manifold F , r ≥ 1. We let Z act on R×F in the followingway:

Z× (R× F ) −→ R× F

(k, (t, x)) 7−→ (t− k, fk(x)).

This action is Cr, proper, free and its orbits are the same as the equivalence classes of therelation

(t, x) ∼f (s, y)⇔ t− s = k ∈ Z and y = fk(x).

The quotient spaceMf := (R× F )∼f = (R× F )Z

is a Cr manifold of dimension dimF + 1 [40, Theorem 7.10], which we call the the mappingcylinder of f . It has a fibre bundle structure F → Mf

π−→ S1, where π : Mf → RZ ≈ S1

is the well defined map induced from the projection R × F → R. Namely, π maps the class[(t, x)] ∈Mf to the class t+Z of S1. Given t0+Z ∈ S1, we let ε be smaller than minbt0c, 1−bt0c


and consider the neighbourhood U = (t0 − ε, t0 + ε) + Z. This neighbourhood is such thatπ−1(U) = [t, x]; t ∈ U and x ∈ F is Cr diffeomorphic to U × F via ϕ : [t, x] 7→ (t+ Z, x) andhence

π−1(U) U × F

U

ϕ

π p(1.5)

commutes, so it is indeed a locally trivial fibration with base space S1 and fibre F .We can endow Mf with a particular dimension 1 foliation F depending on f , which makes(Mf ,F , π) a foliated bundle. In order to do so, consider in R× F the product foliation R×xx∈F . The leaves are clearly invariant under the action Z, in the sense that the image of aleaf under the action is again a leaf (as was the case for Reeb foliation, example 1.2.15). Thismeans the quotient map q : R× F →Mf induces a dimension 1 foliation F in Mf , called thesuspension of the diffeomorphism f . If U is as in 1.5 then any leaf of F ∩ π−1(U) is of the form[t, x]; t ∈ U, and is mapped by ϕ onto (t+ Z, x); t ∈ U, so that F is indeed transverse tothe fibration.

The importance of this construction lies in that it gives us a simple way to look atholonomy groups: given any loop γ in S1 and point x in the fibre π−1(γ(0)), the lifting propertyof the fibration F → Mf

π−→ S1 guarantees γ lifts to a unique path in Mf starting at x. Thecondition that ϕ carries F ∩ π−1(U) onto the product foliation of U × F implies the lifting ofγ|γ−1(U) can not leave the leaf of F ∩ π−1(U) containing x, thus the entire lifting is a leaf curveγ : I → F(x). Let us identify π−1(γ(0)) with F . If the degree of γ is n, then γ will wrap itselfaround F n times before ending, and the holonomy map hγ takes x into fn(x).The diffeomorphism fn ∈ Diffr(F ) is denoted by hγ and called the total holonomy of γ. As wecan perform this construction to any loop in S1 and the degrees depend only on the homotopytypes, we can define the homomorphism

h : π1(S1) −→ Diffr(F )

[γ] 7−→ hγ,

which describes the holonomies acting on the complete transversal F of Mf . This is called thetotal holonomy homomorphism of the fibre bundle (Mf ,F , π). It should be clear the holonomypseudogroup of Mf is G(f).

This can be generalised by the following procedure, a construction called the suspen-sion of a homomorphism h (or of the representation h): Let G be a finitely generated group,endowed with the discrete topology in order to become a Lie group, and B connected manifoldwhose fundamental group is isomorphic to G (by using Seifert-van Kampen’s theorem one can


construct a smooth 4-dimensional, connected and closed – that is, compact and boundaryless –manifold B with this property 1). Consider the universal covering

p : B → B

of B and a homomorphism h : G → Diffr(F ), where Diffr(F ) denotes the set of all globalCr-diffeomorphisms F → F of an arbitrary manifold F , that is, a representation of G byCr-diffeomorphisms of F .G acts freely and properly on the product B × F by means of the fundamental group: if gcorresponds to the class [γ] we define the left action of G via deck transformations:

G× (B × F ) −→ B × F

(g, (x, y)) 7−→ (γx, h(g)(y)), (1.6)

where γx = γ(1), with γ being the lifting of γ starting at x. The quotient of B×F by the orbitsof this action is a Cr-manifold M := (B × F )G whose dimension is dim B + dimF , and thequotient projection q : B →M is a submersion, hence a local diffeomorphism.As in the previous example, the mapping ρ : (x, y) 3 B×F 7→ p(x) ∈ B induces a Cr-submersionπ : M → B such that ρ = π q. In this setting F →M

π−→ B is a locally trivial fibration. Indeed,for a p-admissible open neighbourhood U in B, its preimage under p is a disjoint union

p−1(U) = tiUi ∈ B

and p restricted to each Ui is a smooth diffeomorphism. The family of open sets Ui × F isinvariant under the action of G, in the sense that g · (Ui × F ) = Uj × F . It follows that thequotient projection q maps each Ui × F diffeomorphically onto the same subset of M . Using therelation ρ = π q one can check this image under q is just π−1(ρ(Ui×F )) = π−1(U). If we writeϕi for the inverse (q|Ui×F )−1 and p1 : B × F → B for the projection on the first coordinate weget a commutative diagram

π−1(U) ϕi //

π%%

Ui × Fρ

p×id// U × F

p1yy

U

,

thus ϕ = (p× id) ϕi : π−1(U)→ U × F is a Cr trivialisation for the neighbourhood U .As in example 1.2.37 the product foliation B×yy∈F on B×F is invariant under the actionof G, thus it induces a foliation F on M , called the suspension of h. This foliation makes(M,F , π) into a foliated bundle: for U as above a leaf of F|π−1(U) is carried by ϕ onto a leaf1 This construction is a classical exercise in Algebraic Topology texts. It can also be found in a more general

setting in [36, Theorem 1], a construction due to Kervaire.


U × y in U × F . For fixed points x0 ∈ B and z0 ∈ p−1(x0) we can identify a fibre π−1(x0)with F via

y ↔ [z0, y]

Considering the lifting of a path γ based at x0 to a path in M starting at [z0, y], we see it endsat [z0, h([γ])(y)]. Hence the holonomy pseudogroup of (M,F) is G(Im h), and h is the totalholonomy of (M,F , π).

48

Chapter 2

Entropy

2.1 The growth and entropy of a pseudogroup

2.1.1 Orbit and expansion growth

In order to define the concepts of growth and entropy we intend to deal with in thischapter we need first establish a way to measure the growth of a sequence whose entries areitself sequences. So let

T := t : N→ [0,∞); t(n) ≤ t(n+ 1) ∀ n ∈ N

be the set of all increasing sequences with nonnegative values in R, and denote by T the set ofincreasing sequences with entries in T , by which we mean

T := (tj)j∈N; tj ∈ T and tj(n) ≤ tj+1(n+ 1) ∀ n, j ∈ N.

A element t ∈ T can be identified with the constant sequence (t, t, ...) of T whose entries aret, so we shall consider T as a subset of T . We define a binary relation in T by declaring(tj)j∈N (τk)k∈N if and only if there is b ∈ N such that for every j ∈ N the inequalities

tj(n) ≤ aτk(bn)

hold for all n ∈ N and for some fixed a > 0 and k ∈ N. In this case we say (τk)k∈N dominates(tj)j∈N. This relation is reflexive and transitive, and is therefore a preorder in T . It induces aequivalence relation ' in T given by

(tj) ' (τk)⇔ (tj) (τk) and (τk) (tj).

Definition 2.1.1 (Growth type). The elements of E := T' are called growth types. Given(tj) ∈ T (respectively t ∈ T ), its growth type is denoted by [(tj)] (respectively [t]). By settingE := [t]; t ∈ T we can also talk about the growth type of a usual monotone sequence.

Chapter 2. Entropy 49

The preorder in T induces a partial order in E , which we shall also denote by .In this notation one has, for instance, the following chain

[0] [1] [lnn] [n] [n2] [(1, n, n2, n3, ...)] [2n] [(1.2n, 3n, ...)].

The growth type of any polynomial of degree d is equal to [nd] and called polynomial growth(of degree d). For any constant a > 0 we have [an] = [en]. This is called exponential growth. Agrowth type [(tj)] is quasi-exponential when

limj→∞

lim supn→∞

1n

ln(tj(n)) > 0, (2.1)

and in particular [t] is quasi-exponential when lim supn→∞

1n

ln(t(n)) > 0. If 2.1 does not hold thegrowth type [(tj)] is said subexponential or yet quasi-polynomial. When

limj→∞

lim infn→∞

1n

ln(tj(n)) ≤ 0

the growth type [(tj)] is called nonexponential.

Our next objective is to define concepts of growth of orbits and a topological entropyfor regular pseudogroups, as introduced in 1.1.2. Remark that if a foliated manifold (M,F) iscompact then regular covering always admit finite subcoverings and the holonomy pseudogroupalways have a regular set of generators, that is, holonomy pseudogroups of compact manifoldsare always regular. This will allow us to define the entropy of the holonomy pseudogroup ofany compact foliated manifold M . The condition on the generating set is necessary in order toobtain a well-defined concept of growth types for pseudogroup orbits. A more detailed discussioncan be found in [56], Section 2.3.

Given a topological space X, let Γ ∈ Homeo(X) be a regular set of generators,as in Definition 1.1.2. We denoted by Γn the set of all elements of G = G(Γ) consisting of acomposition of n elements in Γ. Note that, since idX ∈ Γ, every element g ∈ Γn can be identifiedwith idX g ∈ Γn+1, so if we think about these compositions as words in the alphabet Γ, then Γnbecomes just the closed ball of radius n centred at the identity idX , in the word metric. For ourpurposes, let us define Γ0 = idX so that we can write down a sequence

Γ0 ⊂ Γ1 ⊂ Γ2 ⊂ · · · ⊂ Γn ⊂ · · · ⊂ Γ∞ := U∞i=0Γi.

Remark that Γ1 is simply Γ itself. Note as well that Γ∞ is not, in general, the entire pseudogroupG, since the sets in the sequence above are constructed based only in compositions, and notin amalgamations. Nevertheless, working locally is enough for our objectives of defining theexpansion growth and topological entropy of a pseudogroup, so that dealing with the simplersets Γi will be enough. Of course, when G happens to be a group of global homeomorphismsX → X then Γ∞ = G. We consider the following sequences:

tG,x(n) := #g(x); g ∈ Γn and x ∈ Dg.


Definition 2.1.2 (Orbit growth). The growth type of the sequence tG,x(n) is called the growthtype of G at x or the growth type of the orbit Gx, and denoted by gr(G, x).

This is a well defined object, depending neither on the point x nor on the regulargenerating set Γ chosen. Indeed, if y = g(x), g ∈ G, is another point in the orbit of x, thengr(G, y) = gr(G, x). Indeed, since our pseudogroup is regular there is g′ ∈ Γk, for some fixednatural number k, and a neighbourhood U of x such that g|U = g′|U . Thus tG,x(n) ≤ tG,y(n+ k)for all naturals n, hence gr(G, y) ≤ gr(G, x). The converse inequality holds by symmetry. It doesnot depend on the choice of regular generating set for if both Γ and Γ generate G then there issome k such that Γ ⊂ Γk and therefore

tG,x(n) ≤ tG,x(kn), n ∈ N.

Again by symmetry a similar inequality must hold with tG,x being dominated by tG,x and weconclude the growth types are the same.

When (M,F) is a compact Riemannian manifold and H is the holonomy pseudogroupof F , the growth of the orbits is the same as the growth of the leaves, in the following sense:for a complete Riemannian manifold (M, g), we fix a point x and define tM(n) := volB(x, n).This sequence’s growth type does not depend on x and is called the growth type of (M, g). Wedenote it by gr(M). Given any other point y ∈M , then the constant r = d(x, y), where d is theRiemannian distance in M , is such that B(y, n) ⊂ B(x, n + r). If t′M is defined as above butusing y instead of x, for any n > n0 ≥ r we have

t′M(n) ≤ tM(n+ n0) ≤ tM(2n),

and the growth type is therefore well defined, as claimed.

Proposition 2.1.3. 1 Let (M,F) be a foliated compact boundaryless Riemannian manifold,and H its holonomy pseudogroup. Considering in each leave the Riemannian structure inducedfrom M , we have gr(H, x) = gr(F(x)) for all x ∈M .

Proof. Let g be Riemannian metric of M , and denote by dF the Riemannian distance inducedon the leaves by g. Let U be the regular covering defining H, extract a regular covering Γ and letT be the complete transversal on which H acts. By the continuity of the metric induce by g, thefiniteness and regularity of U and the compactness ofM there is 0 < ε < 1 such that dF(z, y) > 2εfor any two distinct points z, y ∈ F(x)∩T . Moreover, v(ε) := minvolBF(y, ε); y ∈M is positiveand δ := maxdiam(P );P is a plaque of U is finite, as is V (δ) := maxvolBF(y, ε); y ∈ M.Then for every n ∈ N and h ∈ Γn we have dF(x, h(x)) ≤ nδ and thus1 Melhorar a demonstração dessa bagaça


v(ε)tH,x(n) ≤∑

h∈Γn∩h;x∈DhvolBF(x, h(x)) ≤ volBF(x, nδ + ε) ≤ tF(x)(n([δ + 1] + 1)),

where [·] denotes the floor function. Let λ be a Lebesgue number for the covering of M byplaques: if two points belong to the same leaf and their leaf distance is less than λ then there isa plaque of that leaf containing both of them. For any point y in the leaf F(x), and any naturalnumber n we have

tF(x)(n) = volBF(y, n) ≤∑

h∈Γn∩h;y∈DhvolBF(h(y), δ) ≤ V (δ)tH,x

([n

λ

]+ 1

).

The inequalities above show that tF(x)(n) ' tH,x(n), giving us the desired result.

In particular, the growth of a leaf does not depend on the chosen Riemannian manifold.

Proposition 2.1.4. A proper leaf of a compact foliated manifold (M,F) has nonexponentialgrowth.

Proof. Let U be a regular covering for F and F(x) be the leaf in question. Since it is proper,each of its plaques P has a neighbourhood whose intersection with F(x) its the plaque P itself.Using the compactness of M we can construct a finite covering U ′ = Uiki=1 such that theintersection of F(x) with an element of U ′ is either empty or a single plaque, so there is no lossof generality in assuming U has such properties. Let H be the holonomy pseudogroup. ThentH,x(n) is eventually constant equal to k! and is therefore nonexponential.

Definition 2.1.5. Two Riemannian metrics g, g′ on a manifoldM are said to be quasi-isometricif there are positive constants a, b such that the norms |·| and |·|′. induced respectively by g andg′, satisfy

a|v| ≤ |v|′ ≤ b|v|, for all v ∈ TM.

A diffeomorphism f : M → (N, g) is called a quasi-isometry if g and f ∗g are quasi-isometric.Naturally, in this case the manifolds M and N are called quasi-isometric as well.

As a corollary for the last result, we have

Corollary 2.1.6. If N is quasi-isometric to a leaf of a foliation F of a compact manifold Mthen gr(N) [en].

Equality above may hold, as is the case for resilient leaves:


Proposition 2.1.7. If F is a transversely oriented codimension one foliation of a the compactmanifold M and L is a resilient leaf of F then gr(L) = [en].

Proof. Since codimF = 1 the transversal T can be identified with an interval in R. Let x, y andf be like in Definition 1.2.33. Put a = x and let h be a holonomy map satisfying h(a) = fk(y) < y

for some k ∈ N, which exists because a and fk(y) belong to the same leaf. We fix a pointc ∈ (fk(y), y) and find b satisfying b > a, [a, b] ⊂ Df ∩ Dh and h(b) < c. We choose nowm ∈ N sufficiently large such that hm(y) < b and set g = fm. The numbers a, b, c and localhomeomorphisms g, h satisfy

h : [a, b]→ (b, c], g : [a, c]→ [a, b) and g(a) = a

For any x ∈ [a, b] and natural n all the point of the form

(gen h gen−1 h . . . h ge1 h)(x), with ei ∈ 1, 2,

are different because the images of g and h are disjoint, consequently for each n there are atleast 2n such points. If we consider the subpseudogroup H′ of H generated by g and h then

tH,x(3n) ≥ tH′,x ≥ 2n

and consequently [exp] gr(L), with together with 2.1.6 gives the desired result.

as two immediate corollaries we have

Corollary 2.1.8. If an orbit Gx is resilient then gr(G, x) = [en].

Corollary 2.1.9. If a Riemannian manifold N is quasi-isometric to a resilient leaf of a compactfoliated manifold (M,F) then gr(N) [en].

Besides the growth of orbits, we would like a way to measure how much a pseudogroupseparates distinct elements of X. We consider then (X, d) to be a compact metric space, andagain fix a regular generating set Γ for G acting on X

Definition 2.1.10. Given ε > 0, two points x, y ∈ X are said to be (n, ε)-separated with respectto G if there is g ∈ Γn such that d(g(x), g(y)) ≥ ε.

A subset A of X is (n, ε)-separated if any two distinct points in A are (n, ε)-separated withrespect to G.

Remark that, because id ∈ Γ, (n, ε)-separated points are also (m, ε)-separated forany m ≥ n, since we can always compose the element g ∈ Γn which separates them with


the identity, how many times necessary. For the same reason, if d(x, y) ≥ ε then x and y are(n, ε)-separated for every natural n. This means the intersection of any open ball of radius ε2with a (n, ε)-separated set contains at most one element, and so every (n, ε)-separated set isfinite, due to the compactness of X. This allows us to define the function

s(n, ε,Γ) := max#A;A ⊂ X is (n, ε)-separated, (2.2)

known as the expansion growth function of the pair (G,Γ). By fixing a natural k we get asequence

(s(n,

1k,Γ))

n∈N, so that

(s(·, 1k,Γ))

k∈Nis an element of T .

Proposition 2.1.11. The growth type of(s(·, 1k,Γ))

k∈Ndoes not depend on the generating

set Γ.

Proof. Suppose Γ and Γ′ are both regular generating sets for the pseudogroup G. For anyg ∈ Γ and x ∈ Dg there is a neighbourhood Ug(x) of x and a element g′ of Γ′m(g,x) satisfyingg′|Ug(x) = g|Ug(x). We cover the compact Dg by a finite covering Ug(x1), ..., Ug(xr(g)) and let λgbe a Lebesgue number for this covering. Let λ = minλg; g ∈ Γ and m = maxm(g, xi); g ∈Γ and i = 1, ..., r(g). These constants are such that for any distinct points x and y of X whosedistance is less than λ there are maps g ∈ Γ and g′ ∈ Γ′m satisfying x, y ∈ Dg ∩Dg′ , g(x) = g′(x)and g(y) = g′(y). In particular, for ε < λ any (n, ε,Γ)-separated set is (nm, ε,Γ′)-separated, andthus

s(n,

1k,Γ)≤ s

(nm,

1k,Γ′)

for any k such that 1k< λ.

Since the same argument can be applied exchanging the roles of Γ and Γ′, a similar inequalityholds where

(s(·, 1k,Γ))

k∈Ndominates

(s(·, 1k,Γ′))

k∈N, hence they have the same growth

type.

We can then make the following definition:

Definition 2.1.12 (Expansion Growth). The type of growth egr(G) :=[(s(·, 1k,Γ))

k∈N

]is

called the expansion growth of the pseudogroup G.

Remark 2.1.13. Isomorphic pseudogroups of diffeomorphisms have the same expansion growth:as we showed in Example 1.1.8 for a regular pair (G,Γ) acting on a compact set X and isomorphicto another pseudogroup G ′ acting on another compact set Y , we can construct a regular generatorΓ′ for G ′ such that if g ∈ cG is related via the isomorphism to g′ ∈ cG′ then the lengths of gwith respect to Γ and g′ with respect to Γ′ are equal. For each ϕi in the family Φ ⊂ Diff(X, Y )there is a constant Ci such that

1CidX(x, y) ≤ dY (ϕi(x), ϕi(y)) ≤ CidX(x, y)


Take a natural number C ≥ maxiCi, and let λ be the minimum between the Lebesgue numbers

of the coverings of X and Y by domains and ranges of the homeomorphisms ϕi, respectively.For any ε < λ

2 , if x, y are (n, ε)-separated by an element g ∈ G written as ϕj h ϕ−1i , h ∈ G ′,

then ϕi(x), ϕi(y) are(n,

ε

C

)-separated by the associated element h in G ′. In particular we have

s(·, 1k,Γ)≤ s

(·, 1Ck

,Γ′)

so that egr(G) egr(G ′). The same argument can be used to show the opposite inequality, sowe have egr(G) = egr(G ′).

In light of this fact we define the expansion growth of a C1 foliation F on a compact manifoldM , which we denote by egr(F), as the expansion growth of its holonomy pseudogroup withrespect to any regular covering.

Proposition 2.1.14. For a subpseudogroup G ′ of a regular pseudogroup G the inequality

egr(G ′) egr(G)

holds.

Proof. Let Γ′ be any finite generator for G ′. Then Γ′ ⊂ Γm for some natural m, and is thereforealso regular. Furthermore, any subset of the spaceX ′ ⊂ X on which G ′ acts that is (n, ε)-separatedwith respect to Γ′ is (mn, ε)-separated with respect to Γ. This means s(n, ε,Γ′) ≤ s(mn, ε,Γ)and, in particular,

s(n,

1v,Γ′)≤ s

(mn,

1k,Γ)∀n, k ∈ N.

This means exactly thats(·, 1v,Γ′)

k∈Ns

(·, 1j,Γ)

j∈N, hence egr(G ′) egrG.

Example 2.1.15 (Expansion Growth of a Local Isometries Pseudogroup). If G is a finitelygenerated pseudogroup of local isometries as in Example 1.1.3 (d) then each g in its generatingset and x ∈ Dg there is a neighbourhood U ⊂ Dg of x such that g|U : U → g(U) is an isometry.We use the compactness of M to find a finite covering U of M by open sets such that forevery generator g and U ∈ U the restriction g|U is an isometry. If we let G be set of all theserestrictions (and shrinking the opens U if necessary) then Γ = G ∪ G−1 ∪ id is a regulargenerating set for G. For any natural number n and g ∈ Γn we have

d(g(x), g(y)) = d(x, y) for all x, y ∈ Dg.


This means the sequences s(n,

1k,Γ)are all constant with respect to n, that is, s

(·, 1k,Γ)is

the constant sequence (s(

0, 1k,Γ), s(

0, 1k,Γ), s(

0, 1k,Γ), ...)

Denoting by (1)j the sequence where each entry is constant equal to 1 is easy to check thats(·, 1k,Γ)k (1)j and (1)j s

(·, 1k,Γ)k. This amounts to saying

egr(G) =[(s(·, 1,Γ), s(·, 1

2 ,Γ), ...)]

= [1].

Hence all pseudogroups of local isometries in compact metric spaces have constant expansiongrowth.

Definition 2.1.16 (Riemannian Foliation). A foliation F whose complete transversal has aRiemannian structure and whose holonomy pseudogroup consists of local isometries is called aRiemannian foliation.

For a Riemannian foliation F the distance between the leaves is locally constant. From whatwas exposed in the last example it follows immediately:

Proposition 2.1.17. Riemannian foliations of compact manifolds have constant expansiongrowth.

It is often useful to bound the expansion growth of a pseudogroup or foliation. Ashappened for the growth of orbits, the growth type of the exponential sequence en will be ourupper bound:

Proposition 2.1.18. If M is a compact manifold and G is a finitely generated pseudogroup ofC1 diffeomorphisms action on M then

egr(G) [en]

Proof. G can always be assumed regular: given a generating set fi : Di → Rini=1 consider afinite covering by open balls of diameter less than a Lebesgue number for the covering ∪iDi. IfG is the set of all possible restrictions of generating elements fi to open sets in this covering,then Γ = G ∪G−1 ∪ id is a regular generator. Take a finite covering U = U1, ..., Uk of M bydomain of charts ϕi : Ui → Im ⊂ Rm, where Im is the unit cube. Let

a := inf|d(ϕi g ϕ−1j )x|; g ∈ Γ, i, j ≤ k and x ∈ Dϕigϕ−1

j.

It follows from the regularity of Γ and the continuity of the derivative that a is strictly positive.The mean value theorem guarantees the existence of second positive constant b such that

1b|ϕi(x)− ϕi(y)| ≤ dM(x, y) ≤ b|ϕi(x)− ϕi(y)|


Given a subset A of Im (n, ε)-separated with respect to the pseudogroup generated on Im byϕi g ϕi; g ∈ Γ, 1 ≤ i ≤ k, the mean value inequality implies |x− y| ≥ ε

anfor any two points

x, y ∈ A, and consequently #A ≤(an

ε

)m. Finally, we choose a Lebesgue number for U . Given

ε ≤ λ each (n, ε)-separated set B ⊂M the subset ϕ(Ui ∩B) is (n, bε)-separated and

#ϕ(Ui ∩B) ≤(an

bε

)m∀1 ≤ i ≤ k.

Therefores(n, ε,Γ) ≤ k

(bε)mamn, (2.3)

and thus egr(G) ≤ [en], as we wanted.

Corollary 2.1.19. If F is a C1 foliation of a compact manifold M then egr(F) [en].

This upper bound can be reached. Here’s an example:

Example 2.1.20 (The Hirsch Foliation). Hirsch [27] constructed a foliation whose expansiongrowth is exponential: we consider a C1 diffeomorphism f : S1 → S1 and a embeddingg : S1 ×D2 → S1 ×D2 of the solid torus into its interior, such that

S1 ×D2 g//

p

S1 ×D2

p

S1f

// S1

commutes, where p denotes the projection on the first coordinate. This is equivalent to askingthe first coordinate function of g to be f . Let V := S1 × D2 \ int g(S1 × D2). Remark thatthe boundary of V is the disjoint union of two tori T 2. V admits a foliation F0 induced bythe submersion q := p|V where each leaf is a disc q−1(x0), x0 ∈ S1. We can generate in V anequivalence relation ∼ by identifying points in the boundary via g, that is, if x, y ∈ ∂V thenx ∼ y if and only if y = g(x). Then the quotient space M := V∼ is a manifold and, sincef q = q g, it admits a codimension one foliation F induced by F0. The complete transversalis diffeomorphic to S1 and its holonomy is given by the diffeomorphism f , that is, H = G(f).

For a more concrete example, consider a degree two mapping f : S1 → S1, namelyz 7→ z2. Let g be given by

g : S1 ×D2 −→ S1 ×D2

(z, w) 7−→(z2,

w

2 + z

4

)


Figure 7 – One of the leaves of the foliation F0 of V .

The image of g is a torus that wraps around itself two times without self intersection. Theresulting foliation F of the manifold M = V∼ is known as the Hirsch foliation.

Let us see that its expansion growth is exponential: consider the pseudogroup G(f).The set Γ = f, f−1, id is a regular generator for it. For any ε < π = diamS1 we haved(x, y) < ε =⇒ d(f(x), f(y)) = 2d(x, y). This means that interval C ⊂ S1 whose length is ε hasa (n, ε)-separated subset consisting of 2n+ 1 elements. Indeed, just partition C in 2n subintervalsof length ε

2n and consider the set of all the endpoints. Consequently we have s(n, ε,Γ) > 2n forall n ∈ N and ε < π, hence egr(G(f)) [en]. Follows from 2.1.18 that egr(G(f)) = [en], hencethe Hirsch foliation has exponential expansion growth.

A more general result in this sense is the following, analogous to Corollary 2.1.8:

Proposition 2.1.21. If a regular pseudogroup of C1-diffeomorphisms on a closed intervalJ ⊂ R admits a resilient orbit then its exponential growth is [en].

Proof. Remember from the proof of Proposition 2.1.7 that there exists a, b, c ∈ J with a < b < c

and holonomies g, h defined in open intervals containing [a, c] and [a, b], respectively, such that

g : [a, c]→ [a, b], h : [a, c]→ [a, b], and g(a) = a.


If we choose sufficiently large natural numbers k, l, and define f1 := gk h and f2 = gl, theneach holonomy fi is defined in an open interval containing a closed interval I satisfying

fi : I → Ii, i = 1, 2, with Ii ⊂ I, I1, I2 closed and I1 ∩ I2 = ∅

The subpseudogroup G(f1, f2) admits a regular generating set Γ′ = id, f1, f2, f−11 , f−1

2 . Weclaim s(n, ε,Γ′) > 2n for ε small enough. Indeed, choose a fixed x0 ∈ I. Given any naturalnumber n and finite sequence (i1, ..., in) ∈ 1, 2n, let y(i1, ..., in) := fi1 · · · fin(x0). For anyε < dist(I1, I2) the set

An := y(i1, ..., 1n); (i1, ..., 1n) ∈ 1, 2n

is (n, ε)-separated. Indeed, let y 6= y′ be point in An with y = y(i1, ..., ik, ik+1, ..., in) and y′ =y(i1, ..., ik, jk+1, ..., in), where ik+1 6= jk+1. Both y and y′ are elements in the domain of the mapf = f−1

ik· · ·f−1

i1 , but f(y) = fik+1 · · ·fin(x0) ∈ Iik+1 while f(y′) = fjk+1 · · ·fin(x0) ∈ Ijk+1 ,

hence d(f(y), f(y′)) ≥ dist(I1, I2) > ε. Clearly #An = 2n. As a consequence, if Γ is a regulargenerator for G and f1, f2 ∈ Γm then

s(mn, ε,Γ) ≥ s(n, e,Γ′) ≥ 2n,

thus egr(G) [en]. The result now follows from Proposition 2.1.18.

Corollary 2.1.22. Let F be a C1 foliation of codimension 1 on a compact manifold M . If Fhas a resilient leaf, then its expansion growth is exponential.

2.1.2 The topological entropy of a pseudogroup

2.1.2.1 The motivation from discrete dynamics

A topological entropy is defined for a pseudogroup on a compact X in much the sameway a topological entropy can be defined for a continuous function f : X → X on a compact X,as was done by Adler, Konheim and McAndrew [1]. We motivate our work in this section bygiven a brief summary of their results in the paper cited above.

We fix a compact Hausdorff space X and f : X → X a continuous transformation.Given any two open coverings U and V, we say V refines U , written as U ≺ V, if every openset in V is a subset of an open set of U . The relation ≺ is a partial order on the set of allopen coverings of X. Furthermore, given two open coverings U and V , one can construct a newcovering U ∨ V := U ∩ V ;U ∈ U , V ∈ V, called the join of U and V, which refines both ofthem. One has:


(i) If U ≺ U ′ and V ≺ V ′, then U ∨ V ≺ U ′ ∨ V ′.Proof : Indeed, given U ′ ∩ V ′ ∈ U ′ ∨ V ′, there are by hypothesis U ∈ U and V ∈ V withU ′ ⊂ U and V ′ ⊂ V , hence U ′ ∩ V ′ ⊂ U ∩ V .

For any covering U of X we denote by N(U) the minimal cardinality of a finite subcovering ofU :

N(U) := min#U ′;U ′ ⊂ U is finite and covers X.

(ii) The function N is increasing with respect to ≺, that is, if U ≺ V , then N(U) ≤ N(V).Proof : Let V1, ..., VN(U) be a minimal subcovering for V. Then U ≺ V implies theexistence of a subset U1, ..., UN(U) ⊂ V such that Vi ⊂ Ui. This subset is also a coveringof X, therefore N(U) ≤ N(V).

(iii) If U ≺ V , then N(U ∨ V) = N(V).Proof : Indeed, since V ≺ U ∨ V, we have N(V) ≤ N(U ∨ V), due to (ii). On the otherhand, U ∨ V ≺ V due to (i) and the hypothesis, then another application of property (ii)gives the desired equality.

Our most important property is the following:

(iv) N(U ∨ V) ≤ N(U)N(V).Proof : Take minimal subcoverings U1, ..., UN(U) of U and V1, ..., VN(V) of V. The setUi ∩ Vj; 1 ≤ i ≤ N(U), 1 ≤ j ≤ N(V) ⊂ U ∨ V is a covering of X with N(U)N(V)elements.

For the function f and an open covering U , define f−n(U) := f−n(U);U ∈ U. Forn = 1 this is again an open covering of X due to the continuity of f , and by induction it is anopen covering for every natural n. We write

U (n)f := U ∨ f−1U ∨ · · · ∨ f−(n−1)U

and letN(f,U , n) := N(U (n)

f ).

Property (iv) implies then that N(f,U , n+m) ≤ N(f,U , n)N(f,U ,m) and consequently thelimit

htop(f,U) := limn→∞

1n

lnN(f,U , n)

exists for any covering U . What is more, Property (ii) implies that whenever V is a refinementof U then htop(f,U) ≤ htop(f,V). The supremum

htop(f) := supUhtop(f,U)


is called the topological entropy of the transformation f .

It turns out, as was shown by Bowen [7], that for a compact metric space (X, d) thetopological entropy of a transformation can be defined in terms of the metric d, an approachoften easier to work with than the one with open coverings. Given a continuous transformationf : X → X, we define in X a countable family of new metrics dnn∈N by the rule

dn(x, y) := max0≤i≤n−1

d(f i(x), f i(y)).

A subset E of X is (n, ε)-separated with respect to f if dn(x, y) ≥ ε for every two points x, y ∈ E.Such two points are said to be (n, ε)-separated with respect to f as well. The complement of a(n, ε)-separated subset is said to be (n, ε)-spanning with respect to f . Given any x belonging toa (n, ε)-spanning there must be y in the same set satisfying dn(x, y) < ε. The points x and y aresaid two be (n, ε)-close with respect to f .

The compactness of X implies both the finiteness of every (n, ε)-separated set andthe existence of finite (n, ε)-spanning sets. We define

s(n, ε, f) := max#E;E is (n, ε)-separated

andr(n, ε, f) := min#E;E is (n, ε)-spanning ,

and lets(ε, f) := lim sup

n→∞

1ns(n, ε, f) and r(ε, f) := lim sup

n→∞

1nr(n, ε, f).

With this setting one has

Theorem 2.1.23. [Bowen] For a metric space (X, d) and a continuous transformation f : X →X the equalities

htop(f) = limε→0+

s(ε, f) = limε→0+

r(ε, f)

all hold.

Example 2.1.24 (Homeomorphisms of the Circle). Any homeomorphism ϕ : X → X of acircle onto itself has vanishing entropy. Consider (X, d) a circle of length 1, d being the usualdistance via arc length, and take ε > 0 such that d(ϕ−1(x), ϕ−1(y)) ≤ 4−1 whenever d(x, y) ≤ ε.Such a constant exists because ϕ is uniformly continuous. We will use induction to show that

r(n, ε, ϕ) ≤ n(⌊1ε

⌋+ 1

)for all natural n.


The case n = 1 is straightforward, since in this case the distance d1 is just the usualdistance d. Suppose E is a (n, ε)-spanning set realising the cardinality r(n, ε, ϕ) ≤ n (b1/εc+ 1) .Consider now the set ϕn(E) and all the arcs determined by two of its consecutive points. Sincefor every point of ϕn−1(E) there is another point in the same set such that the distance betweenthem is less than ε, we conclude that no more than b1/εc+ 1 of arcs determined by consecutivepoints of ϕn(E) has length greater than ε. Therefore, the exists a subset A ⊂ X of cardinality atmost b1/εc+ 1 such that all the arcs determined by ϕn(E)∪B have length less than ε. We claimE ′ := E ∪ ϕ−n(A) is a (n+ 1, ε)-spanning set whose cardinality is at most (n+ 1)

(⌊1ε

⌋+ 1

).

Let x ∈ E ′. For any y ∈ E we have dn(x, y) ≤ ε. If d(ϕn(x), ϕn(y)) ≤ ε thendn+1(x, y) ≤ ε. If no points y ∈ E satisfies the last inequality, then we can find y ∈ E and z ∈ E ′

such that d(ϕn(x), ϕn(z)) ≤ ε, and one the arcs I with endpoints ϕn(x) and ϕn(y) containsϕn(z) and is mapped by ϕ−1 onto an arc I1 of length less or equal to ε. This means ϕn−1(z) ∈ I1

and therefore d(ϕn−1(x), ϕn−1(z)) ≤ ε. The arc I2 is mapped by ϕ−1 onto an arc I2 of lengthless than 1

4 . Since dn(x, y) ≤ ε, the length of I2 is at most ε, and since ϕn−2(z) ∈ I2 we concluded(ϕn−2(x), ϕn−2(z)) ≤ ε. Proceeding like this, we conclude by induction that

max0≤i≤n

d(ϕi(x), ϕi(z)) ≤ ε,

which means exactly that dn+1(x, z) ≤ ε, hence E ′ is (n+ 1, ε)-spanning. Finally,

htop(ϕ) = limε→0+

lim supn→∞

1n

ln r(n, ε, ϕ)

≤ limε→0+

lim supn→∞

1n

lnn(⌊1ε

⌋+ 1

)≤ lim

ε→0+lim supn→∞

1n

lnn(1ε

+ 1)

= limε→0+

0 = 0.

Similar constructions can be used to see any homeomorphism ϕ : J1 → J2 between closedintervals has zero entropy.

2.1.2.2 The pseudogroup case

Now let G be a regular pseudogroup acting on a compact space X, and Γ be a regulargenerating set for G. Inspired by what was exposed above, we would like to define a topologicalentropy to pseudogroups. The most natural idea is to let the coverings g(U∩Dg);U ∈ U and g ∈Γn take the role of f−n(U). This does not work, however, because Γ contains the identityfunction of X, and so the sequence of minimal cardinalities could often end up being constantequal to N(U). In order to avoid this we define the sets

Γ∗n := gi1 · · · gin ∈ G; gij ∈ Γ \ idX, 1 ≤ j ≤ n, and gij gij+1 6= id, 1 ≤ j ≤ n− 1.


Γ∗n is the set of all “real” length n compositions of elements in Γ. For n = 1 the set Γ∗1 = Γ∗ isjust Γ \ id. If our regular generator is Γ = id, f1, f2, f

−11 , f−1

2 , for instance, then f1 f1 f2

would be an element in Γ∗3, while id f1 f−11 f2 ∈ Γ4 would not.

Due to the regularity we can, without loss of generality, assume that

X =⋃g∈Γ∗

Dg ∪Rg.

Given a covering U = Ui of X and a natural n, we define the following family

Γ∗nU = g(U ∩Dg);U ∈ U and g ∈ Γ∗n.

Γ∗U is another covering of X, since X is the union of the domains and ranges of maps in Γ∗. Asbefore, we set

U (n)Γ := U ∨ ΓU ∨ · · · ∨ Γn−1U

and letN(G,Γ,U , n) := N

(U (n)

Γ

).

By considering the subcovering U ′ ∨ V ′ of U ∨ V, where U ′ and V ′ are subcoverings of U andV with respectively N(U) and N(V) elements, we can see that N(U ∨ V) ≤ N(U)N(V) forany two covering of X. In particular, N(G,Γ,U , n+m) ≤ N(G,Γ,U , n)N(G,Γ,U ,m) and as aconsequence the limit

h(G,Γ,U) := limn→∞

1n

lnN(G,Γ,U , n) (2.4)

always exist.

Definition 2.1.25 (Topological entropy of a pseudogroup). The real number

h(G,Γ) := supUh(G,Γ,U) (2.5)

is called the topological entropy of G with respect to Γ.

Remark 2.1.26. Note that this does not generalise the concept of entropy for a single transforma-tion, for it is defined in dependence of the regular generating set. Even if f is an homeomorphismand we choose for G = G(f) the simple regular generating set Γ = id, f, f−1, in generalΓU 6= f−1U .

Proposition 2.1.27. If G ′ is a subpseudogroup of a regular pseudogroup G, then for any choicesof regular generating sets Γ′ and Γ for G ′ and G, respectively, there is a constant K ∈ N suchthat

h(G ′,Γ′) ≤ Kh(G,Γ).

In particular, K = 1 when G ′ ⊂ G.


Proof. There is a K ∈ N such that Γ′ ⊂ ΓK . This means Γ′nU ⊂ (Γk)nU for every natural n,therefore U (n)

Γ′ ⊂ U(n)ΓK and consequently

N(G ′,Γ′,U , n) ≤ N(G,ΓK ,U , n). (2.6)

On the other hand, since (n− 1)K ≤ nK − 1 for every naturals n and K, the open coveringU (Kn)

Γ = U ∨ ΓU ∨ · · · ∨ ΓnK−1 is a refinement of U (n)Γk = U ∨ ΓKU ∨ · · · ∨ Γ(n−1)K , hence

N(G,ΓK ,U , n) ≤ N(G,Γ,U , Kn). Together with 2.6 this yields

N(G ′,Γ′,U , n) ≤ N(G,Γ,U , Kn),

and therefore

h(G ′,Γ′,U) = limn→∞

1n

lnN(G ′,Γ′,U , n) ≤ K limn→∞

1Kn

lnN(G,Γ,U , Kn) = Kh(G,Γ,U),

from where the desired result follows.

As happens for the entropy of continuous transformations, we can give anotherdescription of the entropy of a regular pair (G,Γ) when the compact X is metrisable . For ametric space (X, d) the concept of (n, e)-separated points and sets has already been given inDefinition 2.1.10. We have a dual concept of (n, ε)-closeness:

Definition 2.1.28. Let (G,Γ) be a regular pair on the metric space (X, d). We say two pointsx and y are (n, ε)-close with respect to G if d(g(x), g(y)) < ε for every g ∈ Γn whose domaincontains both x and y.A subset A of X is (n, ε)-spanning if for every y ∈ X there is x ∈ A such that x and y are(n, ε)-close with respect to G.

Given x ∈ X the set BΓ(x;n, ε) := y ∈ X, y is (n, ε)-close to x with respect to G is calleda dynamic ball around x. It is an open set in X: indeed, if y ∈ BΓ(x;n, ε) then ∩Dg; g ∈Γn and x, y ⊂ Dg is an open neighbourhood of y entirely contained in BΓ(x;n, ε). Inparticular, when X is compact and A is (n, ε)-spanning, then

X =⋃x∈A

BΓ(x;n, ε)

is an open covering of X, so there exists finite (n, ε)-spanning sets for every choice of n and ε,and we can therefore define, analogously to 2.2, the sequence

r(n, ε,Γ) := min#A;A ⊂ X is (n, ε)-spanning. (2.7)


Consider then

s(ε,Γ) := lim supn→∞

1ns(n, ε,Γ), (2.8)

r(ε,Γ) := lim supn→∞

1nr(n, ε,Γ). (2.9)

Remark that for ε < ε′ every (n, ε′)-separated set is also (n, ε)-separated, as any (n, ε)-spanningset is also (n, ε′)-spanning. This implies s(n, ε,Γ) ≥ s(n, ε′,Γ) for every natural n. Similarly,r(n, ε,Γ) ≥ r(n, ε′,Γ) for every n. Therefore for ε < ε′ we have s(ε,Γ) ≥ s(ε′,Γ) and r(ε,Γ) ≥r(ε′,Γ) and the limits

limε→0+

s(ε,Γ)

andlimε→0+

r(ε,Γ)

both exist. We claim they are both equal the topological entropy h(G,Γ).

Proposition 2.1.29. For any compact metric space (X, d) and regular pair (G,Γ) acting on Xthe equalities

h(G,Γ) = limε→0+

s(ε,Γ) = limε→0+

r(ε,Γ)

hold.

Proof. Any maximal (n, ε)-separated set A is also (n, ε)-spanning: for all y ∈ X there is x ∈ Asuch that d(g(x), g(y)) < ε for every g ∈ Γn with x, y ⊂ Dg, otherwise A ∪ y would be(n, ε)-separated. As a consequence, r(n, ε,Γ) ≤ s(n, ε,Γ) and r(ε,Γ) ≤ s(ε,Γ). On the otherhand, given A (n, ε)-separated and B

(n,ε

2

)-spanning, consider a mapping σ : A→ B which

sends x ∈ A to σ(x) ∈ B such that x and σ(x) are(n,ε

2

)-close to x. This application is

injective due to the triangle inequality and the fact A is (n, ε)-separated, thus #A ≤ #B andconsequently (n, ε,Γ) ≤ r

(n,ε

2 ,Γ)and s(ε,Γ) ≤ r

(ε

2 ,Γ). It follows that

limε→0+

s(ε,Γ) = limε→0+

r(ε,Γ).

Let now U be a covering of X by open balls of radius ε and A a (n, ε)-separated set. Any openof U (n)

Γ can contain at most one point of A, hence N(G,Γ,U , n) ≥ s(n, ε,Γ), h(G,Γ,U) ≥ s(ε,Γ)and h(G,Γ) ≥ s(ε,Γ), for every positive ε. Finally, let λ be a Lebesgue number for U and A a(n, ε)-spanning set where 0 < ε <

λ

2 . Since each open ball B(x, ε) is contained in some open of

U , each dynamic ball BΓ(x;n, ε) ⊂ B(x, e) is contained in some open set of U (n)Γ . The covering

X =⋃x∈A

BΓ(x;n, ε)


is a finite refinement of U (n)Γ whose cardinality is exactly #A, so #A ≥ N

(U (n)

Γ

). It follows

immediately that r(n, ε,Γ) ≥ N(G,Γ,U , n) and r(ε,Γ) ≥ h(G,Γ), for every ε > 0. So we concludethat r(ε,Γ) ≥ h(G,Γ) ≥ s(ε,Γ) for all positive ε, and taking the limit completes the proof.

In particular, the entropy as defined for metric spaces does not depend on the chosen metric.

The topological entropy of a pseudogroup is strongly dependent on the choice ofgenerating set, but by the same argument used in the proof of Proposition 2.1.11 one can showthat if Γ′ is another regular generating set for G, then there is a constant C ∈ N such that

s(n, ε,Γ) ≤ s(Cn, ε,Γ′) and s(n, ε,Γ′) ≤ s(Cn, ε,Γ),

and as a consequence1Ch(G,Γ) ≤ h(G,Γ) ≤ Ch(G,Γ′).

This is useful because if the entropy of G is positive for one regular pair, then it must be forall of them. In the same away, if it vanishes for one generating set then it will be 0 for all thegenerators as well. We can therefore distinguish pseudogroups with vanishing entropy from thosewith nonvanishing entropy.

We could define a topological entropy for a foliation F as the topological entropy ofits holonomy pseudogroup, but since it depends on the regular generating set, we should makemention to the chosen regular covering as well. For a regular covering U of a foliated manifold(M,F), let us denote by ΓU and HU , respectively, the regular generating set and holonomypseudogroup associated to U . The entropy of the regular pair (HU ,ΓU) is called the topologicalentropy of F with respect to U . It can also be described in terms of separated and spanningsubsets. Remember that the power set of a metric space can be made into a metric space viathe Hausdorff metric: given two subsets A and B a metric space (X, d), we define D(A,B)to be sup

x∈Ainfy∈B

d(x, y). Remark that in general D(A,B) 6= D(B,A), as one can see considering

A = [0, 1] and B = [0, 2], for instance, for which D(A,B) = 0 and D(B,A) = 1. The Hausdorffdistance between the sets A and B is

ρ(A,B) := maxD(A,B), D(B,A).

Given a foliated manifold (M,F), we consider a Riemannian metric in M and the Hausdorffmetric ρ induced by the distance in M on the complete transversal T of a regular covering U .This Hausdorff metric makes T into a metric space (T , ρ), so the topological entropy with respectto U can be worked out using spanning and separated sets. Geometrically, saying to pointsx, y of M are (n, ε)-separated with respect to U means that either x and y belong to different


charts or there is a chain u = (U1, ..., Un) of charts Ui ∈ U , and chains of plaques (P1, ..., Pn)and (Q1, ..., Qn), such that x ∈ P1, y ∈ Q1, Pi ∪Qi ⊂ Ui for every i and ρ(Pn, Qn) ≥ ε.

As we just saw in Proposition 2.1.29, for G acting on a compact metric space (X, d)the topological entropy is


s(ε,Γ) = lim supn→∞

1n

ln s(n, ε,Γ),

which is just the limit from 2.1, that is, the limit that characterises the growth type of s(·, 1k,Γ).

This is how the topological entropy was first defined for pseudogroups acting on compact metricspaces [26]. In particular, the equality above shows that the entropy will be nonvanishing if andonly if egr(G) is quasi-exponential, so whether a foliation F has positive topological entropy ornot depends only on its expansion growth egr(F) . With that in mind, the next two propositionsfollow immediately from Proposition 2.1.21 and its Corollary:

Proposition 2.1.30. Let G be a regular pseudogroup of C1 diffeomorphisms acting on a closedinterval of R and Γ a regular generator. If G admits a resilient orbit then h(G,Γ) is nonvanishing.

Proposition 2.1.31. If a C1 foliation F of a compact manifold M has a resilient leaf then itstopological entropy with respect to any regular covering U is positive.

It also follows directly from previous examples:

Example 2.1.32. (i) In accordance with Example 2.1.15 pseudogroups of local isometrieshave all constant expansion growth, therefore their topological entropies are all zero. As aconsequence, every Riemannian foliation F has vanishing topological entropy.

(ii) The topological entropy of the Hirsch foliation (Example 2.1.20) is nonvanishing.

Example 2.1.33. 2 For a metric space (X, d) and a homeomorphism f : X → X, consider thepseudogroup G(f) and the regular generator Γ = id, f, f−1. Note that if g ∈ Γn then g = fk

for some k ∈ Z such that |k| ≤ n. If a subset A ⊂ X is (n, ε,Γ)-separated then given any twopoints x, y ∈ A the is a |k| ≤ n such that d(fk(x), fk(y) ≥ ε. Hence, the points f−n(x) andf−n(y) are such that d(fn+k(f−n(x)), fn+k(f−n(y))) ≥ ε, thus f−n(A) is (2n, ε, f)-separated.Conversely, f−n(A) being (2n, ε, f)-separated is equivalent to A being (n, ε, f)-separated, whichclearly implies that A is s(n, ε,Γ)-separated. One arrives then at the conclusion that

A is s(n, ε,Γ)-separated ⇐⇒ f−n(A) is s(2n, ε, f)-separated2 could this be adapted to work in general, without the metric d?


and henceforths(n, ε,Γ) = s(2n, ε, f),

which implies s(e,Γ) = 2s(ε, f) and therefore

h(G(f),Γ) = 2htop(f),

emphasising Remark 2.1.26.

2.1.3 Pseudo-orbits and topological entropy

Pseudo-orbits can be traced back up until at least Birkhoff [3] – under the name ofδ-chains – and have by now established themselves as an extremely useful conceptual tool inthe theory of Dynamical Systems, being particularly useful for numerical computations andestimates. Intuitively, a pseudo-orbit of a dinamical system f : X → X is an “orbit withcontrolled errors”: a sequence of points whose distance to the points of a orbit of f does notexced a given fixed real number. In this paragraph we exhibit a definition of pseudo-orbit forpseudogroups acting on compact metric spaces, and show how they can be used to calculate thetopological entropy. All these results were obtained by Biś and Walczak in [5].

Nothing in our definitions so far says the entropy of the pair (G,Γ) has to be finite,but there are certain conditions one can impose in order to achieve such finiteness. In particular,if G acting on a metric space is a pseudogroup of locally Lipschitz homeomorphisms then wecan reproduce the steps in the proof of Proposition 2.1.18 in order to get an estimate like theone in 2.3, which implies that the topological entropy is finite. With that in mind we fix, for therest of this section, a compact metric space (X, d) and a regular pair (G,Γ) acting on X.

Recall that the orbit Gx of point x ∈ X was defined as the set of all possible imagesof x under elements of G, and remark that this is equivalent to saying that Gx = g(x); g ∈Γ∞, x ∈ Dg, where by Dg we denote the domain of g, as usual. We consider the sets

Gx := g ∈ Γ∞;x ∈ Dg ⊂ G,

so that we can writeGx = g(x); g ∈ Gx.

A point x can then be thought of as an evaluation mapping g 7→ g(x) with domain Gx andrange X. Its image is exactly the orbit Gx, as previously defined. By rewriting the relationx ∈ Dhg ⇔ g(x) ∈ Dh we can see the map x satisfies the condition

h g ∈ Gx ⇔ x(g) ∈ Dh

and the equalityx(h g) = h(x(g)).


This point of view motivates the definition of a pseudo-orbit as a particular type of map from Gto X, as follows.

Definition 2.1.34 (α-pseudo-orbit). Let α be a nonnegative real number. An α-pseudo-orbitof the pair (G,Γ) is a mapping

x : Dx → X

satisfying:

(i) idX ∈ Dx;

(ii) Dx ⊂ Gx(idX);

(iii) for any h ∈ Γ and g ∈ Dx one has

h g ∈ Dx ⇔ x(g) ∈ Dh.

Furthermore, in this case the inequality

d(h(x(g)), x(h g)) ≤ α (2.10)

holds.

Let us say that x(idX) = p ∈ X. When α = 0, condition (iii) and induction on thesets Γk can be used to conclude that if g ∈ Gp then g ∈ Dx. Together with condition (ii) weconclude immediately that Dx = Gp. In particular, we can set g = idX in 2.10 to obtain theequality x(h) = h(p) for every h ∈ Γ, hence x is just the usual G-orbit of the point p. Besidesthat, if G = G is a group of global transformations then Γ∞ = G and condition (iii) impliesDx = G. In particular, if G = Z is generated by a single transformation f : X → X then themap x : Z→ X is just the sequence of iterated images of p, and 2.10 becomes the condition

d(f(xj), xj+1) ≤ α ∀j ∈ Z,

that is, we recover the classical definition of pseudo-orbit for the discrete dynamical system(X, f).

We denote by Υα the set of all α-pseudo-orbits of the pair (G,Γ). It follows immedi-ately from 2.10 that Υα ⊂ Υβ whenever α < β. What is more, from the above discussion weconclude that Υ0 = ∩αΥα is the space of all G-orbits.

Next, we give each Υα a suitable topology and define a natural pseudogroup actingon each of these topological spaces.


Definition 2.1.35. Given α-pseudo-orbits x, y ∈ Υα, note that the intersection idX ∈ Dx ∩Dy ∩ Γk ⊂ Γk is nonempty and finite for every natural k, so we can define

d0(x, y) :=∞∑k=0

12k maxd(x(g), y(g)); g ∈ Γk ∩ Dx ∩ Dy.

Moreover, we can consider all the possible finite sequences in Υα beginning at x and ending aty and define

d1(x, y) := inf

n−1∑j=0

d0(zj, zj+1); (zj)nj=1 ∈ Υα, z0 = x and zn = y

.Proposition 2.1.36. For every α ≥ 0 the space (Υα, d1) is a compact metric space.

Proof. Let us show that d1 is a distance function. Note first that since d is a metric on X

the application d0 : Υα × Υα → [0,∞) is symmetric, satisfies the triangle inequality and thecondition d0(x, x) = 0. It is immediate that d1 is nonnegative.

Symmetry. Given a finite sequence zj with z0 = x and zn = y, consider the sequence zj := zn−j.Then zj is a finite sequence from y to x and, since d0 is symmetric, one has

∑j

d0(zj, zj+1) =∑j

d0(zj, zj+1). From this we can conclude that d1(x, y) = d1(y, x).

Triangle inequality. Follows from the fact that d0 satisfies the triangle inequality and theelementary property inf(A+B) ≤ inf A+ inf B.

Identity of indiscernibles The first thing to notice is that d1(x, x) = 0 holds. This is a consequenceof the fact that d0(x, x) = 0 and the constant sequence zj = x is such that

∑j

d0(zj, zj+1) = 0.

To conclude the proof that the identity of indiscernibles holds, and hence d1 is indeed a metric,we have yet to show that d1(x, y) = 0⇒ x = y. It is sufficient to prove that

d1(x, y) = 0⇒ Dx ∩ Γk = Dy ∩ Γk and x = y on Γk ∩ Dx ∩ Dy ∀k ∈ N. (2.11)

We shall do so by induction on k. Since d0 is defined is a maximum, if z0 = x andzn = y then ∑

j

d0(zj, zj+1) ≥∑j

d(zj(idX), zj+1(idX)) ≥ d(x(idX), y(idX)).

Hence d1(x, y) = 0 implies x(idX) = y(idX) = p, showing that 2.11 holds for k = 0.

Now, suppose it holds for k − 1. Since Dx ∩ Γk is exactly the set of pseudogrouptransformations of length k in the domain of x, we can describe it in terms of Dx ∩ Γk1 as usingcondition (iii) of Definition 2.1.34:

Dx ∩ Γk = h g;h ∈ Γ, g ∈ Dx ∩ Γk−1 and x(g) ∈ Dh.


Of course, the same applies to y, giving

Dy ∩ Γk = h g;h ∈ Γ, g ∈ Dy ∩ Γk−1 and y(g) ∈ Dh.

The induction hypothesis implies immediately the equality Dx ∩ Γk = Dy ∩ Γk.

On the other hand, since all the sets involved are finite, we can find ε > 0 such that

B(x(g), ε) ⊂ Dh

for every g ∈ Dx ∩ Γk−1 and every h ∈ Gx(g). As d1(x, y) = 0 there must be a sequencex = z0, ..., zn = y in Υα for which

∑i

d0(zi, zi+1) < ε/2. Then for every g ∈ Dx ∩ Γk−1 and h ∈ Γ

whose domain contains x(g) we have

d(x(g), zj(g)) ≤∑i≤j

d(zi(g), zj+1(g)) ≤∑i≤j

d0(zi, zi+1) < ε

2 ,

implying that zj(g) ∈ Dh, or equivalently, that g = h g ∈ Dzj , for every 1 ≤ j < m. Inparticular,

d(x(g)), y(g) ≤n−1∑i=0

d(zi(g), zi+1(g)) ≤n−1∑i=0

d0(zi, zi+1) < ε, ∀g ∈ Dx ∩ Dy ∩ Γk.

Since ε can be taken arbitrarily small, this means x(g) = y(g) for all g ∈ Dx ∩ Dy ∩ Γk, as wewanted.

All we have to show now is that (Υα, d1) is compact. Given a sequence (xn)n∈N inΥα, we construct a cluster point using a standard diagonal argument. Due to the compactnessof X there is a point p and a subsequence (x0

n)n of (xn)n such that

x0n(idX)→ p.

Define x(idX) := p. Now consider the set Dx ∩ Γ := Gp ∩ Γ. Once again due to the compactnessof X we can find a subsequence (x1

n)n of (x0n)n and a family of points ph ∈ X;h ∈ Dx ∩ Γ

such thatx1n(h)→ ph.

We define x on Dx ∩ Γ by x(h) := ph and the set

Dx ∩ Γ2 := h h;h ∈ Dx ∩ Γ and h ∈ Gph ∩ Γ.

Proceeding in this manner one can construct, for every natural k, a set Dx ∩ Γk, a family ofpoints pg ∈ X indexed by the elements of Dx ∩ Γk and a subsequence (xkn)n of (xk−1

n )n such that

xkn(g)→ pg for every g ∈ Dx ∩ Γk.


We define then Dx := ∪k (Dx ∩ Γk) ∪ idX and the map

x : Dx −→ X

g 7−→ pg

so that the diagonal subsequence (xnn)n of (xn)n converges to x.

We claim that x ∈ Υα. Conditions (i), (ii) and the first part of condition (iii) fromDefinition 2.1.34 are satisfied by construction. All we need to prove is that x satisfies inequality2.10. For h ∈ Dx ∩ Γ there is n0 ∈ N such that x0

n(idX) ∈ Dh for every n > n0, as the sequencex0n(idx)→ p ∈ Dh, by construction. Hence x0

n(idX) ∈ Dh, and consequently h idX = h ∈ Dx0n

for sufficiently large n. Each term of the sequence x0n is an α-pseudo-orbit, thus

d(x0n(h), h(x0

n(idX))) < α for every n > n0.

By letting n → ∞ we conclude that d(x(h), h(x(idX))) < α for every h ∈ Γ whose domaincontains p. The same argument works for an element of Dx ∩ Γ2, and indeed for any k. Ifh g ∈ Dx ∩ Γk+1, with g ∈ Dx and h ∈ Gpg ∩ Γ, then for every n large enough the point xkn(g)belongs to Dh and the the inequality

d(xkn(h g), h(xkn(g))) < α

is satisfied, hence 2.10 holds for h g, and x ∈ Υα. We just shown that every sequence in Υα

admits a cluster points, so that (Υα) is a compact metric space, as we wanted.

Now that the spaces Υα have been conveniently topologised, given a pseudogrouptransformation g ∈ G, let us take a closer look at the set

Ug := x ∈ Υα; g ∈ Dx.

A mild modification of the induction argument used to prove that identity of indiscernibles ford1 applies here to show that Ug is open. Indeed, if g = idX then Ug = Υα is open. Suppose Ug isopen for every pseudogroup transformation g of length k − 1, and consider x ∈ Ug for some g oflength k. Then g is an element of the set Dx ∩ Γk = h g; g ∈ Dx ∩ Γk, h ∈ Gx(g) ∩ Γ. As Ug isopen by assumption, there is δ0 > 0 such that d1(x, y) < δ0 implies y ∈ Ug. Moreover, we canfind δ1 > 0 for which B(x(g), δ1) ⊂ h for every h ∈ Gx(g) ∩ Γ. Let δ < minδ0, δ1 and we havethat if d1(x, y) < δ < δ0 then g ∈ Dy, and more than that,

d(x(g), y(g)) ≤ d0(x, y) ≤ d1(x, y) < δ < δ1,

implying that y(g) ∈ Dh, and consequently g = h g ∈ Dy. Thus y ∈ Ug and the latter is open.


Each g ∈ G induces a local transformation of Υα whose domain is the open setUg. The image of each point x of Ug is again a pseudo-orbit of Υα with domain Dg,x := f ∈Γ∞; f g ∈ Dx. More accurately, the transformation induced by g is given by

Θg : Ug −→ Υα

x 7−→ Θgx : Dg,x −→ X

f 7−→ x(f g).

From the definition of Dg,x and the fact x is an α-pseudo-orbit it follows immediatelythat Θgx is also an α-pseudo-orbit. Remark that ΘidX is defined on all of Υα, that the domainsDidX ,x of its images are just the sets Dx and that it satisfies

ΘidXx(f) = x(f idX) = x(f)

for every x ∈ UidX and f ∈ Dx, hence ΘidX = idΥα . Note as well that when Θgx ∈ Uh thenh ∈ DΘgx = Dg,x and we can write

(Θh Θg)x(f) = Θh(Θgx)(f) = Θgx(f h) = x(f h g) = Θhgx(f).

As the pseudogroup element f was arbitrarily chosen in the domain of definition of (Θh Θg)x,we conclude that (Θh Θg)x = Θhgx for every x on the pre-image Θ−1

g (Θg(Ug) ∩ Uh), andtherefore

Θh Θg = Θhg.

In particular, by definition the α-pseudo-orbit Θgx has domain f ∈ Γ∞; f g ∈ Dx, whichcontains g−−1 for any x ∈ Ug. This means Θg(Ug) ⊂ Ug−1 so that we can consider thecomposition Θg−1 Θg for every x in Ug. Same reasoning applies to Θg Θg−1 , so that we haveinclusions Θg(Ug) ⊂ Ug−1 and Θg−1(Ug−1) ⊂ Ug, together with equalities

Θg−1 Θg = Θg−1g = idΥα = Θgg−1 = Θg Θg−1 ,

meaning each transformation Θg has right and left inverses, both being equal to Θg−1 . Thereforeeach Θg is bijective, has range Ug−1 and the relation

Θg−1 = Θ−1g

holds true for every g ∈ G.

Last but not least, if h ∈ Γn then

d0(Θhx,Θhy) =∞∑k=0

12k maxd(Θhx(f),Θhy(f)); f ∈ Γk ∩ Dh,x ∩ Dh,y

=∞∑k=0

12k maxd(x(f h), y(f h)); f ∈ Γk ∩ Dx ∩ Dy

≤∞∑k=0

12k maxd(x(f), y(f)); f ∈ Γn+k ∩ Dh,x ∩ Dh,y

≤ 2nd0(x, y),


and therefore

d1(Θhx,Θhy) = inf

n−1∑j=0

d0(zj, zj+1); (zj)nj=1 ∈ Υα, z0 = Θhx and zn = Θhy

≤ inf

n−1∑j=0

d0(Θhzj,Θhzj+1); (zj)nj=1 ∈ Υα, z0 = x and zn = y

≤ 2nd1(x, y).

In other words, the map Θh associated to an element h ∈ Γ∞ of length n is Lipschitz withLipschitz constant 2n. In particular each one of these maps is continuous, so every transformationΘg is, at least locally, a composition of continuous maps, and therefore each Θg is continuous.As they all have inverses, which are also continuous, they are all local homeomorphisms of Υα.

Summing up the whole discussion, what we have just shown is that the application

Θ : G −→ Homeo(Υα)

g 7−→ Θg

is an action of the groupoid G on the space Υα of α-pseudo-orbits. What is more, the image ofthe generating set Γ is a symmetric subset of Lip(Υα) (recall item (iv) of Example 1.1.3).

Definition 2.1.37. Let Γ(α) := Θ(Γ) = Θh;h ∈ Γ ⊂ Lip(Υα) be the finite symmetric familyof Lipschitz homeomorphisms induced by Γ on Homeo(Υα). Define

G := G(Γ(α)) ⊂ Lip(Υα)

to be the pseudogroup of locally Lipschitz maps acting on the space of all α-pseudo-orbits.

The pair (G, Γ(α)) is always regular, as (Υα, d1) is always compact. Our first resultis to associate the entropy of G acting on X to that of G acting on the space Υ0 of G-orbits.

Proposition 2.1.38. Given the regular pairs (G,Γ) acting on X and (G, Γ(0)) acting on Υ0,then

h(G,Γ) = h(G, Γ(0)).

Proof. Let x, y be points of X and g ∈ G a transformation whose domain contains both of them.Denote also by x and y the evaluation maps associated to the points x, y of X. Then both mapsare elements of Υ0 and

d(g(x), g(y)) = d(x(g), y(g)) ≤ d1(Θgx,Θgy).

Thus (n, ε,Γ)-separated points of X give rise to (n, ε, Γ(0))-separated orbits of Υ0. It followsthat s(n, ε,Γ) ≤ s(n, ε, Γ(0)) and consequently

h(G,Γ) ≤ h(G, Γ(0)).


On the other hand, let x, y be two (n, ε, Γ(0))-separated G-orbits of Υ0, and g ∈ Γnsuch that d1(Θgx,Θgy) ≤ ε. Remark that x, y ⊂ Ug implies x(idX), y(idX) ⊂ Dg, as well asd1(Θgx,Θgy) ≤ ε implies d0(Θgx,Θgy) ≤ ε. Fix l(ε) 0 such that

∑k>l(ε)

12k <

ε

2 diamX.

Note that maxd(Θgx(f),Θgy(f); f ∈ Γk ∩ Dh,x ∩ Dh,y is a monotone increasing function of k,so that it holds

ε ≤ d0(Θgx,Θgy) =∞∑k=0

12k maxd(Θgx(f),Θgy(f)); f ∈ Γk ∩ Dg,x ∩ Dgy

=l(ε)∑k=0

12k maxd(Θgx(f),Θgy(f)); f ∈ Γk ∩ Dg,x ∩ Dg,y

+∑k>l(ε)

12k maxd(Θgx(f),Θgy(f)); f ∈ Γk ∩ Dg,x ∩ Dg,y

< maxd(Θgx(f),Θgy(f)); f ∈ Γl(ε) ∩ Dg,x ∩ Dg,yl(ε)∑k=0

12k + (diamX)

∑k>l(ε)

12k

< maxd(Θhx(f),Θhy(f)); f ∈ Γl(ε) ∩ Dg,x ∩ Dg,y+ ε

2 ,

implying thatmaxd(Θhx(f),Θhy(f)); f ∈ Γl(ε) ∩ Dg,x ∩ Dg,y >

ε

2 .

In particular, the pseudogroup transformation h ∈ Γl(ε) realising the maximum aboveis such that h g ∈ Γn+l(ε) ∩Dx ∩Dy, and we have

d(h g(x(idX)), h g(y(idX))) = d(x(h g), y(h g)) = d(Θgx(h),Θgy(h)) ≥ ε

2 .

Hence x(idX) and y(idX) are (n + l(ε), ε/2,Γ)-separated in X. It follows that s(n, ε, Γ(0)) ≤s(n+ l(ε), ε/2,Γ), then

s(ε, Γ(0)) = lim supn

1n

ln s(n, ε, Γ(0)) ≤ lim supn

1n

ln s(n+ l(ε), ε/2,Γ) = s(ε/2,Γ)

and finallyh(G,Γ) ≤ h(G, Γ(0)).

Next we show that Υ0 can be used as an upper bound for the infimum of entropiesh(cG, Γ(α)). More accurately, we have the following result.

Lemma 2.1.39. Given n ∈ N and a positive ε there exists a strictly positive α and naturall = l(ε) ∈ N such that s(n+ l, ε/6,Γ) ≥ s(n, ε, Γ(α)).


Proof. Let L be a common Lipschitz constant for all the elements g ∈ Γ (recall that we assumedG ⊂ Lip(X)). Let us first show a general fact about the spaces Υα. Given any finite subsetA = ximi=1 of Υα, associate to each of its elements xi is the G-orbit x0

i ∈ Υ0 of the point xi(idX). We claim that

d(xi(g), x0i (g)) ≤ α(1 + L+ · · ·+ Lk−1) ∀1 ≤ i ≤ m, ∀g ∈ Γk ∩ Dxi .

This is a simple induction on k. Fixed a i, for k = 0 we have xi(idX) = x0i (idX) and therefore

d(xi(idX), x0i (idX)) = 0 < α. If the inequality is true for g ∈ Γk ∩ Dxi , then given h ∈ Γ ∩ Gxi(g)

we have

d(xi(h g), x0i (h g)) = d(xi(h g), h(g(xi(idX))))

≤ d(xi(h g), h(xi(g))) + d(h(xi(g))), h(g(xi(idX))))

< α + Ld(d(xi(g), g(x0i (idX))))

= α + Ld(d(xi(g), x0i (g))) ≤ α(1 + L+ · · ·+ Lk).

Now choose l = l(ε) satisfying ∑j>l

2−j < ε

3 diamX,

and fix α > 0 small enough so that it satisfies

α(1 + L+ · · ·+ Ln+l−1) ≤ ε

12 .

Let A be an (n, ε, Γ(α))-separated subset of Υα. Then given two distinct α-pseudo-orbits xi, xjof A, our choice of l implies the existence of a pseudogroup element g ∈ Γn+l such thatd(xi(g), xj(g)) ≥ ε

3 (by the exact same argument as we used to find h in the proof of Proposition2.1.38 above). For such g one has

d(x0i (g), x0

j(g)) ≥ d(xi(g), xj(g))− d(x0i (g), x0

j(g))− d(x0i (g), x0

j(g))

≥ ε

3 − 2α(1 + L+ · · ·+ Ln+l−1)

≥ ε

3 − 2 ε

12 = ε

6 .

This means the set A0 = x0im

i=1 is an (n + l, ε/6, Γ(0)) subset of Υ0, hence s(n + l, ε/6,Γ) ≥s(n, ε, Γ(α)), as we wanted.

We can see from the definition of d0 that the existence of g such that d(x(g), y(g)) = ε

is not a necessary condition for the inequality d0(x, y) > ε to hold, though it is a sufficient one.It could then happen that pseudo-orbits x, y ∈ Υα are (n, ε, Γ(α))-separated without any of thepoints x(g), y(g) being so.


Definition 2.1.40 (Strongly separated pseudo-orbits). We say two pseudo-orbits x, y ∈ Υα

to be (n, ε,Γ)-strongly separated when d(x(g), y(g)) ≥ ε for some g ∈ Γn ∩ Dx ∩ Dy. A subsetA ⊂ Υα whose elements are pairwise (n, ε,Γ)-strongly separated is called an (n, ε,Γ)-stronglyseparated set. Let sα(n, ε,Γ) denote the maximal cardinality of (n, ε,Γ)-strongly separated subsetof Υα.

Remark that this is a entirely new definition and has, a priori, nothing to do with the pseudogroupG acting on Υα, specially because, in general, this strong separation is not equivalent to theseparation of orbits by the action of G on Υα. They do, however, share some features. Forinstance, compactness of X implies that every (n, ε,Γ)-strongly separated set is finite. Theseparation with respect to Γ(α) bounds the strong separation in following sense:

Lemma 2.1.41. sα(n, ε,Γ) ≤ s(n, ε, Γ(α)) for every n ∈ N, ε > 0 and α ≥ 0.

Proof. Comes straightforward from the definition: if pseudo-orbits x, y ∈ Υα are (n, ε,Γ)-stronglyseparated and g is one element in their domains realising this separation, then

ε ≤ d(x(g), y(g)) = d(Θgx(idX),Θgy(idX)) ≤ d1(Θgx,Θgy),

from where the result follows immediately.

Strong separation can be used to define yet another topological entropy for (G,Γ):

Definition 2.1.42 (Pseudo-orbit entropy of the pair (G,Γ)). For any decreasing sequenceδ = (δn)n∈N of positive real numbers such that δn → 0. Set

sδ(ε) := lim supn→∞

1n

ln sδn(n, ε),

sps(ε) := infδsδ(ε),

and finallyhps(G,Γ) := lim

ε→0+sps(ε).

This section’s main result is to prove that this always coincide with the usual entropy of G, sothat what we came up, in the end, as just another way to calculate the topological entropy ofthe pair (G,Γ).

Theorem 2.1.43 (Biś - Walczak). For any pseudogroup G of Lipschitz transformations, actingon a compact metric space X, and any regular generator Γ, the equality

h(G,Γ) = hps(G,Γ)

holds true.


Proof. We begin by noting that strong-separation in Υ0 and Γ-separation in X are the sameconcept: orbits x, y are (n, ε,Γ)-strongly separated if and only if the points x(idX) and y(idX)are (n, ε,Γ)-separated. As Υ0 ⊂ Υα for every positive α this means

s(n, ε,Γ) = s0(n, ε,Γ) ≤ sα(n, ε,Γ),

for every n ∈ N, ε > 0 and α ≥ 0. Taking the appropriate limits gives us

h(G,Γ) ≤ hps(G,Γ).

As for the converse inequality, let L be a common Lipschitz constant for all theelements of Γ and l(ε) a function defined as in Lemma 2.1.39. Given any ε > 0, let δ = (δn) be asequence bounded above by the sequence(

ε

12

) 11 + L+ · · ·+ Ln+l(ε)−1 .

Lemmas 2.1.39 and 2.1.41, together, yield

sps(ε) ≤ sδ(ε) ≤ lim supn

1n

ln s(n, ε, Γ(δn))

≤ lim supn

1n

ln s(n+ l, ε/6, Γ(0))

≤ lim supn

1n+ l

ln s(n+ l, ε/6, Γ(0))

= s(ε/6, Γ(0)).

Taking limits we conclude that hps(G,Γ) ≤ h(G, G(0)), and consequently, due to Proposition2.1.38,

hps(G,Γ) ≤ h(G,Γ),

providing us with the desired result.

The proof above also provides a more convenient way of calculating a pseudogroup’s entropy, interms of its pseudo-orbits.

Corollary 2.1.44. Fix L a Lipschitz constant for the family Γ and l as in Lemma 2.1.39. Letαn := (1 + L+ · · ·Ln+l−1)−1. Then


lim supn→∞

sεαn(n, ε,Γ).


2.1.4 A topological entropy for pseudogroups acting on noncompact sets

Bowen [8] defined a notion of entropy for transformations between noncompact sets,based on the concept of Hausdorff dimension. In this section we adapt the definitions anddemonstrations given by Bowen in order to define a topological entropy for pseudogroups actingon any topological space. We also show this notion coincides with the one we gave earlier forcompact sets.

Let X be any topological space and Y ⊂ X. For the moment, fix an finite opencovering A = Aii of X. For a subset E of X, we write E ≺ A if E ⊂ Ai for some i, and for afamily E = Ei we write E ≺ A if Ei ≺ A for every i. Suppose (G,Γ) is a regular pair actingon X.

Definition 2.1.45 (Subordinated chain). Let E ⊂ X. An element g ∈ G \ idX is said to be achain subordinated to A for E of length n if

(i) E ⊂ Dg;

(ii) g = gn gn−1 · · · g1, where each gi ∈ Γ;

(iii) gi gi−1 · · · g1(E) ≺ A for every i = 1, ..., n.

To each subset E we associate the nonnegative integer

nΓ,A(E) := max|g|+ 1; g is a subordinated chain for E.

In other words, if nΓ,A(E) = k then E admits a subordinated chain of length k − 1, while forany other g ∈ G whose domain contains E and length |g| ≥ k we have g(E) ⊀ A. If E ⊀ Athen nΓ,A(E) = 0. If there is no element of Γ with E in its domain other than the identity thenwe convention that nΓ,A(E) = 0 as well. When E admits subordinated chains of length n forevery natural n we say nΓ,A(E) =∞. The topological diameter of E with respect to (G,Γ) is

DΓ,A(E) := e−nΓ,A(E).

For a countable family E = Ei define

DΓ,A(E , λ) :=∑i

DΓ,A(Ei)λ =∑i

e−λnΓ,A(Ei).

Then for Y ⊂ X we set

mΓ,A(Y, λ) := lim infε→0

DΓ,A(E , λ);DΓ,A(Ei) < ε for every i and ∪i Ei ⊃ Y ,

HA(G,Γ, Y ) := infλ;mΓ,A(Y, λ) = 0,


H(G,Γ, Y ) := supAhA(G,Γ, Y ).

We say H(G,Γ, Y ) is the topological entropy of G acting on Y . Our main objective isto show it is the same as 2.5 when X is compact, providing thus a suitable generalization forthe noncompact case. We are going to use the following lemma:

Lemma 2.1.46. If E = Ei is a family of subsets of X such that DΓ,A(E , λ) < a then thereexists a family E ′ = E ′i of open sets E ′i ⊃ Ei such that DΓ,A(E ′, λ) < a.

Proof. We construct an open set E ′i associated to each Ei while keeping track of the numbersnΓ,A.Consider first the case nΓ,A(Ej) = k. Then there is some g ∈ G whose length is k − 1 writtenas g = gk−1 · · · g1 such that gi · · · g1(Ej) ≺ A for all i ≤ k − 1. Each gi · · · g1 is anhomeomorphism between open sets with relatively compact domain, so there is an open Ui suchthat gi · · · g1(Ej) ⊂ Ui ⊂ Ai ∈ A. Define

E ′j =k−1⋂i=1

(gi · · · g1)−1(Ui).

E ′j is open, Ej ⊂ E ′j and nΓ,A(E ′j) = nΓ,A(Ej) since for any g ∈ G with |g| ≥ k we haveg(Ej) ⊂ g(E ′j) and g(Ej) ⊀ A.Now note that there is a partition E = E0 ∪ E∞ consisting of the families E0 of sets Ej forwhich nΓ,A(Ej) is finite and E∞ = Ei1 , ..., Ein , ... for which nΓ,A(Ein) =∞. Then the followingequality holds

DΓ,A(E , λ) =∑Ej∈E0

DΓ,A(Ej).

We choose any sequence ki1 , ..., kin , ... of natural numbers such that∑j

eλkij < a−DΓ,A(E , λ).

Since nΓ,A(Ein) =∞ for each Ein we can choose a chain g subordinated to A for Ein of lengthkin − 1, and for this g we construct an open E ′in as we did before. Then nΓ,A(Ein) ≥ kin and thefamily E ′ of all the constructed open sets satisfies

DΓ,A(E ′, λ) =∑Ej∈E0

DΓ,A(E ′j) +∑Ej /∈E0

DΓ,A(E ′j) < DΓ,A(E , λ) + a−DΓ,A(E , λ) = a,

as we wanted.

Theorem 2.1.47. For a regular pair (G,Γ) acting on a compact X the equality H(G,Γ, X) =h(G,Γ) holds.


Proof. Let E be a subcovering of A(n)Γ with N(G,Γ,A, n) elements. Then for any Ei ∈ E one has

g(Ei) ≺ A for all g ∈ G with |g| = n− 1 and Ei ⊂ Dg, thus nΓ,A(Ei) ≥ n for all i. This implies

DΓ,A(E , λ) =∑i

DΓ,A(Ei)λ =N(G,Γ,A,n)∑

i=1e−λnΓ,A(Ei) ≤ N(G,Γ,A, n)e−λn,

and therefore

mΓ,A(X,λ) = lim infε→0

DΓ,A(E , λ);DΓ,A(Ei) < ε,X ⊂ ∪iEi

≤ lim infe→0

N(G,Γ,A, n)e−λn

= limn→∞

N(G,Γ,A, n)e−λn

= limn→∞

(e(

1n

lnN(G,Γ,A,n)−λ))n.

If we assume λ > h(G,Γ,A) = limn→∞

1n

lnN(G,Γ,A, n) then there exists a strictly positive real

number ρ such that 1n

lnN(G,Γ,A, n)− λ < −ρ < 0 for all n ∈ N. Consequently

limn→∞

(e(

1n

lnN(G,Γ,A,n)−λ))n≤ lim

n→∞e−ρn = 0,

thus mΓ,A(X,λ) = 0. It follows that

HA(G,Γ, X) = infλ;mΓ,A(X,λ) = 0 ≤ h(G,Γ,A). (2.12)

In order to finish the proof, we claim mΓ,A(X,λ) = 0 implies λ ≥ h(G,Γ,A). Indeed, ifmΓ,A(X,λ) = 0 we can find a covering E of X with DΓ,A(E , λ) < 1. We can assume E is anopen covering without loss of generality, due to Lemma 2.1.46. Let I = E1, ..., Em be a finitesubcovering. For a sequence (Ej1 , ..., Ejs) of elements of I, define the numbers

nΓ,A(Ej1 , ..., Ejs) :=s∑r=1

nΓ,A(ejr).

We construct open sets U(Ej1 , ..., Ejs) in the following way: for each Ejr , 1 ≤ r ≤ s, let gr be amaximal subordinated chain for Ejr , that is, a subordinated chain whose length is nΓ,A(Ejr)− 1.If Rgr−1 ∩Dgr 6= ∅ for all 2 ≤ r ≤ s− 1, let g := gs−1 · · · g1. Define U(Ej1 , ..., Ejs) := ∪Dg forall g constructed in this way. If no such g exists, then U(Ej1 , ..., Ejs) := ∅. Each g is a chain of

lengths−1∑r=1

nΓ,A(Ejr) subordinated to A for U(Ej1 , ..., Ejs), and it has the property that for each

r the local homeomorphism gr · · · g1 has its image contained in Ejr .By construction U(Ej1 , ..., Ejs) ≺ A

(n)Γ for every n ≤ nΓ,A(Ej1 , ..., Ejs). Since the composition

gs · · · g1 is an subordinated chain for U(Ej1 , ..., Ejs), we conclude nΓ,A(U(Ej1 , ..., Ejs)) ≥nΓ,A(Ej1 , ..., Ejs).Fix M = maxnΓ,A(Ei);Ei ∈ I. For any natural n the family

U = U(Ej1 , ..., Ejs); s ≥ 1 and n ≤ nΓ,A(Ej1 , ..., Ejs) ≤ n+M


is an open covering of X. Indeed, given x ∈ X, there is Ej1 such that x ∈ Ej1 . Take a maximalsubordinated chain g1 for Ej1 and consider now g1(x). It lies in some Ej2 ∈ I, for whichwe choose a maximal subordinated chain g2. We now consider an open Ej3 of I containingg2(g1(x)), and so fort. Proceeding like this, at some point after a finite amount of steps thesum nΓ,A(Ej1) + · · · + nΓ,A(Ejs) will be a natural number in the interval [n, n + M ], and forg = gs · · · g1 we have x ∈ Dg ⊂ U(Ej1 , ..., Ejs).The open covering U is subordinated to A(n)

Γ , thus N(G,Γ,A, n) ≤ #U and

N(G,Γ,A, n)e−λ(n+M) ≤ #Ue−λ(n+M)

≤ DΓ,A(U , λ)

≤∑e−λnΓ,A(Ej1 ,...,Ejs );n ≤ nΓ,A(Ej1 , ..., Ejs) ≤ n+M,

that is,

N(G,Γ,A, n)e−λn) ≤ eλM∑e−λnΓ,A(Ej1 ,...,Ejs );n ≤ nΓ,A(Ej1 , ..., Ejs) ≤ n+M. (2.13)

In the beginning of the construction we chose E such that DΓ,A(E , λ) < 1. This implies thatDΓ,A(I, λ) < 1 and

∞∑s=1

m∑j1,...,js=1

e−λnΓ,A(Ej1 ,...,Ejs ) =∞∑s=1DΓ,A(I, λ)s <∞.

In particular the right side of 2.13 is bounded in n, hence

limn→∞

1n

lnN(G,Γ,A, n)e−λn)

≤ limn→∞

1n

ln(eλM

∑e−λnΓ,A(Ej1 ,...,Ejs );nΓ,A(Ej1 , ..., Ejs) ∈ [n, n+M ]

)≤ lim

n→∞

K

n

The left side of the equation above goes to h(G,Γ, n)−λ, while the right side goes to 0. It impliestherefore that h(G,Γ, n) ≤ λ, as we claimed. In particular, HA(G,Γ, X) = infλ;mΓ,A(X,λ) =0 ≥ h(G,Γ,A). Together with 2.12 the two inequalities give HA(G,Γ, X) = h(G,Γ,A) andconsequently H(G,Γ, X) = h(G,Γ).

In his work [4], Biś gives yet another more general way of calculating the entropy ofa pseudogroup, based on the work of Pesin and the so-called C-structures.

2.2 The geometric entropy of a foliationIn this section, we define yet another entropy for a foliation F on a compact manifold

M , the geometric entropy, as was first done by Ghys, Langevin and Walczak [26]. It has the


advantage of being independent of the choice of regular covering, but it depends on a choiceof Riemannian metric for M . We will, however, be able to distinguish between foliations withvanishing and nonvanishing entropy like it happens for the topological entropy of a pseudogroup.

Fix a Riemannian metric g for the compact foliated manifold (M,F). Let d be themetric induced on M by g, and denote by dF the metric induced on the leaves by the restrictionon each leaf of the metric tensor g. For a real number r we denote by B(x, r) the open ball of din M . We assume the foliation F is smooth. Consider the exponential map expx : TxM →M ,which maps a vector v to the endpoint of the unique geodesic passing through x with velocityvector v. We denote by expFx : TxF → F(x) the exponential map associated to the restrictiong|F(x), that is, to the metric tensor gF . An open ball of radius r in TxF will be denoted byBF(vx, r). By T⊥x F we denote the orthogonal complement TxF(x)⊥ of TxF(x) in TxM . Theopen ball of radius r centred in vx in this normal space is B⊥F (vx, r).

Due to the compactness of M there is a global injectivity radius δ1 for the expo-nential map of g. This means that for each point x ∈ M the carries the open ball B(0x, δ1)diffeomorphically onto the open B(x, δ1) of M . A classical theorem by Whitehead [58] andthe compactness of M guarantee the existence of a second positive constant (possibly infinity)δ2 such that expFx maps the open ball BF(0x, δ2) diffeomorphically onto an convex subset ofF(x), which we denote by BF(x, δ2). For δ < δ1 the open normal ball B⊥F (0x, δ) is mappeddiffeomorphically onto an embedded open q-ball, where q = codimF . By shrinking B⊥F (0x, δ) ifnecessary, we can consider the image B⊥F (x, δ) := expx(B⊥F (0x, δ)) transverse to each leaf of F itintersects. The supremum δ3 of all such δ still has this property and is therefore the maximalsuch. 3 We define δ0 := minδ1, δ2, δ3.

Given points x, y ∈ M , suppose B⊥F (x, δ0) and BF(y, δ0) intersect. Since they aretransverse and codimB⊥F (x, δ0)+codimBF(y, δ0) = dimM the intersection B⊥F (x, δ0)∩BF(y, δ0)is a discrete collection of points. We can then choose a sufficiently small ε such that d(x, y) < ε

implies the intersection consists of a single point py(x). The supremum ε of all such choices for εis also a maximal choice.

We fix now, for the moment, a regular covering U for (M,F). By choosing a Lebesgue

number λ for the covering U and setting ε0 := minδ0, ε,

λ

2

we guarantee that for each

x ∈ U,U ∈ U , and y ∈M such that d(x, y) < ε0 there is a unique orthogonal projection py(x)of x onto the plaque P (y) of U passing through y, satisfying

B⊥F (x, δ0) ∩BF(y, δ0) = py(x).

All the objects involved vary continuously in M , so this orthogonal projection is a continuousmapping.3 Esclarecer mais?


For a leaf curve γ : I → F(x) starting at x = γ(0), choose a chart U containing x andany point y ∈ B(x, ε0). There is a t1 ∈ I such that d(γ(t), y) < ε0 for every t ≤ t1. This means theentire segment of curve γ([0, t1]) can be projected onto the leaf F(y). If d(γ(t1), py(γ(t1))) < ε0

we can again (choosing a new chart if necessary) project another segment γ([t1, t2]) onto theleaf F(y). By proceeding like this until we reach the end point of γ or the distance betweenγ(tn) and py(γ(tn)) exceeds ε0, we construct a leaf curve pyγ : [0, tn] → F(y) starting in theplaque of U containing the point y. In short, this orthogonal projection of the leaf curve γ canbe described as

pyγ(t) = expγ(t) X(t),

whereX : M → T⊥F is a continuous vector field along γ such that |X| < ε0, expγ(0) X(0) ∈ P (y)and expγ(t) X(t) ∈ F(y) for all t for which X is defined.

To define the geometric entropy we will consider how much the foliation F separatespoints in the complete transversal, much in the same way we looked at how much the elementsof the pseudogroup separated the points of a topological space in Definition 2.1.10. Here, insteadof the length of a pseudogroup element we will consider the length l of leaf curves with respectto the Riemannian metric tensor g. As each point in the determines its leaf completely, thisseparation of points by the leafs of F can be thought of as a measure of how much the leavesthemselves are separating from one another as we walk along then. To simplify things a little, weshall always assume a leaf curve γ is entirely projectible, shrinking it if necessary. The definitionthen reads as follows.

Figure 8 – Points x and y separated by F .

Definition 2.2.1 (Geometric separation of leaves). For ε ∈ (0, ε0) and R ∈ R, two pointsx, y ∈ M are said to be (R, ε)-separated by F if either d(x, y) ≥ ε or there is a leaf curve


γ : I → F(x) such that γ(0) = x, l(γ) ≤ R and d(γ(1), pyγ(1)) ≥ ε.

A set A ⊂M is (R, ε)-separated if every pair of distinct points in A is (R, ε)-separated.

As for (n, ε)-separated sets with respect to a pseudogroup, the (R, ε)-separated subsets of Mare always finite due to the compactness of M . Let s(R, ε,F) be the maximal cardinality of a(R, ε)-separated subset of M , and

s(ε,F) := lim supR→∞

1R

ln s(R, ε,F).

Definition 2.2.2 (Geometric Entropy of F). The limit

hg(F) := limε→0+

s(ε,F)

is called the geometric entropy of the foliation F on the compact Riemannian manifold (M, g),with respect to the metric g.

There is the dual concept of (R, ε)-closeness with respect to F : two points x, y are(R, ε)-close when they are not (R, ε)-separated, that is, when d(x, y) < ε and for every leaf curveγ : I → F(x) and ε < ε0 one have d(γ(1), pyγ(1)) < ε. In the same way as before, we can defineF-open balls

BF(x;R, ε) := y ∈M ;x, y are (R, ε)-close with respect to F.

This is indeed open in the topology of M . A set A is said to be (R, ε)-spanning if for everyy ∈M there is a point x ∈ A such that y ∈ BF(x;R, ε). Of course, when this is the case one has

M =⋃x∈A

BF(x;R, ε),

so that for every positive choices of R and ε there are finite (R, ε)-spanning sets. We set r(R, ε,F)be the minimal cardinality of a (R, ε)-spanning subset of M , and

r(ε,F) := lim supR→∞

1R

ln r(R, ε,F).

Once we replace the notation accordingly, the exact same proof as Proposition 2.1.29 gives usthe equality

hg(F) = limε→0+

s(ε,F) = limε→0+

r(ε,F). (2.14)

The regular covering U was used only to set the constant ε0 and plays a minor rolein the construction. Another choice of regular covering would impact only on this constant andhave no effect on the geometric entropy. Indeed, the geometric entropy is indeed independent ofthis choice of regular atlas – which is why the notation hg(F) makes no mention to the regular


covering – as will be become clear once we prove this section’s main result, namely Theorem2.2.10. Until then, assume we are using a fixed regular covering U .

The role of the metric tensor g has less to do with the general distance betweenpoints of M and more with the distance induced in the leaves. Indeed, the geometric entropyof F is only sensitive to the leafwise metric gF , that is, the important role is played by therestriction of g to subbundle TF of the vectors tangent to the leaves. To see this, assume wehave to distinct Riemannian tensors g and g′ on M that are homothetic when restricted to TF :

Proposition 2.2.3. If c ∈ R is a positive constant such that the metric tensors g and g′ on(M,F) satisfy

g′(v, v) = c2g(v, v) ∀v ∈ TF ,

then hg′(F) = 1chg(F)

Proof. Let d be the distance induced by g and d′ the distance induced by g′. Remark thatd′ = cd. Let γ : I → F(x) be a leaf curve starting at x. For ε smaller than both constants ε0and ε′0 associated to g and g′, denote by pyγ and p′yγ the projections of γ with respect to g andg′, respectively. For possible even smaller values of ε, there is a constant a > 1 such that

d(γ(1), p′yγ(1)) ≤ ad(γ(1), pyγ(1)).

Then points (R, ε)-separated with respect to g are(cR,

ε

a

)-separated with respect to g′, and

consequently

s(R, ε,F) ≤ s′((cR,

ε

a

),F)

and s(ε,F) ≤ 1cs′((

ε

a

),F).

It follows that hg(F) ≤ 1chg′(F). Now, using that g = c−2g′, the exact same construction gives

us hg′(F) ≤ chg(F), which completes the proof.

Now, since our manifold M in question is compact, any two Riemannian metrics are quasi-isometric, that is, there is a positive real constant c such that

1c2g(v, v) ≤ g′(v, v) ≤ c2g(v, v) (2.15)

for all v ∈ TM . By using the exact same argument as in proof of Proposition 2.2.3 we get thefollowing result:

Proposition 2.2.4. If two metric tensors g and g′ are such that their restrictions to TF ⊗TFare quasi-isometric, that is, they satisfy 2.15 for all v ∈ TF then

1chg(F) ≤ hg′(F) ≤ chg(F).

In particular, if the Riemannian metrics coincide in TF ⊗ TF then the associated geometricentropies are the same.


This means that even though the geometric entropy is ultimately dependent on the choice ofRiemannian metric, it is always possible to distinguish between foliations with positive entropyand those with vanishing entropy, as was the case with pseudogroups and choices of regulargenerating sets.

The parallels and correlations between the topological entropies of holonomy pseu-dogroups and the geometric entropy of F run even deeper. Our first result in this directionreads as the following proposition:

Theorem 2.2.5. Given a foliation F on a compact Riemannian manifold (M, g), let U bea regular covering of F . Denote by diam(U) the maximum of the diameters of its plaques,measured with respect to the metric induced on the leaves by g. Then

h(HU ,ΓU) ≤ diam(U)hg(F).

In order to prove this statement we will need the following lemma and subsequentproposition:

Lemma 2.2.6. For a chart (ϕ,U) ∈ U and points x, y ∈ U there exists a positive constant csuch that

ρ(P (x), P (y)) ≤ cd(x, y),

where ρ denotes the Hausdorff metric in TU , and by P (x) and P (y) we mean the plaques of Ucontaining x and y, respectively.

Proof. Let ϕ = (ϕ′, ϕ′′). Due to regularity, ϕ(U) is a unitary cube in Rp × Rq and ϕ has anextension (ϕ, U) with U ⊂ U . Denote by µ the standard Euclidian metric in Rp × Rq and g theRiemannian metric ofM . By employing bump functions we can construct a Riemannian metric g′

forM such that g′|U = ϕ∗µ and g′|M\U = g, that is, g coincides with the pullback of the standardEuclidian metric in a neighbourhood of U and with g outside of U . By construction, the Hausdorffmetric ρ′ of T – induced by the Riemannian metric g′ – coincides with the Euclidian metric inRq, in the sense that for any two plaques P and Q of U we have ρ′(P,Q) = µ(ϕ′′(P ), ϕ′′(Q)).Moreover, for any two points x, y ∈ U we have

ρ′(P (x), P (y)) ≤ d′(x, y),

where d′ is the distance on M induced by g′. Finally, the compactness of M implies g and g′

are quasi-isometric, hence d′ ≤ Bd for some constant B. The quasi-isometry between g and g′

implies the induced Hausdorff metrics ρ and ρ′ are quasi-isometric as well, therefore there isanother constant A such that ρ ≤ ρ′. If we let c = AB then

ρ(P (x), P (y)) ≤ cd(x, y),


which is exactly what we wanted.

Proposition 2.2.7. There is a positive constant c such that, given ε small enough, any twopoints (n diam(U), ε)-close with respect to F are also (n, cε)-close with respect to ΓU .

Proof. Let ε ≤ ε0 and x, y be two points (n diam(U), ε)-close with respect to F . In particular,x, y belong to a same chart U1. Consider any given chain u = (U1, ..., Un) with chains ofplaques (P1, ..., Pn) and (Q1, ..., Qn), such that x ∈ P1, y ∈ Q1 and Pi ∪ Qi ⊂ Ui, consider asequence xi ∈ Pi ∩ Pi+1, i = 1, 2, ..., n− 1, and let x0 = x, xn ∈ Pn. Connect each point x1 toxi+1 by a segment yi of leafwise minimising geodesic curve. Then l(γi) ≤ diam(U) for everyi. The concatenation of all these curves gives us a leaf curve γ : I → F(x) whose length isl(γ) ≤ n diam(U).By construction ε is a Lebesgue number for U and by hypothesis d(γ(1), pyγ(1)) = d(xn, pyγ(1)) <ε, thus pyγ(1) belongs in Un. By the precedent Lemma we have

ρ(Pn, Qn) ≤ cd(xn, pyγ(1)) ≤ cε.

The choice of holonomy h containing P1, Q1 in its domain was arbitrary, hence x, y are indeed(n, cε)-close with respect to ΓU .

In particular, last proposition means r(n diam(U), ε,F) ≤ r(n, cε,ΓU), from where we use 2.14to conclude immediately the result stated in Theorem 2.2.5. A important result that is a directcorollary for Theorem 2.2.5 and Proposition 2.1.31 is the following:

Theorem 2.2.8. If a C1 foliation F on a compact Riemannian manifold (M, g) has a resilientleaf then hg(F) is positive.

The converse of this statement is also true. The next chapter is entirely dedicated to itsdemonstration.

Example 2.2.9. The Hirsch Foliation has positive geometric entropy.

At last, we strengthen Proposition 2.2.5 by showing the geometric entropy is theactual supremum of all the topological entropies of holonomy pseudogroups.

Theorem 2.2.10 (Ghys-Langevin-Walczak). Let F be a C1 foliation on a compact Riemannianmanifold (M, g). Then

hg(F) = supU

1diam(U)h(HU ,ΓU),


where the supremum ranges over all the possible choices of regular atlas U for F .

Proof. Due to Theorem 2.2.5 one has

supU

1diam(U)h(HU ,ΓU) ≤ hg(F).

To complete the demonstration we construct a regular covering which approximates the geometricentropy as precisely as we wish. The domains of the distinguished charts of this covering havethe shape of “coin”, meaning they are built in such a way that all of their plaques are discs ofsame radius: they are stacks of discs. It follows that the diameter of the covering is the exactdiameter of any of its plaques, and the properties of these coins are regular enough so thatusing them to cover leaf curves will provide enough data to estimate the geometric entropy interms the holonomy pseudogroup’s topological entropy. This estimate will, in turn, implicatethe desired result.

Consider 0 < 4∆ < ε0. For x ∈M the sets

BF(x,∆) ⊂ BF(x, 2∆) ⊂ BF(x, 4∆)

are geodesically convex balls centred at x. Take δ < ∆ and define also the transversals

Tx := B⊥F (x, δ) and Sx := B⊥F (x, 2δ).

Remark that Tx ⊂ Sx and define

Ux :=⋃y∈Tx

BF(y,∆) and Vx :=⋃y∈Sx

BF(y,∆).

We can choose δ as small as necessary in order to assure both Ux and Vx have compact closurescontained in the domain of a distinguished chart (that is why we choose 4∆ < ε0). We havethen Ux ⊂ Vx and Vx in the domain of a distinguished chart, for every x ∈M . Furthermore, wecan make δ even smaller so that

(i) For every z ∈ Ux, the ball B⊥F (z, ε0) meets each plaque of Vx in exactly one point, whichis point realising the distance between the plaque of Vx through z and that point;

(ii) If Ux ∩ Uz 6= ∅ then Uz ⊂ Vx;

(iii) Distinct plaques of Uz belong to distinct plaques of Vx.

It is clear that, for such families, if a plaque P ⊂ Ux intersects a plaque Q ⊂ Uz then P ∪Q is aplaque in Vx. In this way conditions (i) and (ii) imply that any finite covering Uxini=1 de M isa regular covering. The plaques of this regular covering are geodesically convex balls in theirrespective leaves, all of radius ∆. Consequently, any such covering has diameter 2∆.


Consider now an arbitrary real number η ∈ (0, 1) and the function

ν : R ∈ [0,∞) 7→⌊

R

2(1− η)∆

⌋+ 1 ∈ N.

For them we have the following result.

Lemma 2.2.11. For each η ∈ (0, 1) there is a choice of regular atlas U∆ as above such that, forsufficiently small choices of ε > 0 and every positive real number R, given two points x, y ∈Mthat are (R, ε)-separated by F , then either x, y are (ν(R), ε)-separated by ΓU∆ or d(x, y) ≥ ε.

Proof. To see this holds, we fix η′ < η and choose µ sufficiently small so that for every pair x, zwith d(x, z) < µ we have

(a) z ∈ Ux;

(b) BF(z, (1− η′)∆) ⊂ P (z), where P (z) is the plaque of Ux through z;

(c) If P ′(z) is the plaque of Vx through z′ ∈ B⊥F (z, ε0) such that d(z, z′) < µ, then for anyw ∈ BF(z, (1− η)∆) one has d(pz′(w), z′) < (1− η)∆.

For such µ we consider a µ-dense subset Z ∈M and define

U∆ := Uxx∈Z

andV∆ := Vxx∈Z .

Fix ε <µ

2 . For x, y (R, ε)-separated with respect to F , if d(x, y) ≥ ε there is nothing todemonstrate. Assume then d(x, y) < ε and consider a piecewise smooth curve γ : [0, r]→ F(x)parametrised by arclength and such that r < R, γ(0) = x and d(pyγ(t), γ(t)) ≥ ε for somet ∈ [0, r]. We choose a partition 0 = t0 < t1 < · · · < tk = r of [0, r] such that each segmentγi = γ([ti−1, ti]) satisfies l(γi) ≤ 2(1 − η)∆. Such a partition exists for some k < ν(R) sincel(γ) = r < R. For each i ∈ 1, 2..., k choose xi ∈ Z such that d(xi, γ(ti)) <

µ

2 and write Ui = Uxiand Vi = Vxi . Condition (a) implies γ(ti) ∈ Ui for every i. Let Pi denote the plaque of Uicontaining γ(ti). The fact that l(γi) ≤ 2(1− η)∆ < 2(1− η′)∆ together with condition (b) implythat γi ⊂ Pi1 ∪ Pi, hence (P1, ..., Pk) is a chain of plaques, and consequently (P ′0, ..., P ′k), whereP ′i is the plaque of Vi containing γ(ti), is also a plaque chain.

Since d(x, y) < µ

2 , it follows from the triangle inequality that the chosen pointx0 ∈ Z satisfies d(x0, y) < µ, then by (a) we conclude y ∈ U0. Denote by Q0 the plaque of U0

through y and by Q′0 the corresponding plaque of V0. Property (i) of the coverings U∆ andV∆ implies that py(x) ∈ Q′0. Now note that by the definition of the Hausdorff metric ρ we


have d(py(x), x) ≤ ρ(P ′0, Q′0), hence d(py(x), x) ≥ ε implies ρ(P ′0, Q′0) ≥ ε and consequentlyρ(P0, Q0) ≥ ε. On the other hand, shrinking γ we can assume it is completely projectable andthat d(pyγ(t), γ(t)) < ε for every t ∈ [0, r), that is, that the endpoint is exactly the point wherethe distance reaches ε. Since we chose ε < µ

2 , condition (c) implies that py(γ(ti)) is a pointin Qi and the segment py(γi) has length at most 2(1 − η)∆. Moreover, d(xi, py(γ(ti))) < µ.Condition (b) assures that py(γi) ⊂ Qi−1 ∪ Qi and (Q0, ..., Qk) is a chain of plaques. Sinced(py(γ(r)), γ(r)) = ε, the definition of the Hausdorff metric ρ implies ρ(P ′k, Q′k) ≥ ε, proving ourlemma.

To finish the proof of Theorem 2.2.10, take a subset A ⊂M (R, ε)-separated withrespect to F , with ε < µ

2 . Choose a finite covering BkN(ε)k=1 of M by open balls of diameter ε

and let Ak := A ∩Bk, 1 ≤ k ≤ N(ε). Each Ak is (ν(R), ε)-separated with respect to ΓU∆ , hencedue to the assertion we just proved one has that

s(R, ε,F) ≤ N(ε)s(ν(R), ε,ΓU∆),

and therefore1R

ln s(R, ε,F) ≤ N(ε)R

+ ν(R)R

1ν(R) ln s(ν(R), ε,ΓU∆). (2.16)

On the other hand, limR→∞

ν(R)R

= limR→∞

1R

(⌊R

2(1− η)∆

⌋+ 1

)≤ 1

2(1− η)∆ , thus 2.16 implies

s(ε,F) ≤ 12(1− η)∆s(ε,ΓU∆), and therefore

hg(F) ≤ 12(1− η)∆h (HU∆ ,ΓU∆) = 1

(1− η) diam(U∆)h (HU∆ ,ΓU∆) .

Finally, since 0 < η < 1 is arbitrary, it follows that hg(F) ≤ supU

1diam(U)h(HU ,ΓU), which

together with Proposition 2.2.5 gives us the desired equality.

We remark that if G is a group of homeomorphisms of a compact metric space (X, d),then for any finite generating set Γ for G the length of an element is well defined, and we candefine an entropy for G, with respect to Γ, as

h(G,Γ) := h(G(Γ),Γ).

Example 2.2.12 (Geometric Entropy of Suspensions). Let (M,F , π) be the suspension arepresentation h = G→ π1(B), as in Example 1.2.37 and the subsequent discussion. Fix a finiteset Γ generating G. We use the action of the fundamental group in M in order to think of G as


a group of transformations, so that it makes sense to calculate its topological entropy. Let gbe a Riemannian metric in M such that the projection π : M → B is isometric on the tangentspaces TxF , x ∈M . Given an element z ∈ Γ, denote by l(z) the infimum of the lengths of allthe loops γ in the class associated to z. Let a and b be, respectively, the positive constantscorresponding to the minimum and maximum over z ∈ Γ \ e of the lengths l(z), e being theidentity element of G. Then

ahg(F) ≤ h(G,Γ) ≤ bhg(F).

Indeed, any curve γ : I → B is lifted to a leaf curve γ : I → L. We lose no generality in assumingthe fibres are orthogonal to leaves since this can be done by modifying the Riemannian metriconly in the direction transverse to the leaves. Then the orthogonal projection of γ onto a leaf L′

coincides with a suitable lift of γ to L′. Therefore, any two points of F which are (n, ε)-separatedwith respect to Γ become (bn, ε)-separated with respect to F , and any two points two pointswhich are (R, ε)-separated with respect to F become (bR/ac+ 1, ε)-separated with respect to Γ,from the inequalities above follow.

Example 2.2.13 (Geometric Entropy of the Reeb Foliation). Recall that one way to constructa three-dimensional Reeb component as to choose a suitable function f : (−1, 1)→ R, namely asmooth, symmetric, increasing, even function whose lateral limits are both equal to ∞. We mayassume 0 to be a fixed point. The graph of this function was then translated up and down to fillthe rectangle [−1, 1]× [0, 1], which was then rotated to create a foliated cylinder, and a finalidentification of the border discs of this cylinder resulted in the foliated Reeb component.

The first thing to note is that the holonomy pseudogroup of this codimension 1foliation is the same as the holonomy pseudogroup of the codimension 1 foliation F of thecylinder C obtained by identifying the sides [−1, 1]× 0 and [−1, 1]× 1 of the foliated rectangle[−1, 1]× [0, 1] before rotating it. Let f = sgn(t)f(t) : (−1, 1)→ R be the odd function coincidingwith f on [0, 1). If F is the foliation of C obtained in the same way as F above by using finstead of f , then a pair of points is (R, ε)-separated by F if and only if it is (R, ε)-separated byF . The foliation F can be obtained as the suspension of a homeomorphism ϕ : [−1, 1]→ [−1, 1].As we noted in Example 2.1.24, the topological entropy of ϕ vanishes. On the other hand, if weconsider the set Γ = id, ϕ, ϕ−1, then it follows from Example 2.1.33 that

h(G(ϕ),Γ) = 2htop(ϕ) = 0.

Together with Example 2.2.12 above this gives us hg(F) = 0, hence the Reeb foliation of thesolid torus has vanishing geometric entropy. As we shall see, this happens because it has noresilient leaves.

We finish this section analysing how the geometric entropy is affected by the modifi-cations and constructions we exposed in Chapter 1. First of all, for pullbacks between manifolds


with leafwise isometric Riemannian tensors the entropy can never grow:

Proposition 2.2.14. Let f : (N, g′)→ (M,F , g) be an immersion transverse to the foliationF , and assume that f is a leafwise isometry, that is, that |dfxv|g = |v|g′ for every v ∈ TxF ′ andx ∈ N (in particular, that is the case when g′ = f ∗g). If N is given the foliation F ′ = f ∗F , then

hg′(F ′) ≤ hg(F).

Proof. Consider a regular covering U ′∆ of N , constructed as we did in the proof of Theorem2.2.10. We have seen that it is possible to choose U ′∆ such that 1

diam(U ′∆)h(H′U ′∆ ,Γ

′U ′∆

)is as

close to hg′(F ′) as we wish. We can then construct a regular covering U of M such that

(i) For every U ′ ∈ U ′∆ there is U ∈ U such that f(U ′) ⊂ U ;

(ii) diam(U)diam(U ′∆) is as close to 1 as we wish.

Then the isomorphism constructed in Proposition 1.2.26 is an equivalence between the holonomypseudogroup HU ′ of F ′ and a subpseudogroup of HU generated by a subset of ΓU . Since theisomorphism consists of local isometries, the constants Ci appearing in Remark 2.1.13 are allequal to 1 and therefore both pseudogroups have the same entropy. Together with Proposition2.1.27 this implies that h(HU ′ ,ΓU ′) ≤ h(HU ,ΓU), and the result now follows from Theorem2.2.10.

Proposition 2.2.15. Let (M,F) be a foliation obtained by tangential gluing of foliatedcomponents (Mi,Fi), i = 1, 2, each one of them infinitesimally Cr-trivial along each componentof Si ⊂ ∂τMi. Assume that gi is a Riemannian metric on Mi and that the identification mappingf : S2 → S1 is an isometry with respect to the induced metrics. Then the geometric entropy ofF is equal the maximum of the geometric entropies of the components Fi.

Proof. In a neighbourhood of Si the metrics gi can be modified in directions transverse to thefoliation Fi in order to get two new metrics, which we denote again by g1 and g2, matchingon the boundary, so that we can use them to produce a new Riemannian structure g on Msatisfying g|Mi

= gi. This does not change the geometric entropies of F1 and F2, since themodifications occur only the transverse direction (Proposition 2.2.4). Now given any subsetA ⊂M (R, ε)-separated with respect to F , the restrictions A|Mi

are also (R, ε)-separated withrespect to Fi, hence

s(R, ε,F) ≤ s(R, ε,F1) + s(R, ε,F2)

and consequentlyhg(F) ≤ maxhg1(F1), hg2(F2). (2.17)


On the other hand any subset A ⊂ Mi which is (R, ε)-separated with respect to Fi is also(R, ε)-separated with respect to F when considered a subset of M . This implies

hg(F) ≥ maxhg1(F1), hg2(F2),

which together with 2.17 gives the desired result.

Example 2.2.16 (The Reeb Foliation of S3). As we saw in Example 2.2.13, each Reeb componentof the Reeb foliation of S3 has vanishing entropy so that the last result implies immediatelythat the Reeb foliation of S3 has geometric entropy equal to zero.

A weaker version of the last result holds for transverse gluing. Again, we let gi be aRiemannian metric on the space Mi, modified in a tubular neighbourhood Ni of Si ⊂ ∂tMi inorder to make the identification induced by f : S2 → S1 an isometry. In this way we can producea new Riemannian structure g on M which coincides with gi on Mi \ Ni. This modificationcan be performed in the transverse direction to Fi in such a way that g(v, v) ≤ gi(v, v) forevery v ∈ TxFi and x ∈Mi. This time the (R, ε)-separated subsets of (Mi,Fi) become (R, cε)-separated sets of (M,F), for some positive constant c that depends only on the modificationmade on gi. Consequently, we have s(R, ε,F) ≥ maxs(R, cε,F1), s(R, cε,F2) and therefore

hg(F) ≥ maxhg1(F1), hg2(F2).

The other inequality is not necessarily true, so we have the following proposition:

Proposition 2.2.17. Let (M,F) be a foliation obtained by transverse gluing of foliatedcomponents (Mi,Fi), i = 1, 2. Assume that gi is a Riemannian metric on Mi and that theidentification mapping f : S2 → S1 is an isometry with respect to the induced metrics. Thenthe geometric entropy of F is not less than the maximum of the geometric entropies of thecomponents Fi, that is,

hg(F) ≥ maxhg1(F1), hg2(F2).

Equality does not need to hold, as two foliations with vanishing geometric entropycan be glued via a map f which has itself positive topological entropy, and in this case, theresulting foliation will have positive geometric entropy as well. This is what happens withsuspensions, for instance:

Example 2.2.18. Let M be a compact manifold and ϕ : M → M be a diffeomorphism withpositive entropy, and consider Mi = M × I, i = 1, 2. For any choice of Riemannian metricgi for Mi the product foliation Fi of Mi by segments x × Ix∈Mi

has geometric entropyequals zero. If we set Si = ∂Mi = M × 0 ∪M × 1 and f : S2 → S1 to be f(x, 0) = (x, 0)and f(x, 1) = (ϕ(x), 1), then the gluing (M1 ∪f M2,F1 ∪f F2) is just the suspension of ϕ


(Example 1.2.37). Hence the holonomy pseudogroup H of F1 ∪f F2 is G(ϕ) and thereforeh(H, id, ϕ, ϕ−1) = 2htop(ϕ) > 0.

At last, turbulisation changes the geometric entropy in rather an arbitrary way(especially because the Reeb component’s foliation is rather arbitrary as well), but it turns outthat turbulising a foliation can not make it go from a vanishing entropy foliation to a foliationwith positive entropy. Strictly speaking:

Proposition 2.2.19. If a foliation F ′ of M is obtained from F by turbulisation, then

hg(F ′) ≤ hg(F) ≤ chg(F ′),

where the constant c is strictly greater than 1 and depends only on the geometries of M andthe Reeb component introduced during the processes of turbulisation. Consequently, F ′ hasvanishing entropy if and only if F does.

Proof. This is Proposition 3.5.8 from [56]

2.2.1 Pseudoleaves and geometric entropy

In [33], Inaba gives a definition of pseudoleaf analogue to the one from pseudogroupsand transformations. In their paper [5] (which we already encountered in Section 2.1.3), Biś andWalczak used this definition to relate pseudoleaves and pseudo-orbits, and proposed a way tocompute the geometric entropy of F in terms of pseudoleaves. In this section we display theresults obtained by them. All the notation regarding pseudo-orbits will be as in Section 2.1.3. Inparticular, recall that a pseudo-orbit of (G,Γ) was defined as a map x : Γ∞ → X whose domainwas denoted by Dx.

Given an n-dimensional vector space V , one can define the distance between twosubspaces W1,W2 ⊂ V of same dimension by

dist(W1,W2) := maxw1∈W1‖w1‖=1

minw2∈W2‖w2‖=1

‖w1 − w2‖

.Following [33], we define

Definition 2.2.20 (α-pseudoleaf of a foliation F). Given a foliation F of a Riemannian manifold(M, g) and a nonegative real number α, we say a complete immersed submanifold N of M is anα-pseudoloeaf of F if dimN = dimF and for every point p of N one has

dist(TpN,TpF) ≤ α. (2.18)


Of course, N is a 0-pseudoleaf if and only if it is a leaf of F . Let us fix a compactfoliated Riemannian manifold (M,F). We choose a regular atlas U for F whose chart domainsare small enough so that if Ui ∩ Uj is nonempty then the plaques of Wij (recall Definition 1.2.4)are strongly convex subsets of their leaves, that is, they are intersections of open balls of a fixedradius. We consider the regular generating set Γ for the holonomy pesudogroup H, consistingof all the holonomy maps hij = hUiUj . For each distinguished chart U of U we can identify thespace TU with a transverse section of U , and thus consider the complete transversal T as asubset of M . In what follows, keep in mind that when a holonomy transformation is applied toa point p ∈ T it is being applied to the unique plaque containing p, as in Remark 1.2.27. Weendow T with the Riemannian structure induced by M . The compactness of T implies thatthe induced Riemannian metric (or more precisely, its pullback under the identification doneabove) and the Hausdorff metric of T are quasi-isometric, and therefore strongly equivalent.Thus, they both induce the same topology and it then follows from Proposition 2.1.29 that theirtopological entropies are the same.

Any embedding of T in M induces a natural embedding of the space of pseudoleavesinto the pseudo-orbits of H. To produce such an embedding, consider the subbundle E ⊂ TM ,complementary to TF , given by

Ep = Tp T

for any p ∈ T .

Lemma 2.2.21. There exists positive constants C, C ′ and α0 such that for every α ∈ (0, α0),any α-pseudoleaf N and any point p of N ∩ T there is a unique Cα-pseudo-orbit x of HU forwhich

(i) x(idT ) = p;

(ii) x(Dx) ⊂ N ;

(iii) for any holonomy transformation g ∈ Dx and any h ∈ ΓU whose domain Dh contains x(g)one has

dN(x(h g), x(g)) ≤ C ′dF(h(x(g)), x(g)).

Proof. We proceed by induction on Dx ∩ Γk as we did in the proof of Proposition 2.1.36. Fork = 0 we have Dx ∩ Γ0 = idT , and set x(idT ) = p. Assume x is define on Dx ∩ Γk. Given aholonomy transformation from Dx∩Γk and h ∈ Γ such that x(g) lies in Dh, we define x(h g) inthe following way. The holonomy h is of the form hij for some intersecting distinguished chartsUi and Uj of U . We let γ : [0, L] → Wij be the unique leaf geodesic joining x(g) and h(x(g)),and construct a tubular neighbourhood πγ : Eγ → F(γ) using the fibres Ep, p ∈ F(γ). From the


Unique Lifting Property follows that γ has a unique lifting to a curve γ : [0, L] with γ(0) = x(g)and γ(t) ∈ N for every t ∈ [0, L]. Set x(h g) = γ(L).

Figure 9 – The lifting of γ and an illustration of how x(h g) is defined.

All the global constants exist due to the compactness of M . As diamM <∞, thelengths of all the geodesics γ are uniformly bounded, so that we can find α0 small enough suchthat the construction above holds for every α < α0. We can make α0 as small as we need toassure the tubular neighbourhood construction is unique, that is, that for all the geodesics γjoining p to h(p), with p ∈M and h ∈ Γ, the projection πγ|N is a bijection. This gives us theuniqueness of x. Also due to the manifold’s compactness, its geometry is bounded, meaningall the geometric related objects like sectional curvature and injectivity radius have uniformbounds. Thus, we can find constants C and C ′ such that for every α < α0, g ∈ Dx and geodesicγ from x(g) to h(x(g) we have

l(γ) ≤ C ′l(γ) = C ′L,

and more importantly,dN(x(h g), h(x(g))) ≤ Cα,

meaning the unique x defined above is indeed an Cα-pseudo-orbit of the pair (H,Γ), one thatsatisfies

dN(x(h g), x(g)) ≤ C ′dF(h(x(g)), x(g)),

as we wanted.


Since (M, g) is compact the orthogonal projection described in the beginning ofSection 2.2 is defined here as well, so that we can talk about separation of pseudoleaves. We fixthe same notation involving the orthogonal projection as in the last section, then the definitionreads as follows.

Definition 2.2.22 (Geometric separation of pseudoleaves). For ε ∈ (0, ε0), α ∈ (0, α0) andR ∈ R, two points x, y belonging to α-pseudoleaves N1 and N2, respectively, are said to be(R, ε)-separated by α-pseudoleaves if either d(x, y) ≥ ε or there is a curve γ : I → N1 such thatγ(0) = x, l(γ) ≤ R and d(γ(1), pyγ(1)) ≥ ε.

A set A ⊂ M is (R, ε)-separated by α-pseudoleaves if every pair of distinct points in A is(R, ε)-separated by α-pseudoleaves.

As for (n, ε,F)-separated sets, the subsets of M (R, ε)-separated by α-pseudoleaves are alwaysfinite due to the compactness of M . Let sα(R, ε,F) be the maximal cardinality of a (R, ε)-separated by α-pseudoleaves subset of M . In an analogous way to what we did for the entropyof pseudo-orbits, we let now α : (0,∞)→ (0,∞) be any function satisfying lim

R→∞α(R) = 0, and

setsα(ε,F) := lim sup

R→∞

1R

ln sα(R)(R, ε,F).

Definition 2.2.23 (Geometric Entropy of the pseudoleaves of F). The limit

hpsg (F) := limε→0+

s(ε,F)

is called the geometric entropy of the pseudoleaves of the foliation F on the compact Riemannianmanifold (M, g), with respect to the metric g.

Now, analogous to Theorem 2.1.43, we have

Theorem 2.2.24 (Biś - Walczak). For any compact foliated Riemanian manifold (M,F , g),the equality

hg(F) = hpsg (F)

holds true.

Proof. Any leaf of F is also an α-pseudoleaf for any α ≥ 0, hence s(R, ε,F) ≤ sα(R, ε,F) forany possible choice of α,R and ε, and consequently inequality

hg(F) ≤ hpsg (F)

holds.

All the work now is to show the converse inequality also holds. The arguments hereshare a lot of similarities with the ones from Theorem 2.2.10, so we fix notation to be like there.


Fix a regular covering U∆ of M by “coins” of diameter 2∆ as the one constructed before, andconsider a finite set Z = z1, ..., zN indexing U∆, in the sense that U∆ = UzjNj=1. Using theconstruction from Lemma 2.2.11, we can fix a small number η and assume that Z is sufficientlydense in M so that any point in M is at most η/2 apart in leaf distance from a point of thetransversal T and η/2 apart from in M distance from a point in Z.

For α sufficiently small for Lemma 2.2.21 to hold and 0 < ε < ε0, consider two(R, ε,F)-separated α-pseudoleaves (Ni, pi), pi ∈ Ni ∩ T , and the correspondent Cα-pseudo-orbits xi of the holonomy pseudogroup HU∆ . If the distance in M between p1 and p2 is alreadygreater than ε0 then the pseudo-orbits x1 and x2 are (n, ε,Γ)-strongly separated. This happensbecause idT ∈ Dx1 ∩ Dx2 ∩ Γn for every n and

dT (x1(idT ), x2(idT )) ≥ dM(p1, p2) ≥ ε0 > ε.

If, on the other hand, dM (p1, p2) ≤ ε0, then there is a pseudoleaf curve γ1 : [0, 1]→ N1 of lengthl(γ) ≤ R, starting at p1 and whose projection pp2γ onto N2 is such that dM(γ(1), pp2γ(1)) ≥ ε.We shrink γ if necessary in order to assume, without loss of generality, that its end point isexactly the first time for which dM(γ(1), pp2γ(1)) = ε.

Consider the natural number

n =⌊1 + R

2(∆− η)

⌋

and split the curve γ into n segments γj of length l(γj) ≤ 2(∆− η). Set p′j to be the midpointof the segment γj , j ∈ 1, ..., n, and put p′0 = p1 and p′n+1 = γ(1). Due to our choice of Z thereare indeces ij ∈ 1, ..., N such that

dM(p′j, zij) ≤η

2 ∀j ∈ 0, ..., n+ 1.

We denote the chart Uzij simply by Uj . If our choice of α is sufficiently small then we guaranteethat γj is entirely contained in Uj.

By construction, Uj ∩ Uj+1 6= ∅. Let hj denote de holonomy transformation hUj+1Uj ,and define the transformations

gj := hj hj−1 · · ·h0.

As a consequence of the construction of the pseudo-orbits in Lemma 2.2.21, if only α is smallenough, the inequality dN1(p′j, x1(gj)) ≤ η is satisfied for every j ∈ 0, ..., n+1. Then x1(gj) ∈ Ujfor every j, as long as η min∆, ρ, and, in particular, gn is an element of Dx1 . In a similarway, one can see that, as long as ε is sufficiently small, gn ∈ Dx2 . What is more, as the inequalities

dN1(x1(gn), γ(1)) ≤ η and dN2(x2(gn), pp2γ(1)) ≤ η


both hold and our spaces are all compact, we can find a global constant K depending only onρ,∆, η and the geometry of M (and by that we mean on its sectional curvature, injectivityradius and so forth), for which

dM(x1(gn), x2(gn)) ≥ Kε,

meaning the to pseudo-orbits x1 and x2 ofHU∆ are (n+1, Kε,Γ)-strongly separated. Consequently

sα(R, ε,F) ≤ sCα(n+ 1, Kε,Γ) = sCα(b2 +R/(2∆− 2η)c, Kε,Γ),

for sufficiently small choices of ε and α. Passing now to suitable limits we obtain the relation

hpsg (F) ≤ 12(∆− η)hps(HU∆ ,Γ).

From Theorem 2.1.43 then comes

hpsg (F) ≤ 12(∆− η)h(HU∆ ,Γ),

and from Theorem 2.2.10hpsg (F) ≤ 1

2(∆− η)hg(F).

As the constant η can be chosen arbitrarily small, we conclude that hpsg (F) ≤ hg(F), whichconcludes the proof.

100

Chapter 3

Entropy and Dynamics of C1 Foliations

Smale horseshoes are the embodiement of chaos in the theory of Dynamical Systems.A horseshoe map has periodic orbits of arbitrarily long period, and an infinite number of periodicpoints. The number of periodic orbits of a horseshoe map grows exponentially as we increasethe periods and close to any point of the fractal invariant set there is a point of a periodic orbit.We know the existence of a Smale horseshoe implies positive topological entropy since horseshoemaps are topological conjugate to full shifts, and a result due to Katok [35] says that for C1+α

diffeomorphisms positive topological entropy implies the existence of a Smale horseshoe. In thestudy of dynamics of codimension one foliations, resilient leaves will have a role similar to thatof horseshoes as representatives of chaotic behaviour, in the sense that they characterise theexistence of positive geometric entropy..

Theorem 2.2.8 states that a resilient leaf imposes positive geoemtric entropy ontothe foliation it belongs to. This chapter is mostly based in the homonymous paper [28], and it issolely devoted to proving the converse of Theorem 2.2.8. The C1 condition on the foliation Ftraces back to Proposition 2.1.21, where it was a fundamental hypothesis for the constructionof the suitable subpseudogroup that implied exponential expansion growth. After defining thegeometric entropy in [26], the authors proved a partial converse of 2.2.8 for C2 foliations:

Theorem 3.0.1 (Ghys-Langevin-Walczak [26]). If F is a C2 foliation of codimension 1 andhg(F) > 0 then F has a resilient leave.

The proof provided in [26] is a rather topological one, using the concepts of depth and growth ofthe leaves, and is based on a thorough discussion about the structure of open F -saturated setswith trivial holonomy on which the C2 hypothesis is indispensable. We will instead, followingthe work of Hurder, prove the statement by means of a more analytical approach. By usingthe C1 structure of F we can define a flow on the unitary tangent bundle of F , so that anymeasure invariant under such flow can be seen, in a way, as an invariant measure under the

Chapter 3. Entropy and Dynamics of C1 Foliations 101

foliation F . It turns out that when codimF = 1 the existence of such a measure, which can beguaranteed when hg(F) > 0, implies the occurrence of a phenomenon in the dynamics of theholonomy pseudogroup known as a ping-pong game, which in turn is equivalent to the existenceof a resilient leaf. At last, we are left with the following result:

Theorem 3.0.2 (Hurder [28]). A codimension 1 transversely C1 foliation on a compact orientedRiemannian manifold has positive geometric entropy if and only if it has a resilient leaf.

As the price for avoiding structure theory for C2 foliations, the proof from [28] is avery technical one. So while the C2 hypothesis allows for a considerably easier and less extentproof, it is not really necessary. For the continuous case however, that is, for C0-foliations, thetheorem no longer holds. We provide a counterexample in Section 3.6.

3.1 The frameworkThe expression “transversely C1” in Theorem 3.0.2 means that the functions γ

appearing in 1.3 are C1. It is often the case that the leaves of the foliation are “smoother” assubmanifolds than the holonomy cocycles are as functions. For this sort of scenario we introducethe following definition:

Definition 3.1.1. A foliation F is said to be of class Cr,k, where r > k ≥ 0, if the correspondingtransition maps

ψ ϕ−1 : (x, y) 7→ (α(x, y), γ(y))

are Ck but α : Rp×Rq → Rp is Cr with respect to the first coordinate x and its mixed x partialsof orders less or equal to r are Ck in the coordinates (x, y).

Throughout this chapter M will be a compact orientable smooth Riemannian man-ifold, with normalised metric g so that diamM = 1. We endow M with a C∞,1 foliationF . Such foliations are also called C1-smooth foliations. We fix p = dimF and q = codimF .Assume U = (Ui, ϕi)i∈A is a regular atlas, where each chart ϕi has range (−1, 1)n anda extension ϕi : Ui → (−2, 2)n such that Ui ⊂ Ui. We would like to embed the completetransversal of U in M so we can work with the holonomy group H of F acting on points ofM rather than in the abstract space of plaques. We shall do this as Hurder does in [28]: Forz ∈ (−2, 2)q, let Pi(z) := ϕi

−1((−2, 2)p × z). Accordingly, a plaque of U will be denoted byPi(z) = ϕ−1

i ((−1, 1)p × z). Besides, we assume the Riemannian metric in M is such that eachPi is convex, so that the intersection of any two plaques is either empty or also a convex set.

For each i ∈ A we can define the smooth embedding

ti := ϕi(0 × ·) : (−2, 2)q → Ui.


Ti := Im ϕi is a submanifold ofM . A slight alteration of the charts of U can be made, if necessary,to guarantee that all the Ti are pairwise disjoint. We will also assume that each Ti is actuallyperpendicular to every leaf it meets, which can be achieved by modifying the Riemannian metricof M in a tubular neighbourhood of Ti, and again imposes no loss of generality. We will assumediam Ti ≤ 1.

The space of plaques Ti (recall Definition 1.2.22) is identified with the transversesection

ϕ−1i (0 × (−1, 1q)) ⊂ Ti.

The local coordinate ti := ti|(−1,1)q : (−1, 1)q → Ti identifies Ti to (−1, 1)q. In this the completetransversal becomes

T = ti∈ATi ⊂M.

By a mild abuse of notation we denote by Pi(x) the plaque of Ui containing x.

If Ui ∩ Uj 6= ∅, then Dji := x ∈ Ti;Pi(x) ∩ Uj 6= ∅ corresponds to the sets DUjUi

of Definition 1.2.23, and our holonomy maps are functions hji : Dji → Tj mapping x to theunique y ∈ Tj such that Pi(x) ∩ Pj(y) 6= ∅. Since our foliation C1, each local representationt−1j hji ti : (−1, 1)q → (−1, 1)q is C1, hence the holonomy pseudogroup H of F is indeed apseudogroup of C1 transformations. Making analogous definitions for Ti we conclude that eachhji is extensible to a C1 transformation hji : Dji → Tj. The compacity of M implies that H isregular.

Let Z be a subset of Ui. Fix

ZP :=⊔P ;P is a plaque of Ui and P ∩ Z 6= ∅,

Zt := ZP ∩ T .

In particular, if Z is singleton x then xt = xP ∩Ti is the only point of Ti such that x ∈ Pi(xt).If π : Ui → Ui/F|Ui is the natural projection then Zt corresponds to π(Z) and ZP is thesaturation of Z with respect to F|Ui . Of course, both Zt and ZP are open whenever Z is open.

Rather than use the Hausdorff metric, we equip T with the distance dt induced onTi by the Riemannian metric of M . If x and y belong to distinct transverse sections Ti andTj, then dt(x, y) = ∞. The open ball of radius r with respect to dt and centre in x will bedenoted by Bt(x, r). We fix a Lebesgue number εU for U , that is, every open ball of radius εUis entirely contained in a element of U , and let εt be such that for all Ui ∈ U and all x ∈ Uiwe have Bt(xt, εt) ⊂ (B(x, εU) ∩ Ui)t. The number εt is called a transverse Lebesgue numberassociated to εU , or t-Lebesgue number, for short.


3.2 The holonomy groupoid and foliation cocyclesIn general Dynamical Systems theory, one has the concept of a cocycle over an action.

Suppose α : G×X → X is an action of a group G on a topological space X. A cocycle over theaction α into a Lie group L is a function

c : G×X → L

satisfying the following law, known as cocycle equation:

c(g1g2, x) = c(g2, α(g1, x))c(g1, x),

with the products taking places in the respective groups where the elements belong, of course.Usually the maps involved are structure preserving and satisfy more conditions. The conceptcan also be extended to actions of groupoids and other more general structures.

The holonomy pseudogroup is basically a groupoid with some additional structurethat allows us to glue transformations together when their domains intersect. This property isno longer relevant, however, if one consider only germs of holonomy transformations, in whichcase we have left only the groupoid structure.

Definition 3.2.1 (Topological holonomy groupoid of F). Let F be a foliation with holonomypseudogroup H. The holonomy groupoid of F is the space of germs

GF := [h]x;h ∈ H and x ∈ Dh,

where by [h]x we denote the germ of h at x.

This groupoid comes with two very natural applications s, r : GF → T , the sourceand range mappings, respectively, given by

s : [h]x 7−→ x,

r : [h]x 7−→ h(x).

These maps define subsets

GxF := s−1(x) = θ ∈ GF ; s(θ) = x,

Gx,yF := s−1(x) ∩ r−1(y) = θ ∈ GF ; s(θ) = x and r(θ) = y.

Remark that Gx,xF is nothing other than the local germinal leaf holonomy group HF(x) from

Definition 1.2.29.

A cocycle over a foliation is a cocycle over the natural action of the groupoid GF onthe complete transversal.


Definition 3.2.2 (Cocycle over a foliation [30]). An L-cocycle over the foliation F is a mappingc : GF → L satisfying the cocycle equation

c(θ2θ1) = c(θ2)c(θ1), for all θ1 ∈ Gx,yF and θ2 ∈ Gy,z

F .

In the particular case fixed at the beginning of the chapter, we are dealing with a C1

foliation yielding a holonomy pseudogroup of C1 transformations, so it makes sense to study thederivatives of such maps. Let us denote by e1(x), ..., eq(x) the standard basis induced on Tx Tby the local coordinates ti : (−2, 2)q → Ti. Given a chain of plaques P = (Pi0(x0), ..., Pik(xk)),with x = x0 and y = xk, we have an associated C1 holonomy h whose derivative

dh|x : Tx Ti0 → Ty Tik

defines a matrixDh(x) ∈ GLq(R)

with respect to the standard bases of Ti0 and Ty Tik . This depends only on the germ of h ate x(that is, the choice of chain of plaques does not change the germ, see Proposition 1.2.28), andthus the application

DF : GF × T −→ GLq(R)

[h]x 7−→ Dh(x),

is a well defined cocycle over the groupoid GF . Indeed, given composable chains of plaquesP = (Pi0(x), ..., Pik(y)) and Q = (Qi0(y), ..., Qil(z)) we have hQP = hQ hP , hence the chainrule gives us

dhQP |x = dhQ|hP (x)dhP |x = dhQ|ydhP |x

and thereforeDhQP (x) = DhQ(y)DhP (x).

This cocycle is known, naturally enough, as the derivative cocycle of h.

3.3 Ping-Pong gamesRecall from the discussion about resilient leaves at the end of Section 1.2.3 that a

point y is asymptotic to x if there is a closed chain of plaques P about x such that y belongs tothe domain of hP and hnP (y)→ x.

Definition 3.3.1 (Contractive Holonomy). A holonomy map hP is said to be a contraction atx if there is some δ > 0 such that every y ∈ Bt(x, δ) is asymptotic to x. Specialising this a little


for the case when codimF = 1, we say the map hP is a one side contraction at x if there is a ysuch that every point in the interval [x, y) ∈ T is asymptotic to x.

In particular, x is a fixed point of hP . The map hP is called a hyperbolic contractionat x if the matrix Dh(P, x) is a linear contraction, that is, if its norm is less than 1.

Fixed a chain P , the derivative cocycle is continuous with respect to the second coordinate, sincehP is C1. Hence, due to continuity, there must be a small ε > 0 such that Dh(P, y) is a linearcontraction for every y ∈ Bt(x, ε). We shrink this neighbourhood of x a little, if necessary, toguarantee that C = sup|Dh(P, y)|; y ∈ Bt(x, ε) < 1. Then the mean value inequality implies

dt(x, hP (y)) = dt(hP (x), hP (y)) ≤ Cdt(x, y) < dt(x, y),

and therefore every point of this neighbourhood is asymptotic to x. Furthermore, there is0 < δ < ε such that

hP (Bt(x, δ)) ⊂ Bt(x, δ).

Recall that a point x ∈ T is resilient for H if there is a closed plaque chain around xand another point y ∈ DP ∩ F(x) asymptotic to x. If in addition the map hP is a hyperboliccontraction at x, then x is said to be a hyperbolic resilient point.

Definition 3.3.2 (Ping-Pong Game). We say the holonomy pseudogroup H of a C1 foliationF has a ping-pong game (p.p.g.), or yet, that the dynamics of F admits a p.p.g., if there aretwo distinct points x, y in a transverse section Ti of T and closed chains of plaques P and Q,around x and y, respectively, satisfying

(i) hP is a hyperbolic contraction at x;

(ii) hQ is a hyperbolic contraction at y;

(iii) y ∈ DP is asymptotic to x and x ∈ DQ is asymptotic to y.

The existence of p.p.g is equivalent to that of a hyperbolic resilient point:

Proposition 3.3.3. F admits a p.p.g. if and only if T has a hyperbolic resilient point.

Proof. Let us assume first that H has a p.p.g., with the notation fixed as above. The holonomymap hP is a hyperbolic contraction at x, so x has a neighbourhood Bt(x, ε) where every pointis asymptotic to x via hP . Since hnP (y)→ x and hP is a local diffeomorphism of T , we can findN large enough and a sufficiently small δ such that

hNP (Bt(y, δ)) ⊂ Bt(x, ε).


Now we choose K large enough so that hKQ (x) ∈ Bt(y, δ). Let z = hNP hKQ (x) ∈ Bt(y, δ). ThenPN QK is a chain of plaques joining x to z, hence x and z lie in the same leaf. What is more,z ∈ Bt(y, δ) implies that z is asymptotic to x via hP , so that x is a resilient point, one that ishyperbolic, by hypothesis.

Conversely, assume x is a hyperbolic resilient point. Suppose P is a closed chainaround x, Dh(P, x) is a linear contraction and y0 ∈ DP ∩ F(x) asymptotic to x. Consider aneighbourhood Bt(x, ε) of x where C = sup|Dh(P, y)|; y ∈ Bt(x, ε) < 1 and δ small enoughso that for large enough K one has

x ∈ hKP (Bt(y0, δ)) ⊂ Bt(x, ε).

Let R be a chain of plaques joining x to y0 and Q = R PK . Consider the local diffeomorphism

hQ : Bt(y0, δ)→ Bt(y0, δ)

and note that h−1Q (y0) = x, implying that x ∈ Bt(y0, δ) and that y0 is not a fixed point for hQ.

Moreover, for any z ∈ Bt(y0, δ) one has

|Dh(Q, z)| ≤ |Dh(R, hKP (z))||Dh(P k, z)| ≤ CK |Dh(R, hKP (z))|,

so that for sufficiently large choices of K the derivative of Dh(Q, z) is a linear contraction forevery z in Bt(y0, δ). Now, Banach Fixed Point Theorem implies the existence of a fixed pointy ∈ Bt(y0, δ) for the map hQ. Q is a closed chain of plaques at y, and y 6= x since hQ(x) = y0 6= y.By construction, y (along with every other point of Bt(y0, δ)) is asymptotic to x via hQ. On theother hand, since hQ(x) ∈ Bt(y0, δ) and the iterated images under hQ of every point there tendto y (also due to Banach Fixed Point Theorem), we have that x is asymptotic to y. The pointsx and y, together with the chains of plaques P and Q, constitute a p.p.g. for the dynamics of F .

3.4 Foliated geodesic flow and transverse hyperbolicityIn order to study the dynamical consequences of positive geometric entropy we once

more take advantage of the Riemannian structure of M to define another object: the geodesicflow of the foliation F . First introduced independently by Hurder [30] and Walczak [55], thegeodesic flow main advantage is that it allows us to use techniques from the ergodic theory offlows in the study of the recurrence properties of the plaque chains of F .

Let π : V →M be the Sp−1-sphere bundle of unit vectors tangent to the leaves of F ,that is, V = T1F = tx∈Mx × Sp−1, with π : (x, v) 7→ x, naturally. Given a vector v ∈ T1F ,


denote by γ(x,v) the unit-speed geodesic of F(x) (with respect to dF) starting at x with velocityv, that is,

γ(x,v)(t) := expFx (tv).

The foliated geodesic flow is a mapping Φ : V ×R→ V making the following into a commutativediagram

V × R V

M

Φ

γ π,

where γ : (x, v, t) 7→ γ(x,v)(t). In other words, the characterising property of the geodesic flow isthat γ(x,v)(t) = π(Φ(x, v, t)). The second coordinate corresponds to the velocity vector of thegeodesic at time t, so that we can write

Φ(x, v, t) = (γ(x,v)(t), γ′(x,v)(t)) =(

expFx (tv), dds

expFx (sv)∣∣∣∣∣s=t

).

The foliation F is C1-smooth, the leaves have differentiability class C∞ and the differentialequations describing the leafwise geodesics feature only derivatives in the leaf direction. Thegeodesic flow is therefore C1, or, more accurately, it is C1 with respect to x, and smooth withrespect to v and t. Moreover, note that

Φ(x, v, 0) = (x, v)

andΦ(x, v, t+ s) = Φ(Φ(x, v, s), t),

so that Φ is a flow indeed (that is, an action of R on V ). The second equality is due to theunicity of the geodesic curves: after time s the unique geodesic through x with velocity v arrivesat the point y = γ(x,v)(s) with velocity u = γ′(x,v)(s). There is only one geodesic passing throughy with velocity vector u, so that after time t this geodesic ends up at γ(y,u)(t) = γ(x,v)(t + s)with velocity γ′(y,u)(t) = γ′(x,v)(t+ s). As customary, for fixed t we write Φt = Φ(·, t) : V → V .With this notation the two identities above become Φ0 = idV and Φt+s = Φt Φs, and one caneasily see that each Φt is a C1-diffeomorphism with inverse Φ−t.

We endow V with the restriction gV of the Sasaki metric on TM , and denote bydV the associated distance function. The Sasaki metric is a natural metric for the secondtangent bundle, in the sense that the natural projection TM →M is a Riemannian submersionwith totally geodesic fibres. It induces a distance between points (x, v) and (y, w) of TM thatdepends on the distance between x and y in M and also on the angle between w and the paralleltransport of v along the geodesic from x to y. We give a brief construction, for more detailedtreatments see [44, 23, 20] and the second exercise of Chapter 3 in [18].


Let p be the projection TM → M . For each z ∈ TM , let V (z) := ker dpz be thespace of vertical vectors at z. We consider a connection K : T(TM) → TM defined by thefollowing procedure: for ξ ∈ Tz TM let c(t) = (γ(t), X(t)) be any curve in TM such that

c(0) = z and c′(0) = ξ.

Such a curve is said to be adapted to ξ. Let ∇ denote the be the covariant derivative associatedwith the Levi-Civita connection and define K(z, ξ) := (∇γ′X)(0). This application is well-definedand each mapping Kz := K|Tz(TM) is linear on the fibre Tz(TM). The space H(z) := kerKz iscalled the space of vertical vectors at z. We have Tz(TM) = V (z) ⊕H(z) for every z in thetangent bundle, which provides a splitting

T(TM) = V ⊕H

of the second tangent bundle into the vertical and horizontal subbundles of M , respectively.Both dp|H : H → TM and K|V : V → TM are linear isomorphisms on each fibre, and theSasaki metric g on T(TM) is defined by equipping both these subspaces with pullbacks of theRiemanian metric of M and declaring them to be orthogonal to one another. What we do, morespecifically, is define

gz(ξ, ζ) := gp(z)(dpzξ, dpzζ) + gpi(z)(Kzξ,Kzζ), (3.1)

making the spaces H(z) and V (z) orthogonal for every z.

Remark 3.4.1. We would like to emphasise one particular property of the Sasaki metric in V .Note that if γ is any curve in M , then for any parallel unity vector field v along γ we can definea curve γ := (γ, v), and by Equation 3.1 above the length of γ is just the same as that of γ. Inparticular, if γ : [0, L]→M is a geodesic between x = γ(0) and y = γ(L) then γ′ is parallel toγ and γ = (γ, γ′) is a curve of length l(γ) = L between (x, γ′(0)) and (y, γ′(L)). This means

dV ((x, γ′(0)), (y, γ′(L))) ≤ d(x, y).

Now we return to the natural projection π : V →M . It is a submersion, and thereforeπ t F . Hence, it can be used to define a foliation F = π∗F of V . By doing this (V, F) becomesa foliated manifold whose leaves are the unit tangent bundles to the leaves of F . The conditionγ = π Φ implies that for fixed z ∈ V each curve t→ Φ(z, t) is contained in one leaf L of F ,namely F(z). This means that for every t ∈ R the diffeomorphism Φt maps the leaves of F intothemselves, that is, the foliation F is invariant under the flow Φ. Let us use this fact to define asuitable action of R on the normal bundle of the foliation F , and eventually a cocycle over theflow Φ.


Consider the normal bundle to F

Q :=⊔z∈V

Tz VTz F −→ V

(z, ξ + Tz F) 7−→ z,

with ξ + Tz F being the class of ξ ∈ TV . Due to the pullback construction where each leaf of Fis the pre-image under π of a leaf of F , we have the relation

Tz F = dπ−1z (F(π(z))).

As TV has a Riemannian structure, each Tz V /Tz F is a finite dimensional vector space equippedwith an inner product gz, and can therefore be identified with (Tz F)⊥ ⊂ V (z), so that wehave an identification of Q with T⊥ F . The Riemannian structure of TV induces a Riemannianstructure on Q, which we also denote by g. From now on we will handle the vectors in Q asorthogonal vectors to the foliation F rather than equivalence classes of vector in TV . For eachpoint x ∈ T , the tangent space Tx T inherits the Sasaki metric from TM , so that Tx T isnaturally isometric to the fibre T⊥z F for any z ∈ π−1(x) via an identification induced by π.

Recall we choose our transversal T to be orthogonal to F , so that (TxF)⊥ = Tx T and we havea surjection

dπz : Tz V = Tz F ⊕ (Tz F)⊥ → TxF ⊕ Tx T = TM.

Given a vector v = vF + v⊥ ∈ TxM , there is a vector ξ = ξF + ξ⊥ ∈ Tz V such thatdπzξ

F + dπzξ⊥ = vF + v⊥, and these decompositions are all unique. Now, if v ∈ Tx T then

vF = 0, which implies dπzξF = 0, and therefore ξF ∈ ker dπz ⊂ V (z) is a vertical vector. As thevertical and orthogonal spaces are orthogonal in the Sasaki metric and ξ⊥ is orthogonal to avertical space, it must be an element of H(z), and this subspace is mapped isometrically ontoTxM by the linear mapping dπz.

More concisely, we have an identification I of T⊥z F and Tπ(z) T given by

(z, ξ) 7→ (π(z), dπzξ)

and satisfyinggz(ξ, ζ) = gz(dπzξ, dπzζ).

What this means, in other words, is that the normal bundle of F is simply the pullback under πof the normal bundle of F .

Proposition 3.4.2. [30, Proposition 5.1] Let F be a foliation on M and c : gF → L a cocycleover F . Then there is a canonical lift of c to a cocyle c : GF → L over F , which in turn inducesa cocyle

cΦ : R× V → L

over the foliation geodesic flow Φ.


Proof. Each plaque of F is a the pullback under π of a plaque of F . This means any chainof plaques of F can be pulled back via π to a chain of plaques of F , so that any holonomy hdefined by F can be lifted to a holonomy h defined by F , and this is how the lifted cocycleis defined. Formally, if h is a holonomy defined by the chain P = (P1, ..., Pk), we let h be theholonomy defined by the chain (π(P1), ..., π(Pk)) and set c(h) = c(h). Furthermore, each pair(t, z) ∈ R× V defines a leafwise curve Φ(z, ·) : [0, t]→ F(z), which in turn defines a holonomyh, to each we associate an element of L as above. This gives an action cΦ : R× V → L, as wewanted.

In particular, the derivative cocycle of F can be lifted to a cocycle over the foliation geodesicflow. Let us see how we can describe this cocycle in terms of Φ.

The diffeomorphisms Φt preserve the foliation F , therefore the derivative d(Φt)∗preserves T F , and consequently also Q. So one can consider the action of R on Q induced bythe derivatives dΦt|z : T⊥z F → T⊥Φt(z) F , that is, the mapping

dΦt : Q −→ Q

(z, ξ) 7−→ (Φt(z), dΦt|zξ).

For each point z ∈ V and positive real t the application dΦt : Q → Q can beassociated to the matrix DΦt(z) ∈ GLq(R) which describes the linear operator Φt|z in thecanonical basis defined by the distinguished charts of U . Hence, the above action of R on Qinduces a cocycle over the leafwise geodesic flow Φ

DΦ : R× V −→ GLq(R)

(t, z) 7−→ DΦt(z).

Whenever two points x, y ∈ T are such that Φ(x, v, t) = (y, w) we have a curveγ(s) = π(Φ(x, v, s)) : [0, t]→M joining x to y. The germ of the holonomy hγ is an element ofGx,yF , and as segment of flow orbit Φ((x, v) × [0, t]) is exactly the lift of γ under π, the same

reasoning as in Proposition 3.4.2 gives us

Proposition 3.4.3. [31, Proposition 18.2] Suppose Φ(x, v, t) = (y, w), x, y ∈ T and γ =π(Φ(x, v, ·))|[0,t]. Then

Dhγ(x) = DΦt(x, v).


When the codimension q of F is equal to 1, the maps defined above are maps onintervals of the real line, and the 1-matrices corresponding to the cocycles can be thought of asreal functions instead of operators in vector spaces, since we have

Dh(x) · w = dh|xw = h′(x)w,

and similarly,DΦt(x, v) · w = dΦt|(x,v)w = (Φt)′(x, v)w.

Hence equality of Proposition 3.4.3 can be rewritten, for codimension 1, as

dhγ|x = dΦt|(x,v). (3.2)

For codimension 1 foliations we define the following additive cocycle over the flow Φ,called the transverse expansion cocycle, or yet, the Radon-Nikodym cocycle [31]:

ν : R× V −→ R

(t, x, v) 7−→ ln(dΦt|(x,v)).

This is smooth with respect to t and v and continuous with respect to x. It is indeed a cocyclesince the chain rule gives us

ν(t+ s, x, v) = ln d(Φt Φs)|(x,v) = ln dΦt|Φs(x,v) + ln dΦs|(x,v) = ν(t,Φs(x, v)) + ν(s, (x, v)).

Definition 3.4.4 (Infinitesimal transverse expansion). The infinitesimal transverse expansionfunction of the geodesic flow Φ associated to a C1 foliation F is the continuous function

ψ : V −→ R

(x, v) −→ d

dtν(t, x, v)

∣∣∣∣∣t=0

.

Lemma 3.4.5. Let ν and ϕ be, respectively, the Radon-Nikodym cocycle of the geodesic flowΦ and its infinitesimal transverse expansion function.

(i) ν(t, x, v) =∫ t

0ψ(Φs(x, v))ds.

(ii) The relation ψ(x,−v) = −ψ(x, v) holds for every point of (x, v) ∈ V.

(iii) Let γ : [0, t]→ F(x, v) ⊂ V be a leafwise curve consisting of the concatenation of segmentsof orbit γ1 = Φ(x, v, ·)|[0,t1] and γ2 = Φ(Φt1(x, v), ·)|[t1,t]. Then

ν(t, γ(0)) = ν(t1, γ1(0)) + ν(t− t1, γ2(t1)).


Proof. (i) The cocycle property of ν yields the relation

ν(r,Φs(x, v)) = ν(r + s, x, v)− ν(s, (x, v)).

If we denote by ′ the derivative with respect to the first coordinate the equality abovegives usd

drν(r,Φs(x, v)) = ν ′(r+ s, x, v) and d

dsν(r,Φs(x, v)) = ν ′(r+ s, x, v)− d

dsν(s, (x, v)).

Therefore d

drν(r,Φs(x, v)) = d

dsν(r,Φs(x, v)) + d

dsν(s, (x, v)). As one can see from the

very definition of ν and the fact that Φ0 = idV , the application ν(0, x, v) is identicallyzero. From this we conclude that d

drν(r,Φs(x, v))|r=0 = d

dsν(s, (x, v)), and then it follows

immediately that∫ t

0ψ(Φs(x, v))ds =

∫ t

0

d

drν(r,Φs(x, v))|r=0s =

∫ t

0

d

dsν(s, (x, v))ds = ν(t, x, v).

(ii) Consider in V the symmetry S = (idM ,− idSP−1) : V → V . Remark that dS∗ = (id,− id)is an isomorphism at every point of V , being its own inverse. Besides that, the relation

Φ−t S = S Φt (3.3)

holds for every real number t, since, by the chain rule,

Φ−t(x, v) =(

expFx (−tv), dds

expFx (sv)∣∣∣∣∣s=−t

)

=(

expFx (−tv),− d

dsexpFx (−sv)

∣∣∣∣∣s=t

)= Φt(x,−v).

Differentiating Equation 3.3 and using the fact dS∗dS∗ is the identity, we get

dΦt|(x,v) = dΦ−t|(x,−v). (3.4)

On the other hand, we have

ν(−t,Φt(x, v)) = ν(0− t,Φt(x, v)) = −ν(t, x, v).

Together with 3.4, this last equation yields

ν(t, x, v) = ln dΦt|(x,v)

= ln dΦ−t|x,−v= ln[dΦt|Φ−t(x,−v)]−1

= −ν(t,Φ−t(x,−v)) = ν(−t, x,−v).

From here one has

ψ(x,−v) = d

dtν(t, x,−v)

∣∣∣∣∣t=0

= d

dtν(−t, x, v)

∣∣∣∣∣t=0

= − d

d(−t)ν(−t, x, v)∣∣∣∣∣t=0

= −ψ(x, v),

as desired.


(iii) Just note that

ν(t, γ(0)) = ν(t− t1 + t1, γ(0))

= ν(t− t1,Φt1(γ(0))) + ν(t1, γ(0))

= ν(t1, γ1(0)) + ν(t− t1, γ2(0))).

Property (i) is the one the interests us the most. We are particularly interested in estimatinghow much a holonomy hγ along a leafwise curve γ separates the points in its domain. If γ is asin Proposition 3.2, we identify its derivative with the derivative of a flow diffeomorphism, andare left with

|h′γ(x)| = |dΦt|(x,v)| = |eν(t,x,v)| = e∫ t

0 ψ(Φs(x,v))ds. (3.5)

The number eν(t,x,v) is called the transverse logarithmic expansion of F along the geodesicγ = π(Φ(x, v, ·)) : [0, t]→ F(x).

Remark 3.4.6. Note that for any unit speed leafwise geodesic segment γ : [a, b] → M withendpoints in T , the point (γ(a), γ′(b)) ∈ V satisfies the conditions in Proposition 3.2 for thediffeomorphism Φb−a. Moreover, the cocycle properties of ν imply that ν(b− a, γ(a), γ′(a)) =ν(b, γ(a), γ′(a))−ν(a, γ(a), γ′(a)), which means eν(b−a,γ(a),γ′(a)) = e

∫ baψ(Φs(γ(a),γ′(a)))ds. If γ = γ0∗γ1

is a concatenation of two segments with endpoints in T , then for some c ∈ [a, b] we have

γ0 = π (Φ(γ0(a), γ′0(a)), ·)|[0,c−a] and γ1 = π (Φ(γ1(c), γ′1(c)), ·)|[0,b−c].

Hence

hγ = hγ1 hγ0 = dΦb−c|(γ1(c),γ′1(c))dΦc−a|(γ0(a),γ′0(a))

= eν(b−c,γ1(c),γ′1(c))eν(c−a,γ0(a),γ′0(a))

= e∫ caψ(Φs(γ0(a),γ′0(a)))ds+

∫ bcψ(Φs(γ1(c),γ′0(c)))ds

In order to make proper use of the properties stated in the above Remark andequality 3.5, we define a suitable lift of piecewise geodesic leaf curves in M .

Let γ : [a, b] → F(x) be a piecewise geodesic leaf curve, and consider a partitiona = t0 < · · · < tN = b of [a, b] such that each restriction γ|[ti,ti+1] is a geodesic fragment. Forevery such index i there is a point (xi, vi) ∈ V satisfying

π(xi) = γ(ti) and π(Φ(xi, vi, t)) = γ(t) ∀t ∈ [ti, ti+1]

We denote by γ : [a, b]→ V the curve obtained by concatenating the segments of orbit

Φ(xi, vi, t); ti ≤ t ≤ ti+1i=N−1i=0 .


The leaf curve γ is a lifting of γ in the sense that π γ = γ. This curve may have “corners” –that is, points of its domain where it fails to be differentiable – at each of the points of thepartition. We will also consider piecewise geodesic rays γ : [a,∞) → M , where we allow aninfinite countable number of corners t0 = a < t1 < · · · . Note that, although we call them corners,it may happen that γ and γ are both smooth at these points, though the only points where thedifferentiability of these curves is certain are those inside the intervals [ti, ti+1].

Definition 3.4.7. The transverse expansion rate along the piecewise geodesic curve γ : [a, b]→M , or t-expansion rate, for short, is the real number

λ(γ) := 1b− a

∫ b

aψ(γ(t))dt,

where ψ is the infinitesimal transverse expansion function of the flow Φ.

A piecewise geodesic ray γ : [a,∞)→M is t-hyperbolic with exponent λ(γ) if

λ(γ) := lims→∞

1s

∫ s

aψ(γ(t))dt,

exists and is nonzero. The t-hyperbolic geodesic ray γ is t-stable if λ(γ) < 0, and t-unstablewhen λ(γ) > 0.

Fixed a positive ε, an ε-regular value for a t-stable geodesic ray γ is a real number s0 in thedomain of γ such that ∫ s

s0(ψ(γ(t)) + ε)dt < 0

for every s > s0.

Intuitively, ε-regularity at the point s0 means that the geodesic ray’s average variationin the transverse direction, from s0 forward, does not exceed ε. Of course, this means that in aneighbourhood of the geodesic ray the leaf in which it is contained does not pulse transverselyin average more than ε, and as our foliation is transversely C1, neighbouring leaves will presenta similar behaviour. First order of business is to show that ε-regular values do indeed exist:

Lemma 3.4.8. If a piecewise geodesic ray γ : [a,∞)→M is t-stable then for any ε ∈ (0,−λ(γ))there exists an increasing unbounded sequence (sn) of ε-regular values for γ.

Proof. We can assume, without loss of generality, that a > 0. The function fa : [a,∞) → Rgiven by

fa(s) = 1s

∫ s

a(ψ(γ(t)) + ε)dt = 1

s

∫ s

a(ψγ(t))dt+

(1− a

s

)ε

is continuous, assigns the value 0 to a, and satisfies lims→∞

fa(s) = λ(γ) + ε. Thus, there must exista greatest number s1 ∈ [a,∞) such that fa(s1) = 0. Such s1 is clearly ε-regular, by construction.


We now use the same argument with s1 +1 instead of a to encounter another ε-regularnumber s2 ≥ s1 + 1. Proceeding like this, given a sequence s1 < · · · < sn of ε-regular values,we repeat the construction above for the function fsn+1 in order to obtain an ε-regular valuesn+1 ≥ sn, so that the monotone sequence sn →∞ is constructed inductively.

For the next results and in the subsequent sections we will be interest in the behaviourof holonomies along piecewise geodesic rays. To study such holonomies, we define a particularcovering by plaque chains and a monotone function ε1 : R+ → R+ that will be used in estimatesfor the derivatives of holonomy maps throughout the rest of the chapter.

Remember that, while the germs of holonomies along curves are well defined, the domains ofsuch maps depend on the choice of plaques involved. We will fix a standard choice of plaquesinvolving the Lebesgue number of the regular covering: let γ : [s0, s∗] → M be a piecewisegeodesic leaf curve, and suppose all of its corners belong to the complete transversal T . Givenan index i0 such that the open ball B(γ(s0), εU) ⊂ Ui0 , consider t0 = s0 and let z0 be the pointof Ti0 whose plaque contains γ(s0). Now we define t1 to be the infimum of the values of t forwhich γ(t) /∈ Pi0(z0) and take an index i1 for which B(γ(t1), εU) ⊂ Ui1 . As before, t1 is thepoint of the transversal section Ti1 such that γ(t1) ∈ Pi1(z1). We proceed in this manner untilwe have a chain of plaques (Pij(zj))Nj=1, which we use to define hγ. As we are always choosingthe distinguished charts containing neighbourhoods of radius at least εU , this choice assures acertain maximality of domain sizes.

Now let σ0 be a geodesic segment joining z0 to x0 = γ(s0), and note that, by construction,

l(σ0) ≤ diamU := supdiamF(Pi(x));x ∈ T , i ∈ A.

Take s1 to be a point in [s0, s∗], preferably a corner, if possible, such that the γ([s0, s1]) ⊂Pi0(z0) ∪ Pi1(z1). Join z1 to x1 = γ(s1) through a geodesic segment σ1 entirely contained inPi1(z1). Our choice of convex plaques when setting the regular covering U now implies that theshortest geodesic τ0 from z0 to z1 is entirely contained in Pi0(z0) ∪ Pi1(z1).

This process is repeated along γ until we have reached the last plaque PiN (zN ), wherezN is connected to xN = γ(s∗) via σN . Due to our construction, each segment γi := γ|[si,si+1], 0 ≤i ≤ N − 1 is a leafwise geodesic segment. This means

γ = γ0 ∗ γ1 ∗ · · · ∗ γN−1

and

τ := τ0 ∗ · · · ∗ τN−1 = (σ0 ∗ γ0 ∗ σ−11 ) ∗ (σ1 ∗ γ1 ∗ σ−1

2 ) ∗ · · · ∗ (σN−1 ∗ γN−1 ∗ γ−1N ). (3.6)


Of course, the holonomy along τ is the same as the holonomy along γ. The significant differenceis that τ has all of its corners laying at the complete transversal T .

Figure 10 – Holonomy along γ.

We want the function ε1 to be such that for every leafwise geodesic γ : [a, b]→M ina ε1(δ)-neighbourhood of γ0 : [a, b]→M one has |ψ(γ(t))− ψ(γ0(t))| < δ, for every t ∈ [a, b].

Remark 3.4.9. In particular, such condition implies that∣∣∣∣∣ 1b− a

∫ b

aψ(γ(t))dt− 1

b− a

∫ b

aψ(γ0(t))dt

∣∣∣∣∣ ≤ 1b− a

∫ b

a|ψ(γ(t))− ψ(γ0(t))| dt ≤ δ,

and hence λ(γ) lies in the interval (λ(γ0)− δ, λ(γ0) + δ). This has two important consequencesfor us:

(i) For any piecewise geodesic curve in ε1(δ)-neighbourhood of γ0 the inequality ψ(γ(t)) <ψ(γ0(t)) + δ holds, hence

∫ b

aψ(γ(t))dt <

∫ b

a(ψ(γ0(t)) + δ)dt

(ii) Whenever γ0 is a t-stable geodesic ray and we choose 0 < δ < −λ(γ0), all the geodesicrays in a ε1(δ)-neighbourhood of γ0 will be t-stable as well, since their expansion exponentwill be less than λ(γ0) + δ < 0 .

We construct the function ε1 in the following way. The bundle V is compact andthe infinitesimal transverse expansion function ψ : V → R is continuous, hence it is uniformlycontinuous. Associated to the same fixed Lebesgue number εU used for the construction ofholonomy maps above, let εt be the a t-Lebesgue number and δ0 be such that

dV (w1, w2) < δ0 ⇒ |ψ(w1)− ψ(w2)| < εt.


Given δ ∈ (0, εt) there exists a number ε2(δ) such that

dV (w1, w2) < ε2(δ)⇒ |ψ(w1)− ψ(w2)| < δ, for any w1, w2 in V.

We can assume ε2 : (0, δ0) → (0, εt) to be a increasing function, without loss of generality.Similarly, since V × [−4 diamU , 4 diamU ] is compact there is a monotone increasing positivefunction ε3 : (0, δ0)→ (0, εt), such that for every w1, w2 ∈ V and t ∈ [−4 diamU , 4 diamU ] wehave

dV (w1, w2) < ε3(δ)⇒ dV (Φ(w1, t),Φ(w2, t)) < ε2(δ). (3.7)

Now, for each (Ui, ϕi) ∈ U we pull back the restriction g|Ui of the Riemannian metric g on Mto a Riemannian metric gi on (−2, 2)n, which is expressed as a function associating to eachreal vector in (−2, 2)n a nonzero symmetric positive definite matrix. Using the usual Euclidianmatrix norm given by

‖A‖ := maxx∈Sn−1

|Ax|,

let ‖gi(x)‖ denote the maximum between the norms of gi(x) and gi(x)−1. Define

‖gi‖ := sup‖gi(x)‖;x ∈ (−1, 1)n

and‖g‖ := max

i∈A‖gi‖.

Since for any matrix A the relation ‖A‖‖A−1‖ ≥ 1 holds, this constant is no less than 1. Let

ε1(δ) := 12ε3(δ)‖g‖

2

. Now, if γ and γ0 are two curves whose distance to one another is always

less than ε1(δ), then Remark 3.4.1 implies that γ is in a ε1(δ)-neighbourhood of γ0, meaning|ψ(γ(t))− ψ(γ0(t))| < δ as we wanted.

Lemma 3.4.10. For every (y, z) ∈ (−1, 1)n−1 × (−1, 1), i ∈ A and δ > 0, the curve

c : (−ε1(δ), ε1(δ)) −→ T

t 7−→ ϕ−1i (y, z + t)

has length at most ε3(δ)‖g‖

< ε3(δ).

Proof. Since gi is a pullback, the mapping ϕi : (Ui, g|Ui)→ ((−1, 1)n, gi) is an isometry. Hencethe length of c is the same as the length of the curve ϕ c(t) = (y, z + t), whose velocity isconstant equal to 1. On the other hand, for any inner product matrix A we have |x|A ≤ |x|‖A‖,since

xTAx

|x|2= xT

|x|

(Ax

|x|

)≤ ‖A‖,


hence |x|A ≤ |x|√‖A‖. Thus

l(c) =∫ ε1(δ)

−ε1(δ)|1|g(t)dt ≤

∫ ε1(δ)

−ε1(δ)‖g‖dt = 2ε1(δ)‖g‖ = ε3(δ)

‖g‖

Let us put all these technicalities to use and prove the first of our results aboutholonomies along geodesic rays. It concerns the existence of stable submanifolds for the dynamicsof F , that is, of attracting geodesic rays:

Theorem 3.4.11. Let γ : [a,∞)→M be a t-stable, piecewise geodesic ray whose corners areall points of the complete transversal T . There exists a regular value s0 of γ, a point z ∈ Tiwhose plaque contains γ(s0), and a positive constant ε1 such that for every s∗ ≥ s0 the holonomyh∗ = hγ∗ along the curve γ∗ = γ|[s0,s∗] is defined on the interval I = (z − ε1, z + ε1) ⊂ Ti.Moreover, for sufficiently large s∗ the holonomy transformation h∗ is a contraction satisfying

0 < h′∗(y) < e12 (s∗−s0)λ(γ), ∀y ∈ I.

Proof. By hypothesis λ(γ) < 0. Set ε = −9λ(γ)10 and let s0 be an ε-regular value for γ, whose

existence is guaranteed by Lemma 3.4.8. Let δ = −λ(γ)10 and fix the constants εi = εi(δ) for

i = 1, 2, 3. We show that these set of constants, together with s0 and z = z0, satisfy all of theTheorem’s requirements. For each curve γ∗ let P∗ = (Pi0(z0), ..., PiN (zN )) be the chain of plaquescovering γ∗ constructed as discussed above, and denote by U∗ = (Ui0 , ..., Uin) and h∗ : D∗ → R∗,respectively, the chain of distinguished charts and the holonomy transformation associated toP∗. The proof is done by induction on the plaque chain’s length N .

For N = 1 we only have to consider a holonomy hi1i0 . Write ϕi0 = (0, z0). Sincethe set ϕ−1

i0 (0 × (z0 − ε1, z0 + ε1)) ⊂ Ti0 has length almost ε3/‖g‖ ≤ ε3 < εt, it is entirelycontained in the domain of hi1i0 , and therefore the holonomy transformation hi1i0 is defined onI = (z0 − εl, z0 + ε1). Moreover, since hi1i0(z0) = z1, there are positive constants ε′, ε′′ such thathi1i0(z0 − ε1, z0 + ε1) = (z1 − ε′, z1 + ε′′) ⊂ Ti1 .

Recall the decomposition of the curve t0 given in Equation 3.6, and note thathi1i0 = hτ = h−1

σ1 hγ0 hσ0 . Furthermore, we can look at the holonomies hσ1 and hσ0 aschanges of coordinates, since all they really do is relabel the plaques in their respective charts bymapping their domains into another transverse sections, namely those passing through the pointsx0 = γ(s0) and x1 = γ(s1). In other words, if we let ϕi0(x0) = (x0, z0) and ϕi1(x1) = (x1, z1)then the Transverse Uniformity Theorem says hσ1 maps (z1 − ε′, z1 + ε′′) diffeomorphically ontoI1 = ϕ−1

i1 (x1×(z1−ε′, z1 +ε′′)), while the image of hσ0 is the set I0 = ϕ−1i0 (x0×(z0−ε1, z0 +ε1)),


whose diameter is also less than ε3, once again due to Lemma 3.4.10, so that the holonomyhy0 : I0 → I1 satisfy our ε estimate.

Each leafwise geodesic γ joining a point x of I0 to a point x′ of I1 is entirely containedin Ui0 ∪ Ui1 . Moreover, due to our choice of constants, the lift of every such geodesic lies in anε2-neighbourhood of γ0. By the Mean Value Theorem, the transverse separation of points in I0

can be estimated as

dt(hγ0(z0 − ε1), hγ0(z0 + ε1)) ≤ supx∈I0|h′γ0(x)|2ε1 = sup

x∈I0|h′γ0(x)| ε3

‖g‖. (3.8)

Now on the one hand, each of these geodesic segments determines the same holonomy mapbetween I0 and I1, that is, hγ(x) = hγ0(x) for every x ∈ I0. Thus, using Equation 3.5 andRemark 3.4.9, we have

|h′γ0(x)| = |h′γ(x)| = |dΦs1−s0|(x,γ′(s0))|

= |eν(s1−s0,x,γ′(s0))|

= e∫ s1s0

ψ(γ(t))dt

< e∫ s1s0

(ψ(γ0(t))+δ)dt

= e∫ s1s0

(ψ(γ0(t))−λ(γ)10 )dt

,

while on the other hand, since we chose s0 to be ε-regular, we have∫ s1

s0ψ(γ0(t)dt) < −(s1 − s0)ε = (s1 − s0)9λ(γ)

10 .

Together, these equalities yield|h′γ0(x)| < e(s1−s0) 8

10λ(γ),

which we apply to 3.8 to conclude that

dt(hγ0(z0 − ε1), hγ0(z0 + ε1)) ≤ ε3‖g‖

e(s1−s0) 810λ(γ) <

ε3‖g‖

, (3.9)

since λ(γ) < 0. Now h∗ = hi1i0 = hτ0 = hγ0 , and this concludes the case N = 1.

The induction step is the same as above, but the estimate now involves a integralranging from s0 to sN , and we apply the cocycle properties of ν to get the desired result, due tothe same reasoning as in Remark 3.4.6.


3.5 Flow invariant measuresConsider now a probability measure m on V . This probability is to be thought of as

a linear functional

m : C0(V ;R) −→ R

f 7−→∫Vfdm.

The measure m can be extended to the set of all Borel functions on V (the extension is themeasure associated to m by the Riesz Representation Theorem, see the proof of Theorem 2.14in [48] for a detailed construction), and the measure of a Borel subset of V is defined as themeasure of its characteristic function, i.e., m(B) := m(χB). As usual, we say the measure m isinvariant under the geodesic flow, or Φ-invariant, for short, if it satisfies

m(f) = m(f Φ−t)

for every continuous function f and real number t. A Borel set B is Φ-invariant whenever itscharacteristic function is so, that is, if m(Φ−t(B)) = m(B) for every t ∈ R.

We say m has support in K if m(f) = 0 whenever f is a function whose restrictionto K is zero. The support of m is the closed subset

|m| := ∩K;K is closed and m has support in K.

Since V is compact the support of every measure in V is compact. It is a non-empty set ifthe measure m is not identically zero. It is a Borel set, being closed, and will be Φ-invariantwhenever m is Φ-invariant, since g Φ−t|K ≡ 0 implies g|Φ−t(K) ≡ 0.

MΦ will denote the set of all Φ-invariant measures of V . Recall that a Φ-invariantmeasure m is ergodic if m(B) ∈ 0, 1 whenever B ⊂ V is a Φ-invariant subset, or equivalently,if every Borel function f satisfying f Φ−t m-a.e. for every t ∈ R is constant m-a.e. The setof all ergodic measures of V will be denoted by Mε

Φ. The Ergodic Decomposition Theoremdictates that every m ∈MΦ is expressed as an integral

m =∫Mε

Φ

dτm.

Remark that if m∗ is a ergodic measure appearing in the decomposition of m∗ then its support|m∗| is contained in the support |m| of m.

To each Φ-invariant measure m ∈MΦ of a C1 foliation F , we associate the number

Λ(m) := m(ψ) =∫Vψdm

, called the t-hyperbolicity coefficient of the measure m, where ψ is the infinitesimal transverseexpansion function.


Definition 3.5.1 (Transverse hyperbolic measure). A Φ-invariant ergodic measure m∗ is saidto be t-hyperbolic when

Λ(m∗) 6= 0.

A Φ-invariant measure m ∈ MΦ is t-hyperbolic when Λ(m∗) 6= 0 for τm-almost all everymeasure m∗ appearing in its ergodic decomposition. What is more, if there is a positive constantc such that Λ(m∗) ≥ c for all such measures, then m is called a uniformly t-hyperbolic measure.

In particular, t-hyperbolic measures are nontrivial.

Remark 3.5.2. Recall from the proof of Lemma 3.4.5 (ii) that the symmetry S : (x, v) 7→ (x,−v)conjugates the diffeomorphism Φt to its inverse Φ−t, which ends up causing ψ to satisfy therelation ψ S = S ψ. Therefore, for any m ∈MΦ the pullback m− := S∗m is also a measureinMΦ, one the satisfies the relation Λ(m−) = −Λ(m). As a consequence, we can always assumeΛ(m) ≤ 0 whenever Λ(m) 6= 0, and this imposes no lost of generality. From now on, wheneverwe say a measure is t-hyperbolic we will assume its t-hiperbolicity coefficient is negative.

Before we start proving results involving such Φ-invariant measures we would liketo show the actual existence of nontrivial such measures. We do so for the particular case weare interested in this work, namely, for codimension one C1 foliations with positive entropy.Positive entropy implies the existence of invariant measures, and as we will see the existence ofinvariant measures ultimately results in the existence of resilient leaves. The main argument iswell known in Ergodic Theory and consists basically in constructing a sequence of probabilitymeasures, whose weak-∗ limit in the space of all Borel probability measures exist, due to thecompacity of this space, and is Φ-invariant. First we define the main objects and notation to beused in the proof, as well as the rest of the chapter.

Let Γ = ΓU be the regular generating set for the holonomy pseudogroup of thecovering U we set at the beginning of this chapter. Assume E = hg(F) > 0 and chooseε4 < εt ≤ 1 (recall M is normalised) such that for any ε ∈ (0, ε4) we have s(ε,F) > 3

4E.Since h(H,Γ) is positive, its expansion growth egr(H) is [exp] and therefore there exists amonotone increasing sequence (nk)k∈N of natural numbers such that the associated sequenceek := s(nk, ε4,Γ) satisfies

ek > enkE2 . (3.10)

There exists therefore, for every natural number k, a set Ak = xkjekj=1 which is (n, ε4)-separated

by Γ. Since U is finite we can assume without loss of generality the existence of an index i0 suchthat every Ak is a subset of Ti0 . We will also use the fact that Ti0 ≈ R is orientable and assumethat for each k we have

xk1 < xk2 < · · · < xkek ,


which can be achieved by relabelling the set Ak, if necessary.

Given k ∈ N and l ∈ 1, ..., ek − 1 there exists a chain of distinguished chartsuk,l = (Ui0 , ..., Uin), n < nk, whose associated holonomy map hk,l separates the points xkl andxkl+1:

dt(hk,l(xkl ), hk,l(xkl+1) > ε4.

The chain uk,l shall always be chosen as one whose length is minimal with respect to thisproperty. We denote by Dk,l the maximal domain of the holonomy hk,l. The Mean ValueTheorem guarantees the existence of a point ykl ∈ (xkl , xkl+1) such that

h′k,l(ykl ) ≥ ε4xkl+1 − xkl

.

The “expansiveness” of a subset J ⊂ Ti0 in Ak is a measure of how much the points of J ∩ Akare separated by the holonomy hk,l, defined as

E(J, k) := maxxkl,xkl+1∈J

supy∈(xk

l,xkl+1)

h′k,l(y)

.Using this sequence we designate to each J ⊂ Ti0 the following real number:

Λ(J) := lim supk→∞

1nk

ln E(J, k).

Our construction gives us a estimate for Λ(Ti0) in terms of the foliation’s entropy.Recall that we chose U such that diamTi ≤ 1 for every index i, hence for every natural numberk one can find l ∈ 1, ..., ek − 1 such that xkl+1− xkl ≤ e−1

k (otherwise xkl − xk1 > eke−1k = 1). For

such l the Mean Value Theorem yields h′k,l(ykl ) ≥ ε4ek and our choice of ek in 3.10 implies

E(Ti0 , k) > ε4enk

E2 ,

from where it follows immediately that

Λ(Ti0) > ε4E

2 .

Theorem 3.5.3. If the geometric entropy of a codimension 1 foliation F is positive, then itsgeodesic flow admits an ergodic t-hyperbolic probability measure m∗.

Proof. The notation is as above. For any ε ∈ (0,Λ(Ti0)) let λ = Λ(Ti0)− ε > 0. Since Λ(Ti0) isdefined as a supremum, we can find a sequence (lk)k∈N of natural numbers such that lk < ek

and for each lk there exists a point yk = yklk ∈ (xklk , xklk+1) satisfying the inequality

h′k,lk(yk) ≥ enkλ.


The chain uk,lk gives rise to a holonomy transformation hk = hk,lk . Associated to this holonomythere is a chain of plaques Pk starting at the plaque of Ui0 containing yk and ending at a plaquecontaining hk(yk). This chain of plaques determines a piecewise geodesic leaf curve τk, obtainedby concatenating the geodesic segments joining points in the transverse sections intersectingPk. Then the length l(τk) of this curve is at most nk diamU . Furthermore, as the Riemannianmetric induced on each leaf is complete, there exists a leafwise geodesic γk : J → F(yk) joiningyk to hk(yk) and homotopic to τk, relatively to the endpoints. For this curve we also havel(γk) ≤ nk diamU , and since γk ∼ τk rel(∂J) Proposition 1.2.28 implies that their germs, andconsequently also their derivatives, are the same. Thus

h′γk(yk) = h′k(yk) ≥ enkλ, ∀k ∈ N.

If we let vk = γ′k(0) then we can write γk(t) = π Φ(yk, vk, t) = π γk(t) fort ∈ [0, l(γk)]. This gives us a sequence of linear functionals

Mk : C0(V ;R) −→ R

f 7−→ 1l(γk)

∫ l(γk)

0f(γk(t))dt,

to which we apply the Riesz Representation Theorem to construct a sequence of probabilitymeasures mk on V . The same theorem allows to identify the space of all Borel probabilitymeasures of V with the subset of all the nonnegative linear functionals of norm 1 in C0(V ;R),which is compact when endowed with the weak-∗ topology. Therefore, we can extract from(mk)k∈N a subsequence whose weak-∗ limit m exists. Let us assume, for simplicity’s sake, thatmk itself converges to m. Then m is a Φ-invariant measure and

Λ(m) = m(ψ) =∫Vψdm

= limk→∞

∫Vψdmk

= limk→∞

1l(γk)

∫ l(γk)

0ψ(γk(t))dt

= limk→∞

1l(γk)

∫ l(γk)

0ψ(Φt(yk, vk))dt

= limk→∞

1l(γk)

ν(l(γk), yk, vk)

= limk→∞

1l(γk)

ln(d Φl(γk)

∣∣∣(yk,vk)

)

= limk→∞

1l(γk)

ln h′γk(yk) ≥ limk→∞

nkλ

l(γk)≥ Λ(Ti0)− ε

diamU > 0.

Finally, since m is nontrivial and has nonempty support, there must be an ergodic measure m∗appearing in the ergodic decomposition of m such that m∗(ψ) 6= 0, which is exactly the measurewe wanted.


3.5.1 Hyperbolic measures and the existence of resilient leaves

With the warm felling gave by the reassurance that invariant measures do indeedexist, we proceed to showing how the existence of a especial type of invariant measure impliesthe occurrence of resilient leaves for F . From now on we will be interested in the structure ofthe supports of invariant measures, more specifically in how can these supports be covered byplaques of F . We will work the saturation of their images under π : V →M and study how thedistribution along T of the plaques covering π(|m|) gives us information about m itself.

Definition 3.5.4 (Transverse support). The intersection

|m|t := π(|m|)P ∩ T

is called the t-support of the measure m.

Given x ∈ Ti, denote by Pi(x) its unit tangent bundle π−1(Pi(x)). Then

|m| ⊂ π−1(π(|m|)P) =⋃

x∈|m|t

Pi(x)

Definition 3.5.5 (Transversely discrete measure). A Φ-invariant measure m is said to bet-discrete if its t-support is a finite set

|m|t = y1, ..., yk.

We also say, in a clear abuse of language, that m is supported on the set Pi1(y1), ..., Pik(yk).

Remark 3.5.6. Given a point (x, v) in the support |m| of a t-discrete measure, the leafwisegeodesic γ(x,v) = π Φ(x, v, ·) can not remain inside a single plaque of F(x) forever, and sincethe leaf F(x) is covered by a finite amount of plaques, the chain of plaques P covering γ(x,v)

must have repeated plaques. In this way one sees that associated to each of these geodesics γ(x,v)

there is at least one closed plaque chain, the “subchain” starting and ending at the repeatedplaque of P . As the set |m| is Φ-invariant, the entire curve γ(x,v) is contained in π(|m|)P , sothis chain P can be chosen a subset of the supporting set Pi1(y1), ..., Pik(yk), and thus we canassume without loss of generality that the support of the t-discrete measure m is a finite unionof closed plaque chains. Note that this does not imply, however, that the geodesics γ(x,v) arethemselves periodic curves.

On the other hand we have the following situation:


Definition 3.5.7. The measure m has uncountable t-support, and is said to be t-nondiscrete,if |m|t is uncountable.

For a t-discrete measure each x ∈ π(|m|) has an associated index ix and one pointyix ∈ Tix for which B(x, εU) ⊂ Uix and x ∈ Pix(yix). We loose therefore no generality in assuming,as we will do from now on, that our choice of distinguished charts Uij containing the plaquesPi1(y1), ..., Pik(yk) is made in such a way that each point yij ∈ Tij has a neighbourhood ofdiameter at least εU contained in Tij .

The measure m is said to be transversely non-atomic if m(Pi(x)) = 0 for everyx ∈ |m|t. If a nonzero measure is transversely non-atomic then it must be t-uncountable. Whenthe Hausdorff dimension of |m|t is positive then m is t-uncountable as well.

Now suppose m∗ ∈MεΦ. Birkhoff’s Ergodic Theorem (see Theorem 1.14 in [57], and

the subsequent discussion) guarantees the existence of a total measure subset V0 ⊂ |m∗|, suchthat for every continuous function f : V → R and (x, v) ∈ V0 one has

lims→∞

1s

∫ s

0f(Φt(x, v))dt =

∫Vfdm∗ = Λ(f) (3.11)

The elements in the set V0 are called generic points for m∗. Another consequence of Birkhoff’sErgodic Theorem is that the forward orbit σ∞(x, v) = Φt(x, v); t > 0 of every generic pointis dense in the support |m∗|. This happens because |m∗| is the smallest closed set in whichthe support of m∗ is contained, so every open set B ⊂ |m∗| must have positive measure. Thecharacteristic function of B is continuous and on the one hand we have

lims→∞

1s

∫ s

0χB Φt(x, v)dt = Λ(B) > 0,

while on the other hand, if Φt(x, v); t > 0 ∩B = ∅, then

lims→∞

1s

∫ s

0χB Φt(x, v)dt = lim

s→∞

1s

∫ s

00dt = 0,

a contradiction. Employing a little more ingenuity we can find a increasing sequence sk of pointsreturning to any open B. Just let BV (y, r1) be a ball contained in B, whose centre y doesnot belong in σ∞(x, v). Let s1 be the infimum of times for which Φs1(x, v) ∈ BV (y, r1). Nowtake r2 < r1 such that Φs1(x, v) /∈ BV (y, r2) and let s2 be the infimum of the times for whichΦs2(x, v) ∈ BV (y, r2), an so on.

Proposition 3.5.8. If V admits an ergodic t-hyperbolic ergodic measure m∗, then thereexists a point z ∈ |m∗|t and a contractive holonomy hm∗ at z, defined on the interval (z −ε1(−Λ(m∗)/10), z + ε1(−Λ(m∗)/10)).

Proof. The main idea is to use m∗ in order to construct a suitable t-stable curve γ and thenuse Theorem 3.4.11.


Let (x, v) be a generic point for m∗. Equation 3.11 applied to this point and the in-finitesimal transverse expansion function ψ implies the geodesic ray defined by γ(t) = πΦt(x, v)is t-stable with expansion coefficient λ(γ) = Λ(m∗) < 0. Let z0 denote the point of the completetransversal T whose plaque contains γ(0). Theorem 3.4.11 implies that for ε1 = ε1(−Λ(m∗)/10)and sufficiently large positive s∗, the holonomy h∗ is a hyperbolic contraction defined at(z0 − ε1, z0 + ε1). Now the density of the forward orbit σ∞(x, v) = Φt(x, v); t > 0 implies theexistence of a strictly increasing sequence (sk)k∈N such that Φsk(x, v) ∈ BV (Φs0(x, v), ε1/10) forevery natural k. So choosing a sufficiently large natural s∗ from the sequence above guaranteesus that h∗ : (z0 − ε1, z0 + ε1)→ (z0 − ε1, z0 + ε1) is a hyperbolic contraction, hence it must havea fixed point z. Since the orbit σ∞(x, v) is entirely contained in the closed set |m∗| and theplaques in any plaque chain covering σ∞(x, v) are asymptotic to the plaque containing z, itfollows that z ∈ |m∗|t.

The geodesic ray γ|[s0,∞) does not pulse transversely, in average, more than ε1/10.Hence – by orthogonally projecting, along the transverse sections, the centres zi of the plaquescovering γ onto points of F(z), and then joining these points via geodesic segments – we can liftγ|[s0,∞) to a leafwise ray γ : [s0,∞)→ F(z). The holonomy along γ is clearly the same as theholonomy h∗ in the intersection of their domains. In particular, it fixes the point z. Furthermore,the transverse distance between γ and γ is less than ε1, so that γ is again t-stable. We oncemore make use of Theorem 3.4.11, to get a contracting holonomy hm∗ defined on (z− ε1, z + ε1),of which z is a fixed point.

Remark 3.5.9. In particular, if a family (mb)b∈B of measures satisfy a uniform t-hyperbolicityestimate Λ(mb) > c > 0 for every b ∈ B, then we can construct the holonomies of last propositionsuch that all of them are defined on a interval of width of same length around their fixed points.This lower bound will take a important role in the constructions of Section 3.5.2.

We find ourselves now in a position to prove Theorem 3.0.2 in the particular casewhen V admits a measure m with a uncountable t-support.

Theorem 3.5.10. If there exists a t-hyperbolic, t-nondiscrete measure m∗ ∈MεΦ then F has

a hyperbolic resilient leaf.

Proof. We exhibit a p.p.g. for the dynamics of F and apply Proposition 3.3.3. Assume Λ(m∗) < 0.From Theorem 3.4.11 and Proposition 3.5.8 it follows the existence of z ∈ |m∗|t, a genericpoint (x, v), a positive constant ε1 and 0 < s0 < s1 such that the holonomy hγ along the curveγ = Φ(x, v, ·)|[s0,s1] is a contraction defined on I = (z − ε1, z + ε1), which satisfies

0 < h′γ(x) < e(s1−s0) Λ(m∗)2 , ∀x ∈ I.


As the orbit of any generic points enters every open set of the support, the genericpoint (x, v) can be chosen such that its forward orbit σ∞(x, v) intersects infinitely many plaquesPi(x) of F . Consider the transverse subset (π(σ∞(x, v)))P ∩ T = π(σ∞(x, v))t = |m∗|t. It isclosed as xn → x with xn ∈ |m∗|t implies in the existence of a sequence of points in σ∞(x, v)converging to some point in Pi(x). It also has no isolated points since x ∈ |m∗| isolated wouldmean σ∞(x, v) ∈ Pi(x) and |m∗| = x would be finite. Thus, it is a perfect set in T .

I is connected and z ∈ I ∩ |m∗|t, so the perfectness of |m∗|t implies the existenceof two disjoint connect subsets cI1 and I2 of I whose intersection with the t-support of m∗is nonempty. Each Ij, j = 1, 2 is open, so we can find increasing sequences (sjk)k∈N ∈ (s0,∞)such that π(Φ(x, v, sjk)) ∈ Ij for all k ∈ N. For j = 1, 2, consider the holonomy transformationhj,k : I → I along the curve γj,k = π Φ(x, v·)|[s0,sjk]. For sufficiently large k Theorem 3.4.11guarantees that hj,k(I) ⊂ Ij. Let hj = hj,k for one such k. Then Banach Fixed Point theoremapplies to Ij ⊂ I and assures the existence of a fixed point xj ∈ Ij for hj , j = 1, 2. Furthermore,by construction one has h1(I1)∩h2(I2) = ∅. The points x1 and x2, together with the holonomiesh1 and h2, constitute a p.p.g. for F , and Proposition 3.3.3 gives us the desired result.

3.5.2 Positive entropy and the existence of ping-pong games

In this section we show how positive geometric entropy implies the occurrence ofa ping-pong game, and hence of resilient leaves. As the case when F admits a transverselynondiscrete measure has already been covered in Theorem 3.5.10, it will suffice to analyse thecase when all flow invariant measures are t-discrete. This analysis is an extremely technicalone, and will be divide and several different cases. Periodic geodesic rays will play an importantrole in this section, so we make a couple of definitions and state a result to be used afterwards,before we start to work on the actual problem.

Lemma 3.4.8 guarantees the existence of ε-regular values. In the particular casewhen γ : [a,∞) → M is a periodic function we can go even further and give a result aboutthe distribution of these regular values (or rather the distribution of their periodic analogues,called ε-good values), along [a,∞). Of course, by a periodic ray we mean the existence of anassociated real number tγ such that γ(t + tγ) = γ(t) for every real number t ≥ a. We leta = t0 < · · · < tN = a+ tγ be a partition of the fundamental domain [a, a+ tγ] of γ consistingof all the corners of the segment γ|[a,a+tγ ]. For each 1 ≤ i ≤ N and ε > 0 let

µi :=∫ ti

ti−1(ψ(γ(t)) + ε)dt,


so thatN∑i=1

µi =∫ a+tγ

a(ψ(γ(t)) + ε)dt = (λ(γ) + ε)tγ (3.12)

and|µi| ≤ (|ψ|∞ + ε)(ti − ti−1). (3.13)

Note that since V is compact the norm

|ψ|∞ := sup(x,v)∈V

|ψ(x, v)|

is finite.

The finite sequence (µi) gives rise to a periodic sequence µn+kN = µn, 1 ≤ n ≤ N

and k ∈ N. Similarly, the partition ti can be extended to a sequence of real numbers by definingtn+kN = ktn for 1 ≤ n ≤ N and k ∈ N.

Definition 3.5.11. In the setting presented above, given natural numbers l ≤ n, let

Sl(n) :=n∑i=l

µi.

A corner tk ∈ R is said to be ε-good when Sk(n) < 0 for every n ≥ k. Moreover, a stringti < ti+1 < · · · < tl of consecutive corners such that

µi+1 + µi+2 + · · ·+ µl+1 ≥ 0

is called an ε-bad sequence.

Remark that if the corner tk is ε-good then the point γ(tk) is ε-regular.

We have the following result concerning the ε-good values of a periodic ray γ:

Lemma 3.5.12. Let γ be a t-stable periodic piecewise geodesic ray as above, and ε ∈ (0,−λ(γ)).For the positive constant

c = −(λ(γ) + ε) miniti − ti−1|ψ|∞maxiti − ti−1

at least bcNc of the values in t0, ..., tN−1 are ε-good.

Proof. There exists at least one ε-good value tK that we can assume to be t0 without loss ofgenerality. Indeed, due to Lemma 3.4.8 there exists ε-regular values for γ. Let s be one suchnumber and consider a finite sequence

s < tl < tl+1 < · · · < tl+N .


Take tk realising the maximum

max0≤j≤N

∫ tl+j

s(ψ(γ(t)) + ε)dt

< 0,

so tk is a ε-good value. What is more, if k > N then tk−N is also ε-good, is since γ is periodic.We can therefore assume 0 ≤ k ≤ N1, and by considering tk as the starting point for γ, we canassume k = 0 as well. This allows us to concentrate our attention on the set t0, ..., tN−1.

Since t0 is ε-good, note that tN will also be ε-good whenever µN < 0, since SN (n) =µN + S0(n). By applying this same argument repeatedly, we show that if l is greatest number in1, ..., N such that µl ≥ 0, then tl+1, ..., tN are all ε-good. We can also consider the smallestnumber j in 1, ..., N such that the sequence tj, tj+1, ..., tl is ε-bad. The point tj itself mustsatisfy µj < 0 and be ε-good due to the minimality requirement in our choice. Indeed, we have

µj < −(µj+1 + · · ·+ µl+1) ≤ 0,

otherwise µj + · · ·µl+1 ≥ 0. Furthermore, for n ≥ l + 2 it is clear that

Sj(n) = Sj(l + 1) + Sl+2(n) = µj + µj+1 + · · ·+ µl+1 + Sl+2(n) < 0,

while for j + 1 ≤ n ≤ l + 1 the sum Sj(n) is always negative since Sj(n) ≥ 0 would imply

Sj(l + 1) = Sj(n) + Sn+1(l + 1) ≥ 0,

a contradiction. We now look for the greatest index l′ < j such that µl′ ≥ 0. Then again thecorners tl′+1, ..., tj are all ε-good and we repeat the procedure. Inductively, we continue inthis way until we have grouped all the corners in [a, a+ tγ] into ε-bad sequences and strings ofconsecutive ε good numbers.

Let i1, ..., ir ⊂ 1, ..., N be the set of indices corresponding to all the ε good valuesin [a, a+ tγ]. Due to 3.12 and 3.13 it satisfies

|µi1 + · · ·+ µir | ≤ r|ψ|∞maxiti − ti−1

andµi1 + · · ·+ µir ≤ (λ(γ) + ε)tγ < 0.

On the other hand, N miniti − ti−1 ≤ tγ, so that we have

0 < −(λ(γ)+ε) miniti−ti−1N ≤ −

r∑j=1

µij =

∣∣∣∣∣∣r∑j=1

µij

∣∣∣∣∣∣ ≤ |µi1 + · · ·+µir | ≤ r|ψ|∞maxiti−ti−1,

which implies at once that cN ≤ r, as we wanted.


Now we finally turn our attention to Theorem 3.0.2. As a partial result has alreadybe given in Theorem 3.5.10, all we have to show now is the following.

Theorem 3.5.13. Let F be a codimension one C1 foliation of positive geometric entropy. If allof its t-hyperbolic measures m∗ ∈Mε

ϕ are t-discrete, then the dynamics of F admits a p.p.g.

Let us begin by fixing notation and recalling some of the results we obtained sofar. We denote by mbb∈B the collection of all the t-hyperbolic, t-discrete, flow invariantergodic measures of F . Associated to each such probability measure there is a real numberΛb := Λ(mb) < 0, a point zb ∈ T , a hyperbolic contraction

hb : Ib → Ib

defined on the interval Ib = (zb − ε1(−Λ(m∗)/10), zb + ε1(−Λ(m∗)/10)) and fixing zb (grantedby Proposition 3.5.8), and a finite union of closed plaque chains

π(|mb|)P =Nb⋃j=1

Pibj(ybj)

(that they are closed is a consequence of the discussion in Remark 3.5.6). We observe thateven though these sets are all indexed by the same family B, they do not necessarily sharethe same cardinality, as it could happen that a same holonomy hb is associated to more thanone measure. Let J := ybj ; b ∈ B, 1 ≤ j ≤ Nb ⊂ T be the union of all these plaque’s centres.Associated to each holonomy hb there is a closed plaque chain Pb = (Pi0(z0), ..., Pinb (znb)), withz0, ..., znb ⊂ yb1, ..., ybNb and z0 = zb, as well as a piecewise geodesic leaf curve τb : [0, Lb]→F(zb) obtained through the concatenation of geodesic segments joining zi to zi+1. We consider apartition 0 = s0 < s1 < · · · < snb = Lb of [0, Lb] such that τb(si) = zi ∈ T for every index i. Ofcourse, τb|[si,si+1] is a geodesic curve for every 0 ≤ i < nb, and hτb = hPb = hb. Moreover, since

λb := λ(τb) = 1Lb

∫ Lb

0ψ(τb(t))dt = 1

Lbν(Lb, zb, τ ′b(0)) = 1

Lbln(h′b(zb)),

and hb was defined in Proposition 3.5.8 by means of Theorem 3.4.11, we can choose hb so thatits derivative is small enough to guarantee λb ≤ Λb.

Definition 3.5.14 (Irreducible Chain of Plaques). A chain of plaques P = (Pi0(z0), ..., PiN (zN ))is irreducible if for any 1 ≤ j, k ≤ N it holds

Pij(zj) = Pik(zk)⇒ j = k,

which means basically that no “subchain” of P is closed. A chain is reducible whenever it failsto satisfy this condition.


Remark 3.5.15. It suffices to work with irreducible closed chains. This is because if Pb isreducible and the indices i < j are such that Pij(zj) = Pik(zk), then we can reduce Pb into twoclosed chains

P1 = (Pi0(z0), ..., Pij(zj), Pik+1(zk+1), ..., Pinb (znb)),

P2 = (Pij(zj), ..., Pik(zk)),

and the respective associated closed loops τ1 and τ2. Remark that τb = τ ′1 ∗ τ2 ∗ τ ′′1 , whereτ1 = τ ′1 ∗ τ ′′1 . Together with Lemma 3.4.5 (iii) this decomposition gives us

λbLb = ν(Lb, τb(0)) = ν((snb − sk+1) + sj, τ1(0)) + ν(sk − sj, τ2(sj)),

or, abusing notation,λbLb = ν(τb) = ν(τ1) + ν(τ2),

which implies at least one of the loops τ1 and τ2 has expansion coefficient less than λb, that is,at least one these loops gives raise to a t-stable periodic geodesic ray. We can proceed like thisuntil we have found a irreducible chain P ∗b supported on Pb and such that the related loop τ ∗bhas expansion coefficient λ∗b ≤ λb ≤ Λb. This will be important in Case III.

Our analysis is divided in several cases, based on the cardinality of B, the nature ofthe holonomies hb and the values Λb. For the first three cases, fix a constant Λ∗ > 0 and considerthe set B∗ := b ∈ B; Λb < −Λ∗. Proposition 3.5.8 guarantees, as we noted in the Remarkjust after it, that each of the contractions hb is defined on the interval I ′b = (zb − ε1, zb + ε1)for sε1 = ε1(Λ∗/10). Additionally, due to our exponential estimate in Theorem 3.4.11, we canchoose c ∈ (0, 1) and holonomies hb such that for every b ∈ B∗ the holonomies satisfy a uniformc-estimate on their domains, there is, for every b one has |h′b(x)| < c for every x ∈ I ′b. Of course,for every b the point zb is some of the points ybl in π(|mb|)t.

Case I - The set zb; b ∈ B∗ is infinite

If zb; b ∈ B∗ is infinite it follows immediately from the compacity of M theexistence of an accumulation point z∗ ∈ T . Choose points zα and zβ in the intersectionBt(z∗, ε1/10) ∩ zb; b ∈ B∗ and let I = Iα ∩ Iβ. Then zα, zβ ⊂ I and both hα and hβ aredefined on I. From now on the argument is the same as in the construction of Theorem 3.5.10:for a sufficiently large choice of k ∈ N the images hkα(I) and hkβ(I) are disjoint. We let h1 := hkα

and h2 := hkβ, yielding a p.p.g. for the dynamics of F .

Case II - The set zb; b ∈ B∗ is finite, and every period is bounded

Here by a period we mean the length of the closed plaque chains that form thesupport of |mb|. We show that even though the collection of contractive holonomies hb ⊂ H is


finite, the hypothesis of bounded periods allows us to construct a greater family of contractiveholonomies in H, one that will be infinite and allow us to use the same technique as in CaseI. Suppose that zb; b ∈ B∗ is finite, J∗ = ybj ; b ∈ B∗, 1 ≤ j ≤ Nb is infinite and the setNb; b ∈ B∗ is bounded by N∗.

Each centre ybl can be joined to zb by a plaque chain Pb,l whose length is less thanN∗. This chain defines a piecewise geodesic leaf curve joining ybl to zb, whose corners lie in Tand whose length is at most N∗ diamU . The holonomy hb,l related to this curve is the same asthe one related to the chain Pb,l, and is defined in the whole interval Ib. Besides, using the factthe piecewise geodesics have a uniform bound for the lengths, we can estimate the transverseexpansion of all the holonomies hb,l at once and find a ε∗ > 0 such that (ybl −ε∗, ybl +ε∗) ⊂ h−1

b,l (Ib)for every choice of b ∈ B∗ and l.

Now for sufficiently large k the application

h−1b,l hkb hb,l : (ybl − ε∗, ybl + ε∗)→ (ybl − ε∗, ybl + ε∗)

is a hyperbolic contraction with fixed point ybl . As J∗ is infinite there is an unlimited amount ofsuch contractions, and therefore an accumulation point for the set of all of their fixed points.We repeat the same construction as in Case I, yielding a p.p.g. for F .

Case III - The set zb; b ∈ B∗ is finite with unbounded periods

In the first two cases we constructed sets with unlimited amounts of fixed points.Now we suppose zb; b ∈ B∗ is finite, J∗ is infinite, but Nb; b ∈ B∗ admits no upper bound,and consider the case where the set of fixed points constructed is finite even though J∗ is infinite.

For each b there is a closed chain of plaques constructed from the orbit of a genericpoint (x, v) ∈M as in Theorem 3.5.8. Consider all the irreducible closed chains obtained viathe process described in Remark 3.5.15. Among all these chains, choose the one with maximallength, and denote by τ ∗b the piecewise geodesic associated. The set of all lengths l(τ ∗b ); b ∈ B∗is either bounded or not.

If l(τ ∗b ); b ∈ B∗ is unbounded then there is a sequence of curves τ ∗b whose lengthgoes to infinity. For each of these curves we construct a collection of fixed points using Lemma3.5.12 in the following way: we extend the curve τ ∗b , which is defined by the irreducible chain(Pi0(z0), ..., Pinb (znb)), to a periodic geodesic ray τ ∗b : [0,∞) → F(zb). Due to Lemma 3.5.12,there exists a constant K, depending only on Λ∗ and ε, such that at least bKnbc of the pointsin s0, ..., snb−1 are ε-good. The corresponding points in τ ∗b (s0), ..., τ ∗b (snb) are all ε-regular,and using suitable chains formed from the plaques Pi0(z0), ..., Pinb (znb) we can see that eachone of them is a fixed point of a hyperbolic contraction of H, whose domain contain an intervalof width 2ε1 centred in the fixed point. Now as the sequence of curves τ ∗b was chosen in a way


that the lengths nb →∞, we see that this collection of fixed points and respective holonomytransformations must be infinite. Therefore, we must have an accumulation point, and weproceed as in Case I.

In last paragraph’s construction we used the fact the fixed points were distributedover an increasing sequence of plaques and that the lengths of the curves were going to infinityin order to assure we had enough fixed points to guarantee the existence of a ping-pong table.Basically, even though the collection of holonomies hb and their fixed points zb is finite, wemanaged to construct other holonomies in H that allowed us to use Case I again. This cannothope to succeed in the case those lengths are bounded, as we then meet the problem that allof the measure’s transverse hyperbolicity could be supported in a finite amount of plaques.Furthermore, after the reduction process we are left with closed loops τ ∗b that have little incommon with our original piecewise closed curves τb other than the indexing set. Even thoughthey are bounded, in general the fixed point of the holonomy along τ ∗b is not zb, meaning theargument in Case II would not suffice. In order to deal with this we will have to appeal to thefact hg(F) is positive and its dynamical implications: if l(τ ∗b ); b ∈ B∗ is a bounded set then weproceed to Case IV.

Case IV - The set of centres J is finite

So far we have dealt with all the presented cases in a similar fashion: we constructed aninfinite amount of hyperbolic contractions with domains of the same size, and used compactnessM to find an accumulation point for the contractions’ fixed points, yielding a p.p.g.. The lastcase is when the set of fixed points is indeed discrete, so adapting holonomies in order to use thislimit point argument is no longer an option. Here we will use the full strength of the positivegeometric entropy hypothesis. The idea is fairly simple, though carrying it out will requireseveral technical steps. We motivate it with the following example.

Example 3.5.16. Let f, g : R→ R be given by

f(x) = 12x and g(x) = x+ 1.

The map f is a hyperbolic contraction with 0 as a fixed point, while g clearly has no fixed pointsof its own. However, one of the compositions of these two maps is

g f(x) = 12x+ 1,

which is a hyperbolic contraction and has 2 as a fixed point. Hence f and g f , together withthe points 0 and 2, constitute a p.p.g. for the pseudogroup G(f, g). Intuitively, it is as if theholonomy g acted as a translation of the fixed point of f , moving it from 0 to 2. In this lastcase we show how given a hyperbolic contraction h1 ∈ H with a fixed point one can construct


another holonomy, h2, which translates the fixed point of h1 by a distance as small as we wish.This can therefore be done in such a way that the translated fixed point does not leave thedomain of h1, providing us with the desired ping-pong table in H.

Our main bound in this scenario will be

Λ∗ := min1

2 ,E

96 diamU

.

In this final case we assume the existence of a finite set of plaques

P∗ = Pi1(z1), ..., PiN (zN)

such that every irreducible closed plaque chain P whose associated piecewise geodesic curve τhas coefficient λ(τ) ≤ −Λ∗ is composed by plaques in P∗. We also assume that P∗ is minimalwith respect to this property, meaning that every plaque in P∗ meets at least one such chain ofplaques. The finiteness of P∗ is a necessary condition of both J being finite and l(τ ∗b ); b ∈ B∗being bounded, so that the assumption that P∗ is finite is indeed enough finish all possible cases.

Minimality implies that every centre zn belongs to a closed chain of plaques, so thereis a holonomy transformation hn to which zn is a fixed point. From now on we let ε1 be ε1(Λ∗/10).As we did in Case II, using that all the curves involved have length at most N diamU , we canfind a positive constant ε5 ≤ ε1 such that the collection of intervals In = (zn − ε5, zn + ε5),1 ≤ n ≤ N , is pairwise disjoint, and for every natural number 1 ≤ n ≤ N the holonomy hn isdefined on In and satisfies a uniform bound 0 < h′n(x) < c < 1, for every x ∈ In.

Remark 3.5.17. If F(zn)∩ In 6= zn then the fact hn is contractive implies F(zn) is resilient,and we are done. We can therefore proceed with the assumption that every leaf of F is proper.

The holonomy hn : In → In will assume the role of the map with a fixed point in Example3.5.16. All we have to do now is construct a holonomy to perform the “translation”.

Consider the sets

S(zn, ε) :=⋃

x∈Bt(zn,ε)Pi(x) ⊂M,

which are the union of every plaque containing an element of a transverse ε-neighbourhood ofzn. Define as well

S(ε) :=N⋃n=1S(zn, ε), S(zn, ε) := π−1(S(zn, ε)) and S(ε) := π−1(S(ε)).

The next lemma is a uniform estimate on the transverse expansion outside S(ε), that is, givena geodesic segment whose distance to any plaque of P∗ is no less than ε, we can bound itstransverse expansion by quantities depending only on ε and the geodesic segment’s length.


Lemma 3.5.18. For each positive ε there exists a positive real number L(ε) > 0 such thatwhenever (x, v) ∈ V and L > 0 satisfy the condition Φ(x, v, t); t ∈ [0, L] ∩ S(ε) 6= ∅, then∣∣∣∣∣

∫ L

0ψ(Φt(x, v))dt

∣∣∣∣∣ < |ψ|∞L(ε) + LΛ∗.

In other words, if the leafwise geodesic segment γ = π Φ(x, v, ·) : [0, L]→ F(x) does not meetS(ε) then its transverse expansion satisfies

λ(γ) < Λ∗ + L(ε)|ψ|∞L

.

Proof. Denote by VL,ε the set of all points in V that satisfy the statement’s requirements for εand L, that is,

VL,ε :=

(x, v) ∈ V ; Φ ((x, v) × [0, L]) ∩ S(ε) = ∅.

We note that for L < L′ we haveVL′,ε ⊂ VL,ε. (3.14)

Recall that Lemma 3.4.5 (i) states that for whatever point (x, v) ∈ V one has ν(L, x, v) =∫ L

0ψ(Φt(x, v))dt ≤ L|ψ|∞, hence the continuous real function αε : (0,∞)→ R given by

t 7→ supz∈Vt,ε

∣∣∣∣1t ν(t, x, v)∣∣∣∣

is a well-defined one. It is continuous, since for any sequence tn → t we have

lim infn

Vtn,ε = lim supn

Vtn,ε = Vt,n,

due to 3.14, hence αε(tn)→ αε(t).

The most important of αε’s properties is that lim supt→∞

αε < Λ∗. Indeed, assume bycontradiction this inequality does not hold. Then there must be sequences (xk, vk) and (Lk),k ∈ N, for which

limk→∞

1Lkν(Lk, xk, vk) = 1

Lk

∫ Lk

0(ψ(Φt(xk, vk))dt ≥ Λ∗.

We can then use the fact that ψ(x,−v) = −ψ(x, v) and, changing the sign of the sequence ofvector vk, write down

limk→∞

1Lk

∫ Lk

0(ψ(Φt(xk, vk))dt ≤ −Λ∗.

Each segment of orbit Φ ((xk, vk) × [0, Lk]) defines a measure mk, obtained by applying RieszRepresentation Theorem to the functional

f 7→ 1Lk

∫ Lk

0f Φ(xk, vk, r)dr.


The sequence (mk) must have a convergent subsequence, whose limit m is Φ-invariant andsatisfies Λ(m) ≤ −Λ∗. Any ergodic measure m∗ taking part of m’s ergodic decomposition ist-hyperbolic, since Λ(m∗) ≤ Λ(m) ≤ −Λ∗. Furthermore, due to our construction one has that

|m∗| ⊂ |m| ⊂∞⋃k=1

Φ ((xk, vk) × [0, Lk]) ,

the late union being disjoint from S(ε) by hypothesis. Now, due to the main set of hypothesisof this paragraph, the t-hyperbolic ergodic measure m∗ is also transversely discrete, and hasassociated to it both a irreducible chain of plaques and a piecewise geodesic curve τ∗. The lastsatisfies the inequality λ(τ∗) ≤ −Λ∗, hence it meets at least one plaque of P∗, contradicting thedisjointness hypothesis of the Lemma.

Finally, the upper bound on the limit superior of αε means the set t;αε(t)−Λ∗ ≥ 0is finite. It is nonempty as well, as we can check reasoning by contradiction: if αε(t) < Λ∗ forevery t then every leafwise curve would have expansion coefficient less than Λ∗, and every chainof plaques would be a composition of plaques in the finite set P∗, which can not be. We canthen define the function L to be given by

L(ε) := supt;αε(t)− Λ∗ ≥ 0.

Given any positive L and a pair (x, v) ∈ VL,ε we have two possibilities

(i) 0 < L ≤ L(ε), in which case∣∣∣∣∣∫ L

0ψ(Φt(x, v))dt

∣∣∣∣∣ ≤ L|ψ|∞ ≤ L(ε)|ψ|∞ (3.15)

(ii) L > L(ε). If that’s the case, then αε is strictly less than Λ∗, implying immediately∣∣∣∣∣∫ L

0ψ(Φt(x, v))dt

∣∣∣∣∣ < LΛ∗. (3.16)

We finish the proof by adding LΛ∗ to 3.15 and L(ε)|ψ|∞ to 3.16. Since both these quantitiesare positive, we are left with the desired inequality, which holds for every real number L.

Corollary 3.5.19. Let L > L(ε), s ∈ R be positive and z ∈ V \ S(ε). Then∣∣∣∣∣∫ L

0ψ(Φt(x, v)dt

∣∣∣∣∣ ≥ LΛ∗ ⇒ Φ ((x, v) × [0, L]) ∩ S(ε) 6= ∅.

Proof. An empty intersection would imply that αε(L) < Λ∗, since L > L(ε), which contradictsthe statement.


From now on the techniques involved will make use of the positive geometric entropyand the separating holonomies, in a similar way to Theorem 3.5.3 and the discussion precedingit, so we fix the notation to be as in that section. Recall that hg(F) = E > 0, (ek) and (nk)were sequences of natural numbers such that ek = s(nk, ε4,Γ) > enk

E2 for every natural k, and

Ak = xk1, ..., xekk was a (nk, ε4,Γ)-separated set of maximal cardinality ek. Set

ak = denk E4 e.

Since ak ≤√ek and Ak is of maximal cardinality, it follows from Dirichlet’s Schubfachprinzip

(AKA the Pigeonhole Principle) that for any k ∈ N there is a closed interval Jk of length a−1k

such that ak ≤ #Ak ∩ Jk. We reindex Ak in order to have xkl ; 1 ≤ l ≤ ak ⊂ Ak ∩ Jk. We alsoassume, as we’ve done before, that xkl < xkl+1 for all the indices 1 ≤ l < ak.

For every choice of k ∈ N and 1 ≤ l < ak there is a minimal-length plaque chain Pk,lwhose holonomy map hk,l separates xkl and xkl+1 to a distance no less than ε4. Remark that thischain’s length is at most nk. Set ykl to be the point inside the interval [xkl , xkl+1] which maximisesthe derivative h′k,l. It follows from the Mean Value Theorem that

h′k,l(ykl ) ≥ ε4ak,

since, by construction, dt(xkl , xkl+1) < a−1k and dt(hk,l(xkl ), hk,l(xkl+1)) ≥ ε4. Each point ykl

determines a piecewise geodesic curve τk,l with corners in T , which joins ykl to its image underhk,l. Denote this image by wkl , and note that l(τk,l) ≤ nk diamU . The completeness of theleafwise Riemannian metric implies the existence of a leafwise geodesic curve γk,l from ykl to wkl ,which is homotopic to τk,l relatively to the endpoints. The holonomies along γk,l and τk,l arejust hk,l. Furthermore, since γk,l ≈ τk,l, Proposition 1.2.28 implies the holonomies along themshare the same germ at ykl , hence

h′γk,l(ykl ) = h′τk,l(y

kl ).

Set Lk,lk = l(γk,l) ≤ nk diamU . We parametrise γk,l by arc-length so that it becomes a unitspeed curve, and write vkl := γ′k,l(0). With this conventions the lift γk,l becomes just the segmentof orbit Φt(ykl , vkl ); t ∈ [0, Lk,lk ], that is,

γk,l(t) = Φ(ykl , vkl , t).

Set the family of coefficients λk,l := λ(γk,l). This is nothing more than

λk,l = 1Lk,lk

∫ Lk,l

0ψ(γk,l(t))dt = 1

Lk,lkν(Lk,lk , ykl , vkl ) = 1

Lk,lkln(h′k,l(ykl )),


and therefore the upper bound λk,l ≤ |ψ|∞ holds. From the set of inequalities we have shown sofar it then follows that

|ψ|∞ ≥ λk,l = 1Lk,lk

ln(h′k,l(ykl ))

≥ 1Lk,lk

ln(ε4ak)

≥ 1Lk,lk

ln(ε4enkE4 )

≥ 1nk diamU

(nkE

4 + ln ε4)

k→∞−−−−→ E

4 diamU . (3.17)

In particular, note that the convergence is uniform in the sense that it does not depend on l,but only on k.

Together with Corollary 3.5.19 this implies that for sufficiently large k and any1 ≤ l ≤ ak there exists a real number Fk,l ∈ (0, Lk,l) such that

γk,l(t); t ∈ [0, Fk,l] ∩ S(ε5) = γk,l(t); t ∈ (Fk,l − δ1, Fk,l] for some small positive δ1.

The number Fk,l is an approximation for the first entry-time of γk,l in S(ε5). By ρk,l we denotethe piecewise geodesic curve consisting of the concatenation of γk,l = γk,l|[0,Fk,l] with a leafwisegeodesic segment σ, entirely contained in a single plaque, and joining γk,l(Fk,l) to a point ukl ∈ T .The holonomy along ρk,l will be denoted by fk,l, and it is such that fk,l(ykl ) = ukl . Let us estimateits derivative at the point ykl :

|ln f ′k,l(ykl )| = |ln(h′σ(hγk,l(ykl ))h′γk,l(ykl ))|

= |ln h′σ(hγk,l(ykl )) + ln h′γk,l(ykl )|

= |ν(l(σ), γk,l(Fk,l), γ′k,l(Fk,l)) + ν(Fk,l, ykl , vkl )|

≤∣∣∣∣∣∫ Fk,l

0ψ(γk,l(t))dt

∣∣∣∣∣+∣∣∣∣∣∫ Fk.l+l(σ)

Fk,l

ψ(σ(t))dt∣∣∣∣∣ .

We now choose ε′ small enough such that γk,l does not enter S(ε′). Applying Lemma 3.5.18 tothe first integral above we get

|ln f ′k,l(ykl )| ≤ L(ε′)|ψ|∞ + Fk,lΛ∗ + l(σ)|ψ|∞≤ (L(ε′) + diamU)|ψ|∞ + Fk,lΛ∗.

If only k is sufficiently large, then |ln f ′k,l(ykl )| ≤ (L(ε′) + diamU)|ψ|∞ + Fk,lΛ∗ ≤ 2Λ∗Lk,l, fromwhere it follows

e−2Λ∗Lk,l < f ′k,l(ykl ) < e2Λ∗Lk,l . (3.18)

We know the holonomies along the geodesics γk,l separate points in the transversalto a distance more than ε4 “after k steps”. Intuitively, one may think that the greater k is (or in


other words, the further we must walk along leaves before they peel apart to a distance greaterthan ε4), the closer the points in the transverse were to one another to begin with. So, if inone direction these holonomies are separating elements of T , in the opposite direction thoseelements should be brought together. Let us formalize this intuition and see how we can use it toconstruct a contracting holonomy for the holonomy pseudogroup H. Define γ−1

k,l be the inversethe curve γk,l, given by the relation γ−1

k,l (t) = γk,l(Lk,l − t). It is straightforward to check thatγ−1

k,l(t) = γk,l(Lk,lk − t) and hγ−1k,l

= h−1γk,l

. What is more, λ(γ−1k,l (t)) = −λ(γk,l) = −λk,l, thus the

estimate in 3.17 says that γ−1k,l (t) is t-hyperbolic. We set ε6 = 10Λ∗ < E/4 diamU < λk,l and

use an argument similar to that of Lemma 3.4.8 to construct a specific sequence of ε6-regularvalues for γ−1

k,l (t) satisfying a given equality:

Lemma 3.5.20. Let K1 be any natural number. For sufficiently large k the t-hyperbolicgeodesic γ−1

k,l admits a finite sequence (sjk,l)K1j=1 of ε6-regular values satisfying∫ Lk.l

sjk,l

ψ(γ−1k,l(t) + 10Λ∗)dt = (−λk,l + 10Λ∗)Lk,l + (j − 1)(1 + Λ∗) < 0.

Proof. There must be a greatest real number s1k,l for which∫ s1k,l

0ψ(γ−1

k,l(t) + 10Λ∗)dt = 0.

Then ∫ Lk,l

s1k,l

ψ(γ−1k,l(t) + 10Λ∗)dt =

∫ Lk,l

0ψ(γ−1

k,l(t) + 10Λ∗)dt = (−λk,l + 10Λ∗)Lk,l.

For sufficiently large k one has (−λk,l + 10Λ∗)Lk,l < −(1 + Λ∗). Now we let L1k,l be the smallest

real number for which ∫ L1k,l

s1k,l

ψ(γ−1k,l(t) + 10Λ∗)dt = −(1 + Λ∗),

yielding ∫ Lk,l

L1k,l

ψ(γ−1k,l(t) + 10Λ∗)dt =

∫ Lk,l

0ψ(γ−1

k,l(t) + 10Λ∗)dt

−∫ s1k,l

0ψ(γ−1

k,l(t) + 10Λ∗)dt

−∫ L1

k,l

s1k,l

ψ(γ−1k,l(t) + 10Λ∗)dt

= (−λk,l + 10Λ∗)Lk,l + 1 + Λ∗.

The integral above is again less than zero if k is large enough, so that we can set s2k,l to be the

smallest real number in the curve’s domain such that∫ s2k,l

L1k,l

ψ(γ−1k,l(t) + 10Λ∗)dt = 0,


and then start the procedure all over again. Assuming we have k 0 we can iterate this processK1 times and arrive at a finite sequence (sjk,l)K1

j=1 of ε6-regular values satisfying∫ Lk.l

sjk,l

ψ(γ−1k,l(t) + 10Λ∗)dt = (−λk,l + 10Λ∗)Lk,l + (j − 1)(1 + Λ∗) < 0,

exactly as we wanted.

Now let A = #A be the amount of distinguished charts in the regular atlas U . Onehas the following general result:

Lemma 3.5.21. Given any natural number n, positive constant θ < 2 and B ⊂ T whosecardinality satisfies

#B > n⌊2Aθ

⌋+ 1

there is a subset B′ ⊂ B of cardinality n and a point x ∈ T such that dt(x, y) < θ

2 for ally ∈ B′.

Proof. Yet another application of the Pigeonhole Principle. Each transverse section Ti hasdiameter 1, hence it admits at most bθ/2c points whose distance from one another exceeds θ/2.Now that fact that

#B > n⌊2Aθ

⌋+ 1

implies the existence of a transverse section Ti whose intersection with B has at least n⌊2θ

⌋+ 1

elements. This means we can find an interval (x− θ/2, x+ θ/2) containing at least n elementsfrom B. Choose any of those n elements to form B′ and we have the desired result.

Set K0 := 2|ψ|∞ diamU + 2 and

K1 := K0

⌊4Aε1

⌋+ 1.

Let us use Lemma 3.5.20 with K1 as above to get a sequence (sjk,l)K1j=1 of ε6-regular points

sufficiently close to one another. For each 1 ≤ j ≤ K1 we consider a geodesic segment σjk,l oflength at most diamU , completely contained in one plaque and joining the point γ−1

k,l (sjk,l) to

some point wjk,l in the complete transversal T . By taking the set B of Lemma 3.5.21 to bewjk,l we obtain a subcollection of indices 1 ≤ j1 < · · · < jK0 ≤ K1 and a point x ∈ T forwhich dt(x,wjik,l) < ε1/4 for every i ∈ 1, ..., K0. Hence dt(wjik,l, w

ji′k,l) < ε1/2 for every choice of


i and i′ in 1, ..., K0. Using the restriction γ−1k,l of γ−1

k,l to the segment[sj1k,l, s

jK0k,l

]we define a

new curve joining the points wj1k,l and wjK0k,l by

ςk,l := (σj1k,l)−1 ∗ γ−1k,l ∗ σ

jK0k,l .

Let gk,l be the holonomy along ςk,l. By construction gk,l(wj1k,l) = wjK0k,l . As ε =

9Λ∗/10 < ε6, we have that sjk,l is a ε-regular value as well. Theorem 3.4.11 can then be applied,guaranteeing the domain of gk,l contains the interval I1

k,l = (wj1k,l − ε1, wj1k,l + ε1). The derivative

of gk,l in this interval is estimated in the following lemma:

Lemma 3.5.22. For any y ∈ I1k,l one has

ln g′k,l(y) ≤ −(K0 − 1)(

1 + 910Λ∗

)+ 2|ψ|∞ diamU < −1.

Proof. First, note that

ln g′k,l(wj1k,l) = ln h′

σjK0k,l

(sjK0k,l ) + ln h′γk,l(s

j1k,l) + ln

(h−1σj1k,l

)′(wj1k,l)

≤∫ s

jK0k,l

sj1k,l

ψ(γ−1k,l(t))dt+ 2|ψ|∞ diamU .

Now, on the one hand we have

∫ sjK0k,l

sj1k,l

ψ(γ−1k,l(t))dt =

∫ sjK0k,l

sj1k,l

(ψ(γ−1k,l(t)) + 10Λ∗)dt− (sjK0

k,l − sj1k,l)10Λ∗,

while on the other hand we can use Lemma 3.5.20 to get

∫ sjK0k,l

sj1k,l

(ψ(γ−1k,l(t)) + 10Λ∗)dt =

∫ Lk,l

sj1k,l

(ψ(γ−1k,l(t)) + 10Λ∗)dt−

∫ Lk,l

sjK0k,l

(ψ(γ−1k,l(t)) + 10Λ∗)dt

= (10Λ∗ − λk,l)Lk,l + (j1 − 1)(1 + Λ∗)

− ((10Λ∗ − λk,l)Lk,l + (jK0 − 1)(1 + Λ∗))

= (j1 − jK0)(1 + Λ∗).

Putting these inequalities together yields

ln g′k,l(wj1k,l) ≤ (j1 − jK0)(1 + Λ∗)− (sjK0

k,l − sj1k,l)10Λ∗ + 2|ψ|∞ diamU

≤ (j1 − jK0)(1 + Λ∗) + 2|ψ|∞ diamU .

For an arbitrary y ∈ I1k,l we use the definition of ε1 to get (recall Remark 3.4.9)

ln g′k,l(y) ≤ (j1 − jK0)(1 + Λ∗) + 2|ψ|∞ diamU + (jK0 − j1)Λ∗10 .


Of course, we have jK0 − j1 ≥ K0 − 1, so that we are left with

ln g′k,l(y) ≤ −(K0 − 1)(

1 + 910Λ∗

)+ 2|ψ|∞ diamU .

Finally, recall that K0 := 2|ψ|∞ diamU + 2, so that

−(K0 − 1)(1 + 910Λ∗) + 2|ψ|∞ diamU ≤ −1− 9

5Λ∗|ψ|∞ diamU < −1.

As Λ∗ < 1 this immediately implies ln g′k,l(y) < −Λ∗. Moreover, we have g′k,l(y) < e−1 <12 , and

consequentlygk,l(I1

k,l) ⊂ (wjK0k,l − ε1/2, w

jK0k,l + ε1/2) ⊂ I1

k,l,

because dt(wjik,l, wji′k,l) < ε1/2. This means gk,l has a fixed point z1

k,l ∈ I1k,l, and therefore the

curve ςk,l determines a irreducible closed plaque chain whose associated piecewise geodesic τ∗has expansion coefficient

λ(τ∗) = 1l(τ∗)

ln g′k,l(z1k,l) < Λ∗.

Hence τ∗ must be covered by plaques in P∗ and the fixed point z1k,l lies in the leaf F(zn) for some

1 ≤ n ≤ N (recall that ziNi=1 is the set of all the centres of plaques in P∗). Besides, consideringthe segment γ−1

k,l (t); sj1k,l ≤ t ≤ Lk,l −Fk,l has one of its endpoints inside S(ε5) (recall Fk,l is an

approximate first entry-time of γk,l in S(ε5)), we conclude that the holonomy map along thisgeodesic segment maps z1

k,l to a point of F(zn)∩S(ε5). This, together with our choice of ε5 suchthat F(zn) ∩ Tin = zn (this comes from Remark 3.5.17), imply that gk,l(z1

k,l) = zn. In otherwords, the fixed point of gk,l is one of the centres of plaques in P∗.

Earlier on, we denoted by hk,l the holonomy along the geodesic γk,l. Let hk,l : I1k,l → T

be the holonomy along the geodesic segment γ−1k,l ; s

j1k,l ≤ t ≤ Lk,l (it is somewhat of a

“stable piece” of the holonomy h−1k,l ). Note that this holonomy satisfies hk,l(wj1k,l) = ykl since

γ−1k,l (s

j1k,l) ∈ Pi(w

j1k,l), and that the fixed point zn of gk,l belongs to its domain. Define wn := hk,l(zn).

We estimate the derivative of hk,l in much the same way we did for gk,l in our last Lemma. FromLemma 3.5.20 we have

ln(h′k,l(y)) ≤∫ Lk,l

sj1k,l

ψ(γ−1k,l (t))dt+ Lk,l

Λ∗10 − s

j1k,l

Λ∗10

≤ −λk,lLk,l + 10Λ∗sj1k,l + (j1 − 1)(1 + Λ∗) + Lk,lΛ∗10

≤ −λk,lLk,l + 10Λ∗Lk,l + (K1 − 1)(1 + Λ∗) + Lk,lΛ∗= (−λk,l + 11Λ∗)Lk,l + (K1 − 1)(1 + Λ∗).


Figure 11 – An illustration of all the holonomies and points along the separating curve γk,l.

If only Lk,l is sufficiently large, specifically if Lk,l ≥(K1 − 1)(1 + Λ∗)

Λ∗(remark that K1 does not

bear any dependence on k or l), then

ln(h′k,l(y)) ≤ (12Λ∗ − λk,l)Lk,l.

The asymptotic behaviour of h−1γk,l

is therefore completely determined by the factor12Λ∗−λk,l. Due to the estimate in 3.17, the fact that λk,lLk,l = ln h′k,l(ykl ) and the very definitionof Λ∗, we have

(12Λ∗ − λk,l)Lk,l ≤(

E

8 diamU − λk,l)Lk,l

≤ Enk8 − ln h′k,l(ykl )

≤ 12 ln ak − ln h′k,l(ykl )

= ln( √

akh′k,l(ykl )

), (3.19)

which together with the first estimate above gives us ln(h′k,l(y)) ≤ ln(√ak/h′k,l(ykl )), andconsequently

h′k,l(y) ≤

√ak

h′k,l(ykl ) .

With this and the Mean Value Theorem one can see immediately that

dt(ylk, wn) ≤ h′k,l(y)dt(wj1k,l, z1

k,l) ≤ε1√ak

h′k,l(ykl ) ≤ε1ε4

√akak→ 0. (3.20)


Remark 3.5.23. Summing up our tour so far, we have constructed, for sufficiently large kand any 1 ≤ l < ak, holonomies fk,l, gk,l and hk,l, the last two defined on a transverse ε1-neighbourhood Ik,l of a point wk,l. What is more, the holonomy gk,l contracts the intervalIk,l onto itself and admits a fixed point zk,l, which is really just one of the centres of the setP∗ = Pi1(z1), ..., PiN (zN).

That is about all we need to finally prove the existence of resilient leaves in CaseIV. Before we start with the final step, let us fix a maximal choice between all the curves andholonomies defined for each index k, and in doing so we will also clean up the notation a bit.The dependence on l is eliminated by taking maxima over Ak ∩ Jk, in the following manner.Recall we fixed a certain Ti0 containing all of the separated sets Ak, and that Jk is an intervalof length 1/ak such that #(Ak ∩ Jk) ≥ ak. For each k, we choose the index lk ∈ 1, ..., ak suchthat the point yklk realises the maximum

h′k,lk(yklk

) = max1≤l≤ak

h′k,l(ykl ).

Now, on the one hand the set Jk has length 1/ak, while on the other hand it shares at least ak ofits points with Ak. This means one can find a pair of consecutive points no further apart than a−2

k

from one another, that is, there exists some 1 ≤ l < ak for which xkl+1−xkl ≤ 1/a2k. Consequently,

our choice of yklk , together with the Mean Value Theorem, assures us that h′k,lk(yklk

) ≥ ε4enk

E2 .

We fixhk := hk,lk , yk := yklk , Lk := Lk,lk and λk := 1

Lk,lkln(hk,lk(yklk)).

The curve γk := γk,lk : [0, Lk] → F(yk) is such that γk(0) = yk. Besides, the fixed point z1k,lk

constructed above will be denoted by znk , 1 ≤ nk ≤ N , and the interval I1k,lk

is just Ik. Theregular value sj1k,lk is called sk and the holonomy along the segment γ−1

k (t); sk ≤ t ≤ Lk isdenoted by hk. What is more, vk will be the point of T whose plaque contains γk(sk) (it wasdenoted by wj1k,lk earlier on), so we have hk(vk) = yk and hk(znk) := wnk . As we noted before, allof our estimates so far are uniform on k, so each one of them holds for the maps we just defined.

Defineλ∗ := lim sup

k→∞λk.

As this constant is a supremum we can assume, passing to a subsequence if necessary, that theinequalities

λk ≥(

1− 1k

)λ∗

hold true for every natural number k. Furthermore, λ∗ can be estimated just like we did for thecoefficients λk,l in 3.17, yielding the inequalities

E

2 diamU ≤ λ∗ < λk < |ψ|∞ ∀k ∈ N. (3.21)


Lemma 3.5.24. For sufficiently large k, let znk correspond to the hyperbolic fixed pointconstructed using the holonomy gk of the path γk. There exists a positive constant 0 < ε7 < ε5

and a holonomy transformation

rk : (znk − ε7, znk + ε7)→ (znk − ε5, znk + ε5) = Ink

such that rk(znk) 6= znk .

Proof. Fix k 0 such that our previous estimates hold. From 3.20 comes

dt(yk, wnk) ≤ε1√ak

h′k(yk)=: δk.

Consider the interval Υk = (yk−2δk, yk +2δk) ⊂ Ti0 . By taking its radius to be 2δk we make surethat wnk ∈ Υk. Moreover, we have yk ∈ (xklk , x

klk+1), where xklk and xklk+1 are points in Jk ∩ Ak,

so the cardinality of the intersection Jk ∩Υk depends heavily on δk, and more fundamentally onk itself.

Let mk := #(Υk ∩ Ak). If we assume that mk > 2, than an application of thePigeonhole Principle guarantees the existence of two consecutive points xkl and xkl+1 of Ak suchthat dt(xkl , xkl+1) ≤ 4δk/(mk − 2). Now, the holonomy of the indeces k and l satisfies

h′k,l(ykl ) ≥dt(h′k,l(xkl ), h′k,l(xkl+1))

dt(xkl , xkl+1) ≥ ε4(mk − 2)

4δk.

Due to our maximal choice of hk we have then h′k(yk) ≥ ε4(mk − 2)/4δk, yielding

mk ≤4h′k(yk)δk

ε4+ 2 = 4ε1

ε4

√ak + 2.

From this follows that the complementary set Sck := (Jk \Υk)∩Ak has finite cardinality bk ≤ ak

satisfyingbk ≥ ak −mk ≥ (ak − 2)− 4ε1

ε4

√ak.

What is more, when mk ≤ 2 the above relation is trivially satisfied, so that it holds for everymk. Thus we have

1 ≥ bkak≥ 1− mk

ak≥ 1− 2

ak− 4ε1

ε4

√akak

k→∞−−−−→ 1,

that is, limkbk/ak = 1. This guarantees that for large enough k the complement Sck not only is

nonempty but asymptotically has ak elements, that is, for large k the majority of the elementsin Jk ∩ Ak lie outside Υk. The set Jk \Υk can be written as the disjoint union of two intervalsJ1k and J2

k , with at least one of then nonempty. We can therefore, for sufficiently large k, repeatall the constructions of this section (briefly stated in Remark 3.5.23) outside the interval Υk inorder to obtain a geodesic γk = γk,l, associated holonomies fk, gk, and hk, and points yk and


wnk in the range of the map hk. The point wnk , specifically, is the image under hk of the pointznk – which is fixed under the holonomy gk – and belongs in a δk-neighbourhood of yk (remarkthat δk depends only on k, not on the second index l). One then has that wnk /∈ Υk.

This construction can be repeated N + 1 times if only k is sufficiently large. Sincewe only have N plaque centres in the set P∗, this means at some point the construction abovecan be done in such a way that znk = znk , even though wnk 6= wnk . We assume without loss ofgenerality that is the case. Recall that fk(yk) = uk and that it satisfies, as we have seen in 3.18,the following inequalities

e−2Λ∗Lk < f ′k(yk) < e2Λ∗Lk .

We can write fk = h−1k qk, where qk is the holonomy along the segment γk(t); sk ≤

t ≤ Lk. Thus f ′k(yk) = q′k(vk)(h−1k )′(yk) = [(q−1

k )′(vk)]−1(h−1k )′(yk), from where

(q−1k )′(vk)e−2Λ∗Lk < (h−1)′k(yk) < (q−1

k )′(vk)e2Λ∗Lk .

But the holonomy q−1k is just the holonomy along the segment γ−1

k (t); sk ≤ t ≤ Lk. Hence,from Remark 3.4.6, Lemma 3.5.20 and the fact that sk ∈ [0, Lk], follows that

(q−1k )′(vk) = e

∫ Lksk

ψ(γ−1k,l(t))dt+

∫ 10 ψ(σ(t))dt

≤ e−λkLk+10Λ∗sk+|ψ|∞ diamU

≤ e(10Λ∗−λk)Lk+|ψ|∞ diamU ,

hence(h−1

k )′(yk) < e(12Λ∗−λk)Lk+|ψ|∞ diamU .

Finally, we estimate the separation of Jk under h−1k by applying the Mean Value

Theorem to its endpoints. Since Jk has length a−1k , from the definition of Λ∗, Equation 3.21, the

estimate for (h−1k )′(yk) above, and the fact that Lk ≤ nk diamU , it follows that

ln(diam(h−1k (Jk))) ≤ (12Λ∗ − λk)Lk − ln(ak) + |ψ|∞ diamU

≤(

E

8 diamU −E

2 diamU

)Lk −

E

4 nk + |ψ|∞ diamU

≤ E

8 nk −E

2 nk −E

4 nk + |ψ|∞ diamU

≤ −58Enk + |ψ|∞ diamU k→∞−−−−→ −∞.

This means diam(h−1k (Jk)) → 0, and therefore, there is a small enough ε7 for which the map

rk := hk h−1k is defined on an ε7-neighbourhood of yk. Moreover, it satisfies

rk(znk) = hk(wnk) 6= znk ,


which is exactly as we wanted.

Now, for a sufficiently large choice of natural number l, the image of Ink under hnkis entirely contained in a ε7-neighbourhood of znk . Hence the application

h−lnk rk hlnk

: Ink → Ink

is a hyperbolic contraction of which znk is not a fixed point, and together with hnk it provides aping-pong game for the dynamics of F , as we wanted.

3.6 A continuous counterexampleWe end the chapter showing that the C1 hypothesis on Theorem 3.0.2 is the best

we can do, as the result no longer holds for continuous foliations. In order to prove this, weconstruct a pseudogroup of homeomorphisms which has no resilient orbit, but whose topologicalentropy fails to vanish. This example was provided by Ghys, Langevin and Walczak in [26].

Example 3.6.1. Consider I = [0, 1] and a monotone function f : I → I which fixes theendpoints and satisfies f(x) < x for every x ∈ int I. For some x0 ∈ int I, let y0 = f−1(x0) > andconsider the sequence

xi = f i(x0), i ∈ Z.

Remark that x−1 = y0,, that f i → 0 as i→∞, and that f i → 1 as i→ −∞. What is more, wehave xi = f(xi) ∀i ∈ Z. These simple facts imply that the interval J = (x0, y0] ⊂ int I providesa partition of I, given by the collection of subsets f i(J)i∈Z.

Let Φ : J × R → J be a flow on J such that Φt(x) > x for every positive t andx ∈ int J . Keeping last chapter’s convention that Φ(·, t) = Φt, we define the homeomorphism

g : I −→ I

x 7−→

0 if x = 0,

1 if x = 1,

f i Φ2i f−i if x ∈ f i(J).

Denote by G the pseudogroup G(g−1, f−1, idI , f, g). The extreme points 0 and 1 arefixed by both f and g, thus their orbits are singletons. Moreover, the points of the sequence xiare the only ones in I whose orbit under G have cluster points, and all of these cluster pointsare either 0 or 1, so that none of the orbits Gxi accumulate on xi. That is to say G does nothave any resilient orbit. Now let us show that the topological entropy of G is positive.


Proposition 3.6.2. Given p ∈ int J , let ε = Φ1(p) − p. The set An(p) = Φi(p)2n−1i=0 is

(3n− 2, ε,Γ)-separated, where Γ = g−1, f−1, idI , f, g.

Proof. Due to the manner g was defined we can write Φ2k = f−k g fk for any k ≥ 0. Inparticular, Φ2k is the restriction to J of an element of G whose length with respect to Γ is nomore than 2k + 1. Since any natural i ∈ 0, 1, ..., 2n − 1 can be written as a sum

i = 2k1 + 2k2 + · · · 2kl ,

with ordered exponents k1 < k2 < · · · kl all belonging to the set 0, 1, ..., n− 1, using the flowproperties of Φ we can conclude that

Φi = Φ2k1 · · · Φ2kl = h|J ,

where h is an element of G written as

h = (f−k1 g fk1) (f−k2 g fk2) · · · (f−kl g fkl)

= f−k1 g fk1−k2 g · · · fkl−1−kl g fkl .

In the right hand side of the above equation there are l occurrences of g and k1 + (k2 − k1) +· · ·+ (kl − kl−1) + kl = 2kl occurrences of f . Thus the length of h with respect to Γ is boundedby

2kl + l ≤ 2(n− 1) + n = 3n− 2.

Finally, given any two consecutive points Φi(p) and Φi+1(p) of An(p) we have

|Φ−i(Φi+1(p))− Φ−i(Φi(p))| = |Φ1(p)− p| = ε.

So, composing Φ−i with the identity if necessary, we can always assume it belongs to Γ3n−2,and since it separates the points of Φi(p) and Φi+1(p) to a distance equal to ε, the set An(p) is(3n− 2, ε,Γ)-separated, as we wanted.

Of course, we can take p to be as closed to fixed point 1 as we want, so that wehave p < Φ1(p) < 1 and Φ1(p)− p arbitrarily small. Thus, the construction above implies thats(3n− 2, ε,Γ) ≥ 2n − 1 for every possible choice of ε, from where we conclude that


lim supn→∞

13n− 2 ln s(3n− 2, ε,Γ) ≥ lim

ε→0+lim supn→∞

13n− 2 ln 2n − 1 = 1

3 ln 2 > 0.

To complete the example, we consider a representation π1(T)→ G of the fundamentalgroup of a torus T into G given by mapping one of the generators of π1(T) to f and the other tog. Then the suspension of this representation (see Section 1.2.6) is a foliated manifold (M,F),with M = (R2 × I)G being a fibre bundle with base T and fibre I. Since both fibre and baseare compact, M itself is a compact manifold. Its holonomy pseudogroup is exactly G, andtherefore the geometric entropy of M is positive, while M has no resilient leaves, as resilientleaves correspond to resilient orbits of the holonomy pseudogroup, and G has none.

149

Chapter 4

Generating Entropy

For the dynamics of diffeomorphisms, there are several results (most notably [16])regarding the approximation of a map by maps of positive entropy. These results take advantageof the transitivity of the map one wants to approximate, and usually revolve around techniquesto perturb a map creating homoclinic intersections, from whence one can then apply the resultsfrom [10].

Given all the analogies between classical entropy and the geometric and topologicalentropies for foliations, one may ask oneself wheter something similar could be done forcodimension 1 foliations, using pertubations to create foliations with resilient leaves. It turnsout that, when a codimension 1 foliation F of a 3 manifold (that is, a foliation by surfaces)has a recurrent leaf, a pertubation procedure can be performed with the intent of generatingentropy, as shown in [46]. The main problem here is how to perturb F inside a small domainand then reglue this domain onto F without messing up the foliation’s regularity. In order to dothis, the author introduces the concept of a splitting chart (Definition 4.2.3) which will allow usto perturb foliations in a manner that creates resilient leaves, and therefore positive entropy.

In this chapter, Folrq(M) will denote the set of all possible codimension q Cr foliationsthe manifold M supports. Since we want to "approximate" one foliation of M by others, thefirst thing we do is to give Folrq(M) a topology. The standard choice for such a topology in theliterature is the Epstein topology. In what follows we will fix a 3 manifold M , a Cr-foliation F ofM by surfaces, (that is, F ∈ Folr1(M)) and consider the complete transversal T as a submanifoldof M , as we did last chapter. Besides that, the closed interval [0, 1] shall be denoted by I.

4.1 The Epstein topologyIn his 1976 article named A Topology for the Space of Foliations [21], Epstein proposes

different topologies to the space Folrq(M). One somewhat simple way to topologise such a set

Chapter 4. Generating Entropy 150

is to consider each foliation of M as a section to the bundle of Grassmannians of the tangentspaces of M ; given a foliation F ∈ Folrq(M) and the bundle Ω over M whose fibre at x is the setof all codim q subspaces of TxM , the application x 7→ TxF is a section of Ω uniquely associatedto F . Hence, any topology one endows the function space of all sections F : M → Ω with givesrise to a topology for the space Folrq(M).

The construction done by Epstein in [21] provides a different topology, knownnowadays as the Epstein topology (it was originally referred by Epstein as fine Cr-topology).This topology is, in general, weaker (coarser) than the topology of Grassmannians describedabove, though Epstein shows they both coincide in the case r =∞. The main advantage of theEpstein topology over that of the Grassmannians is that it satisfies the two following properties,the second being generally not true for the topology of the space of sections.

P1 If F ∈ Folrq(M) and ϕ : M →M is a Cr-diffeomorphism which is Cr-close to idM in thefine Whitney topology, then ϕ∗F is Cr-close to F in the Epstein topology;

P2 If F and K are foliations of M close in the Epstein topology, then the holonomy mapsdefined by F are close to the holonomy maps defined by K.

We will not reproduce here Epstein’s construction, but we enunciate Proposition 3.8from [21], of which we shall make use later. If for a given multi-index α = (α1, ..., αn) we denoteby Dα the operator ∂α1

1 · · · ∂αnn and by |α| the sum∑i

αi, then the proposition reads as follows.

Proposition 4.1.1. Let M be a Riemannian manifold, F ∈ Folrq(M), A = ϕj : Uj → Rnj∈Jbe a foliated Cr-atlas and Kjj∈J be a locally finite family of compact sets satisfying

(i) Kj ⊂ Uj ∀j ∈ J ;

(ii) for each j ∈ J the image ϕj(Kj) is a rectangular neighbourhood in Rn;

(iii) M = ∪j intKj.

Then for any family δjj∈J of positive real numbers the set V(F ,A, Kjj, δjj)of all foliations K ∈ Folrq(M) which have transverse projections gj = π2 ϕj : intKj → Rq

satisfying|Dα(gj ϕ−1

j − π2)| < δj on ϕj(intKj),

where π2 : Rn−q ×Rq → Rq is the projection on the second coordinate, is a neighbourhood of Fin the Epstein topology.


4.2 Recurrent leaves and families of jointly splitting charts. In doing so we can describe the orbit Hx of a point x under the holonomy pseu-

dogroup H as the subsetHx = F(x) ∩ T .

In analogy to theory of dynamics of flows, we define the ω-limit set of a point x ∈ T as therelatively compact saturated subset of T given by

ω(x) =⋂

S⊂Hx#S<∞

Hx \ S.

Definition 4.2.1 (Recurrent leaf ). The orbit Hx is recurrent if ω(x) ∩Hx 6= ∅. Otherwise, itis called a proper orbit. If the orbit of the point x ∈ T is recurrent, then F(x) is said to be arecurrent leaf, and x a recurrent point.

In particular, for any minimal setM one has ω(x) =M for every point x inM.

Our goal is to show that a 2-dimensional recurrent leaf in a 3-manifold can beperturbed to create a resilient leaf. In order to do this we will need a specific type of foliatedchart, as pointed out before.

Definition 4.2.2 (Splitting curves). Given way 2-dimensional Riemannian manifold S, we saya curve γ : I → S splits S if there is a positive constant ε0 such that for every ε < ε0 the openball B(γ(I), ε) around the trace of γ has a nonconnected complement. In other words, γ splitsS if

S \B(γ(I), ε)

has more than one connected component for any ε sufficiently small. A splitting curve of S is asimple curve (that is, a curve without self intersections) that splits S.

Figure 12 – Two distinct splitting curves of S.


Definition 4.2.3 (Jointly splitting charts). A finite family Φ := ϕi : Di → I2 × Iq is said tohave the jointly splitting property or be a family of jointly splitting charts with splitting curve γif there exists a compact 2-manifold with boundary S endowed with a splitting curve γ, and aCr-diffeomorphism

f : U → S ×Dq,

where U := ∪ϕi∈ΦDi and Dq is the closed unit disc of R, such that F restricted to U is thepullback under f of the trivial foliation S × cc∈Dq and for each c ∈ Dq the curve f−1(γ(·), c)splits the leaf of F|U in which it is contained.

Remark 4.2.4. Every foliation by surfaces admits trivial families of jointlty splitting charts.We just consider any chart (U,ϕ) of a regular atlas. Then its image ϕ(U) is inside the cubeIm equipped with the foliation I2 × cc∈Iq . Any loop completely inside the square I2 is asplitting curve for (U,ϕ).

The main result of this chapter is the following.

Theorem 4.2.5 (Ponce [46]). Let L be a recurrent leaf of a codimension 1 Cr-foliation Fof a 3-dimensional Riemannian Cr-manifold M . If L admits a homotopically nontrivial loopγ : I → L and a family Φ of jointly splitting charts satisfying

(i) for any ε > 0 there are 0 ≤ t1 < t2 ≤ 1 such that γ(t1) and γ(t2) belong to distinctconnected components of B(Z, ε) \ Z and

γ([t1, t2]) ⊂ B(Z, ε) \ Z,

where Z :=⋃ϕ∩Φ

Dϕ ∩ L;

(ii) there is a ϕ ∈ Φ for which Dϕ ∩ γ(I) 6= ∅.

Then, for any neighbourhood V of F in Folr1(M) in the Epstein topology there is a foliationK ∈ V which has a resilient leaf. Consequently, if M is compact F can be Cr-approximated byfoliations with positive geometric entropy.

The idea of the proof is to construct resilient leaves via a perturbation, and thenapply Theorem 3.0.2. In order to do this we establish, with the help of the family of splittingchart, a particular perturbation procedure on L.


4.3 Perturbing foliations and generating entropy

4.3.1 Water slide functions and the subordinated partition

Let us first set some technical tools we need in our perturbation technique. Theresilient leaves are constructed with the help of what we call a water slide function.

Definition 4.3.1 (Water slide function). Let Im be the unit square foliated by codimension qsquares Im−q × c, and, for any choice of open intervals Jl ⊂ I, 1 ≤ l ≤, let P be the open setP = Im−q × (J1 × · × Jl). Given a Cr diffeomorphism w : J1 × · · · × Jl → J1 × · · · × Jl, we saythat

w = idIm−q ×w : P → P

is a Cr water slide function if

(i) there is a neighbourhood of the set (0 × Im−1) ∩ P on which the restriction of w is theidentity;

(ii) Dα(w− idIm)(x) = 0 for |α| < r and any x ∈ (1 × Im−1) ∩ P .

If w(c) = d thenw(1 × Im−q−1 × c) = 1 × Im−q−1 × d, (4.1)

and in this case we say w slides from height c to height d.

The secret for constructing resilient leaves using water slide functions lies in thefollowing lemma.

Lemma 4.3.2. Let w : I → I be a function Cr close to the identity, with a fixed point t0 ∈ int I.There is a Cr water slide function w : Im−1 × I → Im which is Cr close to the identity andsatisfies

(i) πm w = w, where πm : Im → I is the projection onto the last coordinate;

(ii) πj w = idI for 1 ≤ j ≤ m1;

(iii) for any pair (c, a) ∈ Im−1 × I the function w slides from height (c, a) to height (c, w(a));

(iv) w|Im−1×t0 = idIm−1×t0 .

Proof. Let f : [−2, 2]→ [−2, 2] be a smooth function constant equal to 1 on a neighbourhoodof 0, and for which there is a value t1 such that f |[t1,2] ≡ 0. Define w0 : I2 → I2 by

w0 : (t, x) 7→ (t, f(t)x+ (1− f(t))w(x)).


All of f ’s derivatives are continuous, and therefore uniformly bounded on the compact [−2, 2].Then, by choosing w sufficiently close to the identity we can have w0 as close to idI2 as we wish.Now, for the fixed point t0 one has

w0(t, t0) = (t, f(t)t0 + (1− f(t))t0) = (t, t0),

so that (t, t0) is a fixed point of w0, for every t ∈ I. Our desired water slide function is themapping

w = idm−2I ×w0 : (c, t, x) 7→ (c, w0(t, x)),

which clearly satisfies all the properties (i) – (iv) from the statement.

For codimension one foliations the plaques are all characterised by the last coordinate’svalue. In this case, the function w from last Lemma defines where the each plaque of adistinguished chart (U,ϕ) is mapped by ϕ−1 w ϕ. Note that the image of a plaque under thisapplication is once again a plaque of U . A suitable choice of w is what will allow us to create aresilient leaf, as we shall see later on in the proof of Theorem 4.2.5.

In order to use our water slide functions, we first set a special partition of a neigh-bourhood of the loop γ, where each set is a possible domain for a water slide function. Beforedescribing this partition, though, we would like to show that, when one considers families ofjointly splitting charts, two charts is often enough. To prove this we will need the followingclassical lemma from foliation theory:

Lemma 4.3.3 (Global trivialization lemma). Let F be a codimension q Cr-foliation of them-manifoldM , and γ : I →M a simple continuous leafwise curve. There exists a neighbourhoodU of γ(I) and a Cr-diffeomorphism

ψ : U → int(Dm−q ×Dq)

such that F|U is the pullback under ψ of the trivial codimension q foliation Dm−q × cc∈Dq .

Proof. Let (Ui, ϕi) be a chain of distinguished charts covering γ(I). As γ has no self intersections,we may assume (shrinking the charts, if necessary), that Ui intersects only Ui+1 and Ui−1. LetU := ∪iUi, and V be union of plaques of F(γ(0)) from the charts Ui, that is, V = U ∩ F(γ(0)).We note that V is open in F(γ(0)), and that each point of x ∈ U belongs to a unique transversesection through a point π(x) ∈ V , namely the section

Sx = ϕ−1i (ϕ′i(π(x))× Iq),


where x ∈ Ui and ϕi = (ϕ′i, ϕ′′i ). The mapping π : U → V is a well defined Cr retraction,described by

π(x) = ϕ−1i (ϕ′i(x)× Iq) ∩ F(γ(0)).

The holonomy hγ from the open π−1(γ(0)) to π−1(γ(0)) is defined on an open setDγ ⊂ π−1(γ(0)). Define

U = F|U

(Dγ)

as the saturation of Dγ inside U . There is a well defined Cr submersion from U to Dγ given by

f : x→ F|U(x) ∩Dγ.

Now we may assume (squashing U a little, if necessary), that the intersection U ∩ F(γ(0)) isCr-diffeomorphic to the open disc intDm−q via a diffeomorphism h1 : U ∩ F(γ(0))→ intDm−q.Take h2 : Dγ → intDq to be a Cr-diffeomorphism mapping γ(0) to 0, and set

ψ : U −→ int(Dm−q ×Dq)

x 7−→ (h1(π(x)), h2(f(x))).

Then ψ is a Cr-diffeomorphism whose inverse is given by ψ−1(x, y) = π−1(h−11 (x))∩ f−1(h−1

2 (y))and takes each set intDm−q × c into the leaf f−1(h−1

2 (c)), as we wanted.

Let us now show how we can always assume that our families of jointly splittingcharts have at most two charts.

Proposition 4.3.4. Given a family Φ = ηj : Wj → Imj∈J of jointly splitting charts withsplitting curve γ, there exists an associated family Φ = ψi : Ui → Imni=1 of jointly splittingcharts with splitting curve γ, where n ∈ 1, 2 and Ui ⊂ W := ∪jWj. Moreover, U = U1 ∪ U2

admits foliations Q and τ such that

(i) Q has codimension q + 1 and for each x ∈ U one has Q(x) ⊂ F(x);

(ii) τ has dimension q and for each x ∈ U one has τ(x) t F(x).

Proof. We divide the proof in two cases, considering whether γ is a loop or not.

First, let γ be homeomorphic to I. Then using the Global trivialization Lemmawe can find a foliated chart (U, ψ) around the curve f−1 γ, f being the diffeomorphism fromDefinition 4.2.3. We foliate U by setting

Q = ψ−1(D × c)c∈Dq+1

andτ = ψ−1(c ×Dq)c∈D2 .


Then these foliations satisfy properties (i) and (ii), and Φ = ψ : U → D2×Dq−2 is the familywe wanted.

Now, if γ is homeomorphic to S1, what we do is basically to consider two segments,each homeomorphic to I, and repeat the procedure above in neighbourhoods U1 and U2 of thesesegments. Only we do this in such a way that the foliations Q and τ coincide in the intersectionU1 ∩ U2. To guarantee this property, we define both foliations in terms of flows inside particularneighbourhood of the splitting curve. To be more precise, we proceed as follows.

Let P1 be a rectangular neighbourhood containing f−1(γ(0)) and choose t0 ∈ I suchthat f−1 γ([t0, 1]) ⊂ P1. We may assume, without loss of generality, that P1 ⊂ W1. Using theGlobal trivialization Lemma, fix a foliated chart (U1, ψ1) around the segment f−1 γ|[0,t0]. Weconsider the transverse sections

Ti = ψ−11 (i ×Dq), i = 0, 1,

and, since U1 can be taken arbitrarily thin, we assume both these sections are subsets ofW1. Theopen set U1 has a Cr foliation Q = ψ−1

1 (D × c)c∈Dq+1 , which induces a Cr holonomy-likefunction h : T0 → T1 by mapping x into T1 ∩ Q(x).

On the foliated chart (W1, η1) we define a Cr flow by

Ψ1,0 : (x, t) 7→ η−11 (tη1(h−1(x)) + (1− t)η1(x)).

Remark that Ψ11,0(h(x)) = η−1

1 (tη1(h−1(h(x)))) = x, hence Ψ11,0(T1) = T0. What we are doing

here is to “complete” the leaves of Q outside U by considering the pullback under η1 of the lineinside η1(W1) joining η1(x) to η1(h(x)).

We do the same inside U1 we considering the Cr flow

Ψ0,1 : (x, t) 7→ ψ−11 (tψ1(h(x)) + (1− t)ψ1(x)),

which satisfies an analogue condition Ψ10,1(T0) = T1. Remark that both x and h(x) are at the

same height c in Im−q × Iq, hence the line tψ1(h(x)) + (1− t)ψ1(x) is completely inside a set ofform Im−q × c, and therefore each orbit of Ψ0,1 is completely inside a leaf of Q.

Using smooth bump functions ρi defined on a neighbourhood Vi of Ti and equal tothe identity on these sections, we can define a Cr flow Ψt on the set U1 ∪Ψ1,0(T1 × I) such that

Ψt|W1\(U1∪V1∪V2) = Ψ1,0 and Ψt|U1 = Ψ0,1 ∀t ∈ R.

In other words, the flow Ψ is Ψ0,1 on U1 and Ψ1,0 outside of U1 ∪ V1 ∪ V2.

The second foliated chart (U2, ψ2) is defined as U2 := W1 ∩ (U1 ∪Ψ(T1× I)), and thefoliation Q is the one whose leaves are the sets

Q(x) := Q(x) ∪Ψ((Q(x) ∩ T1)× I).


We define the second foliation τ in a similar manner, by letting

τ(x) :=

ψ−11 (ψ′1(x)×Dq) if x ∈ U1,

Ψ(T0 × t(x)) if x ∈ U2,

where ϕ1 = (ϕ′1, ϕ′′1) and t(x) is the time such that Ψt(x)(T0 ∩Ψ(x)) = x. In other words, theleaves of τ outside U1 are the sets Ψt

1,0(T1), t ∈ I.

The foliated charts U1 and U2 constructed above intersect in such a way that theholonomy h21 : S1 → S2 induced by the chain U1, U2, where Si is the space of plaques of Fbelonging to Ui, is a Cr diffeomorphism between the two transverse sections, that is

h21(S1) = S2 and h−121 (S2) = S1. (4.2)

We fix a neighbourhood

V = V(F , ϕλ : Xλ → Imλ∈Λ, Kλλ∈Λ, δλλ∈Λ) ⊂ Folr1(M)

of F in the Epstein topology, as described in Proposition 4.1.1. In order to perturb F inside theopen set U = U1∪U2 in a manner that produces a foliation F ′ ∈ V , we construct a partition of Uin terms of the family of compacts Kλλ∈Λ, in the following way: since Kλλ∈Λ is locally finite,each p ∈ U has a neighbourhood K(p) intersecting at most a finite number of sets in Kλλ∈Λ.As U is compact, it is covered by a finite amount of such K(p), and there is therefore only afinite amount of sets Kλ1 , ..., Kλl ⊂ Kλλ∈Λ covering U . For simplicity, write Kj = Kλj . Finally,for 1 ≤ j ≤ l we let Kj be the smallest compact set containing Kj which crosses U completely,in the sense that for each i ∈ 1, 2 its image under ψi is a rectangular neighbourhood on whichthe projection on the first coordinate is a submersion over the entire set I. More precisely, foreach 1 ≤ j ≤ l and 1 ≤ i ≤ 2 the compact set Kj satisfies

(i) Kj ⊂ Kj;

(ii) ψi(U i ∩ Kj) = I × J1 × · · · Jl−1, with each Jα ⊂ I being a closed interval.

As a consequence, this guarantees the intersection of ψi(U i∩Kj) with each of the sides (0, 0)×Iq

and (1, 1) × Iq is a nondegenerate rectangular neighbourhood (of dimension q).

For each 1 ≤ j ≤ l, denote by κj the saturation of Kj inside U , that is,

κj := F|U(U ∩ Kj).

Each of these "saturated" sets defines on U a trivial partition given by the two sets κj andU \ κj.


Definition 4.3.5 (The subordinated partition). The subordinated partition of the family of jointlysplitting charts ψi : Ui → Imni=1, n ∈ 1, 2, is the joining of trivial partitions κj, U \ κj

P :=l∧

j=1κj, U \ κj,

that is, the partition given by all the possible finite intersections B1∩· · ·∩Bl with Bj ∈ κj, U\κjfor every 1 ≤ j ≤ l.

Remark that, due to its construction, P is a finite partition of U satisfying

(i) for each B ∈ P and i = 1, 2,

ψi(B) = I2 ×(m∏i=3

[ai, bi])

is a rectangular neighbourhood;

(ii) given two distinct elements B,C ∈ P, either ψi(B ∩ C) = ∅ or ψi(B ∩B) = I2 × J forsome J ⊂ Iq.

Recall that associated to the family of jointly splitting charts Φ = (U1, ψ1), (U2, ψ2)from Proposition 4.3.4 there is a Cr diffeomorphism f : U → Σ×Dq. Let ε be sufficiently smallso that Σ \B(γ(I), ε) is not connected, and consider the two transverse sections

S0 = ψ1−1((0, 0) × Iq)

S1 = ψ1−1((1, 1) × Iq),

where ψi : U i → Iq is the extension of ψi to the boundary of Ui. Given any chain of plaques(P1, P2) of the chain (U1, U2), one can consider the connected components O1

P1,P2 and O2P1,P2 of

the setP1 ∪ P2 \ f−1(B(γ(I)× d)),

where d = π2(f(x)) for any x ∈ P1 ∪ P2. Those two sets are labeled in a way that the inclusions

Sj ∩ (P1 ∪ P2) ⊂ OjP1,P2 , j ∈ 1, 2,

hold. Define the setsOj :=

⋃OjP1,P2 ,

where the union runs over all the possible plaque chains of U , and note that we may assume,decreasing the real numbers from the family δλ if necessary, that for any 1 ≤ j ≤ l the set Kj

does not intersect O1 and O2 simultaneously.

We have constructed all the needed tools to perform our perturbation on F .


4.3.2 The perturbation procedure

It is done in four steps: we perturb the foliation inside U1, then we extend thisperturbation to U2 and finally to every compact set Kj intersecting U . After that, we describethe charts that define the perturbed foliation F ′.

(I) Perturbing the plaques of U1. Recall that, besides F|U , the set U comes we the two foliationsQ and τ from Proposition 4.3.4. Fixed a subordinated partition P = B1, ..., Bs, foreach 1 ≤ i ≤ s let wi = idI ×wi be a Cr water slide function defined on the rectangularneighbourhood ψ1(Bi ∩ U1). Now, since our water slide functions only shift heights, thatis, they only act on the transverse part of the foliation trivial foliation on Im, leaving theleaves untouched (in other words, this is exactly what Equation 4.1 says), we can see that

(i) for any x ∈ U and 1 ≤ i ≤ s, the set ψ−11 wi ψ1(Q(x) ∩Bi ∩ U1) is inside a single

plaque of U1;

What is more, due to the construction of τ ,

(ii) for every x ∈ Bi ∩U1 and 1 ≤ i ≤ s, the application ψ−11 wi ψ1 is a τ -leafwise map

in Bi ∩ U1, that is, ψ−11 wi ψ1(τ(x)) ⊂ τ(x).

Associated to each x ∈ U1 there is a unique natural number 1 ≤ i(x) ≤ s such thatx ∈ Bi(x) ∩ U1. We define a Cr water slide function w : Im → Im by setting

w : Im −→ Im

x 7−→ wi(x)(x).

Note that this newly defined water slide function can be taken as close to the identity as wewish, by assuming that the restrictions wi are sufficiently close to the identity themselves.The perturbation on U1 is the mapping

p1 := ψ−11 w ψ1 : U1 → U1.

(II) Perturbing the plaques of U2. As we remarked in Equations 4.2, given a plaque P2 ⊂ U2,there is a single plaque P1 of U1 such that P1 ∩ P2 6= ∅. We then define the perturbationon P2 to be the Cr function

p2 : U2 −→ U2

x 7−→ F(p1(Q(x) ∩ P1)) ∩ τ(x).


This is indeed a Cr function because all the functions and foliations involved are Cr.Moreover, if x ∈ U1 ∩ U2 then

p2(x) = F(p1(Q(x) ∩ P1)) ∩ τ(x)

= F(ψ−11 w ψ1(Q(x) ∩ P1)) ∩ τ(x)

= F(ψ−11 wi(x) ψ1(Q(x) ∩Bi(x) ∩ P1)) ∩ τ(x)

= ψ−11 wi(x) ψ1(Q(x) ∩Bi(x) ∩ P1)

= ψ−11 wi(x) ψ1(Q(x) ∩Bi(x) ∩ U1) = p1(x).

Thus p1 and p2 can be amalgamated to form a single Cr perturbation p : U → U . Whatis more, by taking the water slide functions wi sufficiently close to the identity we mayassume p arbitrarily close to idU .

(III) Perturbing the sets Kλ. The third step is to extend p to the family Kλλ. The notation islike in the discussion right before the beginning of this Section. As we have assumed thatnone of the sets Kj ∈ Kλλ covering U intersects both O1 and O2, we define functionspKλ : Kλ → Kλ on each possible Kλ by considering separate cases, in the following manner.

(i) if, for any λ ∈ Λ, Kλ ⊂M \ U , that is, if Kλ does not intersect U , then

pKλ = idKλ .

(ii) if Kj ⊂ U \ (O1 ∪ O2), that is, if Kλ is inside U but intersects neither O1 nor O2,then

pKj = p|Kj .

(iii) if Kj ∩O1 6= ∅, that is, if Kλ intersects O1, then

pKj : x 7→

x, if x ∈ Kj \ U

p(x) if x ∈ Kj ∩ U.

(iv) the last case happens when Kj ∩ O2 6= ∅ and x belongs to both U and Kj. In thisscenario there is a unique plaque chain (Q1(x), Q2(x)), Qi ∈ Ui, such that w slidesfrom the ψ1(U1(x) height to the height ψ1(Q(x) ∩ U1) where the Q-leaf trough x

meets U1. If we let η : Kj ∩ U → Kj be the mapping

η : x 7→ τ(x) ∩ (Q1(x) ∪Q2(x)),

then we can set

pKj : x 7→

x, if x ∈ Kj \ U

p(η(x)) if x ∈ Kj ∩ U.


Once again, as p can be as close from the identity as we want, we may construct the mapspKλ arbitrarily Cr close to the identity as well.

(IV) The foliated atlas. We define on M a foliated atlas A′ = (intKλ, θλ)λ∈Λ by setting

θλ := ϕλ p−1Kλ,

where the charts ϕλ : Xλ → Im are the ones from the atlas defining the neighbourhood Vof F . Remark that each θλ is a Cr function, being the composition of Cr maps.

To see that A′ is indeed a foliated atlas, as in Definition 1.2.1, it is sufficient to check thatthe coordinate changes of A′ satisfy Equation 1.3, as we noted in Remark 1.2.5. Now,given indeces λ, λ′ ∈ Λ one has the following possibilities

(i) if intKλ ∩ intKλ′ ⊂M \ U , then

θλ θ−1λ′ = ϕλ p−1

Kλ pKλ′ ϕ

−1λ′ = ϕλ id−1

Kλ idKλ′ ϕ

−1λ′ = ϕλ ϕ−1

λ′ ;

(ii) if intKλ ∩ intKλ′ ⊂ U \ (O1 ∪O2), then

θλ θ−1λ′ = ϕλ p−1

Kλ pKλ′ ϕ

−1λ′ = ϕλ p|−1

Kλ (p|Kλ′ ) ϕ

−1λ′ = ϕλ ϕ−1

λ′ ;

(iii) if Kλ ∩Kλ′ ∩Oe, with e ∈ 1, 2, that is, if they both intersect the same set Oe, theneither e = 1 and

θλ θ−1λ′ |intKλ∩intKλ′ = ϕλ ϕ−1

λ′ |intKλ∩intKλ′

due to an argument similar as the above cases, since for e = 1 the map pKλ is eitherthe identity or the a restriction of p; or e = 2, in which case pKλ is either the identityor we have

θλθ−1λ′ = ϕλη−1p−1

KλpKλ′ ηϕ

−1λ′ = ϕλη−1(p|Kλ)−1p|Kλ′ ηϕ

−1λ′ = ϕλϕ−1

λ′ .

In any of these cases, as ϕλ : Xλ → Imλ∈Λ is itself a foliated atlas and satisfiestherefore Equation 1.3, we conclude immediately that θλ θ−1

λ′ satisfies the same condition.Furthermore, for the remaining cases, we have

(iv) if intKλ ⊂ U and intKλ′ ∩ U 6= ∅, then pKλ = p|Kλ and Kλ′ intersects either O1 orO2, in which case

pKλ′ |Kλ′∩U = p|Kλ′∩U or pKλ′ |Kλ′∩U = p η|Kλ′∩U ,

respectively, while in the remaining of its domain it is equal the identity. Then, if itintersects O1 we have θλ θ−1

λ′ = ϕλ ϕ−1λ′ on intKλ ∩ intKλ′ , as in the first cases,

while if it intersects O2 then on intKλ ∩ intKλ′ we have

θλ θ−1λ′ = ϕλ (p|Kλ)−1 (p|Kλ′ ) η ϕ

−1λ′ = ϕλ η ϕ−1

λ′ ;


(v) if Kλ ∩O1 6= ∅ and Kλ′ ∩O2 6= ∅ then we have

Kλ ∩O2 = ∅ and Kλ′ ∩O1 = ∅,

and consequently intKλ ∩ intKλ′ ⊂ U \ (O1 ∪O2). Then, according to our previousdefinitions, on the set intKλ ∩ intKλ′ the equalities

θλ = ϕλ p−1Kλ

= ϕλ p−1

andθλ′ = ϕλ′ p−1

Kλ′= ϕλ′ η−1 p−1

hold true, from where it follows that

θλ θ−1λ′ |intKλ∩intKλ′ = ϕλ′ η ϕ−1

λ′ |intKλ∩intKλ′ .

So it is sufficient to study whether the map ϕλ η ϕ−1λ′ satisfies or not 1.3. But if you

remark that η preserve plaques, in the sense that if two points of Kj belong to a sameplaque of Kj ∩ U then their images are also in the same plaque, then it is clear thatϕλ′ η ϕ−1

λ′ is a mapping of the form

(x, y) 7→ (α(x, y), γ(y)),

which is exactly what me wanted.

The atlas A′ constructed above gives rise to a foliation F ′ in the neighbourhood Vof F .

Proposition 4.3.6. The foliation F ′ given by the atlas A′ from Step IV belongs to theneighbourhood V = V(F , ϕλ : Xλ → Imλ∈Λ, Kλλ∈Λ, δλλ∈Λ).

Proof. All we have to do is show that A′ satisfies the condition from Proposition 4.1.1. For anyindex λ such that Kλ is not an element of the covering K1, ..., Kl we have θλ = ϕλ id = ϕλ,hence there is nothing to do. On the other hand, for any λ ∈ λ1, ..., λl (recall that Kj := Kλj

we consider the transverse projection gλ = π2 θλ. Given a multi-index α with |α| < r we have

|D(α)(gλ ϕ−1λ − π2)| = |D(α)(π2 θλ ϕ−1

λ − π2)| = |D(α)(π2 (ϕλ p−1Kλ

) ϕ−1λ − π2)|. (4.3)

Now, as we pointed out during the construction of the perturbation procedure, the maps pKλ canbe taken as C−r-close to the identity as we wish, so we can assume they satisfy the condition

|D(α)(ϕλ p−1Kλ ϕ−1

λ − id)| < minδλ1 , ..., δλl.


In particular, such assumption, together with 4.3, implies that

|D(α)(gλ ϕ−1λ − π2)| = |D(α)(π2 ϕλ p−1

Kλ ϕ−1

λ − π2)| < δλ for any λ ∈ λ1, ..., λl,

which is exactly what we wanted to show.

Now we can finally prove Theorem 4.2.5, which we do by, with the aid of Lemma4.3.2, providing the perturbed foliation F ′ with a resilient leaf.

Proof of Theorem 4.2.5: Let V be any neighbourhood of F . Due to Proposition 4.3.4 we mayassume that the family Φ of jointly disjoint charts exactly two charts, (U1, ϕ1) and (U2, ϕ2), andthat γ(0) = a belongs to U1 Let P = P1, ..., Ps be the subordinated partition of U = U1 ∪ U2.By increasing, if necessary, the compact sets of the covering Ki used in the construction of P ,we may assume that a ∈ intPl for some index 1 ≤ l ≤ s. Let Ta denote de transverse sectionpassing through a. The subordinated partition has the property that

ϕ1(Pl ∩ U1) =m∏i=1

[αi, βi]

is a rectangular neighbourhood in Im. We define a rescaling function

r : ϕ1(Pl ∩ U1) −→ Im

(x1, ..., xm) 7−→(α1 − x1

α1 − β1, ...,

αm − xmαm − βm

),

and denote by j : ϕ1(Pl ∩ Ta)→ I the composition j = πmr|ϕ1(Pl∩Ta).We endow ϕ1(Pl ∩ Ta) witha metric ρ obtained from pulling back via j the canonical metric on I. Given x, y ∈ ϕ1(Pl ∩ Ta),by saying x is higher than y we mean that

ρ(x, ϕ1(a)) = ρ(x, y) + ρ(y, ϕ1(a)).

As in last chapter, we use the fact J := Ta ∩ Pl is identified with I via j to use concepts likeorder and maximum between points its points.

The loop γ induces a holonomy transformation ha : Da → Ta defined on an openinterval Da ⊂ Ta around the point a. We construct in J a sequence in the following way. Since Jis open and ha is Cr, we may find b1 6= a inside J such that a1 := ha(b1) ∈ J . Again due to theregularity of ha we can find b2 between a and maxa1, b1 such that a2 := ha(b2) lies between aand a1. More generally, given an and bn, continuity of ha guarantees the existence of a pointbn+1 ∈ (a,maxan, bn) such that ha(bn+1) ∈ (a, an), and we set an+1 := ha(bn+1).

Fix a′ := j(ϕ1(a)), a′n := j(ϕ1(an)), b′n := j(ϕ1(bn)), c := maxa′1, b′1 and choose dsuch that c < d. We consider a function w : I → I characterized by


(i) w(b′n) = a′n for every n ∈ N;

(ii) w|[0,a′] = id[0,a′] and w|[d,1] = id[d,1];

(iii) w is linear inside the intervals [b′n, b′n+1], n ∈ N and [c, d].

We apply Lemma 4.3.2 to w in order to get a Cr-water slide function w : Im → Im whichslides from (x, b′n)-height to (x, a′n)-height and such that w|Im−1×a′ = id. Finally, we definew = r−1 w r on ϕ1(Pl ∩ U1) and extend it on the rest of Im to be the identity. Taking theconstant b1 sufficiently close to a makes w arbitrarily Cr-close to the identity, so we can take w

as close to the identity as necessary in order to make our water slide function w as Cr-closeto the identity as we wish. We now use this water slide function to perturb F as described inSection 4.3.2, yielding a foliation F ′ inside the neighbourhood V we fixed at the beginning.

We claim F ′ has a resilient leaf. To check this, we look at the holonomy g inducedby the holonomy ha on the complete transversal T ′ of F ′. Its domain (a subset of the space ofplaques of F ′) can be identified the transverse section Dg = p(Da), and it acts on this set inaccordance with the law

g(P ′(x)) = P(p ha(x)),

where by P(x) and P ′(x) we mean the plaques of F and F ′ containing x, respectively. Remarkthat we defined g acting on the space of plaques, while ha is acting on the transverse sectionembedded on M . Note as well that, since w slides from (x, b′n)-height to (x, a′n)-height, theimage under ϕ1 of the plaque of F through bn is mapped by w onto the image of plaque of F ′

through an under θ1 = ϕ1 p−1. Hence the holonomy g satisfies:

g(P ′(a)) = P(p ha(a)) = P(p(a)) = P(ϕ−11 w ϕ1(a)) = P(a) = P ′(a),

because a′ is fixed under w. Moreover,

g(P ′(bn)) = P(p ha(bn)) = P(p(an)) = P(ϕ−11 w ϕ1(an)) = P(ϕ−1

1 (ϕ1(bn+1))) = P(bn+1).

In particular, P ′(b1) 6= P ′(a) and gn(P ′(b1))→ P ′(a), hence F ′(a) is a resilient leaf.

Example 4.3.7 (Aproximating Riemannian foliations). Let Φ : T2 × R→ T2 be an irrationalflow on the torus T2, and consider the foliated manifold (M,F0), where M = T2 × I is a solidtorus, and the leaves of F0 are the products F0(x, s) = Φt(x); t ∈ R × I of flow orbits. Theirrational flow on the torus is linear, hence the distance between two orbits is constant. It followsthat F0 is a Riemanian foliation of M , and has therefore vanishing entropy. We embed M in R3

as the set (cos t, cos s, z); (t, s) ∈ [0, 2π)2, z ∈ I, and remove from it the cilinder

C := I × (y, z) ∈ (int I)2; (y − 1/2)2 + (z − 1/2)2 < 1/16,


obtaining a manifold N := M \ C together with a C∞ foliation with zero entropy F := F0|N .Note that, since every flow orbit is dense, every leaf of F has an “infinity number of holes”caused by the removal of C, that is, every leaf has infinity genus. Any loop generating thefundamental group of a leaf can be made into a splitting loop, and thus applying Theorem 4.2.5to this foliation provides an Cr-close foliation with positive geometric entropy.

From the previous example we derive the following corollary to Theorem 4.2.5.

Corollary 4.3.8. There are smooth foliations with zero geometric entropy which can beCr-approximated by foliations of positive entropy.

4.4 On the density of foliations with positive geometric entropyThe question of how common is the property of positive entropy among all the

possible foliations a compact manifold M supports has been open for a while. Our next result isa partial answer to this question, proving that approximation by positive entropy is possible fora certain type of foliations. Given a foliation F , we denote by g(F) the minimum among thegenus of its leaves.

Theorem 4.4.1 (Ponce). Given a compact 3-manifold M , let F ∈ Folr1(M). If either g(F) ≥ 2,or if F is both minimal and has g(F) ≥ 1, then F can be Cr-approximated by foliations ofpositive geometric entropy.

To prove this we ned the following lemma. We recall the result of Epstein, Millettand Tischler [22] that guarantees the existence (and genericness) of leaves without holonomy.

Lemma 4.4.2. Let F be a foliation of M by surfaces, and L a leaf without holonomy. If L hasgenus g then for every k ≤ g we can find k disjoint families of jointly splitting charts (that is,the union of domains from one family does not intersect the union of domains from any other)Φ1, ...,Φk and associated loops γ1, ..., γk, all satisfying the first hypothesis of Theorem 4.2.5.

Proof. As L has genus g ≥ k it is possible to remove from L k loops, each representing adifferent class in L’s fundamental group, in such a way that L \ γi(I) is connected, for every1 ≤ i ≤ k. This can be done in such a way that the loops γi are pairwise disjoint. We can thenfind ε sufficiently small so that for every neighbourhood Σi(ε) = B(γi(I), ε) the set Σi(ε) \ γi(I)is not connected, and the Σi(ε) are also pairwise disjoint.

We fix now one such index i and consider a chain of distinguished charts = (U i1, ...U

iki

)covering γi. Since L is without holonomy, there is a small transverse neighbourhood W aroundγ(0) where the holonomy hγ reduces to the identity. We then consider a foliated tubular


neighbourhood T of γ, given by the “plaque saturation” of W inside U := ∪jU ij , that is,

the union of all plaque chains (P1, ..., Pki) determined by U such that W ∩ P1 6= ∅. Sincethis neighbourhood is tubular, the n-th plaque of every plaque chain as above is completelyprojectible onto the n-th plaque of the plaque chain (Q1, ..., Qki) of L covering γi. We writeVj := U i

j ∩ T ∩ Σi(ε), and the family of jointly splitting charts is then defined to be

Φi = (Vj, ϕj|Vj)kij=1,

where ϕj : Uj → In is the local chart.

Proof of Theorem 4.4.1: Let L be a leaf without holonomy. First, assume F is minimal andg(F) ≥ 1. Then, from Lemma 4.4.2 and Theorem 4.2.5, the result follows. We can then assumethat g(F) ≥ 2, hence from Lemma 4.4.2 we can find in L at least 2 leafcurves covered by disjointfamilies Φ1 and Φ2 of jointly splitting charts, satisfying the first hypothesis from Theorem 4.2.5.We may assume they both are composed by two charts, and write

Φ1 = (U1, ϕ1), (U2, ϕ2)

Φ2 = (V1, ψ1), (V2, ψ2).

We fix a second leaf L′ close enough to L so that it intersects the four sets U1, U2, V1 and V2.

The perturbation along the first curve. Given a plaque P ⊂ L of U = U1 ∪ U2, there is a uniqueassociated plaque P ′ ⊂ L′ associated to P , namely the one belonging to the same distinguishedchart. Now, as we did in Theorem 4.2.5’s proof, we consider a rescaling function r1 identifyingϕ1(U1) with I3, and the transverse section TU1 with (1/2, 1/2) × I. For a := r1(P ) andb := r1(P ′) we consider a water slide function w1 on I3 sliding from a-height to b-height, andpull it back to ϕ1(U1) via r1 to get a water slide function w1 := r−1

1 w1 r1. Using w1 in theperturbation procedure from Section 4.3.2 we get a new foliation F ′. Note that what we didwas to “join” L and L′ in a single leaf N of F ′, that is, as submanifolds of M , the leafs L,L′

and N satisfyN \ U ⊂ L ∪ L′.

The perturbation along the second curve. The procedure here is similar, but we use the singleleaf N of F ′ instead of the leaves L and L′ of F . Let Q and Q′ be two distinct plaques of Nin V1. We proceed as before, only that now we assume our rescaling function r2 : ψ1(V1)→ I3

maps Q to c < d := r2(Q′) in (1/2, 1/2)× I. Now let (xn)n∈N be a strictly decreasing sequenceon the interval (c, d] such that x1 = d and xn → c. We construct on (c, d] a piecewise linearfunction w as in Theorem 4.2.5, mapping xi to xi+1 and fixing c, which we use to derive a water


slide function w2 on I3 satisfying

w2(xi) = xi+1,

w2(x) = c.

The pullback w := r−12 w r2 : ψ1(V1)→ ψ1(V1) is then used to perturb F ′ inside V = V1 ∪ V2,

obtaining a foliation F ′′ of M .

We claim that hg(F ′′) > 0. Indeed, the holonomy h2 along the curve γ2 induces aholonomy h after the perturbation, given by

h(P ′′(y)) = w2 h2(P ′(y)),

where P ′′(y) is the plaque of F ′′ through y, and analogously for P ′(y). Now, as our leaf iswithout holonomy, h2 is just the identity, hence f is just w2 and

fn(Q′) = ψ−11 (xn)→ Q,

f(Q) = Q

and thus the leaf of F ′′ containing Q is a resilient one.

Corollary 4.4.3. Given a minimal foliation F ∈ Folr1(M), then either F can be approximatedby foliations with positive entropy or every leaf of F without holonomy is simply connected.

168

Chapter 5

Open Problems

“It provides so much scope for the imagination!”— Anne Cuthbert Shirley

We finish with a small list of open questions regarding the subjects discussed inprevious chapters. Most of the problems come from the Problem Set of [54] and from [29], bothcompiled by Hurder.

5.1 Topology and geometry of foliationsGeometry of leaves. One of the most basic questions in general foliation theory is whether a givenmanifold M can be realised as leaf of a foliated manifold (M,F). Though simple to formulate,this question is in general very difficult to approach, and attempts to solve it so far have yieldedboth negative and positive answers. For instance, in [14], Cantwell and Conlon showed that everyopen surface is realisable as a codimension one leaf. For general higher dimensional manifolds,the same question has a negative answer, as states the following theorem, due to Étienne Ghys:

Theorem 5.1.1 (Ghys [24]). For every natural number d ≥ 3 there exists a noncompact smoothd-dimensional manifold M which is not homeomorphic to any leaf of any codimension onefoliation F of a compact manifold.

In order to prove this, Ghys constructed manifolds with some very particular properties. Thisbrings to light the following classification problem:

Problem 5.1.2. For a fixed natural number d ≥ 3, which open d-manifolds are homeomorphicto a leaf of some codimension one foliation of a compact manifold?

Chapter 5. Open Problems 169

In dimension greater than 3 one starts to find the so-called exotic manifolds, there is,manifolds that are homeomorphic but not diffeomorphic to a given well known manifold. Thereis, for instance, an extensive literature about exotic spheres (they are known to exist for S7 ,for example) and specially about the exotic R4, since it has been proven to be the only exoticEuclidian space. Given the particular differential structure carried by these manifolds, it is aninteresting question whether or not they can be leaves of foliations. For open 4-dimensionalexotic manifolds it seems unlikely to have a positive answer.

Problem 5.1.3. Show that a smooth manifold L which is an exotic R4 cannot be a leaf of aC1-foliation of a compact manifold.

Another class of interesting questions are the ones about which types of differentleaves can coexist as leaves of the same foliation. A surprisingly interesting problem is thefollowing:

Problem 5.1.4. Let cF be a Cr-foliation of a compact manifold M , 0 ≤ r ≤ ∞. Can F haveexactly one noncompact leaf, while all the others are compact?

The answer is negative in codimension one, and for codimension 2 it was proved by Vogt [53]that if F ∈ Folr2(M), r ≥ 0, then F is either a Seifert fibration or has an uncountable number ofnoncompact leaves. The general case is still open.

More generally, one can ask oneself how are the leaves of a foliation related to oneanother in terms of homeomorphisms between themselves. Ghys formulated this in the followingproblem:

Problem 5.1.5. Is there a foliation F of a compact manifold M such that no two distinctleaves are homeomorphic to each other? In other words, can we construct a foliation F of Msuch that if L and L′ are homeomorphic leaves of F , then L = L′? Moreover, is it possible forthis to happen in codimension one? In general, what is known about the cardinality of distinct(up to homeomorphism) leaves in a codimension one foliation?

Foliation minimal sets. Recall from Definition 1.2.19 that a saturated set on which every leafis dense is called minimal. It is exceptional when it is not the entire manifold nor a singleleaf. Though much is known about the structure of exceptional minimal set for codimension1 foliations [13, 15, 32, 42], in higher codimension there is basically no results so far. In [54],Hurder proposes the following problem:

Problem 5.1.6. Classify the minimal exceptional subsets of codimension q Cr-foliations, forq, r ≥ 2.


5.2 Entropy of pseudogroups and foliationsGeometric entropy. There are several results in the theory of dynamical systems relating theentropy of a map to its dynamics, including the existence of Smale Horseshoes and a numberof other results. There are, however, not many results regarding the dynamical implications ofpositive geometric entropy.

Problem 5.2.1. For a codimension 1 foliation F , exhibit qualitative dynamical properties ofF that are necessary, sufficient, or both, in order to have positive geometric entropy.

One example of such a property is the existence of ping-pong games (and consequently, ofresilient leaves) given by Theorem 3.0.2. Another example is Theorem 5.1 from [26], which statesthat when hg(F) = 0 the complete transversal T admits a measure which is invariant under theholonomy pseudogroup of F . Plante [45] showed that the absence of such measures imply thatall the leaves have exponential growth, but it is unclear whether this implies or not positivegeometric entropy. We can generalise last problem even further and ask

Problem 5.2.2. For any foliation F , exhibit qualitative dynamical properties of F that arenecessary, sufficient, or both, in order to have positive geometric entropy.

The main problem regarding the geometric entropy of a foliation is to give a precisedefinition of a measure theoretic entropy for a foliation F . This problem was posed by Ghys,Langevin and Walczak in [26], but not many advances have been made so far. Several definitionswere proposed, but it is unclear how to related these entropies to the ones already existing. Oneidea, due to Hurder, is to define it in terms of the foliated geodesic flow invariant measures, asthe ones from Chapter 3. Ghys, Langevin and Walczak also offered a potential definition in theirpaper [25], but attempts to relate their definition to the geometric entropy of F have yielded nouseful results. Another possibility is to work with foliated harmonic measures [56, Section 4.7].

Problem 5.2.3. Give a definition of foliation measure entropy. or some other entropy-likedynamical invariant, which can be used to establish positive lower bounds for the geometricentropy.

Once a measure theoretic foliation entropy is defined, the next natural question is toestablish a Variational Principle relating it to the topologial entropy of a foliation.

Problem 5.2.4. Establish the analogue of the variational principle for the entropies of foliations.

Local entropy. Given a pseudogroup G acting on (X, d), one can define its entropy restricted toa subset of X. In this case, a subset A ⊂ Y ⊂ X is (Y, n, ε,Γ)-separated if for any two distinct


points of x, y ∈ A there is a element g ∈ G|Y whose domain contains x, y and such thatd(g(x), g(y)) ≥ ε. Let s(Y, n, ε,Γ) denote the maximal number of points in a (Y, n, ε,Γ)-separatedsubset of Y . The entropy of (G,Γ) restricted to Y is

h(G,Γ, Y ) := limε→0+

lim supn→∞

1n

ln s(Y, n, ε,Γ).

Of course, when Y = X this is exactly what we have defined before in Definition 2.1.25.

Using this and based on the definition of a local measure-theoretic entropy for mapsgiven by Brin and Katok in [9], Hurder [31, Definition 13.3] proposed the following concept

Definition 5.2.5 (Local geometric entropy). Given a point x ∈ X, the local entropy of (G,Γ)at x is the number

hloc(G,Γ, x) := limδ→0

limε→0+

lim supn→∞

1n

ln s(B(x; δ), n, ε,Γ).

If hloc(G,Γ, x) = α we say x is a point of α-entropy.

This is not nearly as well understood as the local entropy of Brin and Katok. Hurdershows [31, Proposition 13.4] how the local entropy determines the global entropy by proving therelations

hloc(G,Γ, x) < h(G,Γ) ∀x ∈ X, and h(G,Γ) = supx∈X

hloc(G,Γ, x).

Denote by Kα the subset of all α-entropy points of X. If h(G,Γ) = E then there arevalues α ∈ (0, E] for which these sets are nonempty. There is a number of questions one can askabout these sets. The following where proposed by Andrzej Biś:

Problem 5.2.6. What is the topology of Kα like?

Problem 5.2.7. Which conditions imply Kα nonempty? Moreover, when G is the holonomypseudogroup of a foliation F , and Kα a subset of the complete transversal, which conditions onF imply Kα nonempty?

Problem 5.2.8. If dimH(X) is the Hausdorff dimension of a set X, what are the characterisingproperties of the map α 7→ dimH(Kα)?

Recall that BΓ(x;n, ε) is a dynamical ball of x with respect to Γ, as in Definition2.1.28. In [4], Biś defines, for a Borel probability µ on X, the quantities

hµ,G(x) := limε→0+

lim infn→∞

1n

lnµ(BΓ(x;n, ε)) and hGµ(x) := limε→0+

lim supn→∞

1n

lnµ(BΓ(x;n, ε)),


called the local lower µ-entropy at x and the local upper µ-entropy at x, respectively. He showsthat X admits decompositions X = tαLα = tαUα, where

Lα := x ∈ X;hµ,G(x) = α and Uα := x ∈ X;hGµ(x) = α.

Problem 5.2.9. Give characterising properties of the families Lα and Uα.

Problem 5.2.10. What is the relation between the families Lα, Uα and kα?

The space of foliations. Let N be a compact boundaryless manifold, and I = [0, 1]. ConsiderCFolr1(N × I) the set of all codimension one Cr foliations of N × I whose all leaves are compactsubmanifolds, equipped with the Epstein topology for r ≥ 1. It is known that this space islocally contractible for r =∞. The following problems were proposed by Paul Schweitzer:

Problem 5.2.11. Show that CFol∞1 (N × I) is connected.

Problem 5.2.12. For Σg the orientable closed surface of genus g, show that CFolr1(Σg × I) iscontractible, for r ≥ 1.

There are as well some problems directly associated to the results of Chapter 4. Themain question, of course, is wheter or not we can generalise the construction of Theorem 4.2.5to foliations of higher dimension. Other problems proposed by Ponce in [46] include:

Problem 5.2.13. Given a compact manifold M , provide examples of foliations F ∈ Folrq(M)which have robustly zero geometric entropy, that is, foliations with a neighbourhood V suchthat hg(K) = 0 for every K ∈ V .

Of course, one might as well try to prove that such foliations do not exist. Since,however, Crovisier showed in [16] that the set of C1-diffeomorphisms with robustly zero entropyis exactly the closure of the set of Morse-Smale diffeomorphisms, one may expect a similarclassification result exists for foliations.

Problem 5.2.14. Given a compact manifold M , classify the set of foliations F ∈ Folrq(M) withC1 robustly zero geometric entropy.

And to finish, in light of Theorem 4.4.1, the following questions are natural:

Problem 5.2.15. Can we classify the manifolds that do not admit foliations satisfying thehypothesis of Theorem 4.4.1?

Problem 5.2.16. If every leaf without holonomy has genus 1, can the foliation be approximatedby ones with positive entropy?


Problem 5.2.17. Could one, making use of surgery theory, generalise the construction used inTheorem 4.4.1, and thus the result itself, to higher dimensions?

For instance, it seems likely that using handle decomposition a similar result could be obtainedfor foliations of arbitrary codimension with a leaf without holonomy and at least two handles,or yet minimal foliations with a leaf without holonomy and at least one handle.

174

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