são paulo, 24 de novembro de 2017€¦ · agradecimentos este trabajo es dedicado a mis dos...

Anosov Families: Structural Stability,

Invariant Manifolds and Entropy for

Non-Stationary Dynamical Systems

Jeovanny de Jesus Muentes Acevedo

Tese apresentadaao

Instituto de Matemática e Estatísticada

Universidade de São Paulopara

obtenção do títulode

Doutor em Ciências

Programa: Matemática

Orientador: Prof. Dr. Albert Meads Fisher

Durante o desenvolvimento deste trabalho o autor recebeu auxílio nanceiro da CAPES e da

CNPq

São Paulo, 24 de novembro de 2017

Anosov Families: Structural Stability,

Invariant Manifolds and Entropy for

Non-Stationary Dynamical Systems

Esta versão da tese contém as correções e alterações sugeridas

pela Comissão Julgadora durante a defesa da versão original do trabalho,

realizada em 24/11/2017. Uma cópia da versão original está disponível no

Instituto de Matemática e Estatística da Universidade de São Paulo.

Comissão Julgadora:

• Prof. Dr. Albert Meads Fisher (orientador) - IME-USP

• Prof. Dr. Sylvain Bonnot - IME-USP

• Prof. Dr. Pedro Salomão - IME-USP

• Prof. Dr. Sergio Augusto Romaña Ibarra - UFRJ

• Prof. Dr. Daniel Smania - ICMC - USP

Agradecimentos

Este trabajo es dedicado a mis dos madres, mi abuelita Herminia Muñoz y a mi mamá Nazly

Acevedo.

Agradezco muchísimo a la Universidad de São Paulo por haberme brindado la oportunidad de

realizar mis estudios de pósgrado. A las agencias CAPES y CNPq por su nanciación durante mis

estudios de Maestría y Doctorado. A mi orientador por su gran apoyo y orientación. A mis familiares

y a mis amigos en Colombia y en Brasil por acompanãrme y apoyarme durante todo este tiempo.

Por útimo:

½½½Gracias Brasil!!!

i

Resumo

ACEVEDO, J. J. M. Famílias Anosov: Estabilidade Estrutural, Variedades Invariantes e

Entropia de Sistemas Dinâmicos Não-Estacionários. 2017. Tese (Doutorado) - Instituto de

Matemática e Estatística, Universidade de São Paulo, São Paulo, 2017.

As famílias Anosov foram introduzidas por P. Arnoux e A. Fisher, motivados por generalizar a

noção de difeomorsmo de Anosov. A grosso modo, as famílias Anosov são sequências de difeomor-

smos (fi)i∈Z denidos em uma sequencia de variedades Riemannianas compactas (Mi)i∈Z, em que

fi : Mi →Mi+1 para todo i ∈ Z, tal que a composição fi+n· · ·fi, para n ≥ 1, tem comportamento

assintoticamente hiperbólico. Esta noção é conhecida como um sistema dinâmico não-estacionário

ou um sistema dinâmico não-autônomo. Sejam M a união disjunta de cada Mi, para i ∈ Z, eFm(M) o conjunto consistente das famílias de difeomorsmos (fi)i∈Z de classe Cm denidos na se-

quência (Mi)i∈Z. O propósito principal deste trabalho é mostrar algumas propriedades das famílias

Anosov. Em particular, mostraremos que o conjunto destas famílias é aberto em Fm(M), em que

Fm(M) é munido da topología forte (ou topología Whitney); a estabilidade estrutural de certa

classe de famílias Anosov, considerando conjugações topológicas uniformes; e várias versões para os

Teoremas de variedades estáveis e instáveis. Os resultados que serão apresentados aquí generalizam

algúns outros resultados obtidos em Sistemas Dinâmicos Aleatórios, os quais serão mencionados

ao longo do trabalho. Além do anterior, será introduzida a entropia topológica para elementos em

Fm(M) e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua

em Fm(M) munido da topología forte. Porém, ela é discontínua em cada elemento de Fm(M)

munido da topología produto. Também apresentaremos um resultado que pode ser uma ferramenta

de muita utilidade no estudo da continuidade da entropia topológica de difeomorsmos denidos

em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao

longo desta pesquisa e que serão analisados em um trabalho futuro.

Palavras-chave: Família Anosov, difeomorsmo de Anosov, sistemas dinâmicos não-estacionários,

sistemas dinâmicos não-autônomos, sistemas dinâmicos aleatórios, entropia topológica, topología

forte.

iii

Abstract

ACEVEDO, J. J. M. Anosov Families: Structural Stability, Invariant Manifolds and En-

tropy for Non-Stationary Dynamical Sytems. 2017. Tese (Doutorado) - Instituto de Matemática

e Estatística, Universidade de São Paulo, São Paulo, 2017.

Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the no-

tion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms (fi)i∈Z

dened on a sequence of compact Riemannian manifolds (Mi)i∈Z, where fi : Mi → Mi+1 for all

i ∈ Z, such that the composition fi+n · · · fi, for n ≥ 1, has asymptotically hyperbolic behavior.

This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system.

Let M be the disjoint union of each Mi, for each i ∈ Z, and Fm(M) the set consisting of families

of Cm-dieomorphisms (fi)i∈Z dened on the sequence (Mi)i∈Z. The main goal of this work is to

explore some properties of Anosov families. In particular, we will show that the set consisting of

these families is open in Fm(M), where Fm(M) is endowed with the strong topology (or Whitney

topology); the structural stability of a certain class of Anosov families, considering uniform topo-

logical conjugacies; and some versions of stable and unstable manifold theorems. The results that

will be presented here generalize some results obtained in Random Dynamical Systems, which will

be mentioned throughout the work. In addition to the above mentioned theorems, the topological

entropy for elements in Fm(M) will be introduced, and we will show some of its properties. We

will prove that this entropy is continuous on Fm(M) endowed with strong topology. However, it is

discontinuous at each element of Fm(M) endowed with the product topology. We will also present

a result that can be a very useful tool in the study of the continuity of the topological entropy of

dieomorphisms dened on compact manifolds. We will nish the work by giving a list of problems

that have arisen throughout this research and that will be analyzed in a future work.

Keywords: Anosov family, Anosov dieomorphism, non-stationary dynamical systems, non-auto-

nomous dynamical systems, random dynamical systems, topological entropy, strong topology.

v

Contents

List of Abbreviations ix

List of Simbols xi

List of Figures xiii

Introduction xv

1 Non-Stationary Dynamical Systems 1

1.1 Non-Stationary Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Uniform Conjugacy Between Non-Stationary Dynamical Systems . . . . . . . . . . . 3

1.3 Compact and Strong Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Some Properties of the Uniform Conjugacy . . . . . . . . . . . . . . . . . . . . . . . 6

2 Entropy for Non-Stationary Dynamical Systems 11

2.1 Entropy for Non-Stationary Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 12

2.2 Properties of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Continuity of Entropy with Product Topology . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Continuity of Entropy for Strong Topology . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Anosov Families 27

3.1 Anosov Families: Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Some Examples of Anosov Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 Lemma of Mather for Anosov Families . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Invariant Cones for Anosov Families . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Openness for Anosov Families 47

4.1 Method of Invariant Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Openness Anosov families with property angles . . . . . . . . . . . . . . . . . . . . . 52

4.3 Openness Anosov families: General case . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.4 Openness for Anosov Families consisting of Matrices . . . . . . . . . . . . . . . . . . 55

5 Stable and Unstable Manifolds 57

5.1 Stable and Unstable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2 Hadamard-Perron Theorem for Anosov Families . . . . . . . . . . . . . . . . . . . . . 58

5.3 Local Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.4 Stable and unstable manifolds for matrix Anosov Families . . . . . . . . . . . . . . . 70

vii

viii CONTENTS

6 Structural Stability for Anosov Families 73

6.1 Openness of A2b(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2 Local Stable and Unstable Manifolds for Elements in A2b(M) . . . . . . . . . . . . . 76

6.3 Structural Stability of A2b(M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7 Other Problems That Arose 83

7.1 Another Classication of Dynamical Systems on the Circle . . . . . . . . . . . . . . . 83

7.2 Entropy for Non-Stationary Dynamical Systems: Further Generalizations . . . . . . . 83

7.3 Existence and classication of Anosov Families . . . . . . . . . . . . . . . . . . . . . 84

7.4 Hölder Continuity of the Subbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Bibliography 87

List of Abbreviations

s.p.a (Satises the property of angles)

n.s.d.s. (Non-stationary dynamical system)

ix

x LIST OF ABBREVIATIONS

List of Simbols

M Disjoint union of the manifolds Mi

d The metric on M

Fm(M) Set consisting of sequences of m-dieomorphisms on M

D(f)p Derivative of f at p

Z Set consisting of integer numbers

R Set consisting of real numbers

N Set consisting of natural numbers

S1 The circle

Tm The m-torus

C Set consisting of complex numbers

Q Set consisting of rational numbers

Esp Stable subspace at p

Eup Unstable subspace at p

TM Tangent bundle of M

TpM Tangent space of M at p

Am(M) The set consisting of Cm-Anosov families on M

Amb (M) Set of Anosov families s.p.a. with bounded second derivative

CFm(M) Set consisting of constant families

Dim(M) Set consisting of Cm- dieomorphisms on M

dm(·, ·) Cm-metric on C(X1, X2) induced by the Riemannian metric on X2

Dmi Set consisting of Cm-dieomorphisms on Mi to Mi+1

Bm(φ, τ) Ball in Dmi with center φ and radius τ

B(x, ε) Ball with center at the point x and radius ε

Bm(φ, (εi)i∈Z) strong basic neighborhood of f

τprod Product topology on Fm(M)

τstr Strong topology on Fm(M)

τunif Uniform topology on Fm(M)

xi

xii LIST OF SIMBOLS

Vs(x, φ) Stable set for φ at x

Vsε(x, φ) Local stable set for φ at x

Vu(x, φ) Unstable set for φ at x

Vuε (x, φ) Local stable set for φ at x

N s(x, (εi)i∈Z) Local stable set for families

N u(x, (εi)i∈Z) Local unstable set for families

f φ Constant family associated to φ

N(A) Number of sets in a nite subcover of A with smallest cardinality

H(A) logN(A)

A ∨ B A ∩B : A ∈ A, B ∈ B∨km=1Am A1 ∩ · · · ∩Ak : Ai ∈ Ai

Hi(f ,A) limn→+∞1nH

(∨n−1k=0(f ki )−1(A)

)H(f ,A) (Hi(f ,A))i∈Z

H(f ) (Hi(f ))i∈Z

Hi(f ) supHi(f ,A) : A is an open cover of Mr[n, i](ε, f ) The smallest cardinality of any (n, ε)-span of Mi with respect to f

r[i](ε, f ) lim supn→+∞

1n log r[n, i](ε, f )

H(f ) (Hi(f ))i∈Z

Hi(f ) limε→0 r[i](ε, f ) for each i ∈ Zs[n, i](ε, f ) The largest cardinality of any (n, ε)-separated subset of Mi with respect f

s[i](ε, f ) lim supn→+∞

1n log s[n, i](ε, f )

H(φ) Topological entropy of a single map φ

expp Exponential application at p

Dr(φ, δ) Set consisting of dieomorphisms ψ such that dr(φ, ψ) ≤ δG(φ) Graph of an application φ

SL(Z,m) Special linear group of mxm matrices with integer entries

Lip(φ) Lipchitz constant of φ

Ksα,f ,p Stable α-cone of f at p

Kuα,f ,p Unstable α-cone of f at p

%p Injectivity radius of expp at p

U(M, 〈·, ·〉) set consisting of Riemannian metric uniformly equivalent to 〈·, ·〉 on M

List of Figures

1.1.1 A n.s.d.s. on a sequence of 2-torus endowed with dierent Riemannian metrics. . . . 3

2.2.1 Graph of φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.1 Exponential application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Shaded regions represent the discs Dr(Ii, ri). G(φ) is the graph of the map φ . . . . . 23

3.1.1 q = φ−1(p) and z = φ(p). D(φ)q(A) = B and D(φ)p(B) = C . . . . . . . . . . . . . . 28

3.2.1 The square [0, 1] × [0, 1] is mapped by A to the parallelogram with vertices (0, 0),

(2, 1), (3, 2), (1, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.1 Stable and unstable α-cones at p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.2 Stable and unstable invariant α-cones. q = f (p) . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 F rp,n =⋂nk=1Dg

±kg±k(p)

(Ksα,f ,g±k(p)

), for r = s, u and n = 1, 2, 3. . . . . . . . . . . . . 52

5.1.1M1,M2,M3,. . . , endowed with the metric given in (3.2.1), for a, b ∈ (λ, 1). . . . . . . 58

5.2.1 G(ψn+1) = fnG(φn). Shaded regions represent the unstable α-cones. . . . . . . . . . . 60

xiii

xiv LIST OF FIGURES

Introduction

Anosov families, which will be dened in Chapter 4, were introduced by P. Arnoux and A.

Fisher in [AF05], motivated by generalizing the notion of Anosov dieomorphisms. Roughly, an

Anosov family is a two-sided sequence of dieomorphisms fi : Mi → Mi+1 dened on a two sided

sequence of compact Riemannian manifolds Mi, for i ∈ Z, having similar behavior to an Anosov

dieomorphism: there is a splitting of the tangent bundle TMi = Esi⊕Eui , invariant by the derivativeD(fi) (that is, D(fi)(E

si ) = Esi+1 and D(fi)(E

ui ) = Eui+1 for any i ∈ Z), and there exist constants

λ ∈ (0, 1) and c > 0 such that for n ≥ 1, p ∈Mi, we have:

‖D(fi+n−1 ... fi)p(v)‖ ≤ cλn‖v‖ for each vector v ∈ Esp

and

‖D(f−1i−n ... f

−1i−1)p(v)‖ ≤ cλn‖v‖ for each vector v ∈ Eup .

The subspaces Esp and Eup are called the stable and unstable subspaces at p, respectively.

The main goal of this work is to verify some properties of Anosov families which are satised

by Anosov dieomorphisms (openness, structural stability and the existence of stable and unstable

manifolds). On the other hand, a notion of topological entropy can be dened for sequences of

dieomorphisms. We will examine the continuity of this entropy at each sequence dened on a

compact Riemannian manifold (see Chapter 2).

Time-dependent dynamical systems are known as non-stationary dynamical systems, non-auto-

nomous dynamical systems, sequence of maps, among others names (see [KL16], [KS96], [KMS99],

[SSZ16], [ZC09], [ZZH06]). We will use these names throughout this thesis. Some results in the

case in which these kinds of systems have hyperbolic behavior can be found in [Ste11], [Bak95a]

and [Bak95b]. Another important approach (and to which the results obtained in this thesis can

be applied) is when the maps fi are small random perturbations of a xed map. This represents a

specic type of random dynamical systems (see [Arn13], [Bog92], [Liu98], [LQ06], [You86]).

One dierence between the notion to be considered in this thesis and the considered in the

above mentioned works is that the Anosov families are not necessarily sequences of Anosov dieo-

morphisms (see [AF05], Example 3). Furthermore, each Mi, although they are dieomorphic, could

have dierent Riemanian structures and therefore the hiperbolicity of the sequence (fi)i∈Z could be

inuenced by the Riemannian metric (see Example 3.2.1).

Other interesting class of examples to consider are the ow families given by non-autonomous

dierential equations, where the orbits are integral curves of time-varying vector elds (see [KR11])

as well as many examples of random dynamical systems (see [Liu98], [Arn13], among other works).

xv

xvi INTRODUCTION

In this thesis we will also give examples obtained from skew product transformations or linear

cocycles (see Denition 3.2.4 and Example 3.2.5), which are a type of random dynamical systems.

LetM be a compact Riemannian manifold and Di1(M) be the set consisting of dieomorphisms

dened on M , endowed with the C1 topology (see [Hir12]). The set A(M) consisting of Anosov

dieomorphisms on M is open in Di1(M), that is, for any Anosov dieomorphism φ : M → M ,

there exists an open set of dieomorphisms O ⊆ Di1(M) such that φ ∈ O ⊆ A(M) (see [Shu13]).

Furthermore, it is possible to take the set O such that for any ψ ∈ O there exists a homeomorphism

h : M → M (depending on ψ), such that φ h = h ψ, that is, φ and ψ are conjugate (this fact

was proved by D. Anosov in [Ano67]). We will obtain analogous versions of these facts for Anosov

families. Let us to talk a few about these results. Consider

M =∐i∈Z

Mi =⋃i∈Z

Mi × i.

The Mi's will be called the components of M, the total space. In (1.1.1) we will give a metric for

M. For m ≥ 1, set

Fm(M) = (fi)i∈Z : fi : Mi →Mi+1 is a Cm-dieomorphism for each i.

We consider three dierent topologies on Fm(M): the compact topology, uniform topology and the

strong topology (or Whitney topology) (see Denitions 1.3.2, 1.3.3 and 1.3.4).

In Theorem 4.3.5 we prove that the collection of C1-Anosov families, denoted by A1(M), is

open in F1(M) endowed with the strong topology. As we said above, the set consisting of Anosov

dieomorphisms on a compact Riemannian manifold is open. Theorem 4.3.5 is an analogue of this

fact for Anosov families. The most important implication of this result is the great variety of non-

trivial examples that it provides (we will show many non-trivial examples of Anosov families in

Section 3.2, thus Theorem 4.3.5 proves that, in a certain way, these examples are not isolated),

since we only ask that the family be Anosov and we do not ask for any additional condition. This

author has submitted a paper titled Openness of Anosov families, which contains the mentioned

above result, to the Journal of the Korean Mathematical Society (see [Ace17d]). This work has been

was accepted for publication.

Young in [You86] proved that families consisting of C1+1 random small perturbations of an

Anosov dieomorphism of class C2 are uniformly hyperbolic sequences, that is, are Anosov families

(see Remark 3.2.7). Let A2b(M) be the set consisting of C2 Anosov families whose second derivative

is bounded and such that the angles between the unstable and stable subspaces are bounded away

from 0 (see (6.0.1) and (3.1.4)). In Section 6.1 we will show that for any f = (fi)i∈Z ∈ A2b(M) there

exists δ > 0 such that if g = (gi)i∈Z with d1(fi, gi) < δ for all i ∈ Z, then g is an Anosov family.

That is, A2b(M) is open in F2(M) endowed with the uniform topology. This is a generalization of

the Young's result, since Anosov families are not necessarily sequences of (small perturbations of)

Anosov dieomorphisms.

Non-stationary dynamical systems are classied by uniform conjugacy, which is dened in Def-

inition 1.2.4. Structural stability of non-stationary dynamical systems will be stated in Denition

1.3.5. In Theorem 6.3.9 we prove that all elements of A2b(M) are structurally stable in F2(M)

xvii

endowed with the uniform topology. This result is a generalization of Theorem 1.1 in [Liu98], which

proves the structural stability of random small perturbations of hyperbolic dieomorphisms. This

author wrote a paper, which contains these results, titled Structural stability of Anosov families

and has been submitted to a journal (see [Ace17e]).

Another approach on the stability of non-stationary hyperbolic dynamical systems can be found

in [CRV17] and [Fra74].

Let φ : M → M be an Anosov dieomorphism. Hirsch and Pugh in [HP70] proved that there

exists ε > 0 such that, for each x ∈ M , the stable and unstable sets at x (see Denition 1.4.2),

which will be denoted by Vsε (x, φ) and Vuε (x, φ) respectively, are dierentiable submanifolds of M .

Furthermore we have that TxVsε (x, φ) = Esx and TxVuε (x, φ) = Eux , that is, Vsε (x, φ) and Vuε (x, φ)

are tangent to the stable and unstable subspaces at x, respectively. φ is a contraction on Vsε (x, φ)

and φ−1 is a contraction on Vuε (x, φ). For Anosov families, Example 3.2.2 proves that the above

properties are not always valid. In Denition 5.1.4 we will give a notion of stable and unstable sets

which works better for non-stationary dynamical systems than the sets given in Denition 1.4.2. We

will prove that, with some conditions on the family (see (5.2.2)), these subsets are dierentiable.

In [Pes76], Pesin proved the existence of invariant manifolds for dieomorphisms of a compact

smooth manifold onto a set where at least one Lyapunov characteristic exponent is nonzero (see

[BP07], [KH97], [Via14]). That theory, which is well-known as Pesin's Theory, has been used

to show the existence of families of invariant manifolds for sequences of random dieomorphisms

dened on a compact manifold (see [Arn13], [LQ06], [QQX03]). These results are probabilistic: the

invariant manifolds exist at almost every point in the support of a chosen invariant measure. Results

to be obtained in this thesis are deterministic: we have an Anosov family (fi)i∈Z, where each fi

is xed, and we give conditions (which depend on the derivative of each fi) to obtain invariant

manifolds along the orbit of a given point p ∈M0. Pesin's Theory has been used to build invariant

manifolds for a two-sided sequence of non-uniformly hyperbolic sequences of dieomorphisms (see

Theorem 7.3.9 in [BP07] or Theorem 6.2.8 in [KH97]). This is also known as The Hadamard-Perron

Theorem. In Proposition 5.2.5 we will show a generalization of the Hadamard-Perron Theorem. The

essence of the proof of our result is the same as that given in [BP07] and [KH97], except that we have

weakened the hypotheses (see Remark 5.2.6). We have written an article titled Local stable and

unstable invariant manifolds for Anosov families, containing the above mentioned facts, submitted

for publication (see [Ace17b]).

Topological entropy is a non-negative real number (possibly +∞) associated to a dynamical

system. It was introduced by R. L. Adler, A.G. Konheim and M. H. McAndrew in [AKM65]. It is

a good tool to classify dynamical systems, since it is invariant with respect to topological conju-

gacy. Let us recall now some known results on the continuity of topological entropy (see [AKM65],

[Wal00]). In [New89], Newhouse proved that the topological entropy of C∞-dieomorphisms on a

compact Riemannian manifold is an upper semicontinuous map. Furthermore, if M is a surface,

this map is continuous. The entropy for any homeomorphism of the circle S1 is zero. Therefore,

it depends continuously on homeomorphisms of S1. In contrast, if we consider all the continuous

maps dened on S1, this entropy is not a continuous map (see [Yan80]). It is clear that the entropy

is continuous at each structurally stable dieomorphism (a dieomorphism φ : M → is structurally

stable if there exist an open neighborhood O of φ such that all the elements in O are topologically

xviii INTRODUCTION

conjugate to φ). In Remark 2.3.5 we will demonstrate an interesting observation which could be an

useful tool to prove the continuity of the topological entropy of a single dieomorphism.

Kolyada and Snoha in [KS96] introduced a notion of topological entropy for non-stationary

dynamical systems (see Section 3.1). In [KL16], [KMS99], [Kus67], [SSZ16], [ZC09], [ZZH06], one

can nd some properties, estimations, formulas and bounds on the topological entropy for non-

stationary dynamical systems. We will prove that this entropy depends continuously on each element

of F1(M) endowed with the strong topology (see Theorem 2.4.5). In contrast, if we consider the

product topology on F1(M), then the entropy is discontinuous at each sequence (see Proposition

2.3.1). Other results on the continuity of the entropy on F1(M) with respect to the uniform topology

will be given in Proposition 2.3.4. In summary, we will give properties of the continuity of the entropy

considering three dierent topologies on F1(M) (see Remark 2.4.6). These results were published

by the author in the Bulletin of the Brazilian Mathematical Society, New Series, in an article titled

On the continuity of the topological entropy of non-autonomous dynamical systems (see [Ace17c]).

Some results about the metric entropy for random dynamical systems can be found in [Bog92],

[LY88], [QQX03] and [Rue97a]).

Next, we will describe the structure of this work.

In Chapter 1 we will introduce the class of objects to be studied in this thesis. We dene the law

of composition for a non-stationary dynamical system, the strong, uniform and product topologies

for the set consisting of families of dieomorphisms and uniform conjugacy, which works properly for

classify the non-stationary dynamical systems. We will nish this chapter by giving some properties

preserved by uniform conjugacy.

Chapter 2 will be devoted to examining the topological entropy for non-stationary dynamical

systems. In this case, we will take Mi = M ×i for each i ∈ Z, where M is a compact Riemannian

manifold. This entropy will be built via open partitions of M (see Denition 2.1.3) and also via

separated and spanning sets (see Denitions 2.1.5 and 2.1.6). These denitions coincide, as in the

case of single maps (see Proposition 2.2.1). This fact can be proved similarly to the case of a single

map. Some properties of this entropy will be given in Section 3.2. These properties generalize those

for the entropy of a single map. One of the most important properties to be shown is that this

entropy is an invariant by uniform conjugacy (see Theorem 2.2.5). In Section 3.3 we will see that

this entropy is discontinuous at any sequence if we consider the product topology on Fm(M). In

contrast, it is continuous on Fm(M) endowed with the strong topology if m ≥ 1.

In Chapter 3 we will introduce the notion of Anosov family and we show some examples and

properties of such families. It is important to keep xed the Riemannian metric on each component,

since the notion of Anosov family depends on the metric dened on each Mi (see Example 3.2.1).

However, Proposition 3.3.1 proves that this notion does not depend on uniformly equivalent metrics

dened on the total space (see Denition 1.1.2). Proposition 3.3.4 shows an analogous version

of the Lemma of Mather adapted to Anosov families (see [Shu13]). The Lemma of Mather for

Anosov dieomorphisms on a compact manifold consists of constructing a Riemannian metric on the

manifold such that, with this metric, the expansion (contraction) of the unstable (stable) subspaces

by the derivative of the dieomorphism is seen after only one iteration. By compactness of the

manifold, this metric is uniformly equivalent to the Riemannian metric that was considered a priori.

xix

This metric, obtained in Proposition 3.3.4, is not necessarily uniformly equivalent to the original

metric on M; the total space M is never compact. The uniform equivalence depends on the angles

between the stable and unstable subspaces of the splitting of the tangent bundle on each component

(see Corollary 3.3.5). In the case of Anosov dieomorphisms dened on compact manifolds those

angles are uniformly bounded away from zero. In the case of families, those angles may decrease

arbitrarily.

Openness for the set consisting of Anosov families in F1(M) with the strong topology will be

proved in Chapter 4 (see Theorem 4.3.5). We will show that if (fi)i∈Z is an Anosov family, then

there exists a two-sided sequence of positive numbers (δi)i∈Z such that if (gi)i∈Z ∈ F1(M) and

d1(fi, gi), then (gi)i∈Z is an Anosov family. In that case, it is not always possible to take (δi)i∈Z

bounded away from zero. In Sections 4.4 and 6.1 we will see that, with some conditions on the norm

of the second derivative of the sequence and the angles between the stable and unstable subspaces,

then the sequence (δi)i∈Z of the neighborhood can be taken bounded away from zero, that is, a

neighborhood in the uniform topology. In order to prove this openness, we will use the method of

invariant cones (see [BP07], [KH97]). First we show the particular case in which the angles between

the stable and unstable subspaces are uniformly bounded away from zero and then, in the nal of

Section 4.3, we consider the general case.

The existence of stable and unstable manifolds for Anosov families will be examined in Chapter 5.

We show in Theorems 5.2.10 and 5.2.11 a generalized version of the Hadamard-Perron Theorem.

In our case, stable and unstable subspaces of an Anosov family are not necessarily orthogonal. We

prove that, with some conditions, there exists a family of submanifolds invariant by the derivative

of the family and show that we can control the expansion or contraction of the submanifolds by the

family. The expansion or contraction of these submanifolds depends also on the angle between the

stable and unstable subspaces (see (5.2.12)). In Section 5.4 we will obtain the unstable and stable

manifold theorems for Anosov family (the Theorems 5.3.5 and 5.3.6). In the Lemmas 5.3.2 and

5.3.3 we give conditions with which the submanifolds obtained in the Theorems 5.2.10 and 5.2.11

coincide with the stable and unstable subsets for an Anosov family.

In Chapter 6 we will prove that A2b(M) is uniformly structuraly stable in F2(M), that is, for any

f = (fi)i∈Z ∈ A2b(M) there exists δ > 0 such that, if gi : Mi → Mi+1 is a C2-dieomorphism and

d2(fi, gi) < δ for each i ∈ Z, then (gi)i∈Z is an Anosov family and is conjugate to f (see Theorem

6.3.9). In Section 6.1 we will show that A2b(M) is open in F2(M) endowed with the uniform

topology: we have that for any (fi)i∈Z ∈ A2b(M) there exists δ > 0 such that, if gi : Mi → Mi+1 is

a C2-dieomorphism and d2(fi, gi) < δ for each i ∈ Z, then (gi)i∈Z is an Anosov family satisfying

the property of angles, that is, the basic neighborhood can be taken uniform (see Denition 4.4.1).

In Section 6.2 we will prove that each element in A2b(M) admits stable and unstable manifolds.

We will nish this thesis in Chapter 7, where we will leave a list of problems that arose through-

out this study and that will be analyzed in a future work.

xx INTRODUCTION

Chapter 1

Non-Stationary Dynamical Systems,

Uniform Conjugacy and Strong Topology

In this chapter we will introduce the objects to be studied in this work. We review some well-

known notions from Dynamical Systems, General Topology, Riemannian Geometry and Dierential

Topology. For readers who wish to know more about these topics, the author recommends, for

instances, the texts [dC92], [Eng89], [Hir12], [KH97] and [Shu13].

1.1 Non-Stationary Dynamical Systems

Given a sequence of compact metric spaces Mi, we will consider the disjoint union

M =∐i∈Z

Mi =⋃i∈Z

Mi × i.

The set M will be called total space and the Mi will be called components of M. If di(·, ·) is the

metric on Mi, the total space is endowed with the metric

d(x, y) =

min1, di(x, y) if x, y ∈Mi

1 if x ∈Mi, y ∈Mj and i 6= j.(1.1.1)

We sometimes use the notation (M,d) to indicate we are considering the metric d given in

(1.1.1).

Denition 1.1.1. Two metrics d and d on a topological space M are uniformly equivalent if there

exist positive numbers k and K such that

kd(x, y) ≤ d(x, y) ≤ Kd(x, y) for all x, y ∈M.

It is clear that if d and d are uniformly equivalent metrics on M , then, for d and d dened on

M, the disjoint union of each Mi = M × i for i ∈ Z, obtained as in (1.1.1), generate the same

topology on M and, furthermore, they are uniformly equivalent on M. On the other hand, if di

and di are uniformly equivalent metrics on Mi for each i ∈ Z, then the metrics d and d, dened

as in (1.1.1), generate the same topology on the total space, but they are not necessarily uniformly

equivalent on M (since M is not compact).

1

2 NON-STATIONARY DYNAMICAL SYSTEMS

If Mi is a compact Riemannian manifold with Riemannian metric 〈·, ·〉i for i ∈ Z, we endow the

total space M with the Riemannian metric 〈·, ·〉 induced by 〈·, ·〉i setting

〈·, ·〉|Mi = 〈·, ·〉i for i ∈ Z, (1.1.2)

and we will use the notation (M, 〈·, ·〉) to indicate that we are considering the Riemannian metric

given in (1.1.2). In that case, on each Mi, we will consider the metric di induced by 〈·, ·〉i.

Denition 1.1.2. Let 〈·, ·〉i and 〈·, ·〉?i be Riemannian metrics on Mi and let ‖ · ‖i and ‖ · ‖?i be theRiemannian norms induced by 〈·, ·〉i and 〈·, ·〉?i , respectively. We say that 〈·, ·〉i and 〈·, ·〉?i (or that‖ · ‖i and ‖ · ‖?i ) are uniformly equivalent on Mi if there exist positive numbers ki and Ki such that

ki‖v‖?i ≤ ‖v‖i ≤ Ki‖v‖?i for all v ∈ TMi,

where TMi is the tangent bundle of Mi. If there exist k and K such that

k‖v‖?i ≤ ‖v‖i ≤ K‖v‖?i for all v ∈ TMi, i ∈ Z,

that is, k and K does not depend on i, then we say that 〈·, ·〉 and 〈·, ·〉? are uniformly equivalent on

M, where 〈·, ·〉? is obtained similarly as in (1.1.2) with the Riemannian metrics 〈·, ·〉?i .

Let di and di be the metrics induced by 〈·, ·〉i and 〈·, ·〉?i , respectively, and d and d dened on

M from di and di as in (1.1.1). These metrics are not necessarily uniformly equivalent on M, since,

for instance, ki could converge to zero or Ki could converge to +∞ when i→ ±∞.

Denition 1.1.3. A non-stationary dynamical system (or n.s.d.s.) f on M is a map f : M→M,

such that, for each i ∈ Z, f |Mi = fi : Mi → Mi+1 is a homeomorphism. Sometimes we use the

notation f = (fi)i∈Z. The n-th composition is dened, for i ∈ Z, to be

f ni :=

fi+n−1 · · · fi : Mi →Mi+n if n > 0

f−1i−n · · · f

−1i−1 : Mi →Mi−n if n < 0

Ii : Mi →Mi if n = 0,

where Ii is the identity on Mi.

We will use the notation (M, 〈·, ·〉, f ) for a n.s.d.s. f dened onM endowed with the Riemannian

metric 〈·, ·〉.

Remark 1.1.4. The notion above can be found in the literature under several dierent names: non-

autonomous dynamical systems, non-autonomous discrete systems, sequence of maps, time dependent

dieomorphisms andmapping families (see [AF05], [ZC09], [KR11], [Fra74], [KS96], [SSZ16], [Ste11],

among others).

Since fi is a homeomorphism, the components Mi are homeomorphic metric spaces, however,

they are not the same object (see Figure 1.1.1). For instance, Mi can be the same Riemannian

manifold, and the metrics 〈·, ·〉i can change with i, or theMi can be the same surfaces with dierent

fractal structures, Thurston corrugations, etc. (see [BJLT12]).

UNIFORM CONJUGACY BETWEEN NON-STATIONARY DYNAMICAL SYSTEMS 3

. . .

Mi−1

fi−1−−−→

Mi

fi−−−→

Mi+1

. . .

Figure 1.1.1: A n.s.d.s. on a sequence of 2-torus endowed with dierent Riemannian metrics.

A simple example of a non-stationary dynamical system is the constant family associated to a

homeomorphism:

Example 1.1.5. Let φ : M →M be a homeomorphism dened on a compact Riemannian manifold

M with metric 〈·, ·〉. Take M =∐Mi, where Mi = M × i for each i ∈ Z with the same

metric 〈·, ·〉. We dene the constant family (M, 〈·, ·〉, f ) associated to φ as the n.s.d.s. (fi)i∈Z, where

fi : Mi →Mi+1 is dened by fi(x, i) = (φ(x), i+ 1) for each x ∈M , i ∈ Z.

Another non-stationary dynamical system we consider in this thesis is obtained from a given

family in the following way.

Denition 1.1.6. Let f and f be non-stationary dynamical systems on M and M, respectively.

We say that f is a gathering of f if there exists a strictly increasing sequence of integers (ni)i∈Z

such that Mi = Mni and f i = fni+1−1 · · · fni+1 fni :

· · ·Mni−1

fi−1=fni−1···fni−1−−−−−−−−−−−−−→ Mni

fi=fni+1−1···fni−−−−−−−−−−−−→ Mni+1 · · ·

If f is a gathering of f , we say that f is a dispersal of f .

In [AF05], Proposition 2.5, it is proved that any non-stationary dynamical system has a dispersal,

which has a gathering, which is equal to the constant family associated to the identity on M0.

1.2 Uniform Conjugacy Between Non-Stationary Dynamical Sys-

tems

In this section we will talk about the morphisms between non-stationary dynamical systems.

Denition 1.2.1. Two continuous maps φ1 : X1 → X1 and φ2 : X2 → X2, where X1 and X2 are

topological spaces, are topologically conjugate if there exists a homeomorphism h : X1 → X2 such

that h φ1 = φ2 h. In that case, h is called a topological conjucagy between φ1 and φ2.

Take

N =∐i∈Z

Ni,

where Ni is a metric space with metric xed di. Let d be the metric on N dened as in (1.1.1)

with the di's. Throughout this chapter, f = (fi)i∈Z and g = (gi)i∈Z will denote a non-stationary

dynamical system dened on (M,d) and (N, d), respectively.


Denition 1.2.2. A topological conjugacy between f and g is a map h : M → N, such that, for

each i ∈ Z, h |Mi = hi : Mi → Ni is a homeomorphism and

hi+1 fi = gi hi,

that is, the following diagram commutes:

M−1f−1−−−−→ M0

f0−−−−→ M1f1−−−−→ M2

···yh−1

yh0 yh1 yh2···N−1

g−1−−−−→ N0g0−−−−→ N1

g1−−−−→ N2

It is clear that the topological conjugacies dene an equivalence relation on the set consisting

of the non-stationary dynamical systems on M. However:

Lemma 1.2.3. If M0 and N0 are homeomorphic then there exists a topological conjugacy between

f and g.

Proof. Let h0 be a homeomorphism between M0 and N0. It is clear that the map h : M → N

dened by

hi =

h0 if i = 0

gi−1 · · · g0 h0 f−10 · · · f−1

i−1 if i > 0

g−1i · · · g

−1−1 h0 f−1 · · · fi if i < 0,

is a conjugacy between f and g .

One type of conjugacy that works for the class of non-stationary dynamical systems is uniform

topological conjugacy :

Denition 1.2.4. We say that a topological conjugacy h : M → N between f and g is uniform

if (hi : Mi → Ni)i∈Z and (h−1i : Ni → Mi)i∈Z are equicontinuous families (that is, h and h−1 are

uniformly continuous maps). In that case we will say that the families are uniformly conjugate.

Since the composition of uniformly continuous functions is uniformly continuous, the class con-

sisting of non-stationary dynamical systems becomes a category, where the objects are the non-

stationary dynamical systems and the morphisms are uniform conjugacies.

Another possible denition of conjugacy for non-stationary dynamical systems is the following:

Denition 1.2.5. A positive (negative) uniform conjugacy between f and g is a sequence of

homeomorphisms hi : Mi → Ni for i ≥ 0 (for i ≤ 0) such that (hi)i≥0 and (h−1i )i≥0 ((hi)i≤0 and

(h−1i )i≤0) are equicontinuous and

hi+1 fi = gi hi : Mi → Ni+1, for every i ≥ 0 (for every i ≤ −1).

That is, (fi)i≥0 and (gi)i≥0 ((fi)i≤0 and (gi)i≤0) are uniformly conjugate.

It is clear that the conjugacy given in Denition 1.2.5 is weaker than the conjugacy given in

Denition 1.2.4.

COMPACT AND STRONG TOPOLOGIES 5

Dynamical systems are classied by topological conjugacy. Uniform topological conjugacies are

very suitable for classifying non-stationary dynamical systems, random dynamical systems, discrete

time process generated by non-autonomous dierential equation, among others systems (see [AF05],

[KR11], [Liu98], and [Arn13] for more details).

1.3 Compact and Strong Topologies

Let X1 and X2 be compact Riemannian manifolds. For r = 1, 2, let ‖ · ‖r be the Riemannian

norm on Xr and distr(·, ·) the metric induced by ‖ · ‖r on Xr. Consider two homeomorphisms

φ : X1 → X2 and ψ : X1 → X2. The d0 metric induced by dist(·, ·) on

Hom(X1, X2) = h : X1 → X2 : h is a homeomorphisms

is given by

d0(φ, ψ) = maxx∈X1

dist2(φ(x), ψ(x)) + maxy∈X2

dist1(φ−1(y), ψ−1(y)).

If φ and ψ are dieomorphisms of class C1, the d1 metric on

Di1(X1, X2) = φ : X1 → X2 : φ is a C1-dieomorphism

is given by

d1(φ, ψ) = d0(φ, ψ) + maxx∈X1

‖Dφx −Dψx‖2 + maxy∈X2

‖D(φ)−1y −D(ψ)−1

y ‖1,

where Dφx and Dψx are the derivatives of φ and ψ at x ∈ X1, respectively. If φ and ψ are

dieomorphisms of class C2, the d2 metric on

Di2(X1, X2) = φ : X1 → X2 : φ is a C2-dieomorphism

is given by

d2(φ, ψ) = d1(φ, ψ) + maxx∈X1

‖D2φx −D2ψx‖2 + maxy∈X2

‖D2(φ)−1y −D2(ψ)−1

y ‖1,

where D2φx and D2ψx are the second derivatives of φ and ψ at x ∈ X1, respectively.

Denition 1.3.1. Suppose that Mi is a Riemannian manifold with Riemannian norm ‖ · ‖i and diis the metric induced by ‖ · ‖i. For m ≥ 0 and τ > 0, set:

i. Dmi = φ : Mi →Mi+1 : φ is a Cm-dieomorphism;

ii. Bm(φ, τ) = ψ ∈ Dmi : dm(φ, ψ) < τ, for φ ∈ Dm

i ;

iii. Fm(M) = f = (fi)i∈Z : fi ∈ Dmi for each i ∈ Z.

Note that

Fm(M) =

+∞∏i=−∞

Dmi .


Denition 1.3.2 (Product Topology). The product topology on Fm(M) is generated by the subsets

U =∏i<−j

Dmi ×

j∏i=−j

[Ui]×∏i>j

Dmi ,

where Ui is an open subset of Dmi , for −j ≤ i ≤ j, for some j ∈ N. The space Fm(M) endowed

with the product topology will be denoted by (Fm(M), τprod).

Denition 1.3.3 (Uniform topology). Given f = (fi)i∈Z and g = (gi)i∈Z in Fm(M), take

dmnorm(f , g) = supi∈Z

(mindm(fi, gi), 1).

The uniform topology on Fm(M) is spanned by dmnorm(·, ·). Let τunif be the uniform topology on

Fm(M).

We can also endow Fm(M) with the Cm-strong topology (or Whitney topology): for each f ∈Fm(M) and a sequence of positive numbers (εi)i∈Z, a strong basic neighborhood of f is the set

Bm(f , (εi)i∈Z) = g ∈ Fm(M) : gi ∈ Bm(fi, εi) for all i.

Denition 1.3.4 (Strong Topology). The Cm-strong topology on Fm(M) is generated by the strong

basic neighborhood of each f ∈ Fm(M). Thus, a subset A of Fm(M) is open if for all f ∈ A, thereexists a strong basic neighborhood Bm(f , (εi)i∈Z) of f , such that Bm(f , (εi)i∈Z) ⊆ A. The space

Fm(M) endowed with the strong topology will be denoted by (Fm(M), τstr).

Unless stated otherwise, we are considering the strong topology on Fm(M). For simplicity, the

τstr will be omitted.

Denition 1.3.5. We say that f ∈ Fm(M) is structurally stable in Fm(M) if there exists a strong

basic neighborhood Bm(f , (εi)i∈Z) of f , such that each g ∈ Bm(f , (εi)i∈Z) is uniformly conjugate

to f . A subset A of Fm(M) is structurally stable if every element in A is structurally stable.

Note that

τprod ⊂ τunif ⊂ τstr.

1.4 Some Properties of the Uniform Conjugacy

From now on, if we do not say otherwise, f = (fi)i∈Z and g = (gi)i∈Z will represent two non-

stationary dynamical systems dened on M and N, respectively. By simplicity, we will denote by

d the metric on both M and N.

The following lemma it is clear and therefore we will omit the proof.

Lemma 1.4.1. f and g are positive (negative) uniformly conjugate if, and only if, for any i0 ∈ Zthere exists a family of homeomorphisms (hi)i≥i0 ((hi)i≤i0) such that (hi)i≥i0 and (h−1

i )i≥i0 ((hi)i≤i0

and (h−1i )i≤i0) are equicontinuous and hi+1 fi = gi hi : Mi → Ni+1, for every i ≥ i0 (for every

i ≤ i0).

SOME PROPERTIES OF THE UNIFORM CONJUGACY 7

Denition 1.4.2. Let φ : X → X be a homeomorphism on a metric space X with metric ρ. For

x ∈ X and ε > 0, set:

1. Vs(x, φ) = y ∈ X : ρ(φn(x), φn(y))→ 0 when n→ +∞:= the stable set for φ at x;

2. Vsε (x, φ) = y ∈ X : ρ(φn(x), φn(y)) < ε, for all n ≥ 0:= the local stable set for φ at x;

3. Vu(x, φ) = y ∈ X : ρ(φn(x), φn(y))→ 0 when n→ −∞:= the unstable set for φ at x;

4. Vuε (x, φ) = y ∈ X : ρ(φn(x), φn(y)) < ε, for all n ≤ 0:= the local stable set for φ at x.

Next we prove that stable and unstable sets for non-stationary dynamical systems are preserved

by uniform conjugacy:

Proposition 1.4.3. Suppose h = (hi)i∈Z is a uniform conjugacy between f and g. For each x ∈Mi,

we have

hi(Vs(x, f )) = Vs(hi(x), g) and hi(Vu(x, f )) = Vu(hi(x), g).

Proof. Let y ∈ Vs(x, f ) and ε > 0. There exists δ > 0 such that, for all j ∈ Z, if z1, z2 ∈ Mj with

d(z1, z2) < δ, then d(hj(z1), hj(z2)) < ε. Since d(f ni (x), f ni (y)) → 0 when n → +∞, there exists

n ∈ N such that, for all n ≥ N , d(f ni (x), f ni (y)) < δ. Consequently, for all n ≥ N ,

ε > d(hn+i(fni (x)), hn+i(f

ni (y))) = d(gni hi(x), gni hi(y))).

Thus hi(y) ∈ Vs(hi(x), g). Now, using the equicontinuity of (h−1i )i∈Z, we can prove that

h−1i (Vs(hi(x), g)) ⊆ Vs(x, f ).

The proof of the unstable case is similar and therefore we omit it.

For the local stable and unstable sets we have:

Proposition 1.4.4. Suppose h = (hi)i∈Z is a uniform conjugacy between f and g. Fix x ∈Mi. For

r = s, u, there exist positive numbers εr, δr and εr such that

hi(Vrεr(x, f)) ⊆ Vrδr(hi(x), g) ⊆ hi(Vrεr(x, f)).

Proof. We will prove the stable case. Since (h−1i )i∈Z is an equicontinuous family, given εs > 0, there

exists δs > 0 such that, for any i ∈ Z, if v, w ∈ Ni and d(v, w) < δs, then d(h−1i (v), h−1

i (w)) < εs.

Fix x ∈ Mi. We will show that if y ∈ Vsδs(hi(x), g) then h−1i (y) ∈ Vsεs(x, f ). Indeed, since y ∈

Vsδs(hi(x), g), d(gni hi(x), gni (y)) < δs for all n ≥ 0. Thus, for all n ≥ 0 we have

εs > d(h−1n+ig

ni hi(x), h−1

n+igni (y)) = d(f ni h

−1i hi(x), f ni h

−1i (y)) = d(f ni (x), f ni h

−1i (y)),

and hence, h−1i (y) ∈ Vsεs(x, f ). Therefore,

h−1i (Vsδs(hi(x), g)) ⊆ Vsεs(x, f ).

Since (hi)i∈Z is an equicontinuous family, analogously we can prove that there exists εs > 0 such

that

hi(Vsεs(x, f )) ⊆ Vsδs(hi(x), g),


which proves the proposition.

Take two homeomorphisms φ : X1 → X1 and ψ : X2 → X2 dened on two compact metric

spaces X1 and X2, respectively. We will denote by f φ and f ψ the constant families associated,

respectively, to φ and to ψ. It is clear that if φ and ψ are topologically conjugate then f φ and f ψ

are uniformly conjugate. In the next proposition we prove the converse is not necessarily true.

Take S1 = z ∈ C : ‖z‖ = 1, with Rα : S1 → S1 the circle rotation by a number α ∈ [0, 1], that

is, Rα(z) = e2παiz for z ∈ S1. It is well-known that if α1 ∈ Q ∩ (0, 1) and α2 ∈ [R \Q] ∩ (0, 1) then

Rα1 and Rα2 are not topologically conjugate. However:

Proposition 1.4.5. Given α1, α2 ∈ [0, 1], the constant families associated to Rα1 and Rα2, respec-

tively, are uniformly conjugate.

Proof. It is sucient to prove that, for any α ∈ [0, 1], the constant families associated to Rα and

to the identity on S1, respectively, are uniformly conjugate. Let I be the identity on S1. Consider

the family of homeomorphisms (hk : S1 → S1)k∈Z, where hk(z) = e2kπ(1−α)iz for all k ∈ Z. Thus,for k ∈ Z and z ∈ S1,

Rα(hk+1(z)) = e2παie2(k+1)π(1−α)iz = e2kπ(1−α)iz = hk(I(z)),

i. e., (hk)k∈Z is a conjugacy between f Rα and f I . If z1, z2 ∈ S1 and d(z1, z2) ≤ 12 minα, 1−α, then

d(R(1−α)(z1), R(1−α)(z2)) = d(z1, z2). Consequently, if d(z1, z2) ≤ 12 minα, 1 − α, for all k ∈ Z,

we have

d(hk(z1), hk(z2)) = d(Rk(1−α)(z1), Rk(1−α)(z2)) = d(z1, z2).

This fact proves that (hk)k∈Z is equicontinuous. Analogously we can prove that (h−1k )k∈Z is equicon-

tinuous. Hence, f Rα and f I are uniformly conjugate.

From Proposition 1.4.5 we have also that the uniform conjugacy does not preserve the rotation

number of a homeomorphism, i. e. there exist homeomorphisms on the circle with dierent rotation

numbers whose associated constant families are uniformly conjugate.

Denition 1.4.6. A map φ : X → X, on a metric space X with metric ρ, is called an isometry if

ρ(φ(x), φ(y)) = ρ(x, y) for all x, y ∈M .

Any circle rotation is an isometry. In the following proposition we will see a more general result

than that obtained in Proposition 1.4.5.

Proposition 1.4.7. If φ : X → X is an isometry, then fφ is uniformly conjugate to the constant

family associated to the identity on X.

Proof. Consider the family (hk)k∈Z, where hk(x) = φ−k(x) for every x ∈Mk = X × k. It is clearthat (hk)k∈Z is a topological conjugacy between f φ and f I , where I is the identity on X. Since φ

is an isometry, the family (hk)k∈Z is equicontinuous.

It follows from Proposition 1.4.7 that all the constant families associated to any isometry on M

are uniformly conjugate.

We will nish this chapter with the following proposition.

SOME PROPERTIES OF THE UNIFORM CONJUGACY 9

Proposition 1.4.8. If f and g are uniformly conjugate, then the gatherings f and g obtained,

respectively, from f and g by a sequence of integers (ni)i∈Z (see Denition 1.1.6), are uniformly

conjugate.

Proof. If f and g are uniformly conjugate by h = (hi)i∈Z, then f and g are uniformly conjugate

by the family h = (hni)i∈Z :

Mni−1

fni−1−−−−→ · · ·fni−1−−−−→ Mni

fni−−−−→ · · ·fni+1−1

−−−−−→ Mni+1

···yhni−1

yhni yhni+1 ···

Mni−1

gni−1−−−−→ · · ·gni−1−−−−→ Mni

gni−−−−→ · · ·gni+1−1

−−−−−→ Mni+1

It is clear that h = (hni)i∈Z is equicontinuous.

Chapter 2

Entropy for Non-Stationary Dynamical

Systems

In [AKM65], R. L. Adler, A. G. Konheim and M. H. McAndrew introduced the topological

entropy of a continuous map φ : X → X on a compact topological space X via open covers

of X. Roughly, the topological entropy is the exponential growth rate of the number of essentially

dierent orbit segments of length n. In 1971, R. Bowen dened the topological entropy of a uniformly

continuous map on an arbitrary metric space via spanning and separated sets, which, when the space

is compact, coincides with the topological entropy as dened by Adler, Konheim and McAndrew.

Both denitions can be found in [Wal00].

S. Kolyada and L. Snoha, in [KS96], introduced a notion of topological entropy for non-autono-

mous dynamical systems, which generalizes the notion of entropy for single dynamical systems. They

considered only sequences of type (fi)i≥0 and the entropy for this sequence was a single number

(possibly +∞). In this chapter we will extend this idea to sequences of type (fi)i∈Z. We can dene

a dierent entropy for the same sequence by considering the composition of the inverse of each fi

for i→ −∞ (see Remark 2.2.11).

First, the entropy of a non-stationary dynamical system (fi)i∈Z will be dened as a sequence

of non-negative numbers (ai)i∈Z, where each ai depends only on fj for j ≥ i. Then we will see

that (ai)i∈Z is a constant sequence (see Corollary 2.2.7). Consequently, this common number will

be considered to be the entropy of (fi)i∈Z. As a consequence, we will also see the entropy of a

non-stationary dynamical system can be considered as the topological entropy of a single homeo-

morphism dened on the total space M (see Remark 2.2.9).

The main goal of this chapter is to show that, if m ≥ 1 and we consider the strong topology on

Fm(M), the entropy depends continuously on each non-stationary dynamical system in Fm(M). In

contrast, with the product topology on Fm(M), the entropy is discontinuous for any non-stationary

dynamical system. To prove that the entropy is continuous on Fm(M) with the uniform topology

is equivalent to prove the continuity of the entropy for single maps (see Proposition 2.3.4). The

present chapter is a work published by the author in the Bulletin of the Brazilian Mathematical

Society, New Series (see [Ace17c]).

Throughout this chapter, we will take

Mi = M × i and Ni = N × i, for each i ∈ Z,

11

12 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS

where M and N are compact Riemannian manifolds.

2.1 Denition of Entropy for Non-Stationary Dynamical Systems

In this section we will introduce the notion of topological entropy for a non-stationary dynamical

system (fi)i∈Z, generalizing the topological entropy for a single map (see [Wal00]). Firstly, this

entropy will not be a single positive number but a sequence of non-negative numbers (ai)i∈Z, where

each ai depends only on fj for each j ≥ i. In Corollary 2.2.7 we will see that this sequence is

constant.

We consider the following denitions: an open cover of M is a collection of open subsets of M ,

A = Aλλ∈Λ, such that M =⋃λAλ. In this section, A and B will denote open covers of M . Since

Mi = M ×i, if A is an open cover of M , then Ai = A×i is an open partition of Mi. By abuse

of notation, we will omit the sub index i of Ai for covers of Mi.

Denition 2.1.1. LetN(A) be the number of sets in a nite subcover ofA with smallest cardinality.

The entropy of A is the number

H(A) := logN(A).

The proof of the following statements can be found in [Wal00] for the case of a single map. Such

proofs can be adapted for non-stationary dynamical systems and therefore we omit them. We will

consider the following notations: For each i ∈ Z and n ≥ 0, set

(f ni )−1(A) = (fi+n−1 · · · fi)−1(A) : A ∈ A.

Set

A ∨ B = A ∩B : A ∈ A, B ∈ B.

Inductively we dene∨km=1Am for a collection of open covers A1, ...,Ak of M.

We say B is a renement of A if each element of B is contained in some element of A.

Proposition 2.1.2. The entropy satises the following properties:

i. H(A ∨ B) ≤ H(A) +H(B).

ii. If B is a renement of A then H(A) ≤ H(B).

iii. H(A) = H((f ki )−1(A)) for each i ∈ Z and k ≥ 0.

iv. H(∨n−1k=0(f ki )−1(A)) ≤ nH(A), for each i ∈ Z and n ≥ 1.

v. The limit

Hi(f,A) = limn→+∞

1

nH

(n−1∨k=0

(f ki )−1(A)

)(2.1.1)

exists and is nite, for each i ∈ Z.

Denition 2.1.3 (Topological entropy). We dene the entropy of f relative to A as the sequence

H(f ,A) = (Hi(f ,A))i∈Z.

ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS 13

The topological entropy of f is the sequence

H(f ) = (Hi(f ))i∈Z,

where

Hi(f ) = supHi(f ,A) : A is an open cover of M.

From now on, X will represent a compact metric space. We recall that the topological entropy

of a homeomorphism φ : X → X, which we denote by H(φ), is dened considering open covers of

X. The above denition only makes sense when A is an open cover of M instead of a general open

cover of M. If we consider arbitrary collections of open covers of each Mi, the limit (2.1.1) could

be innite (we can take open covers Ai of each Mi with N(Ai) arbitrarily large, for each i).

Now we introduce the denition of topological entropy using span and separated subsets. That

entropy will be called ?-topological entropy for dierentiate it from the topological entropy. As in

the case of a single homeomorphism, we will see in Theorem 2.2.1 that the topological entropy

coincides with ?-topological entropy for non-stationary dynamical systems.

Denition 2.1.4. Let n ∈ N, ε > 0 and i ∈ Z be given. We say that a compact subset K ⊆Mi is a (n, ε)-spanning of Mi with respect f if for each x ∈ Mi there exists y ∈ K such that

max0≤j<n

d(f ji (x), f ji (y)) < ε, i. e.,

Mi ⊆⋃y∈K

n−1⋂k=0

(f ki )−1(B(f ki (y), ε)),

where B(f ki (y), ε) is the closed ball with center f ki (y) ∈Mi+k and radius ε.

Denote by r[n, i](ε, f ) the smallest cardinality of any (n, ε)-span of Mi with respect f . Since Mi

is compact, we have r[n, i](ε, f ) <∞ for each i ∈ Z and n ≥ 1. Set

r[i](ε, f ) = lim supn→+∞

1

nlog r[n, i](ε, f ).

Denition 2.1.5. The ?-topological entropy of f is the sequence

H(f ) = (Hi(f ))i∈Z, where Hi(f ) = limε→0

r[i](ε, f ) for each i ∈ Z.

Now we dene the entropy using separated subsets and we will prove that the entropy considering

span subsets coincide with the entropy considering separated subsets.

Denition 2.1.6. Let n ∈ N, ε > 0 and i ∈ Z be xed. A subset E ⊆Mi is called (n, ε)-separated

with respect to f if given x, y ∈ E, with x 6= y, we have max0≤j<n

d(f ji (x), f ji (y)) > ε, i. e., if for all

x ∈ E, the setn−1⋂k=0

(f ki )−1(B(f ki (x), ε))

contains no other point of E.


Denote by s[n, i](ε, f ) the largest cardinality of any (n, ε)-separated subset of Mi with respect

to f . Set

s[i](ε, f ) = lim supn→+∞

1

nlog s[n, i](ε, f ).

Proposition 2.1.7. Given ε > 0 and i ∈ Z we have:

i. r[n, i](ε, f) ≤ s[n, i](ε, f) ≤ r[n, i](ε/2, f), for all n > 0.

ii. r[i](ε, f) ≤ s[i](ε, f) ≤ r[i](ε/2, f), for all n > 0.

Proof. The proposition is proved in [Wal00], Chapter 7, for a single map. That proof works for

non-stationary dynamical systems and, therefore, we omit the proof.

From Proposition 2.1.7 we have

Hi(f ) = limε→0

s[i](ε, f ) for all i ∈ Z.

Consequently, Hi(f ) can be dened using either span or separated subsets.

Notice that if f is a constant family associated to a homeomorphism φ : X → X, then it is clear

that

Hi(f ) = H(φ), for all i ∈ Z. (2.1.2)

Therefore, H extends the notion of topological entropy for single homeomorphisms.

Some estimations and properties of the topological entropy for non-stationary dynamical systems

can be found in [ZC09], [Kaw17], [KMS99], [KS96], [SSZ16] and [ZZH06]. In [KL16], C. Kawan

and Y. Latushkin give a formula for the topological entropy of a non-stationary subshift of nite

type, which were introduced by Fisher and Arnoux in [AF05]. Regarding the metric entropy for

non-autonomous dynamical systems the author recommends C. Kawan's works (see [Kaw14] and

references there).

In [Bog92], [Kus67], [LY88], [Liu98], [Rue97a], is dened a measure-theoretic entropy for random

dynamical systems (see [Arn13]). In these papers we can also found analogous versions for n.s.d.s.

of the thermodynamic formalism of dynamical systems (see [Rue97b]). In [QQX03] some relations

between the entropy for random dynamical systems and the Lyapunov exponents and the Pesin's

formula are given.

2.2 Some Properties of Entropy for Non-Stationary Dynamical Sytems

In this section we will see some properties of this topological entropy for non-stationary dy-

namical systems. Some are analogous to the well-known properties of entropy for a single map (see

[Wal00]). For single maps, the topological entropy is invariant for topological conjugacy. The main

result of this section is to prove the analogous version for non-stationary dynamical systems, that is,

this entropy is invariant for uniform conjugacy (see Theorem 2.2.5). This result will be fundamental

to show the continuity of the entropy in the following section (see Theorem 2.4.5).

As we mentioned, the notions of entropy for non-stationary dynamical systems, considering

either open covers or separated subsets, coincide. This fact can be proved analogously to the case

of single homeomorphisms (see [Wal00], Chapter 7, Section 2):

PROPERTIES OF ENTROPY 15

Proposition 2.2.1. For each i ∈ Z we have Hi(f ) = Hi(f ).

From now on, we will use the notation Hi(f ).

The topological entropyH(φ) of a single homeomorphism φ : X → X satisesH(φn) = |n|H(φ),

for n ∈ Z. For non-stationary dynamical systems we have:

Proposition 2.2.2. Suppose f = (fi)i∈Z is an equicontinuous sequence. Fix n ≥ 1. Let f ∈ F1(M)

be the gathering obtained of f by the sequence (ni)i∈Z, that is, Mi = Mni and fi = fn(i+1)−1· · ·fni;

· · ·Mn(i−1)

fi−1=fni−1···fn(i−1)−−−−−−−−−−−−−−→ Mni

fi=fn(i+1)−1···fni−−−−−−−−−−−−−→ Mn(i+1) · · ·

Thus, for each i ∈ Z we have

Hi(f ) = nHin(f ).

Proof. For i ∈ Z, x, y ∈Mni and m > 0, we have

max0≤k<m

d(fki (x), f

ki (y)) = max

0≤k<md(f nkni (x), f nkni (y)) ≤ max

0≤j<nmd(f jni(x), f jni(y)).

This fact proves that, for all ε > 0, each (nm, ε)-span subset K ofMni with respect to f is a (m, ε)-

span subset of Mni with respect to f . Consequently, we obtain r[m,ni](ε, f ) ≤ r[nm, ni](ε, f ).

Hence,

Hi(f ) ≤ nHin(f ).

On the other hand, since f is equicontinuous, we can prove that (fni)i∈Z, (f 2ni)i∈Z, . . . , (f

n−1ni )i∈Z

is an equicontinuous collection of families. Consequently, given ε > 0, there exists δ > 0 such that

max1≤k<nj∈Z

d(f k+1nj (x), f k+1

nj (y)) : x, y ∈Mnj ,d(x, y) < δ < ε.

Now, if K is a (m, δ)-span of Mni with respect to f , then, for all x ∈Mni, there exists y ∈ K such

that

maxd(x, y),d(f nni(x), f nni(y)), ...,d(f(m−1)nni (x), f

(m−1)nni (y)) < δ.

Thus,

max0≤k<n

d(f kni(x), f kni(y)) < ε,

max0≤k<n

d(f kn(i+1) fnni(x), f kn(i+1) f

nni(y)) < ε,

...

max0≤k<n

d(f kn(i+m−1) f(m−1)nni (x), f kn(i+m−1) f

(m−1)nni (y)) < ε.


Consequently, we have

max0≤k<n

d(f kni(x), f kni(y)) < ε,

max0≤k<n

d(f n+kni (x), f n+k

ni (y)) < ε,

...

max0≤k<n

d(f(m−1)n+kni (x), f

(m−1)n+kni (y)) < ε.

Therefore,

maxd(f kni(x), f kni(y)) : k = 0, ...,mn− 1 < ε,

that is, K is a (mn, ε)-span of Mni with respect to f . Consequently, we have r[m,ni](ε, f ) ≥r[nm, ni](ε, f ) and, therefore,

Hi(f ) ≥ nHin(f ),


From the proof of Proposition 2.2.2, we have always the inequality

Hi(f ) ≤ nHin(f ).

In [KS96] can be found an example of a general n.s.d.s. where the above inequality is strict.

Proposition 2.2.3. Suppose f = (fi)i∈Z is a sequence consisting of isometries, that is, fi : Mi →Mi+1 is an isometry for all i. Thus Hi(f) = 0, for all i ∈ Z.

Proof. This follows directly from Denition 2.1.5.

Let S1 ⊆ R2 be the circle endowed with the Riemannian metric inherited from R2. For homeo-

morphisms φ on S1 we have that H(φ) = 0 (see [Wal00]). This property is also valid for n.s.d.s:

Proposition 2.2.4. Suppose that Mi = S1 × i for each i ∈ Z endowed with the Riemannian

metric inherited from R2. If f is a non-stationary dynamical system on M, then Hi(f ) = 0, for all

i ∈ Z.

Proof. See [KS96], Theorem D.

In the following theorem we will see that this entropy for non-stationary dynamical systems is

an invariant for uniform conjugacy. This result generalizes the fact that the topological entropy of

homeomorphisms dened on compact metric spaces is an invariant for topological conjugacy.

Theorem 2.2.5. If f and g are positively uniformly conjugate, then

Hi(f ) = Hi(g) for all i ∈ Z.

Proof. Fix i ∈ Z. It follows from Lemma 1.4.1 that there exists a uniform conjugacy (hj)j≥i between

(fj)j≥i and (gj)j≥i. Since (hj)j≥i is equicontinuous, given ε > 0 there exists δ > 0 such that, for

all j ≥ i, if x, y ∈ Mj and d(x, y) < δ, then d(hj(x), hj(y)) < ε. Let K be a (m, δ)-span of Mi

PROPERTIES OF ENTROPY 17

with respect to f . Thus, for all x ∈Mi there exists y ∈ K such that max0≤j<m d(f ji (x), f ji (y)) < δ.

Consequently, if 0 ≤ j < m,

ε > max0≤j<m

d(hi+j f ji (x), hi+j f ji (y)) = max0≤j<m

d(g ji hi(x), g ji hi(y)).

This fact proves that r[m, i](ε, f ) ≥ r[m, i](δ, g). Hence,

Hi(f ) ≥ Hi(g).

Since (h−1j )j≥i is equicontinuous, analogously we prove

Hi(f ) ≤ Hi(g),

which proves the theorem.

It follows from the proof of the above theorem that if (fi)i≥i0 and (gi)i≥i0 are uniformly conjugate

then Hi0(f ) = Hi0(g). Furthermore, entropy depends only on the future:

Corollary 2.2.6. Suppose that there exists i0 ∈ Z such that fj = gj for all j ≥ i0. Then we have

Hi(f ) = Hi(g) for all i ∈ Z.

Proof. It is clear that (fj)j≥i0 and (gj)j≥i0 are uniformly conjugate (take hj = Id for each j ≥ i0).It follows from Lemma 1.4.1 that (fj)j≥0 and (gj)j≥0 are uniformly conjugate. By Theorem 2.2.5

we have Hi(f ) = Hi(g) for all i ∈ Z.

Corollary 2.2.7. For all i, j ∈ Z we have Hi(f ) = Hj(f ).

Proof. It is sucient to prove that Hi(f ) = Hi+1(f ) for all i ∈ Z. Fix i ∈ Z. Take the family

g = (gj)j∈Z, where gj = Ij : Mj → Mj+1 for each j ≤ i is the identity, modulo the identication

Mi = M (remember that Mi = M × i), and gj = fj for j > i. Thus Hi(f ) = Hi(g). For each

x, y ∈Mi and n ≥ 2 we have

max0≤j<n

d(g ji (x), g ji (y)) = max0≤j<n−1

d(g ji+1(x), g ji+1(y)).

Using this fact we can prove that Hi(g) = Hi+1(g). Consequently, we have that

Hi(f ) = Hi+1(f ),

for any i ∈ Z.

Remark 2.2.8. From now on we will omit the index i of Hi and we will consider the entropy of a

non-stationary dynamical system as a single number, as a consequence of Corollary 2.2.7.

Remark 2.2.9. We can consider the system f = (fi)i∈Z as a homeomorphism f : M→M. If f is

uniformly continuous, then we can calculate the topological entropy of the single map f , H(f ), via

open covers or spanning or separated sets of M. It can be proved that H(f ) = H(f ).


If we consider another metric d uniformly equivalent to d on M , then the identity

I : (M,d)→ (M, d)

p 7→ p

is a uniformly continuous map. It follows from Theorem 7.4 in [Wal00] that the topological entropy

of f considering the metric d on M coincides with the topological entropy of f considering d on M .

Consequently, the entropy for a non-stationary dynamical system on M is the same for equivalent

metrics on M .

We can dene the inverse of f as f −1 = (gi)i∈Z, where gi := f−1i : Mi+1 → Mi for each i. In

this case, for n > 0 we have

(f −1)0i := Ii+1 : Mi+1 →Mi+1

and

(f −1)ni := gi−n+1 · · · gi : Mi+1 →Mi−n+1

In the case of a single homeomorphism φ : M →M , we have H(φ) = H(φ−1) (see [Wal00], Theorem

7.3). The following example proves that, in general, we could have H(f ) 6= H(f −1).

Example 2.2.10. Let I : M →M be the identity on M and φ : M →M be a homeomorphism on

M with non-zero topological entropy. Let fi : Mi →Mi+1 be the dieomorphisms dened as fi = I

for i ≥ 0 and fi = φ for i < 0 and take f = (fi)i∈Z. From Corollary 2.2.6 we have H(f ) = H(I) = 0

and H(f −1) = H(φ) 6= 0, for each i ∈ Z.

The essence of the above example is that entropy of f depends only on the future, while the

entropy of f −1 depends only on the past.

Remark 2.2.11. As a consequence of Example 2.2.10, we can also consider the entropy H(f −1),

which we denote by H(−1)(f ). All the above results for H have analog versions for H(−1).

There are dynamical systems dened on a compact metric space that are not topologically

conjugate but have the same topological entropy. Now, from Theorem 2.2.5 we have that two

constant families associated to homeomorphisms with dierent topological entropies cannot be

uniformly conjugate. On the other hand, Propositions 1.4.5 and 2.2.4 prove that there are constant

families, associated to homeomorphisms with the same topological entropy, that can be uniformly

topologically conjugate. One natural question that arises from this notion of entropy is as follows:

Suppose that f and g are constant families. If H(f ) = H(g) then are f and g always uniformly

conjugate? The answer is negative, as the following example shows:

Example 2.2.12. In this example we consider the stable and unstable sets given in Denition

1.4.2. Proposition 1.4.3 proves that, if h = (hi)i∈Z is a uniform conjugacy between f and g , then,

for each x ∈Mi, we have

hi(Vs(x, f )) = Vs(hi(x), g) and hi(Vu(x, f )) = Vu(hi(x), g).

Let M = S1, pN be the north pole and pS be the south pole of S1. Suppose that φ : M → M

is a homeomorphism with stable set Vs(pN , φ) = M \ pS (see Figure 2.2.1). Let f and g be the

CONTINUITY OF ENTROPY WITH PRODUCT TOPOLOGY 19

constant families associated to φ and to the identity on M , respectively. Then H(f ) = H(g) = 0

for all i ∈ Z, because all the homeomorphisms on the circle has zero entropy (see (2.1.2)). Note

that φ and the identity are not conjugate on S1. On the other hand, we have

Vs((pN , 0), f ) = [M \ pS]× 0 and Vs((pN , 0), g) = (pN , 0).

Since uniform conjugacy preserves the stable sets, we have that f and g can not be uniformly

conjugate.

pS

pN

pN

Figure 2.2.1: Graph of φ.

In the above example we have that both φ and I have zero entropy. The next example proves

that we can have two homeomorphisms φ and ψ with H(φ) = H(ψ) > 0 and such that their

associated constant families are not uniformly conjugate.

Example 2.2.13. Let A : M →M be a homeomorphism with a xed point z0 ∈M and H(A) > 0

(A could be, for instance, an Anosov dieomorphism induced by a 2x2 hyperbolic matrix dened on

the 2-torus) and take φ as dened in the above example. Take ψ = A×φ : M×S1 →M×S1, (p, z) 7→(A(p), φ(z)) and ζ = A× I : M ×S1 →M ×S1, (p, z) 7→ (A(p), z). Thus H(ψ) = H(ζ) = H(A) > 0.

We can prove that the constant families associated to ψ and ζ are not uniformly conjugate.

2.3 On the continuity of Entropy for the Product Topology

As we said in the introduction, the main goal of this chapter is prove that topological entropy

for non-stationary dynamical systems is a continuous map on Fm(M) endowed with the strong

topology. This will be shown in the next section. In contrast, if we consider the product topology

on Fm(M), we have that:

Proposition 2.3.1. Suppose that H(Fm(M)) has two or more elements. Then

H : (Fm(M), τprod)→ R ∪ +∞

is discontinuous at any f ∈ Fm(M).

Proof. Let f = (fi)i∈Z ∈ Fm(M). Since H(Fm(M)) has two or more elements, there exists g =

(gi)i∈Z ∈ Fm(M) such that H(g) 6= H(f ). Let V ∈ τprod be an open neighborhood of f . For some


k ∈ N, the family h = (hi)i∈Z, dened by

hi =

fi if − k ≤ i ≤ k

gi if i > k or i < −k,

belongs to V, by denition of τprod. It is follow from Corollary 2.2.6 that

H(h) = H(g),

which proves the proposition, since (Fm(M), τprod) a metric space.

From Proposition 2.2.4 we have that, if M = S1, then H(Fm(M)) = 0 for every m ≥ 0. In

this case, H : (Fm(M), τprod)→ R ∪ +∞ is continuous.

Below we will see some interesting results that are obtained when we consider only the constant

families while maintaining the product topology on Fm(M).

Set

CFm(M) = f ∈ Fm(M) : f is a constant family,

τstr = τstr|CFm(M) and τprod = τprod|CFm(M).

Proposition 2.3.2. τstr = P(CFm(M)) = A : A ⊆ CFm(M).

Proof. It is sucient to prove that each (fi)i∈Z, with (fi)i∈Z ∈ CFm(M), is open in CFm(M).

Let (εi)i∈Z be a sequence of positive numbers with εi → 0 when |i| → ±∞. Then, consider the

strong basic neighborhood Bm((fi)i∈Z, (εi)i∈Z) ⊆ Fm(M) of (fi)i∈Z. Notice that

(fi)i∈Z = Bm((fi)i∈Z, (εi)i∈Z) ∩ CFm(M).

Consequently, (fi)i∈Z is open in (CFm(M), τstr).

The map

π0 : (Fm(M), τ)→ (Dm0 , d

m)

(fi)i∈Z 7→ f0,

where Dmi = Dim(Mi,Mi+1) for i ∈ Z, is continuous for τ ∈ τstr, τprod, because

π−10 (U) =

∏i<0

Dmi × [U × 0]×

∏i>0

Dmi ,

for U ⊆ Dm0 , thus, if U is an open subset of Dm

0 , then π−10 (U) is open in (Fm(M), τprod). Hence,

the restriction

π0 = π0|CFm(M) : (CFm(M), τ)→ (Dm0 , d

m)

is continuous for τ ∈ τstr, τprod.

We can identify (Dm0 , d

m) with (Dim(M), dm), the space consisting of dieomorphisms on M

endowed with the Cm-metric obtained from the metric d on M (remember that Mi = M × i).

CONTINUITY OF ENTROPY FOR STRONG TOPOLOGY 21

From now on we will make use of this identication. For a Cm-dieomorphism φ : M → M , we

denote the constant family associated to φ by f φ. Notice that π0 is invertible, in fact,

π−10 : (Dim(M), dm)→ (CFm(M), τ)

φ 7→ f φ.

Clearly, if τ = τstr, then π−10 is not continuous (see Proposition 2.3.2). On the other hand, we

have:

Proposition 2.3.3. If τ = τprod, then π−10 is continuous.

Proof. All the open subsets of (CFm(M), τprod) are unions of sets of the form

U =

∏i<−j

Dmi ×

j∏i=−j

[Ui]×∏i>j

Dmi

∩ CFm(M),

where Ui is an open subset of Dmi , for −j ≤ i ≤ j. Notice that

(π−10 )−1(U) = π0(U) =

j⋂i=−j

Ui,

which is an open subset of Dim(M). Thus, π−10 is continuous.

Consequently, we have:

Proposition 2.3.4. H : (CFm(M), τprod) → R ∪ +∞, is continuous if, and only if, H :

(Dim(M), dm)→ R ∪ +∞, is continuous.

Proof. This is clear, because H = H π and π is a homeomorphism.

Remark 2.3.5. Proposition 2.3.4 could be a useful tool to show the continuity of the topological

entropy at some Cm-dieomorphisms: to show that H is continuous at φ ∈ Dim(M), we could try

to prove that H|CFm(M) is continuous at f φ. In order to prove this fact, we have to nd an open

neighborhood U ⊆ Dim(M) of φ, such that each constant family associated to any dieomorphism

in U is uniformly conjugate to f φ. Thus, by Theorem 2.2.5 and (2.1.2), we had that

H(ψ) = H(f ψ) = H(f φ) = H(φ) for any ψ ∈ U .

Remember that Proposition 1.4.5 proves that there exist dieomorphisms φ and ψ which are

not topologically conjugate, however f φ and f ψ could be uniformly conjugate.

2.4 Continuity of Entropy with respect to the Strong Topology

Finally, we will prove the continuity of H : (Fm(M), τstr)→ R∪ +∞ for the strong topology

on Fm(M), for m ≥ 1. More specically, entropy is locally constant, that is, each (fi)i∈Z ∈ Fm(M)

has a strong basic neighborhood in which the entropy is constant. It is sucient to prove the case

when m = 1. Remember that we are considering a compact Riemannian manifold M with metric d

and we will consider the metric d on M given in (1.1.1).


Denition 2.4.1. Let % > 0 be such that, for each p ∈M , the exponential application

expp : B(0p, %)→ B(p, %)

is a dieomorphism and

‖v‖ = d(expp(v), p), for all v ∈ B(0p, %),

where 0p is the zero vector in TpM , the tangent space of M at p. % is called an injectivity radius

0p

%

%

B(0p, %) TpM

B(p, %)

Mp

expp

Figure 2.4.1: Exponential application.

of M at p (see Figure 2.4.1). See [dC92] for more detail.

We will denote by % the injectivity radius of each p ∈M and we will suppose that % < 1/2. We

will x f = (fi)i∈Z ∈ F1(M). For δ > 0 and r = 0, 1, set

Dr(Ii, δ) = h ∈ Hom(Mi) : h is a Cr-dieomorphism and dr(h, Ii) ≤ δ

and D1(fi, δ) = g ∈ Di1(Mi,Mi+1) : d1(g, fi) ≤ δ.

The closure of D1(Ii, δ) on D0(Ii, δ) will be denoted by D1(Ii, δ).

Lemma 2.4.2. There exist two sequences (ri)i≥0 and (δi)i≥0, with ri → 0 when i → +∞, such

that, for each g ∈ D1(fi, δi), the map

Gi+1 : Dr(Ii+1, ri+1)→ Dr(Ii, ri)

h 7→ g−1hfi

is well-dened for each i ≥ 1 and r = 0, 1. (see Figure 2.4.2).

Proof. Note that if g ∈ Di1(Mi,Mi+1) and h ∈ Dir(Mi+1,Mi+1), we have

dr(g−1hfi, Ii) ≤ dr(g−1hfi, g−1fi) + dr(g−1fi, Ii) for i ≥ 0.

If h is Cr-close to Ii+1, then g−1hfi is C

r-close to g−1fi and if g is C1-close to fi, then g−1fi is

Cr-close to Ii. Fix r0 ∈ (0, %/4). There exist r1 ∈ (0, r0/2) and δ0 > 0 such that, if h1 ∈ Dr(I1, r1)

and g0 ∈ D1(f0, δ0), then g−10 h1f0 ∈ Dr(I0, r0). Take r2 ∈ (0, r1/2) and δ1 > 0 such that, if

h2 ∈ Dr(I2, r2) and g1 ∈ D1(f1, δ1), then g−11 h2f1 ∈ Dr(I1, r1). Inductively, we can build two


Mi−1

hi−1 = g−1i−1hi−1fi−1

Mi−1

Gi

G(Ii−1)

G(hi−1)

Mi

hi = g−1i hifi

Mi

Gi+1

G(Ii)

G(hi)

Mi+1

hi+1 = g−1i+1hi+1fi+1

Mi+1 G(Ii+1)

G(hi+1)

Figure 2.4.2: Shaded regions represent the discs Dr(Ii, ri). G(φ) is the graph of the map φ

sequences (ri)i≥0 and (δi)i≥0, with ri ∈ (0, ri−1/2) for each i ≥ 1, such that if hi ∈ Dr(Ii, ri) and

gi−1 ∈ D1(fi−1, δi−1), then g−1i−1hifi−1 ∈ Dr(Ii−1, ri−1), which proves the lemma.

Analogously, we can nd a sequence of positive numbers (δi)i≤0 and (ri)i≤0, with ri → 0 when

i→ −∞, such that for each g ∈ D1(fi−1, δi−1), the map

Gi−1 : Dr(Ii−1, ri−1)→ Dr(Ii, ri)

h 7→ fi−1hg−1

is well-dened for each i ≤ 0 and r = 0, 1.

Lemma 2.4.3. There exist two sequences h = (hi)i≥0 ∈∏i≥0D

0(Ii, ri) and h = (hi)i≤0 ∈∏i≤0D

0(Ii, ri) such that

Gi+1hi+1 = hi for all i ≥ 0 and Gi−1hi−1 = hi for all i ≤ 0.

Proof. For each i > 0, let hi = G1 · · · Gi(Ii). It follows from Lemma 2.4.2 that hi belongs

to D1(I0, r0). Consequently, the sequence (hi)i≥0 is equicontinuous, because each hi is C1 and

the sequence has uniformly bounded derivative. Hence, there exist a subsequence im → ∞ and

h0 ∈ D0(I0, r0) such that him → h0 as m→∞. Note that

G1 : D1(I1, r1)→ G1(D1(I1, r1)) ⊆ D0(I0, r0)

is invertible, where D1(I1, r1) is the closure in D0(I1, r1), and both G1 and G−11 are continuous.

Therefore,

G1(D1(I1, r1)) = G1(D1(I1, r1)).

Since h0 ∈ G1(D1(I1, r1)), we have

h1 = G−11 (h0) ∈ D1(I1, r1) ⊆ D0(I1, r1).

Inductively, we can prove

hi = G−1i · · · G

−11 (h0) ∈ D0(Ii, ri) for each i ≥ 1.


Take h = (hi)i≥0. It is clear that Gi+1hi+1 = hi for all i ≥ 0.

The proof of the existence of h is analogous and therefore we omit it.

Note that h0 is a limit of C1-dieomorphisms, which are %/4-close to I0 in the C1-topology.

Consequently, for each x ∈M0,

[exp−1x h0 expx − exp−1

x I0 expx]|B(0x,%)

is %/4-Lipschitz. Since % < 1, we can prove that h0 is injective. Furthermore, for each i ≥ 0 and

x ∈Mi, we have

d(h−1i (x), x) = d(h−1

i (x), hih−1i (x)) = d(y, hi(y)),

where y = h−1i (x). Hence d0(hi, Ii) = d0(h−1

i , Ii) for each i ≥ 0.

Analogously, we can prove that hi is invertible and d0(hi, Ii) = d0(h−1

i , Ii) for each i ≤ 0.

Lemma 2.4.4. The families (hi)i≥0, (h−1i )i≥0, (hi)i≤0 and (h−1

i )i≤0 are equicontinuous.

Proof. Let ε > 0. Since hi, h−1i ∈ D0(Ii, ri) and ri → 0 when i→ +∞, there exists k > 0 such that,

for each i > k,

maxd0(hi, Ii), d0(h−1

i , Ii) < ε/3.

Hence, if i < k and x, y ∈Mi with d(x, y) < ε/3, then

d(hi(x), hi(y)) ≤ d(hi(x), Ii(x)) + d(Ii(x), Ii(y)) + d(Ii(y), hi(y)) < ε

and

d(h−1i (x), h−1

i (y)) ≤ d(h−1i (x), Ii(x)) + d(Ii(x), Ii(y)) + d(Ii(y), h−1

i (y)) < ε.

On the other hand, it is clear that there exists δ ∈ (0, ε/3) such that, if 0 ≤ i ≤ k, and x, y ∈Mi

with d(x, y) < δ, then

maxd(hi(x), hi(y)),d(hi(x)−1, h−1i (y)) < ε.

The facts above prove that for each i ≥ 0, if x, y ∈Mi and d(x, y) < δ, then

maxd(hi(x), hi(y)),d(h−1i (x), h−1

i (y)) < ε.

Consequently, (hi)i≥0 and (h−1i )i≥0 are equicontinuous. Analogously we can prove that (hi)i≤0 and

(h−1i )i≤0 are equicontinuous.

Finally, we have:

Theorem 2.4.5. For all m ≥ 1,

H : (Fm(M), τstr)→ R ∪ +∞ and H(−1) : (Fm(M), τstr)→ R ∪ +∞

are locally constants.

Proof. Let f ∈ Fm(M). It follows from Lemmas 2.4.3 and 2.4.4 there exists a sequence of positive

numbers (ri)i∈Z such that every g ∈ B1(f , (ri)i∈Z) is positively and negatively uniformly conjugate


to f . Thus, from Theorem 2.2.5 we have

H(g) = H(f ) and H(−1)(g) = H(−1)(f ), for every g ∈ B1(f , (ri)i∈Z),

which proves the theorem.

Theorem 2.4.5 means that if f = (fi)i∈Z and g = (gi)i∈Z are two non-autonomous dynamical

systems such that d1(fi, gi) → 0 very quickly as i → +∞, then H(g) = H(f ). In particular,

if φ : M → M is a xed dieomorphism and d1(fi, φ) → 0 very quickly as i → +∞, then

H(f ) = H(φ). On the other hand, Kolyada and Snoha in [KS96], Theorem E, proved that for

any non-autonomous dynamical system f = (fi)i∈Z such that d0(fi, φ)→ 0 as i→ +∞, we always

have H(f ) ≤ H(φ). They gave an example where the inequality is strict.

Remark 2.4.6. Summarizing the results on the entropy for non-stationary dynamical systems

shown in this Chapter, we have:

1. If H(Fm(M)) has two or more elements, H : (Fm(M), τprod) → R ∪ +∞ is discontinuousat any f ∈ Fm(M) (Proposition 2.3.1);

2. H : (CFm(M), τunif ) → R ∪ +∞ is continuous if, and only if, H : (Dim(M), dm) →R ∪ +∞, is continuous (see Proposition 2.3.4).

3. H : (Fm(M), τstr)→ R ∪ +∞ is a continuous map (Theorem 2.4.5).

Chapter 3

Anosov Families

Anosov families, which will be presented in Denition 3.1.2, were introduced by P. Arnoux and

A. Fisher in [AF05], motivated by generalizing the notion of Anosov dieomorphism (see [KH97],

[Shu13], [Via14]). These families are non-stationary dynamical systems with a similar behavior to

Anosov difeomorphisms: the tangent space at each point in the total space has a splitting in two

subspaces, one stable and the other unstable. Readers may nd, for example, in [Bak95a], [Bak95b],

and more recently, in [Ste11], several approaches and results in non-stationary dynamical systems

in which each dieomorphism in the sequence has a hyperbolic behavior. Example 3.2.9 proves

that the Anosov families do not necessarily consist of Anosov dieomorphisms. Random dynamical

systems with hyperbolic behavior can be found in [Liu98].

In this chapter we will give the denition of Anosov families and we will also see some interesting

examples and some properties that satisfy such families.

3.1 Anosov Families: Denition

In this section we will introduce the notion of Anosov family. Before that we remember the

notion of hyperbolic sets and Anosov dieomorphism:

Denition 3.1.1. Let M be a Riemannian manifold with Riemannian metric 〈·, ·〉 and let ‖ · ‖ bethe norm induced by 〈·, ·〉 on M . Let φ : M →M be a C1-dieomorphism. A compact subset Λ of

M is hyperbolic for φ if:

i. The tangent bundle TΛ has a continuous splitting Es ⊕Eu which is Dφ-invariant, that is, for

each p ∈ Λ, TpΛ = Esp ⊕ Eup with Dpφ(Esp) = Esφ(p) and Dpφ(Eup ) = Euφ(p);

ii. there exist constants λ ∈ (0, 1) and c > 0 such that for each n ≥ 1, p ∈ Λ, we have:

‖Dpφ−n(v)‖ ≤ cλn‖v‖ for every vector v ∈ Eup ,

and

‖Dpφn(v)‖ ≤ cλn‖v‖ for every vector v ∈ Esp.

See Figure 3.1.1.

In the above denition, if Λ = M , then φ is called an Anosov dieomorphism.

27

28 ANOSOV FAMILIES

Euq

Esq

TqM

D(φ)qA

Eup

Esp

TpM

D(φ)p

B

Euz

Esz

TzMC

Figure 3.1.1: q = φ−1(p) and z = φ(p). D(φ)q(A) = B and D(φ)p(B) = C

Denition 3.1.2. From now on, if we do not say otherwise, Mi will be a Riemannian manifold

with xed Riemannian metric 〈·, ·〉i, for each i ∈ Z. We denote by ‖ · ‖i the induced norm by 〈·, ·〉ion TMi and we will take ‖ · ‖ dened on M as ‖ · ‖|Mi = ‖ · ‖i for i ∈ Z. An Anosov family is a

non-stationary dynamical system f = (fi)i∈Z ∈ F1(M) such that:

i. the tangent bundle TM has a continuous splitting Es ⊕ Eu which is Df -invariant, i. e., for

each p ∈M, TpM = Esp ⊕ Eup with

Df p(Esp) = Esf (p) and Df p(E

up ) = Euf (p),

where TpM is the tangent space at p;

ii. there exist constants λ ∈ (0, 1) and c > 0 such that for each i ∈ Z, n ≥ 1, and p ∈Mi, we have:

‖D(f −ni )p(v)‖ ≤ cλn‖v‖ for each vector v ∈ Eup ,

and

‖D(f ni )p(v)‖ ≤ cλn‖v‖ for each vector v ∈ Esp.

Without loss of generality, we can consider c ≥ 1, since otherwise it can be replaced by 1; if we

can take c = 1 we say the family is strictly Anosov.

Remark 3.1.3. The notion of Anosov family depends on the Riemannian metric considered on

each Mi (see Example 3.2.1). We will use the notation (M, 〈·, ·〉, f ) for point out we are xing the

Riemannian metric 〈·, ·〉 on M.

In Proposition 3.3.4 we will show that there exists a Riemannian metric 〈·, ·〉∗, dened on the

total space, equivalent to 〈·, ·〉 on each Mi, such that (M, 〈·, ·〉∗, f ) is an strictly Anosov family.1 In

the case of a dieomorphism on a compact Riemannian manifold this fact is known as Lemma of

Mather (see [Shu13]).

Notice that for an Anosov family, there are many invariant splittings of the tangent bundle

TMi, which are Df -invariant. Simply we can choose a splitting of TM0 and transport it forward

and backward to the other components by Df : for instance, if (M, 〈·, ·〉, f ) is an Anosov family,

1The Riemannian metric 〈·, ·〉∗ to be built in the Proposition 3.3.4 is not necessarily uniformly equivalent to 〈·, ·〉on M.

ANOSOV FAMILIES: DEFINITION 29

then for p ∈M and v ∈ Esp, the family F u = 〈Df np (v)n∈Z〉 ∪ Eu is Df -invariant and its vectors

are expanded by Df n when n → +∞. Analogously, if w ∈ Eup , Fs = 〈Df np (w)n∈Z〉 ∪ Eu is

Df -invariant and its vectors are expanded by Df −n when n→ +∞. However, these expansions donot satisfy the condition (ii) in Denition 3.1.2, that is, these splittings do not satisfy the condition

of hyperbolicity. Actually, in [AF05], Proposition 2.12, it is shown for an Anosov family that the

splitting TpM = Esp ⊕ Eup satisfying the condition of hyperbolicity is unique.

Lemma 3.1.4. Fix p ∈Mi.

i. Let v ∈ Eup . The condition

‖D(f−ni )p(v)‖ ≤ cλn‖v‖ for each i ∈ Z, n ≥ 1,

is equivalent to the condition

‖D(fni )p(v)‖ ≥ c−1λ−n‖v‖ for each i ∈ Z, n ≥ 1.

ii. Let v ∈ Esp. The condition

‖D(fni )p(v)‖ ≤ cλn‖v‖ for each i ∈ Z, n ≥ 1,

is equivalent to

‖D(f−ni )p(v)‖ ≥ c−1λ−n‖v‖ for each i ∈ Z, n ≥ 1.

Proof. See [AF05], Lemma 2.7.

Lemma 3.1.5. For each p ∈Mi we have

i. Esp = v ∈ TpMi : ‖D(fni )p(v)‖ is bounded, for n ≥ 1.

ii. Eup = v ∈ TpMi : ‖D(f−ni )p(v)‖ is bounded, for n ≥ 1.

Proof. Set

Bsp = v ∈ TpMi : ‖D(f ni )p(v)‖ is bounded, for n ≥ 1.

It is clear that Esp ⊆ Bsp. Suppose that there exists a vector v ∈ TpMi such that v /∈ Esp. Thus

v = avs + bvu, with bvu 6= 0. Therefore

‖D(f ni )p(v)‖ ≥ |b|‖D(f ni )p(vu)‖ − |a|‖D(f ni )p(vs)‖ ≥ |b|c−1λ−n‖vu‖ − |a|cλn‖vs‖,

where ‖D(f ni )p(v)‖ → +∞, that is, v /∈ Bsp. Thus B

sp ⊆ Esp.

Analogously we can prove ii.

Remark 3.1.6. One of the conditions in Denition 3.1.2 is the continuity of the splitting Es⊕Eu

of the tangent bundle TM. This means that the subspaces Esp and Eup depend continuously on p:

a family of subspaces Fp ⊆ TpM depends continuously on p ∈M if there are continuous maps

Fk : M→ TM

p 7→ vk(p), for k = 1, . . . , dimFp,

30 ANOSOV FAMILIES

such that v1(p), . . . , vdimFp(p) is a basis of Fp for each p ∈M.

We will see that the continuity of the splitting Es⊕Eu can be obtained from the condition (ii)

in the Denition 3.1.2 and using the Df -invariance of the splitting. First we will prove the following

lemma:

Lemma 3.1.7. The dimensions of the subspaces Eup and Esp are locally constant for p ∈M.

Proof. Let p ∈M and k = dimEsp. Suppose by contradiction that there exists a sequence (pm)m∈N ∈M converging to p such that dimEspm ≥ k + 1. Take a sequence of orthonormal vectors

v1(pm), ..., vk(pm), vk+1(pm) in Espm , for each m.

Choosing a suitable subsequence, we can suppose that

v1(pm)→ v1 ∈ TpM, ..., vk+1(pm)→ vk+1 ∈ TpM as m→∞.

If follows from condition (ii) in Denition 3.1.2 that, for all n ≥ 1,

‖Dp(fni )(vs)‖ ≤ cλn‖vs‖ for each s = 1, . . . , k + 1. (3.1.1)

By Lemma 3.1.5 we obtain v1, ..., vk+1 ∈ Esp. Since v1(pm), ..., vk(pm), and vk+1(pm) are orthonormal

for all n ≥ 1, by the continuity of the Riemannian metric we have that v1, ..., vk+1 are orthonormal,

which contradicts that dimEsp = k.

Similarly we can prove that there is not any sequence pm converging to p with dimEspm < k.

Therefore, the dimension of Esp is locally constant.

Analogously we obtain that the dimension of Eup is locally constant.

The following proposition is a version for non-stationary dynamical systems of Proposition 5.2.1

in [BS02].

Proposition 3.1.8. Let (M, 〈·, ·〉, f ) be a non-stationary dynamical system of class C1. Suppose

that TM has a splitting Es ⊕ Eu which is Df-invariant and satises the property (ii) from the

denition of Anosov family. Thus, the subspaces Esp and Eup depend continuously on p.

Proof. Let (pm)m∈N be a sequence in M such that pm → p ∈ M as m → ∞. Without loss of

generality, we can suppose that (pm)m∈N ⊆ Mi and p ∈ Mi for some i ∈ Z (see (1.1.1)) and

furthermore dimEspm = dimEsp = k for every m ≥ 1 (see Lemma 3.1.7). Let v1(pm), ..., vk(pm) bean orthonormal basis of Espm , for each m ≥ 1, such that v1(pm) → v1 ∈ TpMi, ..., vk(pm) → vk ∈TpMi as m→∞. Thus v1, ..., vk are orthonormal and belong to Esp (see (3.1.1)), which proves that

Esp depends continuously on p. Analogously we can prove that Eup depends continuously on p.

For each i ∈ Z, set

θi = minp∈Mi

θp : θp is the angle between Esp and E

up . (3.1.2)

Since for each p ∈ Mi, the subspaces Esp and Eup are transversal, that is, Esp ⊕ Eup = TpMi, then,

by the compactness of Mi and the continuity of the Riemannian metric and the subspaces Esp and

SOME EXAMPLES OF ANOSOV FAMILIES 31

Eup , we obtain that θi > 0 for each i ∈ Z. That is, the angles between the subspaces are uniformly

bounded away from zero on each component. Hence, there exists 0 < µi < 1 such that

cos(θi) ∈ [µi − 1, 1− µi] for each i ∈ Z. (3.1.3)

Notice that the sequence (µi)i∈Z could have a subsequence converging to 1, that is, there may be a

point p ∈ M0 and a subsequence (nk)k∈Z, such that the angles between the subspaces Esfnk (p) and

Eufnk (p) converge to 0, as k →∞ or k → −∞.

Denition 3.1.9. We say that (M, 〈·, ·〉, f ) satises the property of angles (s.p.a) if there exists

µ ∈ (0, 1) such that

cos(θi) ∈ [µ− 1, 1− µ] for each i ∈ Z. (3.1.4)

In Example 3.2.3 we will show that there exist Anosov families that do not satisfy the property

of angles.

3.2 Some Examples of Anosov Families

An easy example of an Anosov family is the constant family associated to an Anosov dieomor-

phism (see Example 1.1.5). In this section we will see some other examples of Anosov families. The

next example, which is due to Arnoux and Fisher [AF05], shows that suitably changing the metric

〈·, ·〉i on each Mi, the constant family associated to the identity could become an Anosov family,

hence it is important to keep xed the metrics on each Mi.

Example 3.2.1. For each i ∈ Z, take M = T2 = R2/Z2. Let M be the disjoint union of Mi =

M ×i and f = (fi)i∈Z be the constant family on M associated to the identity on T2. If we endow

each Mi with the Riemannian metric 〈·, ·〉 induced by the plane, then it is clear that (M, 〈·, ·〉, f )

is not an Anosov family, where 〈·, ·〉i = 〈·, ·〉 for all i. On the other hand, on each Mi, take the

Riemannian metric obtained by the inner product

〈(x1, y1), (x2, y2)〉∗i = 2−2ix1x2 + 22iy1y2 where (x1, y1), (x2, y2) ∈ R2 ≡ TpMi.

Hence, the norm induced on Mi by 〈·, ·〉i is given by

‖(x, y)‖∗i =√

(2−ix)2 + (2iy)2 for all (x, y) ∈ R2 ≡ TpMi.

Thus, for p ∈ Mi, if (x, y) ∈ TpMi, we have Dp(fi)(x, y) = (x, y) ∈ TpMi+1. Take Esp as being the

x-axis and Eup as being the y-axis for each p ∈Mi, i ∈ Z. Note that, for all p ∈Mi, (x, 0) ∈ Esp and(0, y) ∈ Eufi(p), we have

‖Dp(fi)(x, 0)‖∗i+1 = ‖(x, 0)‖∗i+1 =√

(2−i−1x)2 =1

2

√(2−ix)2 =

1

2‖(x, 0)‖∗i

and

‖Dfi(p)(f−1i )(0, y)‖∗i = ‖(0, y)‖∗i =

√(2iy)2 =

1

2

√(2i+1y)2 =

1

2‖(0, y)‖∗i+1.

Consequently, (M, 〈·, ·〉∗, f ) is an Anosov family, where 〈·, ·〉∗ is obtained by 〈·, ·〉∗i as in (1.1.2).

32 ANOSOV FAMILIES

Notice that 〈·, ·〉i and 〈·, ·〉∗i in the above example are uniformly equivalent on eachMi, however,

〈·, ·〉 and 〈·, ·〉∗ are not uniformly equivalent on M.

Another interesting example is the following one:

Example 3.2.2. Let M be a Riemannian manifold with metric ‖ · ‖ and φ : M →M is an Anosov

dieomorphism with constant c ≥ 1 and λ ∈ (0, 1). Take Mi = M for all i with Riemannian norm

dened as

‖(vs, vu)‖i =

√a2i‖vs‖2 + b2i‖vu‖2 if i ≥ 0

‖(vs, vu)‖ if i < 0,(3.2.1)

where a, b ∈ (λ, 1/λ). Consider M as the disjoint union of each Mi endowed with the metric

obtained above. Let f be the constant family associated to φ. We show that f is an Anosov family

with constants c = c and λ = maxλ, aλ, λ/b < 1. Indeed, let TM = Esφ ⊕ Euφ be the splitting of

the tangent bundle TM corresponding to φ. Let v ∈ Esφ and n ≥ 1. If i ≤ 0 and i+ n ≤ 0, then

‖Df n(v)‖i+n = ‖Df n(v)‖i ≤ cλn‖v‖i ≤ cλn‖v‖i.

If i ≤ 0 and i+ n > 0, then

‖Df n(v)‖i+n = an−i‖Df n(v)‖ ≤ an−icλn‖v‖ = c(aλ)n−iλi‖v‖i ≤ cλn‖v‖i.

If i > 0, then for all n ≥ 1 we have

‖Df n(v)‖i+n = an+i‖Df n(v)‖ ≤ an+icλn‖v‖ = c(aλ)nai‖v‖ ≤ cλn‖v‖i.

Now, suppose that v ∈ Euφ and n ≥ 1. If i > 0 and i− n ≥ 0, then

‖Df −n(v)‖i−n = bi−n‖Df −n(v)‖ ≤ bic(λ/b)n‖v‖ ≤ cλn‖v‖i.

If i > 0 and i− n < 0, then

‖Df n(v)‖i−n = ‖Df n(v)‖ ≤ c(λ/b)iλn−ibi‖v‖ ≤ cλn‖v‖i.

If i ≤ 0, then

‖Df −n(v)‖i−n = ‖Df −n(v)‖i ≤ cλn‖v‖i ≤ cλn‖v‖i.

Consequently, the splitting Esφ⊕Euφ induces a splitting of TM with which f satises Denition

3.1.2.

The following example, where we only suitably changed the Riemannian metric on the 2-torus,

shows that there exist Anosov families that do not satisfy the property of angles.

Example 3.2.3. Take M = T2 and let φ : M →M be the Anosov dieomorphism induced by the

matrix

A =

(2 1

1 1

).

The eigenvalues of A are λ = (3+√

5)/2 > 1 and 1/λ. Consider the eigenvectors vs = ((1+√

5)/2, 1)

and vu = ((1−√

5)/2, 1) of A associated to λ and 1/λ, respectively (see Figure 3.2.1). Let (ζi)i∈Z


A1

A4

A3

A3

A2 A2

A1

Es

Eu

(3, 0)

(3, 2)(0, 2)

Figure 3.2.1: The square [0, 1] × [0, 1] is mapped by A to the parallelogram with vertices (0, 0),(2, 1), (3, 2), (1, 1)

be a sequence in [0, 1). In the basis vs, vu of R2, set

Bi =

(1 ζi

ζi 1

)i ∈ Z.

The eigenvalues of Bi are αi = 1 + ζi and βi = 1 − ζi. Since ζi ∈ [0, 1), the matrix Bi is positive

denite. Thus, it induces an inner product 〈·, ·〉i on R2: if v1 = avs + bvu, v1 = cvs + dvu ∈ R2,

〈v1, v2〉i =(a b

)( 1 ζi

ζi 1

)(c

d

)i ∈ Z.

Notice that the angle between vs and vu with the inner product 〈·, ·〉i is:

θi = arccos

(〈v1, v2〉i√

〈v1, v1〉i · 〈v2, v2〉i

)= ζi.

Furthermore, if ‖ · ‖i is the norm induced by 〈·, ·〉i and ‖ · ‖ is the canonical norm of R2, we have

‖vs‖i = ‖vs‖ and ‖vu‖i = ‖vu‖ for all i ∈ Z (the inner product 〈·, ·〉i change only the angles betweenvs and vu). Consequently, (M, 〈·, ·〉, f ) is an Anosov family, whereM is the disjoint union of theMi

and 〈·, ·〉 is obtained by 〈·, ·〉i as in (1.1.2) and fi(x, i) = (φ(x), i+ 1) for x ∈M , i ∈ Z. If ζi → 0 as

i→∞, then (M, 〈·, ·〉, f ) is an Anosov family that does not satisfy the property of angles.

Before presenting the next example, we consider the following denition:

Denition 3.2.4 (Linear cocycles). Let X be a compact metric space, φ : X → X a homeomor-

phism and A : X → SL(Z,m) a continuous application, where SL(Z,m) is the special linear group

of m ×m matrices with integer entries and with determinant 1. The linear cocycle dened by A

34 ANOSOV FAMILIES

over φ is the transformation

F : X × Rm → X × Rm, (x, v)→ (f(x), A(x)v).

The cocycle F is hyperbolic if there exist λ ∈ (0, 1) and c > 0 such that, for all x ∈ M, there exist

subspaces Esx and Eux of Rm such that Rm = Esx ⊕ Eux , and furthermore,

i. A(x)Esx = Esφ(x) and A(x)Eux = Euφ(x),

ii. ‖An(x)vs‖ ≤ cλn‖vs‖ for vs ∈ Esx and ‖A−n(x)vu‖ ≤ cλn‖vu‖ for vu ∈ Eux ,

for all n ≥ 1, where

An(x) = A(fn−1(x)) · · ·A(x) and A−n(x) = A(f−n(x))−1 · · ·A(f−1(x))−1.

Example 3.2.5. Let F be a hyperbolic linear cocycle dened by A : X → SL(Z,m) over φ : X →X, then for each x ∈ X, the family (A(fn(x)))n∈Z dened on Mi = Rm/Zm, the m-dimensional

torus endowed with the Riemannian metric inherited from Rm, induces an Anosov family.

When m = 2 we have that the linear cocycle F dened by A over φ is hyperbolic if, and only

if, there exist constants σ > 1 and c > 0 such that ‖An(x)‖ ≥ cσn for all x ∈ X and n ≥ 1 (see

[Via14], Proposition 2.1). Let SL(N, 2) be the set of 2 × 2 matrices with entries in N = 1, 2, . . . and with determinant 1. It follows from Proposition 2.7 in [Via14] that if the image of A is in

SL(N, 2), then F is hyperbolic. Notice that, in this case, the image of A is nite, because M is

compact. Consequently:

Proposition 3.2.6. Let Y = F1, F2, . . . , Fk be a subset of SL(N, 2). Any non-stationary dynam-

ical system (Ai)i∈Z with values in Y is Anosov.

Remark 3.2.7. Let φ : M →M be an Anosov dieomorphism of class C2 on a compact Rieman-

nian manifold M and β > 0 such that Lip(Dφ) < β, where Lip(Dφ) is a Lipchitz constant of the

derivative application x 7→ Dφx. For α > 0, take

Ωα,β(φ) = ψ ∈ C1(M) : d(φ, ψ) ≤ α and Lip(Dψ) ≤ β,

where d(·, ·) is the C1-metric on Di1(M). If α is small enough, any sequence (ψi)i∈Z in Ωα,β(φ)

denes an Anosov family in M =∐i∈ZM (see [You86], Proposition 2.2). Consequently, the set

consisting of the constant families associated to Anosov dieomorphisms of class C2 is open in

F2(M).

Using the above fact we have:

Example 3.2.8. Given α ∈ R, consider φα : T2 → T2 dened by

φα(x, y) = (2x+ y − (1 + α) sinx mod 2π, x+ y − (1 + α) sinx mod 2π).

Thus, for all α ∈ [−1, 0), φα is an Anosov dieomorphism (see [BP07]). Notice that φ−1 is the

well-known linear toral automorphism induced by the matrix

A0 =

(2 1

1 1

).


Thus we have that given α? ∈ [−1, 0) there exists ε > 0 such that, if (αi)i∈Z is a sequence in [−1, 0)

with |αi − α?| < ε, then (fi)i∈Z is an Anosov family, where fi = φαi for each i ∈ Z.

The next results provide many examples of Anosov families which do not necessarily consist

of (perturbations of) Anosov dieomorphisms. The following example, which was taken of [AF05],

proves that the Anosov families do not necessarily consist of Anosov dieomorphisms.

Example 3.2.9. For any sequence of positive integers (ni)i∈Z set

Ai =

(1 0

ni 1

)for i even and Ai =

(1 ni

0 1

)for i odd.

The family (Ai)i∈Z is called the multiplicative family determined by the sequence (ni)i∈Z. Since the

entries of the matrices Ai are integers and detAi = 1 for all i, they induce dieomorphisms fi on

the 2-torus T2 = R2/Z2, that is, each fi is dened as πAi = fiπ, where π : R2 → R2/Z2 is the

canonical projection. Take Mi as T2 and let ‖ · ‖ be the Riemannian metric on T2 inherited from

R2. For each i ∈ Z, let si = (ai, bi), ui = (ci, di) and λi ∈ (0, 1) be such that aidi + cibi = 1,

for i even, ai = [nini+1...], bi = 1,dici

= [ni−1ni−2...], and λi = ai,

and

for i odd, bi = [nini+1...], ai = 1,cidi

= [ni−1ni−2...] and λi = bi.

Here, [nini+1...] = 1ni+

1ni+1+···

. Thus, for all i ∈ Z, Aisi = λisi+1 and Aiui = λ−1i ui+1 (see [AF05]).

Therefore,

‖Ani si‖ = λi+n−1...λi‖si+n‖ ≤ cλi+n−1...λi‖si‖

and

‖Ani ui‖ = λ−1i+n−1...λ

−1i ‖ui+n‖ ≥ c

−1λ−1i+n−1...λ

−1i ‖ui‖,

where

c = max

supi,j

‖si‖‖sj‖

, supi,j

‖ui‖‖uj‖

(c <∞ because ‖v‖ ∈ (1/2,√

2) for all v ∈ si : i ∈ Z ∪ ui : i ∈ Z).

Note that, if there exists λ ∈ (0, 1) such that λi < λ for all i, we have

‖Ani si‖ ≤ cλn‖si‖ and ‖Ani ui‖ ≥ c−1λ−n‖ui‖ for all n ≥ 1.

This shows that, if there is a λ ∈ (0, 1) such that λi ≤ λ for all i, then fi : Mi → Mi+1 for i ∈ Z,dene an Anosov family, with constants λ and c as dened above, the stable subspaces are spanned

by si and the unstable subspaces are spanned by ui. However, we will prove that any multiplicative

family is Anosov.

For the rest of this section, we will x a multiplicative family (Ai)i∈Z determined by a sequence

(ni)i∈Z. Furthermore, we consider the values c and λi dened in Example 3.2.9.

Proposition 3.2.10. (Ai)i∈Z is an Anosov family with constant λ =√

2/3 and 2c.

36 ANOSOV FAMILIES

Proof. Notice that, if λj ∈ (2/3, 1) for some j ∈ Z, then λj−1 ∈ (0, 2/3) and λj+1 ∈ (0, 1/2). Indeed,

if λj = 1nj+

1nj+1+···

∈ (2/3, 1) we must have nj = 1 and nj+1 ≥ 2. Hence,

λj−1 =1

nj−1 + 11+···

<1

1 + (1/2)and λj+1 =

1

nj+1 + 1nj+2+···

< 1/2.

Next, by induction on n, we prove that cλi+n−1 . . . λi < 2cλn, for each i ∈ Z and n ≥ 1.

Fix i ∈ Z. It is clear that if n = 1, 2, then cλi+n−1 . . . λi < 2cλn. Let n ≥ 2 and assume that

cλi+m−1 . . . λi < 2cλm for each m ∈ 1, . . . , n. Clearly, if λi+n+1 ≤ 2/3, then cλi+n+1λi+n . . . λi <

2cλn+1. On the other hand, if λi+n+1 > 2/3, then λi+n−1 < 1/2 and by induction assumption we

have

cλi+nλi+n−1 . . . λi < 2cλn−2λi+nλi+n−1 < 2cλn−2 1

2< 2cλn−2λ2.

It follows from the above facts that

‖Ani si‖ =≤ 2cλn‖si‖ and ‖Ani ui‖ ≥ (2c)−1λ−n‖ui‖,

for each i ∈ Z and n ≥ 1, which proves the proposition.

It is clear that:

Proposition 3.2.11. Any gathering of an Anosov family is also an Anosov family (see Denition

1.1.6).

Example 3.2.12. It follows from Proposition 3.2.11 that if φ : M → M is an Anosov dieomor-

phism, then for each sequence of positive integers (ni)i∈Z, if fi = φni , then (fi)i∈Z is an Anosov

family. Moreover, any gathering of a multiplicative family is an Anosov family

If Fi ∈ SL(N, 2), then

Fi =

(1 0

ni,ki 1

)(1 ni,ki−1

0 1

)· · ·

(1 0

ni,2 1

)(1 ni,1

0 1

), (3.2.2)

for some non-negative integers ni,1, . . . , ni,ki , that is, SL(N, 2) is a semigroup generated by

M =

(1 0

1 1

)and N =

(1 1

0 1

)

(see [AF05], Lemma 3.11).

Corollary 3.2.13. Consider a sequence (Fi)i∈Z in SL(N, 2) and the factorization of each Fi as in

(3.2.2). If ni,ki and ni,1 are non-zero for each i ∈ Z, then (Fi)i∈Z is an Anosov family.

Proof. Note that (Fi)i∈Z is a gathering of an multiplicative family. It follows from Proposition 3.2.10

that (Fi)i∈Z is an Anosov family.

Suppose that Mi = M × i for each i, where M is a compact Riemannian manifold. Since we

are considering the total space as the disjoint union of the Mi's, the splitting of the tangent spaces

at the points (p, i) and (p, j) can be dierent for i 6= j, as we will see in the next remark.

LEMMA OF MATHER FOR ANOSOV FAMILIES 37

Remark 3.2.14. Set ni = 1 for each integer i 6= 0 and n0 = 2. Consider

Ai =

(1 0

ni 1

)for i even and Ai =

(1 ni

0 1

)for i odd.

From Example 3.2.9 we have

a0 =1

2 + φ, b0 = 1, c0 = −2 + φ

2, d0 = −φ2 + φ

2,

a1 = 1, b1 =1

φ, c1 =

φ

1 + φ+ φ2, d1 =

φ(1 + φ)

1 + φ+ φ2,

where φ = [111 . . . ]. Thus,

s0 = (1

2 + φ,−1), u0 = (−2 + φ

2,−φ2 + φ

2), s1 = (1,− 1

φ), u1 = (

φ

1 + φ+ φ2,φ(1 + φ)

1 + φ+ φ2).

This fact shows that we can have two dierent splitting of the tangent bundle in each component,

one is obtained by s0 and u0 and other one is obtained by s1 and u1.

Note that in the examples obtained from the results shown above, the families consist of factors

of hyperbolic matrices. We have(1 0

n 1

)(1 m

0 1

)=

(1 m

n nm+ 1

),

which is a hyperbolic matrix for n,m ∈ N. The results to be obtained in Chapter 4 will provide

more general examples of Anosov families, which, I doubt it, it be possible that none of the maps

fi is a factor of a hyperbolic matrix (see Problem 7.3.3): in Theorem 4.4.2 we will prove that if an

Anosov family (Fi)i∈Z consists of matrices, then there exists a ε > 0 such that if a family (gi)i∈Z

is ε-close to (Fi)i∈Z, then the family is Anosov. Consequently, we can choose dieomorphisms in

those neighborhood so that none of them are factors of a hyperbolic matrix.

3.3 Lemma of Mather for Anosov Families

A Riemannian metric is adapted to an hyperbolic set of a dieomorphism if, in this metric, the

expansion (contraction) of the unstable (stable) subspaces is seen after only one iteration. By Lemma

of Mather (see [Shu13], Proposition 4.2 or [BS02], Proposition 5.2.2) each Anosov dieomorphism

on a compact manifold M admits a Riemannian metric adapted to M . By compactness of M , this

metric is uniformly equivalent to the Riemannian metric rstly considered. In Proposition 3.3.4,

whose proof is based on the proof of Proposition 5.2.2 in [BS02], we will obtain an analogous version

of the Lemma of Mather for Anosov families. The Riemannian metric to be obtained in Proposition

3.3.4 is not necessarily uniformly equivalent to the rst metric on M.

In this section, (M, 〈·, ·〉, f ) will represent an Anosov family with constants λ ∈ (0, 1) and c ≥ 1.

Sometimes we will omit the index i of fi if it is clear that we are considering the i-th dieomorphism

of f .

The notion of Anosov dieomorphism on a compact Riemannian manifold does not depend

38 ANOSOV FAMILIES

on the Riemannian metric (see [KH97]). In contrast, the notion of Anosov family depends on the

Riemannian metric taken on each Mi (see Example 3.2.1). However, the next proposition proves

that the notion of Anosov family does not depend on the Riemannian metrics uniformly equivalent

on the total space.

Proposition 3.3.1. Suppose that 〈·, ·〉 and 〈·, ·〉? are Riemannian metrics uniformly equivalent on

M. Thus, (M, 〈·, ·〉, f ) is an Anosov family if, and only if, (M, 〈·, ·〉?, f ) is an Anosov family.

Proof. Let ‖ · ‖ and ‖ · ‖? be the norms induced by 〈·, ·〉 and 〈·, ·〉?, respectively. Let k and K be

such that

k‖v‖? ≤ ‖v‖ ≤ K‖v‖? for all v ∈ TMi, i ∈ Z.

Suppose that (M, 〈·, ·〉, f ) is an Anosov family with constant λ ∈ (0, 1) and c ≥ 1. Thus, for

v ∈ TpM, n ≥ 1,

‖Dp(fni )(v)‖? ≤ (1/k)‖Dp(f

ni )(v)‖ ≤ (c/k)λn‖v‖ ≤ (Kc/k)λn‖v‖?.

On the other hand,

‖Dp(f−ni )(v)‖? ≤ (Kc/k)λ−n‖v‖?, for v ∈ TpM, n ≥ 1.

Therefore, (M, 〈·, ·〉?, f ) is an Anosov family with constant λ e c = Kc/k.

Similarly we can prove that if (M, 〈·, ·〉?, f ) is an Anosov family then (M, 〈·, ·〉, f ) is an Anosov

family.

Denition 3.3.2. Set

U(M, 〈·, ·〉) = 〈·, ·〉? : 〈·, ·〉? is a Riemannian metric uniformly equivalent to 〈·, ·〉 on M.

Proposition 3.3.1 means that if 〈·, ·〉? ∈ U(M, 〈·, ·〉), then (M, 〈·, ·〉, f ) is an Anosov family if,

and only if, (M, 〈·, ·〉?, f ) is an Anosov family. Therefore, we can redene the notion of Anosov

family on M as follows:

Denition 3.3.3. f is an Anosov family on (M, 〈·, ·〉) if there exist 〈·, ·〉? ∈ U(M, 〈·, ·〉) such that

(M, 〈·, ·〉∗, f ) satises the conditions in Denition 3.1.2.

Next, we show the Lemma of Mather for Anosov families.

Proposition 3.3.4. Given ζ > 0, there exists a C∞ Riemannian metric 〈·, ·〉? and uniformly

equivalent to 〈·, ·〉 on each Mi, such that (M, 〈·, ·〉?, f ) is a strictly Anosov family with λ′ = λ + ζ.

Furthermore, for each p ∈ M, we have 〈vs, vu〉? < ε for every unit vectors vs ∈ Esp and vu ∈ Eup .Consequently, (M, 〈·, ·〉?, f ) satises the property of angles.

Proof. For each p ∈M, if (vs, vu) ∈ Esp ⊕ Eup , take

‖(vs, vu)‖∗ =

√‖vs‖∗2 + ‖vu‖∗2, (3.3.1)

where

‖vs‖∗ =

∞∑n=0

(λ+ ζ)−n‖D(f n)pvs‖ and ‖vu‖∗ =∞∑n=0

(λ+ ζ)−n‖D(f −n)pvu‖.

LEMMA OF MATHER FOR ANOSOV FAMILIES 39

Notice that if vs ∈ Esp we have

‖vs‖∗ =∞∑n=0

(λ+ ζ)−n‖D(f n)pvs‖ ≤∞∑n=0

(λ+ ζ)−ncλn‖vs‖ =λ+ ζ

ζc‖vs‖. (3.3.2)

Analogously, ‖vu‖∗ ≤ λ+ζζ c‖vu‖ for vu ∈ Eup . Consequently the series ‖vs‖∗ and ‖vu‖∗ converge

uniformly.

Let us see that the norm ‖ · ‖∗ is uniformly equivalent to the norm ‖ · ‖ on each Mi. It is clear

that ‖vs‖ ≤ ‖vs‖∗ and ‖vu‖ ≤ ‖vu‖∗. Thus,

‖(vs, vu)‖ ≤ ‖vs‖+ ‖vu‖ ≤ 2(‖vs‖2 + ‖vu‖2)1/2 ≤ 2(‖vs‖2∗ + ‖vu‖2∗)1/2 = 2‖(vs, vu)‖∗.

This fact implies

‖v‖ ≤ 2‖v‖∗ for all v ∈ TM. (3.3.3)

Let θp be the angle between two vectors vs ∈ Esp and vu ∈ Eup , for p ∈Mi. Take µi as in (3.1.3).

Since (1− µi)(‖vs‖2 + ‖vu‖2) ≥ 2(1− µi)‖vs‖‖vu‖, we have

‖vs‖2 + ‖vu‖2 + 2(µi − 1)‖vs‖‖vu‖ ≥ µi(‖vs‖2 + ‖vu‖2).

Therefore

‖(vs, vu)‖2 = ‖vs‖2 + ‖vu‖2 − 2 cos θp‖vs‖‖vu‖ ≥ ‖vs‖2 + ‖vu‖2 + 2(µi − 1)‖vs‖‖vu‖

≥ µi(‖vs‖2 + ‖vu‖2).

Consequently,

‖(vs, vu)‖2∗ = ‖vs‖2∗ + ‖vu‖2∗ ≤ (λ+ ζ

ζc)2(‖vs‖2 + ‖vu‖2) ≤ 1

µi(λ+ ζ

ζc)2‖(vs, vu)‖2.

Thus,

‖v‖∗ ≤1

µi(λ+ ζ

ζc)2‖v‖ for all v ∈ TMi. (3.3.4)

From (3.3.3) and (3.3.4) we have that the metrics ‖ · ‖ and ‖ · ‖∗ are uniformly equivalent on each

Mi.

Notice that, if vs ∈ Esp,

‖Df pvs‖∗ =

∞∑n=0

(λ+ ζ)−n‖Df n+1p vs‖ = (λ+ ζ)

∞∑n=1

(λ+ ζ)−n‖Df npvs‖

= (λ+ ζ)∞∑n=1

(λ+ ζ)−n‖Df npvs‖+ (λ+ ζ)‖vs‖ − (λ+ ζ)‖vs‖

= (λ+ ζ)

( ∞∑n=0

(λ+ ζ)−n‖Df np vs‖ − ‖vs‖

)= (λ+ ζ)(‖vs‖∗ − ‖vs‖) ≤ (λ+ ζ)‖vs‖∗.

40 ANOSOV FAMILIES

Similarly, we can prove that, if vu ∈ Eup , then

‖D(f −1)pvu‖∗ ≤ (λ+ ζ)‖vu‖∗.

Notice that the norm ‖ · ‖∗ comes from an inner product 〈·, ·〉∗, which denes a continuous

Riemannian metric on M. Consequently, for each i, we can choose a C∞-Riemannian metric 〈·, ·〉?,isuch that |〈v, v〉?,i − 〈v, v〉∗| < ε for each v ∈ TMi. We take 〈·, ·〉? on M, dened on each Mi as

〈·, ·〉?|Mi = 〈·, ·〉?,i. It is clear that (M, 〈·, ·〉?, f ) satises the property of angles.

By (3.3.3) and (3.3.4) we have that, for each i ∈ Z,(1

µi(λ+ ε

εc)2

)−1

‖v‖∗ ≤ ‖v‖ ≤ 2‖v‖∗ for all v ∈ TMi, (3.3.5)

where µi depends on the angles between the stable and unstable subspaces of the splitting of TMi.

Hence, we have:

Corollary 3.3.5. Suppose that (M, 〈·, ·〉, f ) satises the property of angles. Then, there exists a

C∞-Riemannian metric 〈·, ·〉? ∈ U(M, 〈·, ·〉), such that (M, 〈·, ·〉?, f ) is a strictly Anosov family

which satises the property of angles.

Proof. Let µ be as in (3.1.4). From (3.3.3) and (3.3.4) we have(1

µ(λ+ ε

εc)2

)−1

‖v‖∗ ≤ ‖v‖ ≤ 2‖v‖∗ for all v ∈ TM,

where ‖ ·‖∗ is a norm dened in (3.3.1). Thus, ‖ ·‖ and ‖ ·‖∗ are uniformly equivalent. The corollary

follows from the proof of the Proposition 3.3.4.

The metric obtained in Proposition 3.3.4 is not necessarily uniformly equivalent to the original

metric on M. The uniform equivalence depends on the angles between the stable and unstable

subspaces of the splitting of the tangent bundle on each component. In the case of Anosov dieo-

morphisms dened on compact manifolds those angles are uniformly bounded away from 0. In the

case of Anosov families, those angles may decrease arbitrarily, however, they can never be zero, by

compactness of each component.

3.4 Invariant Cones for Anosov Families

In this section we will prove that f satises the property of invariant cones (see [BP07], [KH97]).

This fact will be useful to prove the openness of the set consisting of Anosov families. From now on,

we will x ζ ∈ (0, 1− λ) and consider the Riemannian metric 〈·, ·〉? on M obtained in Proposition

3.3.4. Hence (M, 〈·, ·〉?, f ) is a strictly Anosov family with constant λ = λ+ ζ and we can suppose

the stable and unstable subspaces are orthogonal (see (3.3.1)). Furthermore, ‖ ·‖? will represent thenorm induced by 〈·, ·〉?.

Denition 3.4.1. For each p ∈M and α > 0, set

Ksα,f ,p = (v, w) ∈ Esp ⊕ Eup : ‖w‖? < α‖v‖? ∪ (0, 0),

Kuα,f ,p = (v, w) ∈ Esp ⊕ Eup : ‖v‖? < α‖w‖? ∪ (0, 0).

INVARIANT CONES FOR ANOSOV FAMILIES 41

Ksα,f ,p is called the stable α-cone of f at p and Ku

α,f ,p the unstable α-cone of f at p (see Figure

3.4.1).

Eup

Esp

TpM

Kuα,f ,p

Ksα,f ,p

Figure 3.4.1: Stable and unstable α-cones at p.

Notice that (v, w) ∈ Kuα,f ,p if and only if ‖v‖? ≤ α‖w‖? and (v, w) ∈ Ks

α,f ,p if and only if

‖w‖? ≤ α‖v‖?.

Lemma 3.4.2. Fix p ∈M and α ∈ (0, 1). We have:

i. Let v ∈ TpM. Then, D(fn)p(v) ∈ Kuα,f,fn(p) for all n ≤ 0 if and only if v ∈ Eup .

ii. Let v ∈ TpM. Then, D(fn)p(v) ∈ Ksα,f,fn(p) for all n ≥ 0 if and only if v ∈ Esp.

Proof. We prove i. It is clear that if v ∈ Eup then D(f n)p(v) ⊆ Kuα,f ,f n(p) for all n ≤ 0. Suppose

that v /∈ Eup . Thus v = vs + vu where vs ∈ Esp \ 0 and vu ∈ Eup . By Lemma 3.1.4 we have

‖D(f n)p(vs)‖ ≥ c−1λn‖vs‖ → +∞ and ‖D(f n)p(vu)‖ ≤ cλn‖vu‖ → 0 as n→ −∞.

Consequently, we can not have that D(f n)p(v) ∈ Kuα,f ,f n(p) for all n ≤ 0.

Lemma 3.4.3. Let α ∈ (0, 1−λ1+λ) and take λ′ = λ1+α

1−α < 1. Thus:

i. Dfp(Kuα,f,p) ⊆ Ku

α,f,f(p). Furthermore, if (v, w) ∈ Kuα,f,p, then

‖Dfp(v, w)‖? ≥ (λ′)−1‖(v, w)‖?.

ii. Df−1f(p)(K

sα,f,f(p)) ⊆ K

sα,f,p. Furthermore, if (v, w) ∈ Ks

α,f,f(p), then

‖Df−1f(p)(v, w)‖? ≥ (λ′)−1‖(v, w)‖?.

See Figure 3.4.2.

Proof. Fix (v, w) ∈ Kuα,f ,p. Thus

‖Df p(v)‖? ≤ λ‖v‖? ≤ λα‖w‖? ≤ λ2α‖Df p(w)‖? ≤ α‖Df p(w)‖?.

Therefore Df p(Kuα,f ,p) ⊆ Ku

α,f ,f (p).

42 ANOSOV FAMILIES

Eup

Esp

TpM

D(f )−1q

Euq

Esq

TqMD(f )p

Figure 3.4.2: Stable and unstable invariant α-cones. q = f(p)

On the other hand,

‖Df p(v, w)‖? ≥ ‖Df p(w)‖? − ‖Df p(v)‖? ≥ (1− α)‖Df p(w)‖? ≥1− α

λ(1 + α)‖(v, w)‖?,

and this fact proves i.

The statement (ii) can be proved analogously.

An equivalent way to prove the above lemma is by using coordinate charts for open sets of Mi,

as we will see below. In this case we will use the exponential charts (see Denition 2.4.1). For each

p ∈M, let %p > 0 be the injectivity radius of expp at p. Take εp = %p/2. Let δp > 0 be small enough

such that

fp = exp−1f (p) f expp : B(0p, δp)→ B(0f (p), εf (p))

is well dened. It is clear that δp depends on both %p and f . Notice that D(f )p = D(fp)0p .

Remark 3.4.4. For each n ∈ Z, consider Sn = maxp∈Mn ‖D(fn)p‖. Notice that if, for each p,

δp ≤ minεp, εf (p)/maxSn, 1,

then, for all x ∈ B(0p, δp), we have fp(x) ∈ B(0f (p), εf (p)). Consequently, if Mn = M × n, whereM is a compact Riemannian manifold, 〈·, ·〉n = 〈·, ·〉, where 〈·, ·〉 is the Riemannian metric on M ,

and (Sn)n∈Z is bounded, then we can nd a uniform δ with which fp is well-dened for each p ∈M,

that is, there exists δ > 0 such that, considering δp = δ for each p ∈M, fp is well-dened.

Denition 3.4.5. For z ∈ TpM, we denote by zs and zu the orthogonal projections of z on Esp

and Eup , respectively, and hence z = (zs, zu). If (zs, zu) ∈ Bs(0p, δp) × Bu(0p, δp), where Bs(0p, δp)

is the ball in Esp and Bu(0p, δp) is the ball in E

up , then

fp(zs, zu) = (ap(zs, zu) +Ap(zs), bp(zs, zu) +Bp(zu)), (3.4.1)


where

ap : Bs(0p, δp)×Bu(0p, δp)→ Es, ap(zs, zu) = (fp(zs, zu))s −D(fp)0p(zs),

bp : Bs(0p, δp)×Bu(0p, δp)→ Eu, bp(zs, zu) = (fp(zs, zu))u −D(fp)0p(zu),

Ap : Bs(0p, δp)→ Es, Ap(zs) = D(fp)0p(zs), and

Bp : Bu(0p, δp)→ Eu, Bp(zu) = D(fp)0p(zu).

Notice that ap(0p) = bp(0p) = D(ap)0p = D(bp)0p = 0. Set

µp = supv∈Esp

‖Apv‖?‖v‖?

and κp = supv∈Eu

f (p)

‖B−1p v‖?‖v‖?

. (3.4.2)

It is clear that maxµp, κp ≤ λ.

Denition 3.4.6. For each p consider εp > 0 small enough such that

(fp)−1 = exp−1

p f −1 expf (p) : Bs(0f (p), εf (p))×Bu(0f (p), εf (p))→ Bs(0p, εp)

is well-dened. For (zs, zu) ∈ Bs(0f (p), εf (p))×Bu(0f (p), εf (p)), set

(fp)−1(zs, zu) = (cp(zs, zu) + Cp(zs), dp(zs, zu) +Dp(zu)),

where

cp : B(0f (p), εf (p))→ Es, cp(zs, zu) = (fp)−1s (zs, zu)− Cp(zs);

dp : B(0f (p), εf (p))→ Eu, dp(zs, zu) = (fp)−1u (zs, zu)−Dp(zu);

Cp : Bs(0f (p), εf (p))→ Es, Cp(zs) = D(fp)−10 (zs);

Dp : Bu(0f (p), εf (p))→ Eu, Dp(zu) = D(fp)−10 (zu).

Notice that

supv∈Esp

‖C−1p v‖?‖v‖?

= supv∈Esp

‖Apv‖?‖v‖?

= µp and supv∈Eu

f (p)

‖Dpv‖?‖v‖?

= supv∈Eu

f (p)

‖B−1p v‖?‖v‖?

= κp.

Denition 3.4.7. Set

σp(δp) = supz∈Bs(0p,δp)×Bu(0p,δp)

‖D(ap, bp)z‖?

and

ρp(εf (p)) = supz∈Bs(0f (p),εf (p))×Bu(0f (p),εf (p))

‖D(cp, dp)z‖?.

Remark 3.4.8. Notice that

(ap(z), bp(z)) = fp(z)−D(fp)0(z) for z ∈ Bs(0p, δp)×Bu(0p, δp).

44 ANOSOV FAMILIES

Hence, for each z ∈ Bs(0p, δp)×Bu(0p, δp) we have

‖D(fp)z −D(fp)0‖? = ‖D[fp −D(fp)0]z‖? = ‖D(ap, bp)z‖? ≤ σp(δp).

For q ∈ B(p, δp), let z = exp−1p (q), Esz = D(exp−1

p )q(Esq), E

uz = D(exp−1

p )q(Euq ),

Ksα,f ,z = (vs, vu) ∈ Esz ⊕ Euz : ‖vu‖? < α‖vs‖? ∪ (0, 0),

and Kuα,f ,z = (vs, vu) ∈ Esz ⊕ Euz : ‖vs‖? < α‖vu‖? ∪ (0, 0).

We have

D(fn)z(Esz) = D(exp−1

f n+1(p)fn expf n(p))zD(exp−1

f n(p))q(Esq)

= D(exp−1f n+1(p)

fn expf n(p) exp−1f n(p))q(E

sq)

= D(exp−1f n+1(p)

fn)q(Esq) = D(exp−1

f n+1(p))fn(q) D(fn)q(E

sq)

= D(exp−1f n+1(p)

)fn(q)(Esfn(q)) = Es

fn(z).

Analogously, we can prove that

D(fn)z(Euz ) = Eu

fn(z).

For the next two lemmas we will suppose that δp > 0 and εp > 0 are small enough such that

σp(δp) <(κ−1p − µp)α(1 + α)2

and ρp(εf (p)) <(µ−1p − κp)α(1 + α)2

.

Lemma 3.4.9. We have

i. D(fp)z(Kuα,f,z) ⊆ Ku

α,f,fp(z), if z ∈ Bs(0p, δp)×Bu(0p, δp);

ii. D(fp)−1

fp(z)(Ks

α,f,fp(z)) ⊆ Ks

α,f,z, if fp(z) ∈ Bs(0f(p), εf(p))×Bu(0f(p), εf(p)).

Proof. If (vs, vu) ∈ Kuα,f ,z, then ‖vs‖? ≤ α‖vu‖?. We have

D(fp)z(vs, vu) = (D(ap)z(vs, vu) +Ap(vs), D(bp)z(vs, vu) +Bp(vu)).

Now,

‖D(bp)z(vs, vu) +Bp(vu)‖? > −σp‖(vs, vu)‖? + κ−1p ‖vu‖? ≥ (κ−1

p − σp(1 + α))‖vu‖?

and therefore

‖D(ap)z(vs, vu) +Ap(vs)‖? ≤ σp‖(vs, vu)‖? + µp‖vs‖? ≤ ((1 + α)σp + αµp)‖w‖?

≤ (1 + α)σp + αµp

κ−1p − σp(1 + α)

‖D(bp)z(vs, vu) +Bp(vu)‖?.

Since σp <(κ−1p −µp)α

(1+α)2, we have

αµp+σp(1+α)

κ−1p −σp(1+α)

≤ α. This fact proves D(fp)z(vs, vu) ∈ Kuα,f ,fp(z)

.

Analogously we can prove that D(fp)−1

fp(z)(Ks

α,f ,fp(z)) ⊆ Ks

α,f ,z.


Lemma 3.4.10. We have

i. if z ∈ Bs(0p, δp)×Bu(0p, δp), then

‖D(fp)z(v)‖? ≥κ−1p − σp(1 + α)

1 + α‖v‖? for each v ∈ Ku

α,f,fp(z);

ii. if fp(z) ∈ Bs(0f(p), εf(p))×Bu(0f(p), εf(p)), then

‖D(fp)−1z (v)‖? ≥

µ−1p − ρp(1 + α)

1 + α‖v‖? for each v ∈ Ks

α,f,fp(z).

Proof. If (vs, vu) ∈ Kuα,f ,z, then

‖D(fp)z(vs, vu)‖? ≥ (κ−1p − σp(1 + α))‖vu‖? ≥

κ−1p − σp(1 + α)

1 + α‖(vs, vu)‖?.

Analogously we can prove (ii).

46 ANOSOV FAMILIES

Chapter 4

Openness for Anosov Families

A well-known fact is that the set consisting of dieomorphisms Anosov on a compact Riemannian

manifold is open (see, for example, [Shu13]). Set

A1(M) = g ∈ F1(M) : g is an Anosov family.

The goal of this chapter is to show the analogous result for Anosov families, that is, we will prove

that A1(M) is an open subset of (F1(M), τstr) (remember that (F1(M), τstr) is the set consisting

of non-stationary dynamical systems on (M, 〈·, ·〉) endowed with the C1-strong topology). It is

important to point out that the openness of A1(M) provides a great variety of non-trivial examples

of Anosov families, because the examples given in Section 3.2, besides are not trivial, they are not

isolated in a certain way. We do not ask for additional conditions on each element f ∈ A1(M) to

prove that there is a strong basic neighborhood of f contained in A1(M). For this reason, that

basic neighborhood is not necessarily uniform (see 4.4.1). In Theorems 6.1.7 and 4.4.2 we will

see that, with some conditions on the family, that neighborhood can be taken uniform, that is, a

neighborhood in the uniform topology.

Young in [You86] proves that families consisting of random small perturbations of an Anosov

dieomorphism of class C2 are Anosov families (see Remark 3.2.7). Theorem 6.1.7 is a generalization

of this result. To prove Theorem 6.1.7 we use the same method to be used in this chapter (the method

of invariant cones).

On the other hand, let X be a compact metric space, φ : X → X a homeomorphism and

A : X → SL(Z,m) a continuous application such that the linear cocycle F dened by A over

φ is hyperbolic. Thus, there exists an ε > 0 such that, if B : X → SL(Z,m) is continuous and

‖A(x)−B(x)‖ < ε for all x ∈ X, then the linear cocycle G dened by B over φ is hyperbolic (see

[Via14]). This fact shows the openness of Anosov families that are obtained by hyperbolic cocycles.

This is another particular case of Theorems 4.3.5 and 6.1.7.

As a particular case, in Section 5.4 we will prove that if the family consists of m×m-matrices

acting on the m-torus, then the stability is uniform (see Denition 4.4.1).

47

48 OPENNESS FOR ANOSOV FAMILIES

4.1 Method of Invariant Cones

In order to prove the openness of A1(M), we use the method of invariant cones (see [BP07] and

[KH97]). The results to be given in this section and Section 6.2 are versions for Anosov families

of some results given in [BP07], Chapter 7, where is considered nonuniformly hyperbolic sequences

of dieomorsms on open subsets of Rm (see Denition 7.7.7 in [BP07]). We have adapted those

ideas to Anosov families. In the case of Anosov families, the injectivity radius of the exponential

application at each point of Mi could decrease when |i| increases, since the Mi's are not necessarily

the same Riemannian manifold. We need a radius small enough such that the inequality in (4.1.3)

be valid, and it depends also on the behavior of each fi in the family.

In this section we will prove that there exists a sequence of positive numbers (ξi)i∈Z and η ∈ (0, 1)

such that: given g ∈ B1(f , (ξi)i∈Z), then g satises the property of invariant cones, that is, if p ∈M,

then

Dgp(Kuα,f ,p) ⊆ K

uα,f ,g(p) and ‖Dgpv‖ ≥

1

η‖v‖ if v ∈ Ku

α,f ,p and

Dg−1p (Ks

α,f ,g(p)) ⊆ Ksα,f ,p and ‖Dg−1

g(p)v‖ ≥1

η‖v‖ if v ∈ Ks

α,f ,g(p),

where Kuα,f ,p and Ks

α,f ,p are the α-cones of f at p (see Denition 3.4.1). These properties will be

proved in the Lemmas 4.1.3-4.1.5.

In the rest of this section, f = (fi)i∈Z will represent an Anosov family on (M, 〈·, ·〉) which

satises the property of angles. We will use the notations given in Section 4.4. We can choose

βi > 0, with βi < min%i−1, %i, %i+1/2, such that, if p ∈Mi,

fi(B(p, 2βi)) ⊆ B(fi(p), %i+1/2) and f−1i (B(fi(p), 2βi+1)) ⊆ B(p, %i/2).

Thus, if g = (gi)i∈Z ∈ B1(f , (βi)i∈Z), we have

gi(B(p, βi)) ⊆ B(fi(p), %i+1) and g−1i (B(fi(p), βi+1)) ⊆ B(p, %i). (4.1.1)

Consider a linear isomorphism τp : TpM → Rm which maps an orthonormal basis of Esp to an

orthonormal basis of Rk and maps an orthonormal basis of Eup to an orthonormal basis of Rm−k,for each p ∈M, where k is the dimension of Esp and m the dimension of each Mi.

Remark 4.1.1. Since f satises the property of angles, the norm

‖(vs, vu)‖∗ =

√‖vs‖∗2 + ‖vu‖∗2, for (vs, vu) ∈ Es ⊕ Eu,

dened in (3.3.1), is uniformly equivalent to the norm ‖ · ‖ on M (Corollary 3.3.5). Furthermore,

we have ‖τq(v)‖∗ = ‖v‖∗ for all v ∈ TqM, q ∈M.

Without losing generality, in this chapter, if we do not say otherwise, we will suppose that f is

strictly Anosov and satises the property of angles with the norm ‖ · ‖ on M.

Denition 4.1.2. For g = (gi)i∈Z ∈ B1(f , (βi)i∈Z) and p ∈Mi we set

gp = τfi(p) exp−1fi(p) gi expp τ−1

p : B(0, βi)→ B(0, %i+1)

METHOD OF INVARIANT CONES 49

and

g−1p = τp exp−1

p g−1i expfi(p) τ

−1fi(p)

: B(0, βi+1)→ B(0, %i),

which are well-dened as a consequence of (4.1.1).

For x ∈ Rm, we denote by x1 and x2 the orthogonal projections of x on Rk and Rm−k, respec-tively. If (v, w) ∈ Bk(0, βi)×Bm−k(0, βi), then

fp(v, w) = (ap(v, w) + Ap(v), bp(v, w) + Bp(w)),

where

ap = τfi(p) ap τ−1p , bp = τfi(p) bp τ

−1p , Ap = τfi(p) Ap τ

−1p and Bp = τfi(p) Bp τ

−1p

(see (3.4.1)).

Analogously, if (v, w) ∈ Bk(0, βi+1)×Bm−k(0, βi+1), we have

f−1p (v, w) = (cp(v, w) + Cp(v), dp(v, w) + Dp(w)),

where

cp = τp cp τ−1fi(p)

, dp = τp dp τ−1fi(p)

, Cp(v) = τp Cp τ−1fi(p)

and Dp(w) = τp Dp τ−1fi(p)

Take

ϑp(βi) = supσp(βi), ρp(βi), (4.1.2)

(see Denition 3.4.7).

Lemma 4.1.3. Fix α ∈ (0, 1−λ1+λ) and ξ > 0. For each i ∈ Z, there exist Xi = p1,i, . . . , pmi,i ⊆Mi

and βi > 0 such that Mi = ∪mij=1B(pj,i, βi) and

ϑi := maxq∈Xi

ϑq(βi) ≤ ξ.

Proof. Since D(ap)0 = 0, D(bp)0 = 0, D(cp)0 = 0, D(dp)0 = 0, each fi is of class C1 and Mi is

compact, it follows that for each i we can choose Xi = p1,i, . . . , pmi,i ⊆ Mi and βi > 0 small

enough such that Mi = ∪mij=1B(pj,i, βi) and maxq∈Xi

ϑq(βi) ≤ ξ.

We will consider Xi ⊆ Mi and βi obtained from Lemma 4.1.3, with βi > 0 small enough such

that

ϑi ≤ min

(λ−1 − λ)α

2(1 + α)2,λ−1(1− α)− (1 + α)α

2(1 + α)

for each i ∈ Z. (4.1.3)

Since α ∈ (0, 1−λ1+λ), the minimum in (4.1.3) is positive.

Set

Ksα = (v, w) ∈ Rk ⊕ Rm−k : ‖w‖ < α‖v‖ and

Kuα = (v, w) ∈ Rk ⊕ Rm−k : ‖v‖ < α‖w‖.


Lemma 4.1.4. Let α ∈ (0, 1−λ1+λ) and βi be such that (4.1.3) be valid. Thus, there exists ξi > 0 for

each i ∈ Z such that, if g ∈ B1(f, (ξi)i∈Z), for all p ∈ Xi we have:

i. D(gp)z(Kuα) ⊆ Ku

α for all z ∈ Bk(0, βi)×Bm−k(0, βi), and

ii. D(g−1p )z(Ks

α) ⊆ Ksα for all z ∈ Bk(0, βi+1)×Bm−k(0, βi+1).

Proof. Let us take ξi < minβi, βi+1, ϑi. Fix z ∈ Bk(0, βi)×Bm−k(0, βi). If (x, y) ∈ Kuα \ (0, 0),

then

‖(D(gp)z(x, y))1‖ ≤ ‖(D(gp)z(x, y))1 − (D(fp)z(x, y))1‖+ ‖(D(fp)z(x, y))1‖

≤ ξi‖(x, y)‖+ ‖D(ap)z(x, y)‖+ ‖Ap(x)‖

≤ ϑi(α‖y‖+ ‖y‖) + ϑi‖(x, y)‖+ λ‖x‖

≤ ϑi(α+ 1)‖y‖+ ϑi(α+ 1)‖y‖+ λα‖y‖

= ((α+ 1)2ϑi + λα)‖y‖.

Therefore

‖(D(gp)z(x, y))1‖ ≤ ((α+ 1)2ϑi + λα)‖y‖. (4.1.4)

On the other hand, we have

‖(D(gp)z(x, y))2‖ ≥ ‖(D(fp)z(x, y))2‖ − ‖(D(gp)z(x, y))2 − (D(fp)z(x, y))2‖

≥ ‖Bpy‖ − ‖D(bp)z(x, y)‖ − ξi‖(x, y)‖

≥ λ−1‖y‖ − ϑi(α+ 1)‖y‖ − ϑi(α+ 1)‖y‖

≥ (λ−1 − 2ϑi(α+ 1))‖y‖,

that is,

‖(D(gp)z(x, y))2‖ ≥ (λ−1 − 2ϑi(α+ 1))‖y‖. (4.1.5)

Since ϑi <α(λ−1−λ)2(1+α)2

, we can prove that (α+1)2ϑi+λαλ−1−2ϑi(α+1)

< α. Consequently, from (4.1.4) and (4.1.5)

we have

‖(D(gp)z(x, y))1‖ < α‖(D(gp)z(x, y))2‖,

that is, D(gp)z(x, y) ∈ Kuα. Thus, D(gp)z(Ku

α) ⊆ Kuα for all z ∈ Bk(0, βi)×Bm−k(0, βi).

Analogously we can prove ii.

Lemma 4.1.5. If ξi < minβi, βi+1, ϑi for each i ∈ Z, there exists η < 1 such that, if g ∈B1(f, (ξi)i∈Z), then, for p ∈ Xi, z ∈ Bk(0, βi)×Bm−k(0, βi), we have

i. ‖D(gp)z(x, y)‖ ≥ η−1‖(x, y)‖ if (x, y) ∈ Kuα;

ii. ‖D(g−1p )z(x, y)‖ ≥ η−1‖(x, y)‖ if (x, y) ∈ Ks

α.

Proof. We will prove i. since the proof of ii. is analogous. Fix p ∈ Xi and take (x, y) ∈ Kuα. By

Lemma 4.1.4 we have

‖(D(fp)z(x, y))1‖ ≤ α‖(D(fp)z(x, y))2‖ for z ∈ Bk(0, βi)×Bm−k(0, βi).

METHOD OF INVARIANT CONES 51

Thus,

‖D(gp)z(x, y)‖ ≥ ‖D(fp)z(x, y)‖ − ‖D(fp)z(x, y)−D(gp)z(x, y)‖

≥ ‖(D(fp)z(x, y))2‖ − ‖(D(fp)z(x, y))1‖ − ξi‖(x, y)‖

≥ (1− α)‖(D(fp)z(x, y))2‖ − ϑi‖(x, y)‖

≥ (1− α)(‖Bp(y)‖ − ‖D(bp)z(x, y)‖)− ϑi‖(x, y)‖

≥ (1− α)(λ−1‖y‖ − ϑi‖(x, y)‖)− ϑi‖(x, y)‖

≥ (1− α)(λ−1

1 + α‖(x, y)‖ − ϑi‖(x, y)‖)− ϑi‖(x, y)‖

= ((1− α)(λ−1

1 + α− ϑi)− ϑi)‖(x, y)‖ =

1

η‖(x, y)‖,

where 1η := (1− α)( λ

−1

1+α − ϑi)− ϑi > 1, because ϑi <(1−α)λ−1−(1+α)

2(1+α) .

For each i ∈ Z, let us take the set of charts

φj,i : Bk(0, βi)×Bm−k(0, βi)→ B(pj,i, βi) where φj,i = exppj,i τ−1pj,i ,

where pj,i ∈ Xi, for j = 1, ...,mi. It follows from Lemmas 4.1.4 and 4.1.5 that there exists a strong

basic neighborhood B1(f , (ξi)i∈Z) of f such that, if g ∈ B1(f , (ξi)i∈Z), then:

Lemma 4.1.6. (B(pj,i, βi), φj,i) : j = 1, . . . ,mi, i ∈ Z is an Euclidean atlas for M such that, for

all i ∈ Z and j = 1, ...,mi:

i. φ−1j,i+1gφj,i(B

k(0, βi)×Bm−k(0, βi)) ⊆ Bk(0, δi+1)×Bm−k(0, δi+1).

ii. φ−1j,i g

−1φj,i+1(Bk(0, βi+1)×Bm−k(0, βi+1)) ⊆ Bk(0, δi)×Bm−k(0, δi).

iii. For all z ∈ Bk(0, βi)×Bm−k(0, βi), if x ∈ Kuα, we have

D(φ−1j,i+1gφj,i)z(K

uα) ⊆ Ku

α and ‖D(φ−1j,i+1gφj,i)z(x)‖ ≥ η−1‖x‖.

iv. For all z ∈ Bk(0, βi+1)×Bm−k(0, βi+1), if x ∈ Ksα, we have

D(φ−1j,i g

−1φj,i+1)z(Ksα) ⊆ Ks

α and ‖D(φ−1j,i g

−1φj,i+1)z(x)‖ ≥ η−1‖x‖.

Since D0expp = IdTpM , gp = τf(p) exp−1f(p) gi expp τ

−1p and τp is an isometry, by choosing

βi suciently small, the Lemmas 4.1.4 and 4.1.5 are valid for g and some η′ ∈ (0, 1) (which we will

continue calling by η). That is, since Mi =⋃mij=1 φj,i(B

k(0, βi)×Bm−k(0, βi)), if p ∈Mi, then

p ∈ φj,i(Bk(0, βi)×Bm−k(0, βi)) for some j ∈ 1, ...,mi,

and therefore:

Lemma 4.1.7. There exists η ∈ (0, 1) such that, if g ∈ B1(f, (ξi)i∈Z), for each p ∈M we have:

i. Dgp(Kuα,f,p) ⊆ Ku

α,f,g(p). Furthermore,

‖Dgp(v)‖ ≥ η−1‖v‖ for any v ∈ Kuα,f,p.


ii. D(g−1)g(p)(Ksα,f,g(p)) ⊆ K

sα,f,p. Furthermore,

‖D(g−1)g(p)(v)‖ ≥ η−1‖v‖ for any v ∈ Ksα,f,g(p).

4.2 Openness for Anosov Families with the Property of Angles

First we prove that the set consisting of Anosov families satisfying the property of the angles

is open. In the next section we will show the general case. We will consider (ξi)i∈Z as in Lemma

4.1.7 and x g ∈ B1(f , (ξi)i∈Z). Using the results obtained in the previous section, we will build

families of subspaces F sp and F up of TpM, for each p ∈M, with which g satises the conditions from

Denition 3.1.2.

Lemma 4.2.1. For each p ∈M, take

F sp =∞⋂n=0

Dg−ngn(p)(K

sα,f,gn(p)) and F up =

∞⋂n=0

Dgng−n(p)(Kuα,f,g−n(p)

). (4.2.1)

Thus, the families F sp and F up are Dg-invariant (see Figure 4.2.1).

Eup

Esp

TpM

Fup,3Fup,2Fup,1

F sp,3

F sp,2

F sp,1

Kuα,f ,p

Ksα,f ,p

Figure 4.2.1: F rp,n =⋂nk=1Dg±k

g±k(p)(Ks

α,f,g±k(p)), for r = s, u and n = 1, 2, 3.

Proof. By Lemma 4.1.4 we have for all p ∈M

Dg−1g(p)(K

sα,f ,g(p)) ⊆ K

sα,f ,p and Dgp(K

uα,f ,p) ⊆ K

uα,f ,g(p).

Therefore

Dg−1g(p)(F

sg(p)) =

∞⋂n=0

Dg−1g(p)(Dg

−ngn+1(p)

(Ksα,f ,gn+1(p)

)) =

∞⋂n=0

Dg−n−1gn+1(p)

(Ksα,f ,gn+1(p)

)

⊆∞⋂n=0

Dg−ngn(p)(K

sα,f ,gn(p)) = F sp .

OPENNESS ANOSOV FAMILIES WITH PROPERTY ANGLES 53

Now,

Dgp(Fsp ) = Dgp(K

sα,f ,p) ∩

∞⋂n=1

Dgp(Dg−ngn(p)(K

sα,f ,gn(p))) ⊆

∞⋂n=1

Dg−n+1gn(p) (Ks

α,f ,gn(p))

=∞⋂n=0

Dg−ngn+1(p)

(Ksα,f ,gn+1(p)

) = F sg(p).

The above facts show that Dgp(Fsp ) = F s

g(p). Analogously we can show that

Dgp(Fup ) = F ug(p),

which proves the lemma.

Inductively we have that for all n ≥ 1

Dgnp (F sp ) = F sgn(p) and Dgnp (F up ) = F ugn(p). (4.2.2)

Since F rp ⊆ Krα,f ,p for r = s, u, it follows from Lemma 4.1.7 that for all n ≥ 1

‖Dg−np v‖ ≥ 1

ηn‖v‖ if v ∈ F sp and ‖Dgnpv‖ ≥

1

ηn‖v‖ if v ∈ F up . (4.2.3)

Lemma 4.2.2. F sp and F up given in (4.2.1) are vectorial subspaces and furthermore TpM = F sp⊕F up ,for each p ∈M.

Proof. Fix p ∈ M. Since Esq ⊆ Ksα,f ,q for each q ∈ M, we can choose an orthonormal basis

vn1,p, . . . , vnk,p of Dg−ngn(p)(E

sf n(p)) ⊆ F sp for each n ≥ 0. We can nd a subsequence ni, with

ni →∞ when i→∞, such that vnii,p → vi,p, where vi,p ∈ F sp , for each i = 1, . . . , k. This fact proves

that F sp contains a k-dimensional vectorial subspace Jsp , the subspace spanned by v1,p, . . . , vk,p.Analogously, we can prove that F up contains a (m−k)-dimensional vectorial subspace Jup . Remember

that α < 1−λ1+λ < 1. Therefore,

Ksα,f ,p ∩Ku

α,f ,p = 0p.

Since Jsp ⊆ Ksα,f ,p and J

up ⊆ Ku

α,f ,p, we have Jsp ∩ Jup = 0p. Consequently,

Jsp ⊕ Jup = TpM.

Next, we prove that Jsp = F sp and Jup = F up . Let v ∈ F sp . Since TpM = Jsp ⊕ Jup , v = vs + vu,

where vs ∈ Jsp and vu ∈ Jup . Notice that v − vs ∈ Ksα,f ,p because Ks

α,f ,p is a cone. It follows from

(4.2.2) and (4.2.3) that, for each n ≥ 1,

‖vu‖ ≤ ηn‖Dgnp (v − vs)‖ ≤ η2n‖v − vs‖.

Since η < 1, we have that vu = 0. Hence v ∈ Jsp and therefore Jsp = F sp . Analogously we can prove

that Jup = F up .

Consequently,

Proposition 4.2.3. If g ∈ B1(f, (ξi)i∈Z), then g ∈ A1(M) and satises the property of angles.


Proof. From Lemmas 4.1.7, 4.2.1 and 4.2.2 we have that, considering the splitting TpM = F sp ⊕F up ,for each p ∈M, g has hyperbolic behaviour. We can prove that this splitting is unique (see Lemma

3.1.5) and depends continuously on p (see Proposition 3.1.8). Consequently, g is an Anosov family.

Finally, since F sp ⊆ Ksα,f ,p and F

up ⊆ Ku

α,f ,p for all p and α < 1−λ1+λ < 1, we have that g s. p. a.

From Proposition 4.2.3 we obtain the set consisting of the Anosov families that satisfy the

property of angles is open in F1(M).

4.3 Openness of Anosov Families: General Case

Finally will show that the set consisting of all the Anosov families is open in F1(M). Indeed, we

will suppose that (M, 〈·, ·〉, f ) ∈ A1(M) does not satisfy the property of angles with the Riemannian

metric 〈·, ·〉. Let 〈·, ·〉? be the Riemannian metric obtained in Proposition 3.3.4. Thus (M, 〈·, ·〉?, f )

is a strictly Anosov family that satisfy the property of angles.

Denition 4.3.1. Let µi be as in (3.1.3). Fix ζ > 0. We will consider α > 0 such that 1−αcλ+ζζ > 0.

For each i ∈ Z, set∆i =

1

µi(λ+ ζ

ζc)2.

By (3.3.3) and (3.3.4) we have

∆−1i ‖v‖? ≤ ‖v‖ ≤ 2‖v‖? for all v ∈ TMi, i ∈ Z.

Let d? be the metric on D1i obtained considering the Riemannian metric 〈·, ·〉? on each Mi (see

Denition 1.3.4). For a sequence of positive numbers (εi)i∈Z, let B?(f , (εi)i∈Z) be the strong basic

neighborhood of f in F1(M) considering the metric d? on each D1i .

From Proposition 4.2.3 it follows that:

Lemma 4.3.2. There exists a sequence (ξi)i∈Z such that, if g = (gi)i∈Z ∈ B?(f, (ξi)i∈Z), then

(M, 〈·, ·〉?, g) is an Anosov family.

In that case, we have that (M, 〈·, ·〉?, g) is an Anosov family with constants c = 1 and λ = η ∈(0, 1) (see (4.2.3)).

Let (ξi)i∈Z, where ξi = ξi/∆i for each i ∈ Z. Notice that if g ∈ B1(f , (ξi)i∈Z), then g ∈B?(f , (ξi)i∈Z). Consequently, if g ∈ B1(f , (ξi)i∈Z), then (M, 〈·, ·〉?, g) is an Anosov family.

Remark 4.3.3. In the following lemma we will show that each non-stationary dynamical system

in B1(f , (ξi)i∈Z) is an Anosov family with the metric 〈·, ·〉. This fact is not immediate, since 〈·, ·〉and 〈·, ·〉? are not necessarily uniformly equivalent on M (Example 3.2.1 proves that the notion of

Anosov family depends on the metric on the total space).

Lemma 4.3.4. If g ∈ B1(f, (ξi)i∈Z), then (M, 〈·, ·〉, g) is an Anosov family.

Proof. Consider the stable subspace Esg ,p of g at p with respect to the metric 〈·, ·〉?. We have from

Lemma 4.1.5 that Esg ,p is contained in the stable α-cone of f at p. If v ∈ Esg ,p, then v = vs + vu,

where vs ∈ Esf ,p and vu ∈ Euf ,p. It follows from (3.3.2) that

‖vs‖ ≤ ‖vs + vu‖+ α‖vs‖? ≤ ‖v‖+ αλ+ ζ

ζc‖vs‖.

OPENNESS FOR ANOSOV FAMILIES CONSISTING OF MATRICES 55

Thus, by (3.3.3),

‖Dgnp (v)‖ ≤ 2‖Dgnp (v)‖? ≤ 2ηn‖v‖? ≤ 2ηn(‖vs‖? + ‖vu‖?) ≤ 2ηn(‖vs‖? + α‖vs‖?)

= 2ηn(1 + α)‖vs‖? ≤ 2ηn(1 + α)λ+ ζ

ζc‖vs‖ ≤ 2ηn(1 + α)

λ+ ζ

ζc(1− αλ+ ζ

ζc)−1‖v‖

= c′ηn‖v‖,

where c′ = 2(1 + α)λ+ζζ c(1− αλ+ζ

ζ c)−1.

Analogously we obtain that if v ∈ Eug ,p then ‖Dg−np (v)‖ ≤ c′ηn‖v‖.Consequently, (M, 〈·, ·〉, g) is an Anosov family with constants η ∈ (0, 1) and c′.

From Proposition 4.2.3 and Lemma 4.3.4 we have:

Theorem 4.3.5. Let f ∈ A1(M). There exists a sequence of positive numbers (ξi)i∈Z such that, if

g ∈ B1(f, (ξi)i∈Z), then (M, 〈·, ·〉, g) is an Anosov family. Consequently, A1(M) is open in F1(M).

Proof. The theorem follows from Proposition 4.2.3 and from Lemmas 4.3.2 and 4.3.4.

Notice that for the basic neighborhood B1(f , (ξi)i∈Z) of f the ξi could be arbitrarily small for

|i| large. In that case, if (gi)i∈Z ∈ B1(f , (ξi)i∈Z), then fi and gi are C1-closer for |i| large.

4.4 Openness for Anosov Families consisting of Matrices

In this section, we will suppose that F = (Fi)i∈Z is an Anosov family satisfying the property of

angles, where Fi : Tm → Tm is a dieomorphism on Tm induced by a matrix Ai : Rm → Rm and

Tm = Rm/Zm is the m-torus endowed with the Riemannian metric inherited from Rm. Hence, thediagram

Rm Ai−−−−→ Rm

π

y yπTm Fi−−−−→ Tm

commutes for each i ∈ Z, where π is the projection map.

Denition 4.4.1. For ε > 0 and r ≥ 0, a uniform basic neighborhood of f = (fi)i∈Z is the set

Br(f , ε) = (gi)i∈Z ∈ Fr(M) : dr(fi, gi) < ε for all i ∈ Z.

The goal of this section is to prove the following theorem:

Theorem 4.4.2. There exists ξ > 0 such that, if g ∈ B1(F, ξ), then g ∈ A1(M).

Since f satises the property of angles, the Riemannian metric ‖·‖∗ obtained in Proposition 3.3.4is uniformly equivalent to ‖ · ‖. It is clear that all the uniform basic neighborhood considering the

metric ‖ · ‖∗ contains a uniform basic neighborhood considering ‖ · ‖∗ and viceversa. Consequently,

without loss of generality, we can suppose that f is strictly Anosov with the metric ‖·‖ (f is strictly

Anosov with ‖ · ‖∗).

Remark 4.4.3. Notice that in this case, ap(z) = bp(z) = cp(z) = dp(z) = 0 for each p ∈ Tm and

z ∈ Rm (see (3.4.1)). Therefore, σp(δ) = ρp(δ) = 0 for each p ∈ Tm and δ > 0 (see Denition 3.4.7).


For α ∈ (0, 1−λ1+λ), set

ϑ = min

(λ−1 − λ)α

2(1 + α)2,λ−1(1− α)− (1 + α)α

2(1 + α)

for each i ∈ Z.

Let X = p1, . . . , pk ⊆ Tm be such that Tm = ∪ki=1B(pi, ϑ).

Lemma 4.4.4. Fix ξ ∈ (0, ϑ). If g ∈ B1(F, ξ), then for all p ∈ X we have:


α for all z ∈ B(p, ϑ), and

ii. D(g−1p )z(Ks

α) ⊆ Ksα for all z ∈ B(p, ϑ).

Proof. See Lemma 4.1.4.

Lemma 4.4.5. If ξ ∈ (0, ϑ), there exists η < 1 such that, if g ∈ B1(F, ξ), then, for p ∈ X,

z ∈ B(p, ϑ), we have

i. ‖D(gp)z(w)‖ ≥ η−1‖w‖ if w ∈ Kuα;

ii. ‖D(g−1p )z(w)‖ ≥ η−1‖w‖ if w ∈ Ks

α.

Proof. See Lemma 4.1.5.

Proof of Theorem 4.4.2. Fix g ∈ B1(F , ξ). Following the Lemmas 4.1.7 and 4.2.1 we have that for

each p ∈M, the subspaces F sp and F up , given in Lemma 4.2.1, are Dg -invariants. By Lemma 4.2.2

we have that F sp and F up are vectorial subspaces and furthermore TpM = F sp ⊕F up , for each p ∈M.

This facts prove Theorem 4.4.2.

There are many examples of Anosov families consisting of sequences of matrices, as we saw in

Section 3. Any multiplicative family satises the property of angles, consequently they are uniformly

stable. On the other hand, since any gathering of an Anosov family is Anosov, it can be proved

that if a family satises the property of angles, any gathering satises the property of angles. Thus,

any family obtained from a gathering of a multiplicative family satises the property of angles.

Chapter 5

Stable and Unstable Manifolds

Let φ : M → M be a dieomorphism on a compact Riemannian manifold M with metric ρ.

Let x ∈ M and ε > 0. Let Vsε (x, φ) and Vuε (x, φ) be the local stable and unstable sets of φ at x,

respectively (see Denition 1.4.2). If φ is an Anosov dieomorphism, there exists ε > 0 such that,

for every x ∈M , Vsε (x, φ) and Vuε (x, φ) are dierentiable submanifolds of M , tangent to the stable

and unstable subspaces at x, respectively (see [Shu13]). The goal of this chapter is to show a similar

version of this property for Anosov families.

5.1 Stable and Unstable Sets

As we mentioned above, if φ : M → M is an Anosov dieomorphism, then there is ε > 0 such

that, for each x ∈M , the stable and unstable sets at x are dierentiable submanifolds ofM tangent

to the stable and unstable subspaces at x, respectively. In that case, φ is a contraction on Vsε (x, φ)

(that is, there exists a ν ∈ (0, 1) such that d(φ(z), φ(y)) ≤ νd(z, y) for all z, y ∈ Vsε (x, φ)) and φ−1 is

a contraction on Vuε (x, φ). Furthermore, φ(Vsε (x, φ)) ⊆ Vsε (φ(x), φ) and φ−1(Vuε (φ(x), φ)) ⊆ Vuε (x, φ)

for each x ∈M . If we consider the stable and unstable subsets for non-stationary dynamical systems

as in Denition 1.4.2, the facts above are not always valid for Anosov families (see Example 3.2.2).

In Denition 5.1.4 we will give a notion of stable and unstable sets which works better for non-

stationary dynamical systems than the sets given in Denition 1.4.2.

Denition 5.1.1. A homeomorphism φ : X → X on the metric space (X, ρ) is expansive on a

subset Y of X if there is ε > 0 such that

supn∈Zρ(φn(x), φn(y)) : x ∈ X, y ∈ Y, x 6= y > ε.

It is well known that if Λ ⊆ X is a compact hyperbolic subset for a C1-dieomorphism φ : X →X, then φ is expansive on Λ. In the following example we will see that there are Anosov families

that are not expansive.

Example 5.1.2. Take a, b ∈ (λ, 1) in the Example 3.2.2. For x, y ∈M0 we obtain d(f n(x), f n(y))→0 as n→ +∞1. Therefore, the stable set of f at any point x in M0 is the whole M0. Thus, for all

1Notice that the volume of each Mi with the Riemannian metric ‖ · ‖i dened in (3.2.1) is decreasing, for i ≥ 1,if a, b ∈ (λ, 1) (see Figure 5.1.1).

57

58 STABLE AND UNSTABLE MANIFOLDS

x ∈M0 and ε > 0, Vsε (x, φ) ∩ Vuε (x, φ) = Vuε (x, φ). On the other hand, if y ∈ Vu(x, φ), we obtain

d(f −n(x), f −n(y)) = d(φ−n(x), φ−n(y))→ 0 as n→ +∞.

Consequently, f is not expansive, because d(f n(x), f n(y))→ 0 as n→ ±∞.

. . .

M1 M2 M3

. . .

Figure 5.1.1: M1,M2,M3,. . . , endowed with the metric given in (3.2.1), for a, b ∈ (λ, 1).

Next we dene a notion about stables and unstable sets which work better for non-stationary

dynamical systems than the sets given in Denition 1.4.2. These sets consist of the points whose

orbits approach exponentially to the orbits of a given point.

Denition 5.1.3. Given two points p, q ∈M, set

Θp,q = lim supn→∞

1

nlogd(f ni (q), f ni (p)) and ∆p,q = lim sup

n→∞

1

nlogd(f −ni (q), f −ni (p)).

Denition 5.1.4. Let ε = (εi)i∈Z be a sequence of positive numbers. Fix p ∈Mi. Set

N s(p, ε) = q ∈ B(p, εi) : f ni (q) ∈ B(f ni (p), εi+n) for n ≥ 1 and Θp,q < 0.

N s(p, ε) will be called the local stable set for f at p;

N u(p, ε) = q ∈ B(p, εi) : f −ni (q) ∈ B(f −ni (p), εi−n) forn ≥ 1 and∆p,q < 0.

N u(p, ε) will be called the local unstable set for f at p.

5.2 Hadamard-Perron Theorem for Anosov Families

In this section we will give conditions for obtain invariant manifolds at each point of the total

space, whose expansion or contraction by each fi can be controlled (see Theorems 5.2.10 and

5.2.11). This result is a generalized version of the Hadamard-Perron Theorem for obtain local stable

and unstable manifold for Anosov families (see [BP07], [KH97]). In our case, stable and unstable

subspaces are not necessarily orthogonal. Therefore, the size of the manifolds to be obtained could

decay along the orbits.

We will x an Anosov family (M, 〈·, ·〉, f ) with constant λ ∈ (0, 1) and c ≥ 1.

Remark 5.2.1. If c > 1, let n be the minimum positive integer such that cλn ≤ λ. Hence the

gathering f obtained of f by the sequence (ni)i∈Z is a strictly Anosov family with constant λ.

Thus, considering a gathering of f if necessary, we can assume that the family is strictly Anosov.

HADAMARD-PERRON THEOREM FOR ANOSOV FAMILIES 59

Let us x p ∈ M. Without loss of generality, we can assume that p ∈ M0 (if p /∈ M0, q =

f n(p) ∈ M0 for some n ∈ Z, then consider q instead of p). By simplicity, throughout this chapter

we will consider the following notations:

Denition 5.2.2. For ε > 0 and n ∈ Z, set

i. Bn(ε) the ball in Tf n0 (p) with radius ε and center 0f n0 (p) ∈ Tf n0 (p)M,

ii. Bsn(ε) the ball in Es

f n0 (p) with radius ε and center 0f n0 (p) ∈ Esf n0 (p),

iii. Bun(ε) the ball in Eu

f n0 (p) with radius ε and center 0 ∈ Euf n0 (p),

iv. an = af n(p), bn = bf n(p), An = Af n(p), Bn = Bf n(p), Cn = Cf n(p), Dn = Df n(p), cn = cf n(p),

dn = df n(p), κn = κf n(p), µn = µf n(p), σn = σf n(p) and ρn = ρf n(p) (see Section 4.4).

For each n ∈ Z, let %n > 0 be the injectivity radius of expf n0 (p) at fn0 (p) (see Denition 2.4.1).

Take εn = %n/2 and let δn > 0 be small enough such that

fn = exp−1fn+1(p)

fn expfn(p) : Bsn(δn)×Bu

n(δn)→ Bn+1(εn+1)

is well dened, for each n ∈ Z.

Denition 5.2.3. Let α ∈ (0, 1) and (γn)n∈Z be a sequence of positive numbers. Dene:

i. Γun(α, γn) = φ : Bun(γn)→ Bs

n(γn) : φ is α-Lipschitz and φ(0) = 0.

ii. Γu(α, (γn)n) = φ = (φn)n∈Z : φn ∈ Γun(α, δn).

If φ = (φn)n∈Z, ψ = (ψn)n∈Z ∈ Γu(α, (γn)n), dene the metric

dΓu(φ, ψ) = supn∈Z

sup

x∈Bun(γn)\0

‖φn(x)− ψn(x)‖‖x‖

.

It is not dicult to prove the following proposition:

Proposition 5.2.4. (Γu(α, (γn)n), dΓu) is a complete metric space.

For an application F : X → Y , we will denote by G(F ) the set (F (x), x) : x ∈ X. Throughoutthis section, we will x α ∈ (0, 1), γ ∈ (λ2, 1) and

σn = min

(κ−1n − µn)α

(1 + α)2,

(γκ−1n − µn)

(1 + α)(1 + γ)

. (5.2.1)

Proposition 5.2.5. Suppose that for each n ≤ −1 we can choose the δn's such that

κ−1n + αµn1 + α

δn ≥ δn+1 for n ≤ −1 and σn < σn. (5.2.2)

Then, there exists a sequence of positive numbers (δn)n≥0 such that, for each n ∈ Z, if φn ∈Γun(α, δn), we have that

fn(φn(w), w) : w ∈ Bun(δn) ∩Bs

n+1(δn+1)×Bun+1(δn+1)

is the G of an application ψn+1 in Γun+1(α, δn+1) (see Figure 5.2.1).


Esp

Eup

Bsn(δn)×Bun(δn)

fn

Esq

Euq

Bsn+1(δn+1)×Bun+1(δn+1)

G(φn) G(ψn+1)

Figure 5.2.1: G(ψn+1) = fnG(φn). Shaded regions represent the unstable α-cones.

Proof. Inductivelly, for each n ≥ 0 we can choose δn > 0 such that σn < ωn and if φn−1 ∈Γun−1(α, δn−1), then fn−1(φn−1(w), w) : w ∈ Bu

n−1(δn−1) ∩Bsn(δn)× Bu

n(δn) is the G of an appli-

cation ψn in Γun(α, δn).

Now, x φn ∈ Γun(α, δn). For w ∈ Bun(δn), let

rn(w) = Bnw + bn(φn(w), w). (5.2.3)

If w, z ∈ Bun(δn) we have

‖rn(w)− rn(z)‖ ≥ ‖Bnw −Bnz‖ − ‖bn(φn(w), w)− bn(φn(z), z)‖

≥ κ−1n ‖w − z‖ − σn(1 + α)‖w − z‖

(from the second inequality in (5.2.2) we have κ−1n − σn(1 + α) > 0). Thus,

‖rn(w)− rn(z)‖ ≥ (κ−1n − σn(1 + α))‖w − z‖ (5.2.4)

and therefore rn is injective. From (5.2.2) and (5.2.4) we obtain

Bun(δn+1) ⊆ rn(Bu

n(δn)). (5.2.5)

Consequently, we can dene the map ψn+1 : Bun+1(δn+1)→ Esn+1, as

ψn+1(w) = Anφn(r−1n (w)) + an(φn(r−1

n (w)), r−1n (w)) (5.2.6)

for w ∈ Bun+1(δn+1). Now, if x = rn(w), y = rn(z) ∈ Bu

n+1(δn+1), it follows from (5.2.4) that

‖ψn+1(x)− ψn+1(y)‖ ≤ ‖An(φn(w)− φn(z))‖+ ‖an(φn(w), w)− an(φn(z), z)‖

≤ αµn‖w − z‖+ σn(1 + α)‖w − z‖

≤ αµn + σn(1 + α)

κ−1n − σn(1 + α)

‖rn(w)− rn(z)‖

≤ α‖rn(w)− rn(z)‖ = α‖x− y‖.


Thus,

‖ψn+1(x)− ψn+1(y)‖ ≤ α‖x− y‖, (5.2.7)

that is, ψn+1 is α-Lipschitz. It is clear that ψn+1(0) = 0 and, since α < 1, it follows from (5.2.7)

that ψn+1(Bun+1(δn+1)) ⊆ Bs

n+1(δn+1). Consequently, ψn+1 ∈ Γun+1(α, δn+1). On the other hand, if

x = rn(w) ∈ Bun+1(δn+1) we have

(ψn+1(x), x) = (Anφn(w) + an(φnw,w), Bn(w) + bn(φnw,w)) = fn(φnw,w). (5.2.8)

Therefore, fn(φn(w), w) : w ∈ Bun(δn) ∩ Bs

n+1(δn+1) × Bun+1(δn+1) is the G of ψn+1. This fact

proves the proposition.

Remark 5.2.6. Proposition 5.2.5 is shown in [BP07], Proposition 7.3.5, when there exists δ > 0

such that, considering δn = δ for all n ∈ Z, σn < σ for a small enough σ > 0. We have adapted that

proof to obtain a more general result, in which δn may vary with n but satisfying the rst condition

in (5.2.2) (this condition means that δn must not decay very quickly when n→ −∞) and σn could

increase but not more than σn, which could be very large. Notice that

κ−1n + αµn1 + α

> 1 for each n ∈ Z.

On the other hand, from Remark 3.4.8 we have that

‖D(fn)z −D(fn)0‖ = ‖D(an, bn)z‖ for each z ∈ Bn(δn) and n ∈ Z.

Therefore, the assumption of Proposition 5.2.5 means that the sequence (δn)n∈Z must not decay

quickly to zero (the decay of δn is controlled by κ−1n +αµn

1+α ) and

‖D(fn)z −D(fn)0‖ < σn for each z such that ‖z‖ < δn.

Remark 5.2.7. Since an(0) = bn(0) = D(an)0 = D(bn)0 = 0, we always can chose a sequence δn

satisfying the second condition in (5.2.2). If each fi is C2 and the second derivative p→ D2f p, for

p ∈ M, is bounded, then we can nd an uniform δ satisfying (5.2.2). We will explain these facts

with more detail in Chapter 7.

From Proposition 5.2.5 we have the application

G : Γu(α, (δn)n)→ Γu(α, (δn)n)

(φn)n∈Z 7→ (ψn−1)n∈Z,

where ψn is given in (5.2.6), is well dened. The following proposition, whose proof is based on the

proof of Proposition 7.3.6 in [BP07], shows that G : Γu(α, (δn)n)→ Γu(α, (δn)n) is a contraction.

Proposition 5.2.8. G : Γu(α, (δn)n)→ Γu(α, (δn)n) is a contraction.

Proof. Fix φ = (φn)n∈Z, ϕ = (ϕn)n∈Z ∈ Γu(α, (δn)n). Let

ψ = (ψn)n∈Z = G(φ) and χ = (χn)n∈Z = G(ϕ).


For x ∈ Bun+1(δn),

ψn+1(x) = Anφn((rφ)−1n (x)) + an(φn((rφ)−1

n (x)), (rφ)−1n (x))

and

χn+1(x) = Anϕn((rϕ)−1n (x)) + an(ϕn((rϕ)−1

n (x)), (rϕ)−1n (x))

where

(rφ)n(w) = Bnw + bn(φn(w), w) and (rϕ)n(w) = Bnw + bn(ϕn(w), w).

for w ∈ Bun(δn) (see (5.2.3)). If x ∈ Bu

n+1(δn+1), there exists w ∈ Bun(δn) such that x = (rφ)nw.

Thus

‖ψn+1(x)− χn+1(x)‖ = ‖ψn+1(rφ)nw − χn+1(rφ)nw‖

≤ ‖ψn+1(rφ)nw − χn+1(rϕ)nw‖+ ‖χn+1(rϕ)nw − χn+1(rφ)nw‖

≤ ‖Anφn(w) + an(φn(w), w)−Anϕn(w)− an(ϕn(w), w)‖+ α‖(rϕ)nw − (rφ)nw‖

≤ ‖Anφn(w)−Anϕn(w)‖+ ‖an(φn(w), w)− an(ϕn(w), w)‖

+ α‖Bnw + bn(ϕn(w), w)−Bnw − bn(φn(w), w)‖

≤ µn‖φn(w)− ϕn(w)‖+ σn‖(φm(w), w)− (ϕn(w), w)‖

+ ασn‖(ϕn(w), w)− (φn(w), w)‖

= (µn + σn(1 + α))‖φn(w)− ϕn(w)‖.

From (5.2.4) we have

‖x‖ = ‖(rφ)nw‖ ≥ (κ−1n − σn(1 + α))‖w‖.

Therefore‖ψn+1(x)− χn+1(x)‖

‖x‖≤ µn + σn(1 + α)

κ−1n − σn(1 + α)

‖φn(w)− ϕn(w)‖‖w‖

.

Since σn <γκ−1−µn

(1+α)(1+γ) (see (5.2.2)), then

µn + σn(1 + α)

κ−1n − σn(1 + α)

< γ < 1.

Consequently,

dΓu(G(φ),G(ϕ)) = supn∈Z

sup

x∈Bun(δn+1)\0

‖ψn+1(x)− χn+1(x)‖‖x‖

≤ supn∈Z

sup

w∈Bun(δn)\0γ‖φn(w)− ϕn(w)‖

‖w‖

= γ supn∈Z

sup

w∈Bun(δn)\0

‖φn(w)− ϕn(w)‖‖w‖

= γdΓu(φ, ϕ).

This fact proves that G is a contraction.


Since Γu(α, (δn)n) is a complete metric space, by the Banach xed-point Theorem we have there

exists a unique φ? ∈ Γu(α, (δn)n) such that

G(φ?) = φ?.

In other words, for each n ∈ Z, there exists a unique φ?n ∈ Γun(α, δn) such that

fn(φ?nw,w) = (φ?n+1rn(w), rn(w)), for all w ∈ Bun(δn)

(see (5.2.8)). Consequently, if

Wn(δn) = (φ?nw,w) : w ∈ Bun(δn), (5.2.9)

we have Wn+1(δn+1) ⊆ fn(Wn(δn)), because Bun+1(δn+1) ⊆ rn(Bu

n(δn)).

Notice that the size of α only changes the diameter of each Wn(δn). This is because the unique-

ness of the xed point of a contraction and, furthermore, if α ≤ α, then

Γu(α, (δn)n) ⊆ Γu(α, (δn)n) and Γs(α, (εn)n) ⊆ Γs(α, (εn)n)

for appropriate δn and εn. We will assume that α = (λ−1 − 1)/2. Fix τ ∈ (1+λ2 , 1). From now on,

we will suppose that, for each n ∈ Z,

σn < min

σn,

2λτκ−1n − 1− λ1 + λ

.

Hence,

τn :=1 + α

κ−1n − σn(1 + α)

=1 + λ

2λκ−1n − σn(1 + λ)

< τ for each n ∈ Z. (5.2.10)

The sets Wn(δn) are topological submanifolds of Mn, because they are graphs of α-Lipschitz

maps. Next, we prove that Wn(δn) is dierentiable (that proof is taken from [BP07], p. 201).

Lemma 5.2.9. Wn(δn) is dierentiable and T0Wn(δn) = Eufn(p).

Proof. For x, y ∈ Bun(δn), with x 6= y, and φ ∈ Γun(α, δn), set

Ψx,y(φ) =(φ(x), x)− (φ(y), y)

‖(φ(x), x)− (φ(y), y)‖,

ωx(φ) = w ∈ T(φ(x),x)Bn(δn) : Ψx,xi(φ)→ w for some sequence xi → x

and $x(φ) = tw : t ∈ R and w ∈ ωx(φ).

We have that φ is dierentiable at x if, and only if, $x(φ) is a dim(Eu(φ(x),x))-dimensional

subspace. Hence, we will prove that $x(φ) is a dim(Eu(φ(x),x))-dimensional subspace.

For z = exp−1f n(p)(q), set

Esz = D(exp−1f n(p))q(E

sq), Euz = D(exp−1

f n(p))q(Euq ),

Ksα,f ,z = D(exp−1

f n(p))q(Ksα,f ,q) and Ku

α,f ,z = D(exp−1f n(p))q(K

uα,f ,q).


Since φ?n is α-Lipschitz, for each x ∈ Bun(δn) we have

$x(φ∗n) ⊆ Kuα,f ,(φ?(x),x).

Suppose that a sequence (xi)i∈N ⊆ Bun(δn) converges to x ∈ Bu

n(δn) as i → ∞. Let wi, w ∈Bun+1(δn+1) be such that (φ∗n+1(wi), wi) = fn(φ∗n(xi), xi) and (φ∗n+1(w), w) = fn(φ∗n(x), x). Thus,

Ψx,xi(φn)→ z as i→∞, if, and only if

limi→∞

(φ∗n+1(wi), wi)− (φ∗n+1(w), w)

‖(φ∗n+1(wi), wi)− (φ∗n+1(w), w)‖= lim

i→∞

fn(φ∗n(xi), xi)− fn(φ∗n(x), x)

‖fn(φ∗n(xi), xi)− fn(φ∗n(x), x)‖

= limi→∞

fn(φ∗n(xi),xi)−fn(φ∗n(x),x)‖(φ∗n(xi),xi)−(φ∗n(x),x)‖

‖fn(φ∗n(xi),xi)−fn(φ∗n(x),x)‖‖(φ∗n(xi),xi)−(φ∗n(x),x)‖

=D(fn)(φ∗n(x),x)(z)

‖D(fn)(φ∗n(x),x)(z)‖.

This fact implies that, for all n ∈ Z, x ∈ Bun(δn) and w ∈ Bu

n+1(δn+1) with (φ∗n+1(w), w) =

fn(φ∗n(x), x), we have

D(fn)(φ∗n(x),x)$x(φ∗n) = $w(φ∗n+1).

Since $x(φ∗n) ⊆ Kuα,f ,(φ?(x),x), by Lemma 3.4.2 we have $x(φ∗n) ⊆ Eu(φ∗n(x),x).

On the other hand, for any v ∈ Bun(δn), we can choose a sequence (tm)m∈Z converging to 0,

such that Φ(φ∗n(x),x)(φ?) converges as m→∞. This fact implies that $x(φ∗n) projects onto Rk. The

above facts imply that $x(φ∗n) = Eu(φ∗n(x),x).

Therefore φ∗n is dierentiable at x. Taking x = 0 we have

T0Wn(δn) = Eu(φ∗n(0),0) = Euf n(p).


The angles between the stable and unstable subspaces will be important to control the contrac-

tion or expansion of the submanifolds by f . Notice that, if the angles θn (see (3.1.2)) decay when

n→ ±∞, the vectors in Bsn(δn) and in Bu

n(δn) are ever closer. From (3.3.5), we have

∆n‖v‖∗ ≤ ‖v‖ ≤ 2‖v‖∗, for v ∈ TMn, (5.2.11)

where ∆n =(

11−cos(θn)(λ+ζ

ζ c)2)−1

. Hence, if w ∈ Bun(δn) for n ∈ Z, by (5.2.4), (5.2.8) and (5.2.11)

we have

‖fn(φ?nw,w)‖ ≥ ∆n+1‖fn(φ?nw,w)‖∗ ≥ ∆n+1‖rn(w)‖∗ ≥ ∆n+1κ−1n − σn(1 + α)

1 + α‖(φ?nw,w)‖∗

≥ ∆n+1

2

κ−1n − σn(1 + α)

1 + α‖(φ?nw,w)‖.

Consequently, since (fn)−1(Wn+1(δn+1)) ⊆ Wn(δn), for any z ∈ Bun+1(δn+1), there exists w ∈

Bun(δn) such that fn(φ?nw,w) = (φ?n+1z, z). Thus we have that

‖(fn)−1(φ?n+1z, z)‖ ≤2

∆n+1τ‖(φ?n+1z, z)‖, for z ∈ Bu

n+1(δn+1)


(see (5.2.10)). Analogously, there exists y ∈ Bun−1(δn−1) such that fn−1(φ?n−1y, y) = (φ?nw,w).

Therefore,

‖(φ?n+1z, z)‖ ≥ ∆n+1τ−1‖(φ?nw,w)‖∗ ≥ ∆n+1τ

−1τ−1‖(φ?n−1y, y)‖∗ ≥∆n+1

2τ−2‖(φ?n−1y, y)‖

=∆n+1

2τ−2‖(fn−1)−1(φ?nw,w)‖ =

∆n+1

2τ−2‖(fn−1)−1(fn)−1(φ?n+1z, z)‖.

Inductively we can prove for k ≥ 0 that

‖(fn−k)−1 · · · (fn)−1(φ?n+1z, z)‖ ≤2

∆n+1τk+1‖(φ?n+1z, z)‖ for z ∈ Bu

n+1(δn+1). (5.2.12)

Theorem 5.2.10. Fix p ∈ M0. Suppose that the Anosov family (M, 〈·, ·〉, f ) admits a sequence of

positive numbers δ = (δn)n∈Z as in Proposition 5.2.5. Thus, there exists a two-sided sequence

Wu(fn0 (p), δ) : n ∈ Z,

where Wu(fn0 (p), δ) is a dierentiable submanifold of Mn with size δn, such that for n ∈ Z:

i. fn0 (p) ∈ Wu(fn0 (p), δ) and Tfn0 (p)Wu(fn0 (p), δ) = Eufn0 (p),

ii. f−1n−1(Wu(fn0 (p), δ)) ⊆ Wu(fn−1

0 (p), δ), and furthermore

iii. if q ∈ Wu(pn+1, δ), where pn = fn0 (p), and k ≥ 0 we have

d(f−(k+1)n+1 (q), f

−(k+1)n+1 (pn+1)) ≤ 2

∆n+1τk+1d(q, pn+1). (5.2.13)

Proof. Let Wn(δn) be as in (5.2.9) and take

Wu(f n0 (p), δ) = expf n(p)(Wn(δn)) for each n ∈ Z.

The statements (i) and (ii) of the theorem are clear.

For (iii); if q ∈ Wu(pn, δ), for each k ≥ 1 there exists a unique vn−k+1 ∈ Tf −kn+1(p)M such that

expf −kn+1(p)(vn−k+1) = f −kn+1(q) and ‖vn−k+1‖ = d(f −kn+1(p), f −kn+1(q)). By (5.2.12) and the invariance

of Wu(f n0 (p), δ) by f we have (5.2.13).

Theorem 5.2.10 is a more generalized version of the Hadamard-Perron Theorem adapted to

Anosov families for the unstable case, since the angles between the stable and unstable subspace

could be arbitrarily small and, furthermore, the sequence (δn) satisfying the condition (5.2.2) is not

necessarily bounded away from zero.

Analogously we can obtain a more generalized version of the Hadamard-Perron Theorem adapted

to Anosov families for the stable case. Indeed, set

ρn = min

(µ−1n − κn)α

(1 + α)2,

(γµ−1n − κn)

(1 + α)(1 + γ),2λτµ−1

n − 1− λ1 + λ

for n ∈ Z. (5.2.14)

Suppose that for each n ≥ 0 there exists εn > 0 satisfying

εn−1 ≤µ−1n + ακn1 + α

εn and ρn(εn+1) < ρn (5.2.15)


(see Denition 3.4.7). Thus, there exists a sequence of positive numbers (εn)n<0 such that, consid-

ering ε = (εn)n∈Z, we have:

Theorem 5.2.11. There exists a two-sided sequence of dierentiable submanifold

Ws(fn0 (p), ε) : n ∈ Z,

such that

i. fn0 (p) ∈ Ws(fn0 (p), ε) and Tfn0 (p)Ws(fn0 (p), ε) = Esfn0 (p),

ii. fn(Ws(fn0 (p), ε)) ⊆ Ws(fn+10 (p), ε), and furthermore

iii. if q ∈ Ws(pn, ε) and k ≥ 1 we have

d(f kn (q), f kn (pn)) ≤ 2

∆nτkd(q, pn). (5.2.16)

Denition 5.2.12. We will call Ws(f n0 (p), ε) : n ∈ Z as a family of admissible (s, α, ε)-manifold

at p and Wu(f n0 (p), δ) : n ∈ Z as a family of admissible (u, α, δ)-manifold at p (see [BP07]).

In the next section we wil see that with some conditions, the family of admissible manifolds

coincide with the sets given in Denition 5.1.4.

The rst inequality (5.2.2) means that the radius δn of the balls Bun(δn) must not decrease very

fast when n→ −∞. This condition is sucient to have the invariance of the admissible manifolds

by f obtained in Theorem 5.2.10n (see (5.2.4)). Remember we have considered exponential charts

to work on the ambient Euclidian and each δn depends on both f and %n, the injectivity radius

of the exponential map at f n0 (p). This fact is of great importance to the construction of unstable

(stable) manifolds, because the expansions (contractions) of each manifold could not be caused by

the family but by the geometry of each component (see Example 3.2.2).

5.3 Local Stable and Unstable Manifolds for Anosov Families

In the previous section we obtained admissible manifolds for Anosov families whose expansion

or contraction are controlled by the ∆k's, τk's and ςk's. In this section we will give certain conditions

with which the stable and unstable sets (see Denition 5.1.4) coincide with the admissible manifolds

(see Lemmas 5.3.2 and 5.3.3).

We had talked about the importance of maintaining the metrics established, because the notion

of Anosov family depends on the Riemannian metrics on each Mn. Changing the metrics on each

component we could get very dierent stable (unstable) sets (see Example 3.2.2). However, these

sets don't depend on the metrics if they are uniformly equivalent on M, only that they can change

the diameter:

Proposition 5.3.1. Let 〈·, ·〉 and 〈·, ·〉′ be uniformly equivalent Riemannian metrics on M. Fix

p ∈ M0. Given a sequence of positive numbers small enough ε = (εi)i∈Z, there exist sequences of

positive numbers ε′ = (ε′i)i∈Z and ε = (εi)i∈Z such that, for r = u, s,

N r(p, ε, 〈·, ·〉) ⊆ N r(p, ε′, 〈·, ·〉′) ⊆ N r(p, ε, 〈·, ·〉).

LOCAL STABLE AND UNSTABLE MANIFOLDS 67

Proof. We will show only the stable case, since the unstable case is analogous. Consider N = M

with the Riemannian metric 〈·, ·〉′, that is, Ni = Mi with the Riemannian metric 〈·, ·〉′i := 〈·, ·〉′|Mi

for each i ∈ Z. Let Ii : (Mi, 〈·, ·〉)→ (Mi, 〈·, ·〉′) be the identity. For each n ≥ 0 we can nd a εn > 0

small enough such that

diam[B(f n0 (p), εn, 〈·, ·〉)] < %n and diam[In(B(f n0 (p), εn, 〈·, ·〉))] < %′n

(we use the notations B(f ni (p), εn, 〈·, ·〉) for the ball in (Mn, 〈·, ·〉) and %′n for the injectivity radius

of the exponential map at f n0 (p) considering the metric 〈·, ·〉′ on M). For each n ≥ 0, take

ε′n =1

2diam[In(B(f n0 (p), εn, 〈·, ·〉))].

Fix q ∈ N s(p, ε, 〈·, ·〉). Thus In(f n0 (q)) ∈ B(f n0 (p), ε′n, 〈·, ·〉′) for all n ≥ 0. Let v ∈ TpM0 be such

that expp(v) = q. Since 〈·, ·〉 and 〈·, ·〉′ are uniformly equivalent, there exist positive numbers k,K

such that k‖v‖′ ≤ ‖v‖ ≤ K‖v‖′, for all v ∈ Tf n(p)Mn, n ≥ 0, where ‖ · ‖ and ‖ · ‖′ are the norms

induced by 〈·, ·〉 and 〈·, ·〉′, respectively. Thus,

1

nlogd(f n(p), f n(q)) =

1

nlog ‖fi+n−1 · · · fi(v)‖ ≥ 1

nlog k‖fi+n−1 · · · fi(v)‖′.

Therefore, q ∈ N s(p, ε′, 〈·, ·〉). Thus, N s(p, ε, 〈·, ·〉) ⊆ N s(p, ε′, 〈·, ·〉′). Analogously we can prove the

existence of the sequence ε = (εi)i∈Z such that

N s(p, ε′, 〈·, ·〉′) ⊆ N s(p, ε, 〈·, ·〉),


From now on we assume that f admits sequences of positive numbers δ = (δn)n∈Z and ε =

(εn)n∈Z which satisfy the conditions of Theorems 5.2.10 and 5.2.11. In the following lemma we

show the inclusions Wu(p, δ) ⊆ N u(p, δ) and Ws(p, ε) ⊆ N s(p, ε). In Lemma 5.3.3 we will see a

condition to obtain the reverse inclusion.

Lemma 5.3.2. For each p ∈M0 we have

Wu(p, δ) ⊆ N u(p, δ) and Ws(p, ε) ⊆ N s(p, ε).

Proof. We will proveWu(p, δ) ⊆ N u(p, δ). Take q ∈ Wu(p, δ). By Theorem 5.2.10, we have f −n0 (q) ∈B(f −n0 (p), δ−n) and

d(f −n0 (q), f −n0 (p)) ≤ 2

∆0τnd(p, q) for each n ≥ 1.

Therefore,

1

nlogd(f −n0 (q), f −n0 (p)) ≤ 1

nlog

2

∆0+

logd(q, p)

n+

1

nlog τn.


Since τ < 1, we have

lim supn→∞

1

nlogd(f −n0 (q), f −n0 (p)) ≤ lim sup

n→∞

1

nlog τn = log τ < 0.

Hence, q ∈ N s(p, δ). Consequently, Ws(p, δ) ⊆ N s(p, δ).

Lemma 5.3.3. Set

Ω = lim supn→−∞

θn Ω = lim infn→∞

(1

nlog(1− cos θ−n))

Θ = lim supn→∞

θn Θ = lim infn→∞

(1

nlog(1− cos θn)).

Thus

i. Assume that we can choose the δn's such that δn ≤ εn for each n ≤ 0. If Ω > 0 and Ω ≥ log τ,

then there exists a sequence of positive numbers δ′ = (δ′n)n∈Z such that N u(p, δ′) ⊆ Wu(p, δ′).

ii. Assume that we can choose the εn's such that εn ≤ δn for each n ≥ 0. If Θ > 0 and Θ ≥ log τ ,

then there exists a sequence of positive numbers ε′ = (ε′n)n∈Z such that N s(p, ε′) ⊆ Ws(p, ε′).

Proof. We will prove (i). Fix ν ∈ (0,Ω). Let (ni)i∈N be a sequence of natural numbers, with

0 = n0 < n1 < · · · < nm < · · · , and θni ≥ ν for each i ≥ 0. Since δn ≤ εn and Ω > 0, we can choose

δ′n ≤ δn/3 small enough such that δ′ = (δ′n)n∈Z satises (5.2.2) and

B(f −ni0 (p), δ′−ni) ⊆ Ws(f −ni0 (p), ε)×Wu(f −ni0 (p), δ), for each i ≥ 0.

By Theorem 5.2.10, we have a family

Wu(f n0 (p), δ′) : n ∈ Z

of admissible (u, α, δ′)-manifold. Next, we prove that N u(p, δ′) ⊆ Wu(p, δ′). Indeed, suppose there

exists q ∈ N u(p, δ′) \Wu(p, δ′). Since f −ni0 (q) ∈ B(f −ni0 (p), δ′−ni), we have

(f−ni)−1 · · · (f−1)−1(exp−1

p (q)) = (x−ni , y−ni) ∈ Ws(f −ni0 (p), ε)×Wu(f −ni0 (p), δ),

for all i ≥ 0, where x−ni ∈ Ws(f −ni0 (p), ε) \ 0 and y−ni ∈ Wu(f −ni0 (p), δ). We can obtain from

(5.2.13) and (5.2.16) that

‖(x−ni ,y−ni)‖ ≥ ‖x−ni‖ − ‖y−ni‖ ≥∆−ni

2τ−ni‖x0‖ −

2

∆0τni‖y0‖.

We have 2∆0τni‖y0‖ → 0 as i→ +∞. Consequently,

lim supi→∞

1

nilog ‖(x−ni , y−ni)‖ ≥ lim sup

i→∞

1

nilog

∆−ni2

τ−ni‖x0‖ = lim supi→∞

(1

nilog ∆−ni − log τ)

= lim supi→∞

(1

nilog ∆−ni)− log τ = lim sup

i→∞(

1

nilog(1− cos θ−ni))− log τ

≥ lim infn→∞

(1

nlog(1− cos θ−n))− log τ ≥ 0.

LOCAL STABLE AND UNSTABLE MANIFOLDS 69

This fact contradicts that q ∈ N s(p, δ′).

From now on we will assume that for each p ∈ M0 we can choose the sequences (δn)n∈Z and

(εn)n∈Z as in Theorems 5.2.10 and 5.2.11, such that

δn = εn, for each n, log λ ≤ minΩ, Θ and 0 < minΩ,Θ. (5.3.1)

Therefore we have by Lemmas 5.3.2 and 5.3.3 that there exist two sequences of positive numbers

δ′ = (δ′n)n∈Z and ε′ = (ε′n)n∈Z such that

Wu(f n0 (p), δ′) : n ∈ Z and Ws(f n0 (p), ε′) : n ∈ Z

are two-sided sequeces of admissible manifolds at p, and furthermore

N u(p, δ′) =Wu(p, δ′) and N s(p, ε′) =Ws(p, ε′).

Next we will prove that N u(p, η) and N s(p, η) depend continuously on p.

Lemma 5.3.4. Let (pm)m∈N be a sequence in M0 converging to p ∈ M0 as m → ∞. If qm ∈N r(pm, η) converges to q ∈ B(p, η0) as m→∞, then q ∈ N r(p, η), for r = s, u.

Proof. We will prove only the stable case. Set

ω = suppm,m≥0

lim supn→∞

1

nlogd(f n0 (q′), f n0 (pm)) : q′ ∈ N s(pm, η).

By compactness of M0, we have ω ≤ 0. Let β ∈ (0, exp(ω)). For each n ∈ N, take mn ∈ N, withm1 < · · · < mn < · · ·, such that

d(f n0 (pmn), f n0 (p)) < βn and d(f n0 (qmn), f n0 (q)) < βn.

For every n we have

1

nlog(d(f n0 (q), f n0 (p))) ≤ 1

nlog[d(f n0 (q), f n0 (qmn)) + d(f n0 (qmn), f n0 (pmn)) + d(f n0 (pmn), f n0 (p))].

Since

lim supn→∞

1

nlog(an + bn) = maxlim sup

n→∞

1

nlog(an), lim sup

n→∞

1

nlog(bn)

for any sequence of positive numbers an and bn, we have

lim supn→∞

1

nlog(d(f n0 (q), f n0 (p))) ≤ maxlog(β), ω = ω.

Consequently, q ∈ N s(p, η).

Finally, by Theorems 5.2.10 and 5.2.11 and Lemmas 5.3.2-5.3.4, we obtain the following local

unstable and stable manifold theorems for Anosov families:

Theorem 5.3.5 (Local unstable manifold for Anosov families). If (M, 〈·, ·〉, f) admits a sequence of

positive numbers (δn)n∈Z satisfying (5.3.1) for each p ∈M0, then there exists a sequence of positive

numbers η = (ηn)n∈Z, such that N u(fn0 (p), η) is a dierentiable submanifold of Mn with:


i. Tfn(p)N u(fn0 (p), η) = Eufn(p);

ii. f−1n−1(N u(fn0 (p), η)) ⊆ N u(fn−1

0 (p), η);

iii. if q ∈ N u(pn+1, δ), where pn = fn0 (p), and k ≥ 0 we have

d(f−(k+1)n+1 (q), f

−(k+1)n+1 (pn+1)) ≤ 2

∆n+1τk+1d(q, pn+1).

iv. N u(p, η) depends continuously on p.

Theorem 5.3.6 (Local stable manifold for Anosov families). If (M, 〈·, ·〉, f) admits a sequence of

positive numbers (εn)n∈Z satisfying (5.3.1) for each p ∈M0, then there exists a sequence of positive

numbers η = (ηn)n∈Z, such that N s(fn0 (p), η) is a dierentiable submanifold of Mn with:

i. Tfn(p)N s(fn0 (p), η) = Esfn(p);

ii. fn(N s(fn0 (p), η)) ⊆ N s(fn+10 (p), η);

iii. if q ∈ N s(pn, ε), where pn = fn0 (p), and k ≥ 1 we have

d(fkn(q), fkn(pn)) ≤ 2

∆nςkd(q, pn),

iv. N s(p, η) depends continuously on p.

5.4 Stable and unstable manifolds for Anosov Families consisting

of matrices

In this section, we will see that any Anosov family F = (Fn)n∈Z, where Fn : Tm → Tm is a

dieomorphism induced by a matrix An : Rm → Rm and Tm is the m-torus endowed with the

Riemannian metric inherited from Rm, admits stable and unstable admissible manifolds. This is

a particular case of the theorems obtained in the previous sections. As we saw in Remark 4.4.3,

ap(z) = bp(z) = cp(z) = dp(z) = 0 for each p ∈ Tm and z ∈ Rm (see (3.4.1)). Therefore, σp(δ) =

ρp(δ) = 0 for each p ∈ Tm and δ > 0 (see Denition 3.4.7).

Inductively we can choose a two-sided sequence δ = (δn)n∈Z of positive numbers such that

Fn(Bs(δn)) ⊆ Bsn(δn+1) for each n ∈ Z. Take

Ws(Fn0 (p), δ) = Fn0 (p) + tvs : vs ∈ EsFn0 (p) and t ∈ (−δn, δn) for p ∈M0. (5.4.1)

The two-sided sequence Ws(Fn0 (p), δ) : n ∈ Z satises the properties i., ii. and iii. from Theorem

5.2.10 (in this case, we can take τ = λ and take 2c instead of 2∆n

). Consequently:

Theorem 5.4.1. If F = (Fn)n∈Z is an Anosov family where Fn : Tm → Tm is a dieomorphism

induced by a matrix An : Rm → Rm, then the local stable manifolds for F consist of vectors obtained

as in (5.4.1).

Analogously we can obtain:

STABLE AND UNSTABLE MANIFOLDS FOR MATRIX ANOSOV FAMILIES 71

Theorem 5.4.2. If F = (Fn)n∈Z is an Anosov family where Fn : Tm → Tm is a dieomorphism

induced by a matrix An : Rm → Rm, then for any p ∈M0 we have

Wu(Fn0 (p), δ) = Fn0 (p) + tvs : vs ∈ EuFni (p) and t ∈ (−δn, δn),

are the unstable manifolds of F at p, for some two-sided sequence of positive numbers (δn)n∈Z.

Chapter 6

Structural Stability for Anosov Families

As noted above, when f is the constant family associated to an Anosov dieomorphism, Theorem

4.3.5 is valid for a uniform basic strong neighborhood of f (see Denition 4.4.1). It is also possible to

nd a uniform neighborhood of f if each fi is a small perturbations of an Anosov dieomorphim. In

general it is not possible to nd a uniform neighborhood of an Anosov family such that each family

in that neighborhood is Anosov. For example, if the injectivity radius of the exponential application

at each point in Mi is arbitrarily small for |i| large, or if the angle between the stable and unstable

subspace of the splitting of the tangent bundle decays, or if we can not get the inequality (4.1.3)

with a uniform βi, etc., it is necessary to take each ξi ever smaller.

Set

A2b(M) = f ∈ F2(M) : f is Anosov, s.p.a. and Sf <∞, (6.0.1)

where

Sf := supi∈Z‖fi‖C2 = sup

i∈Zmax

‖Dfi‖, ‖Df−1

i ‖, ‖D2fi‖, ‖D2f−1

i ‖.

The main goal of this chapter is to prove that A2b(M) is uniformly structurally stable in F2(M),

that is, for each f ∈ A2b(M), there exists ε > 0 such that any g ∈ B2(f , ε) is Anosov and uniformly

conjugate to f (see Theorems 6.1.7 and 6.3.9). This fact generalizes Theorem 1.1 in [Liu98], which

proves the structural stability of random small perturbations of hyperbolic dieomorphisms. Fur-

thermore, in Section 6.2 we will prove another version of Theorems 5.3.5 and 5.3.6 for elements in

A2b(M). We will prove that every element in A2

b(M) admits stable and unstable manifolds, and, fur-

thermore, the size of these stable and unstable manifolds is the same along the orbits (see Theorems

6.2.2 and 6.2.3).

For each n ∈ Z, let %n > 0 be an injectivity radius of each p ∈ Mn. Throughout this chapter,

we will suppose that

% := infn∈Z

%n

is positive.

6.1 Openness of A2b(M)

In Chapter 4 we showed that A1(M) is open in F1(M) endowed with the strong topology,

that is, for each g ∈ A1(M) there exists a strong basic neighborhood B1(g , (ξi)i∈Z) of g such

73

74 STRUCTURAL STABILITY FOR ANOSOV FAMILIES

that B1(g , (ξi)i∈Z) ⊆ A1(M). In that case, it is not always possible to take the sequence (ξi)i∈Z

bounded away from zero, that is, ξi could decay as i → ±∞. The goal of this section is to show

that for the case of Anosov families in A2b(M), this basic neighborhood can be taken uniform (see

Denition 4.4.1), as in the case where the family consists of matrices acting on the torus, as we saw

in Section 4.4. That is, A2b(M) is open in F2(M) endowed with the uniform topology. This fact is

a generalization of the Young's result in [You86] (see Remark 3.2.7).

From now on, f = (fi)i∈Z will be a xed element in A2b(M).

Remark 6.1.1. The Riemannian norm ‖ · ‖∗, given in (3.3.1), is uniformly equivalent to ‖ · ‖ onM, since f s.p.a. By Proposition 3.3.4, f is strictly Anosov with the norm ‖ · ‖∗. Clearly, this normis uniformly equivalent to the norm on TM given by

‖(vs, vu)‖? = max‖vs‖∗, ‖vu‖∗, for (vs, vu) ∈ Esp ⊕ Eup , p ∈M. (6.1.1)

Consequently, there exists C ≥ 1 such that

(1/C)‖v‖? = ‖v‖ ≤ C‖v‖?, for every v ∈ Esp ⊕ Eup , p ∈M. (6.1.2)

From now on we will consider the metric given in (6.1.1). Take r ∈ (0, %/20Sf ). Fix g = (gi)i∈Z ∈B2(g , r). If p, q ∈ Mi and d(p, q) ≤ r, then d(fi(p), gi(q)) < %/2 and d(f−1

i (p), g−1i (q)) < %/2.

Therefore, gi(B(p, r)) ⊆ B(fi(p), %/2) and g−1i (B(fi(p), r)) ⊆ B(p, %/2). Consequently,

gp = exp−1f (p) g expp : B(0p, r)→ B(0f (p), %/2)

and g−1p = exp−1

p g−1 expf (p) : B(0f (p), r)→ B(0p, %/2),

are well-dened for each p ∈M.

Denition 6.1.2. For r ≤ r, consider

σp(r, g) = supz∈B(0p,r)

‖D(D(fp)0 − gp)z‖? and ρp(r, g) = supz∈B(0f (p),r)

‖D(D(f−1p )0 − g−1

p )z‖?.

Proposition 6.1.3. Fix τ > 0. There exist r > 0, δ > 0 and Xi = p1,i, . . . , pmi,i ⊆ Mi for each

i ∈ Z, such that Mi = ∪mij=1B(pj,i, r) for each i ∈ Z and, furthermore, for every g ∈ B2(f, δ), we

have that

ϑ(r, g) = supp∈Xi,i∈Z

σp(r, g), ρp(r, g) < τ.

Proof. Fix p ∈ Mi. Let z ∈ B(0p, r) and (v, w) ∈ Es ⊕ Eu. There exists K > 0 (which does not

depend on p), such that

‖D(fp)0(v, w)−D(fp)z(v, w)‖∗ ≤ K[1 + ‖Dfi‖∗]‖D2fi‖∗‖z‖∗‖(v, w)‖∗

(see [LQ06], p. 50). Thus,

‖D(D(fp)0 − gp)z(v, w)‖∗ = ‖D(fp)0(v, w)−D(gp)z(v, w)‖∗≤ ‖D(fp)0(v, w)−D(fp)z(v, w)‖∗ + ‖D(fp − gp)z(v, w)‖∗≤ K[1 + Sf ]Sf ‖z‖∗‖(v, w)‖∗ + d2(fi, g)‖(v, w)‖∗.

OPENNESS OF A2B(M) 75

Therefore, if δ1 < τ/2 and r1 < τ/2(K(1 + Sf )Sf ), then for each g ∈ B1(f , δ1) and z ∈ B(0p, r1),

we have ‖D(D(fp)0 − gp)z‖∗ < τ .

Analogously we can prove there exist δ2 > 0 and r2 > 0 such that, if g ∈ B2(f , δ2) and

z ∈ B(0fi(p), r2), then ‖D(D(f−1p )0 − g−1

p )z‖∗ ≤ τ .Take r = minr1, r2 and δ = minδ1, δ2. Notice that neither r nor δ depend on p. Since Mi is

compact, we can choose a nite subset Xi = p1,i, . . . , pmi,i ⊆ Mi such that Mi = ∪mij=1B(pj,i, r)

for each i ∈ Z, which proves the proposition.

We will x α ∈ (0, 1−λ1+λ

) and furthermore, we will suppose that δ ∈ (0, %/20Sf ) and r are small

enough such that

ϑ(r, g) < σA := min

(λ−1 − λ)α

2(1 + α)2,λ−1(1− α)− (1 + α)α

2(1 + α)

. (6.1.3)

For each i ∈ Z, let Xi = p1,i, . . . , pmi,i ⊆Mi be such that Mi = ∪mij=1B(pj,i, r). Note that, for

each i ∈ Z,ϑ := max

q∈Xiϑq(r) ≤ ϑ(r, f ) < σA (see Lemma 4.1.3).

The next two lemmas can be showed analogously to the Lemmas 4.1.4 and 4.1.5, respectively.

Lemma 6.1.4. If δ ≤ 12 minr, δ, ϑ, then, for g ∈ B2(f, δ) and p ∈M we have:


α for all z ∈ Bk(0, r)×Bm−k(0, r), and

ii. D(gp)−1z (Ks

α) ⊆ Ksα for all z ∈ Bk(0, r)×Bm−k(0, r).

Lemma 6.1.5. If δ ≤ 12 minr, δ, ϑ, there exists η ∈ (0, 1) such that, if g ∈ B1(f, δ) then, for

p ∈M and z ∈ Bk(0, r)×Bm−k(0, r), we have

i. ‖D(gp)z(v)‖ ≥ η−1‖v‖ if v ∈ Kuα;

ii. ‖D(g−1p )z(v)‖ ≥ η−1‖v‖ if v ∈ Ks

α.

Now, take ξ = 12C minr, δ, ϑ (see (6.1.2)). Using the Lemmas 6.1.4 and 6.1.5 we can prove

that:

Lemma 6.1.6. Let g = (gi)i∈Z ∈ B2(f, ξ). For each p ∈M, take

F sp =∞⋂n=0

D(g−n)gn(p)(Ksα,f,gn(p)) and F up =

∞⋂n=0

D(gn)g−n(p)(Kuα,f,g−n(p)

).

The families F sp and F up are Dg-invariant subspaces with which g satises the conditions in Deni-

tion 3.1.2. Furthermore, g satises the property of angles.

Consequently, we have:

Theorem 6.1.7. For each f ∈ A2b(M), there exists ξ > 0 such that B2(f, ξ) ⊆ A2

b(M). That is,

A2b(M) is open in F2(M) with the uniform topology.


6.2 Local Stable and Unstable Manifolds for Elements in A2b(M)

In Theorems 5.3.5 and 5.3.6 we gave conditions for obtain stable and unstable manifolds at each

point in each component Mi. In that case, the diameter of each manifold (2εi and 2δi) could decay

when i → ±∞. In Theorems 6.2.2 and 6.2.3 we will see that each f ∈ A2b(M) admits stable and

unstable manifold with the same size at each point.

Fix γ ∈ (λ2, 1) and consider σn and ρn as in (5.2.1) and (5.2.14), respectively.

Remark 6.2.1. It is clear that

σ := min

(λ−1 − λ)α

(1 + α)2,

γλ− λ(1 + α)(1 + γ)

,1− λ1 + λ

≤ minσn, ρn.

Take δ small enough such that maxσp(δ, f ), ρp(δ, f ) ≤ σ (see Denition 6.1.2). Thus, f satises

the assumption of Proposition 5.2.5, considering δn = δ for every n ∈ Z.

Therefore, there exist ε > 0 and τ ∈ (0, 1) such that:

Theorem 6.2.2. For each p ∈ M, N u(p, (ε)i∈Z) is a dierentiable submanifold of M and there

exists Ku > 0 such that:

(i) TpN u(p, (ε)i∈Z) = Eup ,

(ii) f−1(N u(p, (ε)i∈Z)) ⊆ N u(f−1(p), (ε)i∈Z),

(iii) if q ∈ N u(p, (ε)i∈Z) and n ≥ 1 we have

d(f−n(q), f−n(p)) ≤ Kuτnd(q, p).

(iv) Let (pm)m∈N be a sequence in Mi converging to p ∈Mi when m→∞. If qm ∈ N u(pm, (ε)i∈Z)

converges to q ∈ B(p, ε) as m→∞, then q ∈ N u(p, (ε)i∈Z).

Proof. Theorems 5.2.10 and 5.3.5.

Analogously, we have:

Theorem 6.2.3. For each p ∈ M, N s(p, (ε)i∈Z) is a dierentiable submanifold of M and there

exists Ks > 0 such that:

(i) TpN s(p, (ε)i∈Z) = Esp,

(ii) f(N s(p, (ε)i∈Z)) ⊆ N s(f(p), (ε)i∈Z),

(iii) if q ∈ N s(p, (ε)i∈Z) and n ≥ 1 we have

d(fn(q), fn(p)) ≤ Ksζnd(q, p).

(iv) Let (pm)m∈N be a sequence in Mi converging to p ∈Mi when m→∞. If qm ∈ N s(pm, (ε)i∈Z)

converges to q ∈ B(p, ε) as m→∞, then q ∈ N s(p, (ε)i∈Z).

Proof. See Theorems 5.2.11 and 5.3.6.

STRUCTURAL STABILITY OF A2B(M) 77

Ku and Ks from Theorems 6.2.2 and Theorems 6.2.3, respectively, depend on the constant

c of f , on the constant C in (6.1.2) and on the minimum angle between the stable and unstable

subspaces of the splitting TM = Es⊕Eu, which is positive because we are supposing that f satises

the property of angles.

6.3 Structural Stability of A2b(M)

In this section we will show that A2b(M) is uniformly structurally stable in F2(M): for each

f ∈ A2b(M) there exists a uniform basic neighborhood B2(f , δ) of f such that, each g ∈ B2(f , δ)

is uniformly conjugate to f . Since ‖ · ‖ and ‖ · ‖? are uniformly equivalent (see (6.1.2)) and f is

strictly Anosov with ‖ · ‖?, we can suppose, without loss of generality, that f is strictly Anosov

with constant λ ∈ (0, 1) considering the norm ‖ · ‖ on M. Furthermore, we can suppose that the

the stable and unstable subspaces are orthogonal.

To prove the structural stability of Anosov families in A2b(M) we have adapted the Shub's ideas

in [Shu13] to prove the structural stability of Anosov dieomorphisms on compact Riemannian

manifolds. We will divide the proof of this fact into a series of lemmas and propositions. Throughout

this section, we will consider r > 0, ξ > 0 and η ∈ [λ, 1) as in Section 4.3.

Denition 6.3.1. For τ > 0 and i ∈ Z, set:

(i) D(Ii, τ) = h : Mi →Mi : h is C0 and d(h(p), Ii(p)) ≤ τ for any p ∈Mi;

(ii) D(τ) = (hi)i∈Z : hi ∈ D(Ii, τ) for any i ∈ Z;

(iii) Γ(Mi) = σ : Mi → TMi : σ is a continuous section;

(iv) Γτ (Mi) = σ ∈ Γ(Mi) : supp∈Mi‖σ(p)‖ ≤ τ;

(v) Γ(M) = (σi)i∈Z : σi ∈ Γ(Mi) and supi∈Z ‖σi‖Γi <∞ ;

(vi) Γτ (M) = (σi)i∈Z ∈ Γ(M) : σi ∈ Γτ (Mi) for each i ∈ Z.

Γ(Mi) is a Banach space with the norm ‖σ‖Γi = supp∈Mi‖σ(p)‖. Therefore:

Lemma 6.3.2. Γ(M) is a Banach space endowed with the norm ‖(σi)i∈Z‖∞ = supi∈Z ‖σi‖Γi .

Let 0 < ε ≤ %/2. We can identify D(ε) with Γε(M) by the homeomorphism

Φ : D(ε)→ Γε(M)

(hi)i∈Z 7→ (Φi(hi))i∈Z,

where Φi(hi)(p) = exp−1p (hi(p)), for p ∈Mi. Note that Φi(Ii) is the zero section in Γ(Mi).

Next we will prove the following lemma:

Lemma 6.3.3. Fix κ > 0. There exist ξ′ ∈ (0, ξ] and r′ ∈ (0, r/3] such that, if g = (gi)i∈Z ∈B2(f, ξ′), then the map

G : D(r′)→ D(κ)

(hi)i∈Z 7→ (gi−1 hi−1 f−1i−1)i∈Z


is well-dened.

Proof. It is sucient to prove that there exist ξ′ > 0 and r′ > 0 such that, for every i ∈ Z, ifgi : Mi →Mi+1 is a dieomorphism with d1(gi, fi) < ξ′ and h ∈ D(Ii, r

′), then gihf−1i ∈ D(Ii+1, κ).

For each continuous map h : Mi →Mi, we have

d(gihf−1i (p), Ii+1(p)) ≤ d(gihf

−1i (p), gif

−1i (p)) + d(gif

−1i (p), Ii+1(p)).

Let S := supi∈Z ‖Dfi‖ <∞. Consider r′ < κ/2(S + 1) and ξ′ < min1, κ/2. For i ∈ Z, if

gi ∈ B1(fi, ξ′) = g ∈ Di1(Mi,Mi+1) : maxd1(g, fi), d

1(g−1, f−1i ) < ξ′

we have

d(gif−1i (p), Ii+1(p)) = d(gif

−1i (p), fif

−1i (p)) ≤ d1(gi, fi) < ξ′ < κ/2.

Furthermore, if h ∈ D(Ii, r′) and gi ∈ B1(fi, ξ

′), then

d(gihf−1i (p), gif

−1i (p)) ≤ ‖Dgi‖d(hf−1

i (p), f−1i (p)) ≤ (ξ′ + S)r′ < κ/2.

Therefore, if h ∈ D(Ii, r′) and gi ∈ B1(fi, ξ

′), we have gi h f−1i ∈ D(Ii+1, κ), which proves

the lemma.

Fix ζ ∈ (0,min1− λ, 1− η, %/2). It follows from Lemma 6.3.3 that there exist

r′ ∈ (0, r/3) and ξ′ ∈ (0,minξ, r′(1− λ− ζ)), (6.3.1)

such that if g = (gi)i∈Z ∈ B2(f , ξ′) then

G : Γr′(M)→ Γζ(M),

(σi)i∈Z 7→ (ΦiGi−1Φ−1i−1(σi−1))i∈Z

is well-dened, where Gi−1(h) = gi−1 h f−1i−1, for h ∈ D(Ii−1, r

′). Consequently,

ΦiGi−1Φ−1i−1(σ)(p) = exp−1

p gi−1 expf−1i−1(p)σ(f−1

i−1(p)) for p ∈Mi, σ ∈ Γr′(Mi−1).

Set

Γs(Mi) = σ ∈ Γ(Mi) : the image of σ is contained in Es

and

Γu(Mi) = σ ∈ Γ(Mi) : the image of σ is contained in Eu

where Es and Eu are the stable and unstable subbundles induced by f . It is clear that

Γ(Mi) = Γs(Mi)⊕ Γu(Mi).


Proposition 6.3.4. The map

F : Γ(M)→ Γ(M)

(σi)i∈Z 7→ (Fi−1(σi−1))i∈Z

is a hyperbolic bounded linear operator, where Fi : Γ(Mi) → Γ(Mi+1) is dened by the formula

Fi(σ)(p) = D(fi)f−1i (p)(σ(f−1

i (p))), for p ∈Mi+1, σ ∈ Γ(Mi).

Proof. It is not dicult to prove that F is a linear operator and

‖F‖ = supi∈Z‖Fi(σi)‖Γi : ‖(σi)‖∞ = 1 ≤ sup

i‖Dfi‖∗ <∞.

Thus, F is a bounded linear operator. It is clear that Fi(Γs(Mi)) = Γs(Mi+1) and Fi(Γ

u(Mi)) =

Γu(Mi+1). Furthermore,

‖Fi(σ)‖Γi+1 ≤ λ‖σ‖Γi for σ ∈ Γs(Mi) and ‖F−1i (σ)‖Γi ≤ λ‖σ‖Γi+1 for σ ∈ Γu(Mi+1).

For t = s, u, set

Γt(M) = (σi)i∈Z ∈ Γ(M) : σi ∈ Γt(Mi) for each i ∈ Z.

We can prove that Γ(M) = Γs(M)⊕ Γu(M) and F is a hyperbolic linear operator with respect to

this splitting.

Lemma 6.3.5. There exist ξ′ ∈ (0, ξ] and r′ ∈ (0, r/3] such that, if g ∈ B2(f, ξ′), then

Lip([F−G]|Γr′ (M)) < ζ,

where Lip denotes a Lipschitz constant.

Proof. For σ ∈ Γr′(Mi−1) and p ∈Mi we have

[Fi−1 − ΦiGi−1Φ−1i−1](σ)(p) = [D(fi−1)f−1

i−1(p) − exp−1p gi−1 expf−1

i−1(p)](σ(f−1i−1(p))).

Let q = f−1i−1(p). Note that

D(fi−1)q = D(exp−1p fi−1 expq)0q where 0q is the zero vector in TqM.

For any v ∈ TqM with ‖v‖ < ξ, we have

D[D(fi−1)q − exp−1p gi−1 expq]v = D[D(exp−1

p fi−1 expq)0q − exp−1p gi−1 expq]v

= D(exp−1p fi−1 expq)0q −D(exp−1

p gi−1 expq)v.

As we saw in the proof of Proposition 6.1.3,

‖D(exp−1p fi−1 expq)0q −D(exp−1

p gi−1 expq)v‖ ≤ K[1 + Sf ]Sf ‖v‖+ d2(fi−1, gi−1).


Hence if σ = (σi)i∈Z, σ = (σi)i∈Z ∈ Γr′(Mi−1) and p ∈Mi we have

‖[Fi−1 − ΦiGi−1Φ−1i−1](σi−1)(p)− [Fi−1 − ΦiGi−1Φ−1

i−1](σi−1)(p)‖ ≤ J ‖σi−1(p)− σi−1(p)‖

≤ J ‖σi−1 − σi−1‖Γi≤ J‖σ − σ‖∞,

where J = K[1+Sf ]Sf ‖v‖+d2(fi−1, gi−1).We can choose r′ and ξ′ small enough such that J < ζ.

Thus

‖[Fi−1 − ΦiGi−1Φ−1i−1](σi−1)− [Fi−1 − ΦiGi−1Φ−1

i−1](σi−1)‖Γi ≤ ζ‖σ − σ‖∞,

and therefore

‖[F−G](σ)− [F−G](σ)‖∞ ≤ ζ‖σ − σ‖∞,


From now on we will suppose that ξ′ and r′ satisfy (6.3.1) and Lemma 6.3.5. Furthermore, we

will x g = (gi)i∈Z ∈ B2(f , ξ′). To prove the following lemma, we have based on the proof of the

Proposition 7.7 in [Shu13].

Lemma 6.3.6. G|Γr′ (M) has a xed point in Γr′(M).

Proof. Since Γ(M) = Γs(M)⊕ Γu(M), each σ = (σi)i∈Z ∈ Γr′(M) can be written as σ = σs + σu,

where σs = (σi,s)i∈Z ∈ Γs(M) and σu = (σi,u)i∈Z ∈ Γu(M). Let G be dened on Γr′(M) as

G(σ) = (G(σ))s + (F−1[σu + F(σu)− (G(σ))u])u.

If σ ∈ Γr′(M) is a xed point of G, we have

(G(σ))s = σs and (G(σ))u = σu,

that is, σ is a xed point of G. Therefore, in order to prove the lemma, it is sucient to nd

a xed point of G. First we prove that G is a contraction. Take σ = (σi,s)i∈Z + (σi,u)i∈Z and

σ = (σi,s)i∈Z + (σi,u)i∈Z in Γr′(M), where σs, σs ∈ Γs(M) and σu, σu ∈ Γu(M). Thus,

‖G(σ)i+1 − G(σ)i+1‖Γi+1 = ‖Gi(σi)− Gi(σi)‖Γi+1

≤ max‖(F−1[(σi,u − σi,u) + ((F−G)(σ))i,u − ((F−G)(σ))i,u])u‖, ‖(G(σ − σ))i,s‖

≤ maxλ(1 + ζ)‖σ − σ‖∞, (λ+ ζ)‖σ − σ‖∞ = (λ+ ζ)‖σ − σ‖∞.

Hence, ‖G(σ) − G(σ)‖∞ ≤ (λ + ζ)‖σ − σ‖∞. Since λ + ζ < 1, G is a contraction. Now we prove

that G(Γr′(M)) ⊆ Γr′(M). If 0 = (0i)i∈Z is the sequence of the zero sections, we have that

‖G(σ)i+1‖Γi+1 ≤ (λ+ ζ)‖σ‖Γi + ‖G(0)i+1‖Γi+1

≤ (λ+ ζ)‖σ‖∞ + max‖(F−1(G(0))u)i+1‖Γi+1 , ‖(G(0))s‖Γi+1

≤ (λ+ ζ)‖σ‖∞ + maxλ‖(G(0)u)i+1‖Γi+1 , ‖(G(0)s)i+1‖Γi+1

≤ (λ+ ζ)‖σ‖∞ + ‖G(0)‖∞,


thus ‖G(σ)‖∞ ≤ (λ+ ζ)‖σ‖∞ + ‖G(0)‖∞. Now, for each i ∈ Z, p ∈Mi, we have

‖Gi(0i)(p)‖ = ‖exp−1p gi expf−1

i (p)(0f−1i (p))‖ = ‖exp−1

p (gif−1i (p))‖

= d(gif−1i (p), p) = d(gif

−1i (p), fif

−1i (p)) < δ′

Consequently, if σ ∈ Γr′(M), then ‖G(σ)‖∞ < (λ+ζ)r′+δ′ < r′, that is, G(σ) ∈ Γr′(M). Therefore,

G has a xed point in Γr′(M).

If (σi)i∈Z is the xed point of G, then, considering hi = Φ−1i (σi) for each i ∈ Z, we have that

(hi)i∈Z is a xed point of G. Hence, Lemma 6.3.6 implies that G|D(r′) has a xed point in D(r′).

Lemma 6.3.7. Fix g = (gi)i∈Z ∈ B2(f, δ′). Let p ∈M and v = (vs, vu), w = (ws, wu) ∈ Bs(0, r)×Bu(0, r). If ‖vs − ws‖ ≤ ‖vu − wu‖, then

‖(gp(v))s − (gp(w))s‖ ≤ (η−1 − ζ)‖vu − wu‖ ≤ ‖(gp(v))u − (gp(w))u‖.

On the other hand, if ‖vu − wu‖ ≤ ‖vs − ws‖, then

‖(g−1p (v))u − (g−1

p (w))u‖ ≤ (η−1 − ζ)‖vs − ws‖ ≤ ‖(g−1p (v))s − (g−1

p (w))s‖.

Proof. See [Shu13], Lemma II.1.

Let v = (vs, vu) ∈ Bs(0, r) × Bu(0, r) and q = expp(v). Suppose that d(gn(p), gn(q)) < r for

n = ±1. Thus, if ‖vs‖ ≤ ‖vu‖, by Lemma 6.3.7 we have

d(q, p) = ‖v‖ ≤ C‖v‖ = C‖vu‖ ≤ C(η−1 − ζ)−1‖(gp(v))u − (gp(0p))u‖

≤ C2(η−1 − ζ)−1‖gp(v)− gp(0p)‖ ≤ 2C2(η−1 − ζ)−1d(g(p), g(q))

≤ 2C2(η−1 − ζ)−1r.

Analogously if ‖vu‖ ≤ ‖vs‖, then

d(q, p) ≤ 2C2(η−1 − ζ)−1r.

Inductively, we can prove that:

Proposition 6.3.8. For each i ∈ Z, if p, q ∈ Mi and d(gni (p), gni (q)) < r for each n ∈ [−N,N ],

then

d(q, p) ≤ 2C2(η−1 − ζ)−N r.

Finally,

Theorem 6.3.9. A2b(M) is uniformly structurally stable.

Proof. Take (gi)i∈Z ∈ B2(f , δ′). It follows from Lemma 6.3.6 that there exists h = (hi)i∈Z, with

hi ∈ D(Ii, r′) for each i, such that

gi hi = hi+1 fi for each i ∈ Z.


We will prove that h is equicontinuous and each hi is injective. Let α > 0. Take N > 0 such that

2C2(η−1 − ζ)−N r < α. Since (fi)i∈Z is equicontinuous, the family of sequences

(f ni )i∈Z : n ∈ [−N,N ]

is equicontinuous. Consequently, there exists β > 0 such that, for each i ∈ Z, if x, y ∈ Mi and

d(x, y) < β, then d(f ni (x), f ni (y)) < r/3 for any n ∈ [−N,N ]. Hence, for each i ∈ Z and n ∈[−N,N ], if x, y ∈Mi and d(x, y) < β, then

d(gni hi(x), gni hi(y)) ≤ d(gni hi(x), f ni (x)) + d(f ni (x), f ni (y)) + d(f ni (y), gni hi(y))

< r′ + r/3 + r′ ≤ r.

It follows from Proposition 6.3.8 that d(hi(x), hi(y)) < α. This fact proves that (hi)i∈Z is an

equicontinuous family. Notice that if hi(x) = hi(y) for some x, y ∈Mi, then d(gni hi(x), gni hi(y)) < r

for any n ∈ Z. Thus x = y and therefore hi is injective. Analogously we can prove that (h−1i )i∈Z is

equicontinuous. Consequently, A2b(M) is uniformly structurally stable.

Chapter 7

Some Other Problems That Arose From

This Work

We will nish this work by presenting some problems that arose from this thesis which we hope

to work in future projects.

7.1 Another Classication of Dynamical Systems on the Circle

In Remark 2.3.5 we gave a motivation to study the classication of constant families by uniform

conjugacies. Now, take S1 = z ∈ R2 : ‖z‖ = 1, endowed with the Riemannian metric inherited

from R2. For a homeomorphism φ : S1 → S1, we denote by f φ the constant family associated

to φ. In Proposition 1.4.5 we prove that there exist homeomorphisms φ and ψ on S1, which are

not topologically conjugate, with f φ and f ψ uniformly conjugate. Uniform conjugacies provide an

equivalence relation. It might be interesting to try to classify the homeomorphisms on the circle

with respect to this new equivalence relation.

Notice that, by the Poincaré Classication Theorem we have that if φ is topologically transitive,

then it is topologically conjugate to a rotation (see [KH97], Theorem 11.2.7). Therefore, every

constant family associated to a topologically transitive homeomorphism on the circle is uniformly

conjugate to the constant family associated to the identity on the circle (see Proposition 1.4.5). For

more details about this problem, see [Ace17a].

7.2 Entropy for Non-Stationary Dynamical Systems: Further Gen-

eralizations

In this work was proved the continuity of the entropy of non-stationary dynamical systems as

long as each dieomorphism fi is of class Cm with m ≥ 1. A very interesting project would be to

study the continuity of this entropy for sequences of homeomorphisms or Hölder continuous maps.

In this case, M could be a general metric space, that is, not necessarily a dierentiable manifold. A

series of results that could be very useful to work on this problem can be found in [ZC09], [KL16],

[KMS99], [KS96], [Yan80], [ZZH06], among others papers.

The entropy built here, was for a xed metric space M , with each map fi : M → M a home-

omorphism. This notion could be extended considering, for each i ∈ Z, a more general metric

83

84 OTHER PROBLEMS THAT AROSE

space Mi, with a xed metric di, and each fi a continuous map on Mi to Mi+1, not necessarily a

homeomorphism. Another interesting project would be to study the properties of this entropy.

7.3 Existence and classication of Anosov Families

The existence of Anosov dieomorphisms φ : M → M imposes strong restrictions on the

manifold M . All known examples of Anosov dieomorphisms are dened on infranilmanifolds (see

[BP07], [Shu13], [Via14]). If M is a parallelizable Riemannian manifold1, suitably changing the

metrics on each component Mi = M × i we can build an Anosov family in the disjoint union of

Mi, taking fi as the identity Ii : Mi →Mi+1 (see Example 3.2.1). As we saw in Example 3.2.9, an

Anosov family does not necessarily consist of Anosov dieomorphisms. We say that a Riemannian

manifoldM with Riemannian metric 〈·, ·〉, admits an Anosov family if there exists an Anosov family

on M =⋃i∈ZM × i, considering on each M × i the Riemannian metric 〈·, ·〉i = 〈·, ·〉 for all

i ∈ Z. A natural question that arises from the above facts is: if M is a parallelizable Riemannian

manifold, is there an Anosov family on M? Since the constant family associated to an Anosov

difeomorphism is an Anosov family, each manifold admitting an Anosov dieomorphism admits an

Anosov family.

It is well-known that there are not Anosov dieomorphisms on the circle S1. Next we prove that

the circle does not admit Anosov families.

Proposition 7.3.1. Take Mi = S1 × i with Riemannian metric inherited from R2 and M the

disjoint union of the Mi. Thus, there is not any Anosov family on M. More specically, it does not

exist a contractive or expansive family on S1.

Proof. Suppose that (fi)i∈Z is an Anosov family on M. Fix p ∈ M0. Since the circle is one-

dimensional, then, either ‖D(f n0 )p(v)‖ ≤ cλn‖v‖ for all n ≥ 1, v ∈ TpS1 or ‖D(f −n0 )p(v)‖ ≤ cλn‖v‖for all n ≥ 1, v ∈ TpS1. Without loss of generality we can assume that ‖D(f n0 )p(v)‖ ≤ cλn‖v‖for all n ≥ 1, v ∈ TpS1. Let n be large enough such that cλn < 1/2. Take φ : S1 → S1 as

φ = fn−1 · · · f0. Thus, if p ∈ S1, then ‖Dφp(v)‖ ≤ (1/2)‖v‖ for all v ∈ TpS1. Since φ is bijective,

this is impossible.

From the previous proposition we get that if M is S1 then the answer to the above question is

no. Non-parallelizable manifolds cannot admit Anosov families. S1, S3 and S7 are the only spheres

which are parallelizable (see [Ker58]). Therefore the n-sphere Sn does not admit Anosov families

for n ∈ 1, 2, 4, 5, 6, 8, 9, .... On the other hand, non-orientable manifolds are non-parallelizable.

Consequently, non-orientable manifolds do not admit Anosov families.

Another natural question is:

Problem 7.3.2. Does M admit an Anosov family if and only if M admits an Anosov dieomor-

phism?

Results obtained in Section 4.4 provide of a great variety of examples of Anosov families. How-

ever, in all these examples, the dieomorphisms fi live in a small neighborhood of an (factor of an)

Anosov dieomorphism. Hence, another problem is the following one:

1A manifold M is called parallelizable if there exist dierentiable vector elds X1, . . . , Xm on M such thatX1(x), . . . , Xm(x) is a basis for the tangent space TxM , for all x ∈M .

HÖLDER CONTINUITY OF THE SUBBUNDLES 85

Problem 7.3.3. Given an Anosov family f = (fi)i∈Z dened on M , for each (or some) i ∈ Z there

exists n > 0 such that f ni is (homotopic to) an Anosov dieomorphism?

Next, it follows from J. Franks [Fra69] and A. Manning [Man74] that:

Theorem 7.3.4. If φ : Tm → Tm is an Anosov dieomorphism dened on the m-torus Tm, thenφ is topologically conjugate to a hyperbolic toral automorphism A : Tm → Tm, which belongs to the

same homotopy class of φ.

Hence all the Anosov dieomorphisms on Tm are classied. It is possible to generalize this result

for Anosov families:

Problem 7.3.5. Let (fi)i∈Z be an Anosov family, where each fi : Tm → Tm is an Anosov dif-

feomorphism in the same homotopy class of a hyperbolic toral automorphism A : Tm → Tm. Are(fi)i∈Z and (A)i∈Z uniformly conjugate?

Theorem 6.3.9 provides a particular case of Problem 7.3.5.

7.4 Hölder Continuity of the Subbundles

Let H be a Hilbert space with norm ‖ · ‖. Given two subspaces E and F of H, set

Γ(E,F ) := supv∈E‖v‖=1

infw∈F‖v − w‖.

Γ(E,F ) is called the aperture between E and F .

Denition 7.4.1. Let X be a metric space with metric d. A family Exx∈X of subspaces of H is

called Hölder continuous with exponent α ∈ (0, 1] and constant L > 0, if for any x, y ∈ X we have

ρ(Ex, Ey) := maxΓ(Ex, Ey),Γ(Ey, Ex) < L[d(x, y)]α.

A rst problem would be to nd conditions to obtain the Hölder continuity of the subbundles

Es and Eu.

On the other hand, the absolute continuity of the stable and unstable manifolds could also be

studied. The author recommends viewing [BS02], Chapter 6 and [LQ06], Chapter 3 to address these

problems.

86 OTHER PROBLEMS THAT AROSE

Bibliography

[Ace17a] Jeovanny de Jesus Muentes Acevedo. Another classication of dynamical systems on thecircle. Preprint, 2017. 83

[Ace17b] Jeovanny de Jesus Muentes Acevedo. Local stable and unstable manifolds for anosovfamilies. arXiv preprint arXiv:1709.00636, 2017. xvii

[Ace17c] Jeovanny de Jesus Muentes Acevedo. On the continuity of the topological entropy ofnon-autonomous dynamical systems. Bulletin of the Brazilian Mathematical Society,

New Series, páginas 118, 2017. xviii, 11

[Ace17d] Jeovanny de Jesus Muentes Acevedo. Openness of anosov families. arXiv preprint

arXiv:1709.00633, 2017. xvi

[Ace17e] Jeovanny de Jesus Muentes Acevedo. Structural stability of anosov families. arXiv

preprint arXiv:1709.00638, 2017. xvii

[AF05] Pierre Arnoux e Albert M Fisher. Anosov families, renormalization and non-stationarysubshifts. Ergodic Theory and Dynamical Systems, 25(3):661709, 2005. xv, 2, 3, 5, 14,27, 29, 31, 35, 36

[AKM65] Roy L Adler, Alan G Konheim e M Harry McAndrew. Topological entropy. Transactionsof the American Mathematical Society, 114(2):309319, 1965. xvii, 11

[Ano67] DV Anosov. Geodesic ows on compact manifolds of negative curvature trudy mat. Inst.VA Steklova, 90, 1967. xvi

[Arn13] Ludwig Arnold. Random dynamical systems. Springer Science & Business Media, 2013.xv, xvii, 5, 14

[Bak95a] Victor Ivanovich Bakhtin. Random processes generated by a hyperbolic sequence ofmappings. i. Izvestiya: Mathematics, 44(2):247279, 1995. xv, 27

[Bak95b] Victor Ivanovich Bakhtin. Random processes generated by a hyperbolic sequence ofmappings. ii. Russian Academy of Sciences. Izvestiya Mathematics, 44(3):617, 1995. xv,27

[BJLT12] Vincent Borrelli, Saïd Jabrane, Francis Lazarus e Boris Thibert. Flat tori in three-dimensional space and convex integration. Proceedings of the National Academy of Sci-

ences, 109(19):72187223, 2012. 2

[Bog92] Thomas Bogenschütz. Entropy, pressure, and a variational principle for random dynam-ical systems. Random Comput. Dynam, 1(1):99116, 1992. xv, xviii, 14

[BP07] Luis Barreira e Yakov Pesin. Nonuniform hyperbolicity: Dynamics of systems with nonzero

Lyapunov exponents, volume 115. Cambridge University Press, 2007. xvii, xix, 34, 40,48, 58, 61, 63, 66, 84

87

88 BIBLIOGRAPHY

[BS02] Michael Brin e Garrett Stuck. Introduction to dynamical systems. Cambridge universitypress, 2002. 30, 37, 85

[CRV17] Armando Castro, F. Rodrigues e P. Varandas. Stability and limit theorems for sequencesof uniformly hyperbolic dynamics. arXiv preprint arXiv:1709.01652, 2017. xvii

[dC92] Manfredo Perdigão do Carmo. Riemannian geometry. Bikhausen, Boston, 1992. 1, 22

[Eng89] Ryszard Engelking. General topology, sigma series in pure mathematics, vol. 6, 1989. 1

[Fra69] John Franks. Anosov dieomorphisms on tori. Transactions of the American Mathemat-

ical Society, 145:117124, 1969. 85

[Fra74] John M Franks. Time dependent stable dieomorphisms. Inventiones mathematicae,24(2):163172, 1974. xvii, 2

[Hir12] Morris W Hirsch. Dierential topology, volume 33. Springer Science & Business Media,2012. xvi, 1

[HP70] Morris W Hirsch e Charles C Pugh. Stable manifolds and hyperbolic sets. Em Global

Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), páginas 133163,1970. xvii

[Kaw14] Christoph Kawan. Metric entropy of nonautonomous dynamical systems. Nonau-

tonomous dynamical systems, 1(1), 2014. 14

[Kaw17] Christoph Kawan. Entropy of nonautonomous dynamical systems. arXiv preprint

arXiv:1708.00815, 2017. 14

[Ker58] Michel A Kervaire. Non-parallelizability of the n-sphere for n> 7. Proceedings of the

National Academy of Sciences, 44(3):280283, 1958. 84

[KH97] Anatole Katok e Boris Hasselblatt. Introduction to the modern theory of dynamical

systems, volume 54. Cambridge university press, 1997. xvii, xix, 1, 27, 38, 40, 48, 58, 83

[KL16] Christoph Kawan e Yuri Latushkin. Some results on the entropy of non-autonomousdynamical systems. Dynamical Systems, 31(3):251279, 2016. xv, xviii, 14, 83

[KMS99] Sergiy Kolyada, Michal Misiurewicz e Lubomír Snoha. Topological entropy of nonau-tonomous piecewise monotone dynamical systems on the interval. Fundamenta Mathe-

maticae, 160(2):161181, 1999. xv, xviii, 14, 83

[KR11] Peter E Kloeden e Martin Rasmussen. Nonautonomous dynamical systems. Number 176.American Mathematical Soc., 2011. xv, 2, 5

[KS96] Sergiy Kolyada e Lubomír Snoha. Topological entropy of nonautonomous dynamicalsystems. Random and Computational Dynamics, 4(2):205, 1996. xv, xviii, 2, 11, 14, 16,25, 83

[Kus67] Anatolii Georgievich Kushnirenko. On metric invariants of entropy type. Russian Math-

ematical Surveys, 22(5):53, 1967. xviii, 14

[Liu98] Pei-Dong Liu. Random perturbations of axiom a basic sets. Journal of statistical physics,90(1-2):467490, 1998. xv, xvii, 5, 14, 27, 73

[LQ06] Pei-Dong Liu e Min Qian. Smooth ergodic theory of random dynamical systems. Springer,2006. xv, xvii, 74, 85

[LY88] François Ledrappier e L S Young. Entropy formula for random transformations. Proba-bility theory and related elds, 80(2):217240, 1988. xviii, 14

BIBLIOGRAPHY 89

[Man74] Anthony Manning. There are no new anosov dieomorphisms on tori. American Journal

of Mathematics, 96(3):422429, 1974. 85

[New89] Sheldon E Newhouse. Continuity properties of entropy. Annals of Mathematics,129(1):215235, 1989. xvii

[Pes76] Ja B Pesin. Families of invariant manifolds corresponding to nonzero characteristic ex-ponents. Mathematics of the USSR-Izvestiya, 10(6):1261, 1976. xvii

[QQX03] Min Qian, Min-Ping Qian e Jian-Sheng Xie. Entropy formula for random dynamicalsystems: relations between entropy, exponents and dimension. Ergodic Theory and Dy-

namical Systems, 23(6):19071931, 2003. xvii, xviii, 14

[Rue97a] David Ruelle. Positivity of entropy production in the presence of a random thermostat.Journal of Statistical Physics, 86(5-6):935951, 1997. xviii, 14

[Rue97b] David Ruelle. Positivity of entropy production in the presence of a random thermostat.Journal of Statistical Physics, 86(5-6):935951, 1997. 14

[Shu13] Michael Shub. Global stability of dynamical systems. Springer Science & Business Media,2013. xvi, xviii, 1, 27, 28, 37, 47, 57, 77, 80, 81, 84

[SSZ16] Hua Shao, Yuming Shi e Hao Zhu. Estimations of topological entropy for non-autonomousdiscrete systems. Journal of Dierence Equations and Applications, 22(3):474484, 2016.xv, xviii, 2, 14

[Ste11] Mikko Stenlund. Non-stationary compositions of anosov dieomorphisms. Nonlinearity,24(10):2991, 2011. xv, 2, 27

[Via14] Marcelo Viana. Lectures on lyapunov exponents, volume 145 of cambridge studies inadvanced mathematics, 2014. xvii, 27, 34, 47, 84

[Wal00] Peter Walters. An introduction to ergodic theory, volume 79. Springer Science & BusinessMedia, 2000. xvii, 11, 12, 14, 16, 18

[Yan80] Koichi Yano. A remark on the topological entropy of homeomorphisms. Inventiones

mathematicae, 59(3):215220, 1980. xvii, 83

[You86] Lai-Sang Young. Stochastic stability of hyperbolic attractors. Ergodic Theory Dynam.

Systems, 6(2):311319, 1986. xv, xvi, 34, 47, 74

[ZC09] Jin-lian Zhang e Lan-xin Chen. Lower bounds of the topological entropy for nonau-tonomous dynamical systems. Applied Mathematics-A Journal of Chinese Universities,24(1):7682, 2009. xv, xviii, 2, 14, 83

[ZZH06] Yuhun Zhu, Jinlian Zhang e Lianfa He. Topological entropy of a sequence of monotonemaps on circles. Journal of the Korean Mathematical Society, 43(2):373382, 2006. xv,xviii, 14, 83

são paulo, 24 de novembro de 2017€¦ · agradecimentos este trabajo es dedicado a mis dos...

Documents