São Paulo, 24 de novembro de 2017€¦ · Agradecimentos Este trabajo es dedicado a mis dos...
Transcript of 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!!!
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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.
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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.
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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
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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.
4 NON-STATIONARY DYNAMICAL SYSTEMS
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 .
6 NON-STATIONARY DYNAMICAL SYSTEMS
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),
8 NON-STATIONARY DYNAMICAL SYSTEMS
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.
10 NON-STATIONARY DYNAMICAL SYSTEMS
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.
14 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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.
16 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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 ),
which proves the proposition.
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 ).
18 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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
20 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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).
22 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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
CONTINUITY OF ENTROPY FOR STRONG TOPOLOGY 23
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.
24 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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
CONTINUITY OF ENTROPY FOR STRONG TOPOLOGY 25
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).
26 ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS
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
SOME EXAMPLES OF ANOSOV FAMILIES 33
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
).
SOME EXAMPLES OF ANOSOV FAMILIES 35
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)
INVARIANT CONES FOR ANOSOV FAMILIES 43
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.
INVARIANT CONES FOR ANOSOV FAMILIES 45
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‖.
50 OPENNESS FOR ANOSOV FAMILIES
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.
52 OPENNESS FOR ANOSOV FAMILIES
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.
54 OPENNESS FOR ANOSOV FAMILIES
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).
56 OPENNESS FOR ANOSOV FAMILIES
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:
i. D(gp)z(Kuα) ⊆ Ku
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).
60 STABLE AND UNSTABLE MANIFOLDS
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‖.
HADAMARD-PERRON THEOREM FOR ANOSOV FAMILIES 61
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).
62 STABLE AND UNSTABLE MANIFOLDS
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.
HADAMARD-PERRON THEOREM FOR ANOSOV FAMILIES 63
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).
64 STABLE AND UNSTABLE MANIFOLDS
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).
which proves the lemma.
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)
HADAMARD-PERRON THEOREM FOR ANOSOV FAMILIES 65
(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)
66 STABLE AND UNSTABLE MANIFOLDS
(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, 〈·, ·〉),
which proves the proposition.
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.
68 STABLE AND UNSTABLE MANIFOLDS
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:
70 STABLE AND UNSTABLE MANIFOLDS
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.
72 STABLE AND UNSTABLE MANIFOLDS
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:
i. D(gp)z(Kuα) ⊆ Ku
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.
76 STRUCTURAL STABILITY FOR ANOSOV FAMILIES
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
78 STRUCTURAL STABILITY FOR ANOSOV FAMILIES
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).
STRUCTURAL STABILITY OF A2B(M) 79
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).
80 STRUCTURAL STABILITY FOR ANOSOV FAMILIES
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‖∞,
which proves the lemma.
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)‖∞,
STRUCTURAL STABILITY OF A2B(M) 81
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.
82 STRUCTURAL STABILITY FOR ANOSOV FAMILIES
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
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