Famílias Anosov: estabilidade estrutural, variedades invariantes, e entropía para sistemas dinâmicos não-estacionários

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(1)Anosov Families: Structural Stability, Invariant Manifolds and Entropy for Non-Stationary Dynamical Systems Jeovanny de Jesus Muentes Acevedo. Tese apresentada ao Instituto de Matemática e Estatística da Universidade de São Paulo para obtenção do título de 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.

(2) 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.

(3) Agradecimentos. Este trabajo es dedicado a mis dos madres, mi abuelita Herminia Muñoz y a mi mamá Nazly Acevedo.. Agradezco muchísimo a la Universidad de São Paulo por haberme brindado la oportunidad de realizar mis estudios de pósgrado. A las agencias CAPES y CNPq por su nanciación durante mis estudios de Maestría y Doctorado. A mi orientador por su gran apoyo y orientación. A mis familiares y a mis amigos en Colombia y en Brasil por acompanãrme y apoyarme durante todo este tiempo.. Por útimo:. ½½½Gracias Brasil!!!. i.

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(5) Resumo. Famílias Anosov: Estabilidade Estrutural, Variedades Invariantes e Entropia de Sistemas Dinâmicos Não-Estacionários. 2017. Tese (Doutorado) - Instituto de. ACEVEDO, J. J. M.. 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 difeomorsmos. (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). M. a união disjunta de cada. Mi ,. para. i ∈ Z,. e. F m(. m denidos na seo conjunto consistente das famílias de difeomorsmos (fi )i∈Z de classe C. 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. M). F m(. F m (M),. em que. 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. F m (M) m em F (. e mostraremos algumas das suas propriedades. Provaremos que esta entropia é contínua. M). munido da topología forte. Porém, ela é discontínua em cada elemento de. F m (M). munido da topología produto. Também apresentaremos um resultado que pode ser uma ferramenta de muita utilidade no estudo da continuidade da entropia topológica de difeomorsmos denidos em variedades compactas. Finalizaremos o trabalho dando uma lista de problemas que surgiram ao longo desta pesquisa e que serão analisados em um trabalho futuro.. Palavras-chave: Família Anosov, difeomorsmo de Anosov, sistemas dinâmicos não-estacionários, sistemas dinâmicos não-autônomos, sistemas dinâmicos aleatórios, entropia topológica, topología forte.. iii.

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(7) Abstract. Anosov Families: Structural Stability, Invariant Manifolds and Entropy for Non-Stationary Dynamical Sytems. 2017. Tese (Doutorado) - Instituto de Matemática ACEVEDO, J. J. M.. e Estatística, Universidade de São Paulo, São Paulo, 2017.. Anosov families were introduced by P. Arnoux and A. Fisher, motivated by generalizing the notion of Anosov dieomorphisms. Roughly, Anosov families are sequences of dieomorphisms. (fi )i∈Z. fi : Mi → Mi+1. for all. dened on a sequence of compact Riemannian manifolds. (Mi )i∈Z ,. i ∈ Z,. has asymptotically hyperbolic behavior.. such that the composition. fi+n ◦ · · · ◦ fi ,. for. n ≥ 1,. where. This notion is known as a non-stationary dynamical system or a non-autonomous dynamical system. Let of. M be the disjoint union of each Mi , for each i ∈ Z, and F m (M) the set consisting of families. C m -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. F m (M),. where. F m (M). is endowed with the strong topology (or Whitney. topology); the structural stability of a certain class of Anosov families, considering uniform topological 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. F m (M). will be introduced, and we will show some of its properties. We. will prove that this entropy is continuous on. F m (M). endowed with strong topology. However, it is. M) endowed with the product topology. We will also present. m discontinuous at each element of F (. 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-autonomous dynamical systems, random dynamical systems, topological entropy, strong topology.. v.

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(9) 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. . . . . . . . . . . . . . . . . . .. 5 Stable and Unstable Manifolds. 55. 57. 5.1. Stable and Unstable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 5.2. Hadamard-Perron Theorem for Anosov Families . . . . . . . . . . . . . . . . . . . . .. 58. 5.3. Local Stable and Unstable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66. 5.4. Stable and unstable manifolds for matrix Anosov Families. 70. vii. . . . . . . . . . . . . . . ..

(10) CONTENTS. viii. 6 Structural Stability for Anosov Families 2 6.1 Openness of Ab (M) . . . . . . . . . . . 6.2 6.3. 73 . . . . . . . . . . . . . . . . . . . . . . . . .. M. 2 Local Stable and Unstable Manifolds for Elements in Ab ( ) . . . . . . . . . . . . . 2 Structural Stability of Ab ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. M. 7 Other Problems That Arose. 73 76 77. 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.

(11) List of Abbreviations. s.p.a n.s.d.s.. Satises the property of angles ) (Non-stationary dynamical system ). (. ix.

(12) x. LIST OF ABBREVIATIONS.

(13) List of Simbols M d. Disjoint union of the manifolds. M. The metric on. M). Mi. F m(. Set consisting of sequences of. D(f )p. Derivative of. Z. Set consisting of integer numbers. R. Set consisting of real numbers. N. Set consisting of natural numbers. S1. The circle. Tm. The. C. Set consisting of complex numbers. Q. Set consisting of rational numbers. Eps Epu. Stable subspace at. TM. Tangent bundle of. Tp M. Tangent space of. f. at. M) Am b (M ) CF m (M) m. Di. (M ). p p. M. M. at. p. m The set consisting of C -Anosov families on. Set consisting of constant families Set consisting of. C m-. D. m Set consisting of C -dieomorphisms on. C(X1 , X2 ). induced by the Riemannian metric on. Ball in. B(x, ε). Ball with center at the point. B m (φ, (i )i∈Z ). strong basic neighborhood of. D. M. dieomorphisms on. m i B m (φ, τ ). τunif. M. Set of Anosov families s.p.a. with bounded second derivative. C m -metric on. τstr. M. m-torus. dm (·, ·). τprod. on. p. Unstable subspace at. Am (. m-dieomorphisms. m i with center. φ. and radius. M) m Strong topology on F (M) m Uniform topology on F (M) m Product topology on F (. xi. x. f. Mi. τ. and radius. ε. to. Mi+1. X2.

(14) LIST OF SIMBOLS. xii. V s (x, φ). Stable set for. Vs. ε (x, φ) V u (x, φ). Local stable set for. Vεu (x, φ) N s (x, (ε. Local stable set for. i )i∈Z ) u N (x, (εi )i∈Z ). fφ. φ. x. at. Unstable set for. φ. φ. at. φ. at. x. x at. x. Local stable set for families Local unstable set for families Constant family associated to. φ. N (A). Number of sets in a nite subcover of. H(A). log N (A). A∨B Wk. with smallest cardinality. {A ∩ B : A ∈ A, B ∈ B} Am. Hi (f , A). {A1 ∩ · · · ∩ Ak: Ai ∈ Ai } Wn−1 k −1 limn→+∞ n1 H ( f ) (A) k=0 i. H(f ). (Hi (f ))i∈Z. m=1. A. H(f , A) Hi (f ). r[n, i](ε, f ). r[i](ε, f ). H(f ) Hi (f ). s[n, i](ε, f ) s[i](ε, f ). (Hi (f , A))i∈Z. sup{Hi (f , A) : A. is an open cover of. The smallest cardinality of any. lim sup n1 n→+∞. log r[n, i](ε, f ). M}. (n, ε)-span. of. Mi. with respect to. (Hi (f ))i∈Z. limε→0 r[i](ε, f ). for each. i∈Z. The largest cardinality of any. lim sup n1 n→+∞. log s[n, i](ε, f ). (n, ε)-separated. subset of. Mi. H(φ). Topological entropy of a single map. expp. Exponential application at. Dr (φ, δ). Set consisting of dieomorphisms. G(φ). Graph of an application. SL(Z, m). Special linear group of mxm matrices with integer entries. Lip(φ). Lipchitz constant of. s Kα, f ,p u Kα, f ,p. Stable. %p. Injectivity radius of expp at. U(M, h·, ·i). f. α-cone. Unstable. of. α-cone. f. of. φ. p ψ. such that. dr (φ, ψ) ≤ δ. φ. φ. at. f. with respect. p at. p p. set consisting of Riemannian metric uniformly equivalent to. h·, ·i. on. M. f.

(15) List of Figures. 1.1.1 A n.s.d.s. on a sequence of 2-torus endowed with dierent Riemannian metrics.. . . .. 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 2.4.1 Exponential application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. r 2.4.2 Shaded regions represent the discs D (Ii , ri ).. . . . . .. 23. . . . . . . . . . . . . . .. 28. 2.2.1 Graph of. 3.1.1. φ.. q = φ−1 (p). 3.2.1 The square. and. z = φ(p). D(φ)q (A) = B. [0, 1] × [0, 1]. (2, 1), (3, 2), (1, 1). is mapped by. α-cones. at. 3.4.2 Stable and unstable invariant. r = Fp,n. Tn. k=1 D. A. is the graph of the map. D(φ)p (B) = C. φ. to the parallelogram with vertices. (0, 0),. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4.1 Stable and unstable. 4.2.1. and. G(φ). s g ±k g (p) (Kα,f ,g ±k. p.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. . . . . . . . . . . . . . . . . . . . . .. 42. α-cones. q = f (p). ±k (p). 33. ),. for. r = s, u. and. n = 1, 2, 3.. . . . . . . . . . . . .. M1 , M2 , M3 ,. . . , endowed with the metric given in (3.2.1), for a, b ∈ (λ, 1). 5.2.1 G(ψn+1 ) = fñ G(φn ). Shaded regions represent the unstable α-cones. . . . . 5.1.1. xiii. 52. . . . . . .. 58. . . . . . .. 60.

(16) xiv. LIST OF FIGURES.

(17) 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 sequence of compact Riemannian manifolds. Mi ,. for. i ∈ Z,. dieomorphism: there is a splitting of the tangent bundle. D(fi ). (that is,. λ ∈ (0, 1). and. s D(fi )(Eis ) = Ei+1. c>0. such that for. and. dened on a two sided. having similar behavior to an Anosov. T Mi = Eis ⊕Eiu , invariant by the derivative. u D(fi )(Eiu ) = Ei+1. n ≥ 1, p ∈ M i ,. fi : Mi → Mi+1. for any. i ∈ Z),. and there exist constants. we have:. kD(fi+n−1 ◦ ... ◦ fi )p (v)k ≤ cλn kvk. for each vector. v ∈ Eps. −1 −1 kD(fi−n ◦ ... ◦ fi−1 )p (v)k ≤ cλn kvk. for each vector. v ∈ Epu .. and. The subspaces. Eps. and. Epu. 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 dieomorphisms (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.

(18) INTRODUCTION. xvi. skew product transformations. In this thesis we will also give examples obtained from. cocycles Let. or. linear. (see Denition 3.2.4 and Example 3.2.5), which are a type of random dynamical systems.. M. 1. be a compact Riemannian manifold and Di. (M ) be the set consisting of dieomorphisms. 1 dened on M , endowed with the C topology (see [Hir12]). The set dieomorphisms on. M. is open in Di. 1. (M ),. there exists an open set of dieomorphisms. O. Furthermore, it is possible to take the set. h:M →M. (depending on. ψ ),. consisting of Anosov. that is, for any Anosov dieomorphism. 1. O ⊆ Di (M ). such that. such that for any. φ ◦ h = h ◦ ψ,. such that. A(M ). ψ∈O. that is,. φ. φ ∈ O ⊆ A(M ). φ : M → M, (see [Shu13]).. there exists a homeomorphism 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=. a. Mi =. i∈Z The. Mi 's. components. will be called the. M. For m ≥ 1, set. of. strong topology. (or. is a. F m (M):. C m -dieomorphism. the. for each. i}.. compact topology, uniform topology. and the. 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. 1 open in F (. Mi × i.. i∈Z. M, the total space. In (1.1.1) we will give a metric for. F m (M) = {(fi )i∈Z : fi : Mi → Mi+1 We consider three dierent topologies on. [. C 1 -Anosov. families, denoted by. A1 (M),. is. 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 nontrivial 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. 2 Anosov dieomorphism of class C are 2 (see Remark 3.2.7). Let Ab (. C 1+1. random small perturbations of an. uniformly hyperbolic sequences, that is, are Anosov families. M) be the set consisting of C 2 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 exists. δ>0. such that if. M). 2 That is, Ab (. g. = (gi )i∈Z. 2 is open in F (. with. d1 (fi , gi ) < δ. for all. i ∈ Z,. f. = (fi )i∈Z ∈ A2b (M) there. then. g. is an Anosov family.. 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 Definition 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. F 2 (M).

(19) 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 exists. φ:M →M. ε>0. be an Anosov dieomorphism. Hirsch and Pugh in [HP70] proved that there. such that, for each. x ∈ M,. the stable and unstable sets at. x. (see Denition 1.4.2),. s u which will be denoted by Vε (x, φ) and Vε (x, φ) respectively, are dierentiable submanifolds of M . s s u u s u Furthermore we have that Tx Vε (x, φ) = Ex and Tx Vε (x, φ) = Ex , that is, Vε (x, φ) and Vε (x, φ) s are tangent to the stable and unstable subspaces at x, respectively. φ is a contraction on Vε (x, φ) −1 is a contraction on V u (x, φ). For Anosov families, Example 3.2.2 proves that the above and φ ε 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. is xed, and we give conditions (which depend on the derivative of each manifolds along the orbit of a given point. p ∈ M0 .. (fi )i∈Z , fi ). where each. fi. to obtain invariant. Pesin's Theory has been used to build invariant. 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. manifolds for a. 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 conjugacy. 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. compact Riemannian manifold is an upper semicontinuous map. Furthermore, if. M. on a. is a surface,. 1 this map is continuous. The entropy for any homeomorphism of the circle S is zero. Therefore, it depends continuously on homeomorphisms of. S1 .. In contrast, if we consider all the continuous. 1 maps dened on S , this entropy is not a continuous map (see [Yan80]). It is clear that the entropy is continuous at each structurally stable dieomorphism (a dieomorphism. stable. if there exist an open neighborhood. O. of. φ. φ:M →. such that all the elements in. O. is. structurally. are topologically.

(20) 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 nonstationary dynamical systems. We will prove that this entropy depends continuously on each element of. F 1 (M). endowed with the strong topology (see Theorem 2.4.5). In contrast, if we consider the. product topology on. F 1 (M),. then the entropy is discontinuous at each sequence (see Proposition. 2.3.1). Other results on the continuity of the entropy on. F 1 (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 by the author in the. F 1 (M). (see Remark 2.4.6). These results were published. 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. manifold. This entropy will be built via open partitions of. M. is a compact Riemannian. (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. M) endowed with the strong topology if m ≥ 1.. F m (M).. In. m contrast, it is continuous on F (. 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..

(21) 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. F 1 (M ). proved in Chapter 4 (see Theorem 4.3.5). We will show that if there exists a two-sided sequence of positive numbers. d1 (fi , gi ), then. (gi )i∈Z. (δi )i∈Z. with the strong topology will be. (fi )i∈Z. is an Anosov family, then. such that if. (gi )i∈Z ∈ F 1 (M). is an Anosov family. In that case, it is not always possible to take. and. (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.. f. In Chapter 6 we will prove that. = (fi )i∈Z ∈ A2b (M). d2 (fi , gi ). <δ. for each. A2b (M) is uniformly. there exists. δ>0. i ∈ Z,. (gi )i∈Z. then. such that, if. structuraly stable in F 2 (M), that is, for any. gi : Mi → Mi+1. is a. C 2 -dieomorphism. is an Anosov family and is conjugate to. M M. M. f. and. (see Theorem. 2 6.3.9). In Section 6.1 we will show that Ab ( ) is open in F 2 ( ) endowed with the uniform 2 topology: we have that for any (fi )i∈Z ∈ Ab ( ) there exists δ > 0 such that, if gi : Mi → Mi+1 is 2 2 a C -dieomorphism and d (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 throughout this study and that will be analyzed in a future work..

(22) xx. INTRODUCTION.

(23) 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 wellknown 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. M=. a. Mi ,. Mi =. i∈Z The set. M will be called total space. metric on. Mi ,. we will consider the. [. disjoint union. Mi × {i}.. i∈Z. and the. Mi. will be called. components of M.. If. di (·, ·). is the. the total space is endowed with the metric.  min{1, d (x, y)} i d(x, y) = 1 We sometimes use the notation. (M, d). if. x, y ∈ Mi. if. x ∈ Mi , y ∈ M j. (1.1.1) and. i 6= j.. to indicate we are considering the metric. d. given in. (1.1.1).. Denition 1.1.1.. Two metrics. exist positive numbers. k. and. dˆ and d˜ on. K. a topological space. dˆ and d˜ are. are. uniformly equivalent. if there. such that. ˜ y) ≤ d(x, ˆ y) ≤ K d(x, ˜ y) k d(x, It is clear that if. M. for all. x, y ∈ M.. uniformly equivalent metrics on. M,. then, for. dˆ 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 î topology on M and, furthermore, they are uniformly equivalent on M. On the other hand, if d ˜i are uniformly equivalent metrics on Mi for each i ∈ Z, then the metrics d ˆ and d ˜ , dened and d 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.

(24) NON-STATIONARY DYNAMICAL SYSTEMS. 2. If. Mi. h·, ·ii. is a compact Riemannian manifold with Riemannian metric. for. M with the Riemannian metric h·, ·i induced by h·, ·ii setting. total space. h·, ·i|Mi = h·, ·ii and we will use the notation. (M, h·, ·i). given in (1.1.2). In that case, on each. Denition 1.1.2.. Let. h·, ·ii. and. we endow the. i ∈ Z,. for. (1.1.2). to indicate that we are considering the Riemannian metric. Mi ,. h·, ·i?i. i ∈ Z,. we will consider the metric. be Riemannian metrics on. Mi. di. induced by. and let. k · ki. h·, ·ii .. and. k · k?i. be the. ? ? Riemannian norms induced by h·, ·ii and h·, ·ii , respectively. We say that h·, ·ii and h·, ·ii (or that k · ki and k · k?i ) are Mi if there exist positive numbers ki and Ki such that. uniformly equivalent on. ki kvk?i ≤ kvki ≤ Ki kvk?i where. T Mi. is the tangent bundle of. Mi .. M. k. and. K. k. K. and. for all. does not depend on i, then we say that. such that. v ∈ T Mi , i ∈ Z,. h·, ·i. h·, ·i?. and. are. uniformly equivalent on. ? , where h·, ·i is obtained similarly as in (1.1.2) with the Riemannian metrics. dî and d˜i be î and d˜i as from d Let. M. v ∈ T Mi ,. If there exist. kkvk?i ≤ kvki ≤ Kkvk?i that is,. for all. for instance,. ki. such that, for each notation. f. dˆ and d˜ dened on in (1.1.1). These metrics are not necessarily uniformly equivalent on M, since, the metrics induced by. could converge to zero or. Denition 1.1.3.. A. f |M. i ∈ Z,. = (fi )i∈Z .. n-th. The. n i. i. is the identity on. h·, ·i?i ,. respectively, and. could converge to (or. +∞. n.s.d.s.) f. on. i → ±∞.. M is a map f : M → M,. is a homeomorphism. Sometimes we use the. composition is dened, for. i ∈ Z,.    f ◦ · · · ◦ fi : Mi → Mi+n   i+n−1 −1 −1 := fi−n ◦ · · · ◦ fi−1 : Mi → Mi−n    I : M → M i. when. i. to be. if. n>0. if. n<0. if. n = 0,. Mi .. We will use the notation metric. and. = fi : Mi → Mi+1. i. Ii. Ki. h·, ·ii. non-stationary dynamical system. f where. h·, ·i?i .. (M, h·, ·i, f ) for a n.s.d.s. f. dened on. M endowed with the Riemannian. h·, ·i.. Remark 1.1.4. The notion above can be found in the literature under several dierent names: nonautonomous dynamical systems, non-autonomous discrete systems, sequence of maps, time dependent dieomorphisms and mapping 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, manifold, and the metrics. h·, ·ii. can change with i, or the. Mi. Mi. can be the same Riemannian. can be the same surfaces with dierent. fractal structures, Thurston corrugations, etc. (see [BJLT12])..

(25) UNIFORM CONJUGACY BETWEEN NON-STATIONARY DYNAMICAL SYSTEMS. .... −−−→. Mi−1. Figure 1.1.1:. .... fi. fi−1. −−−→ Mi. 3. Mi+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.. φ:M →M. Let. be a homeomorphism dened on a compact Riemannian manifold. M = Mi , where Mi = M × {i} for each i ∈ Z with the same metric h·, ·i. We dene the constant family (M, h·, ·i, f ) associated to φ as the n.s.d.s. (fi )i∈Z , where M. with metric. fi : Mi → Mi+1. h·, ·i.. `. Take. 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.. f , respectively. f˜ be non-stationary dynamical systems on M and M ˜ is a gathering of f if there exists a strictly increasing sequence of integers (ni )i∈Z We say that f fi = Mn and f˜ i = fn −1 ◦ · · · ◦ fn +1 ◦ fn : such that M. Denition 1.1.6.. Let. f. and. i. i+1. i. i. f˜i =fni+1 −1 ◦···◦fni. f˜i−1 =fni −1 ◦···◦fni−1. · · · Mni−1 −−−−−−−−−−−−−→ Mni −−−−−−−−−−−−→ 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 Systems In this section we will talk about the morphisms between non-stationary dynamical systems.. Denition 1.2.1.. Two continuous maps. topological spaces, are that. h ◦ φ1 = φ2 ◦ h.. φ1 : X1 → X1. topologically conjugate. In that case,. h. is called a. and. φ2 : X2 → X2 ,. where. if there exists a homeomorphism. topological conjucagy between φ1. X1. and. X2. h : X1 → X2 and. are. such. φ2 .. Take. N=. a. Ni ,. i∈Z where. Ni. with the. is a metric space with metric xed. dî 's.. Throughout this chapter,. dynamical system dened on. (M, d). f. and. dî .. Let. dˆ. be the metric on. = (fi )i∈Z and g = (gi )i∈Z ˆ ), respectively. (N, d. N. dened as in (1.1.1). will denote a non-stationary.

(26) NON-STATIONARY DYNAMICAL SYSTEMS. 4. Denition 1.2.2. each. i ∈ Z,. h |M. i. A. topological conjugacy. = hi : Mi → Ni. between. f. and. g. is a map. h : M → N, such that, for. is a homeomorphism and. hi+1 ◦ fi = gi ◦ hi , that is, the following diagram commutes:. f−1. M−1 −−−−→   ··· yh−1. f0. M0 −−−−→   yh0. g−1. f1. M1 −−−−→   yh1. g0. M2   yh2 ···. g1. N−1 −−−−→ N0 −−−−→ N1 −−−−→ 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 =. is a conjugacy between. f.    h   0. −1 g ◦ · · · ◦ g0 ◦ h0 ◦ f0−1 ◦ · · · ◦ fi−1  i−1   g −1 ◦ · · · ◦ g −1 ◦ h ◦ f ◦ · · · ◦ f 0 −1 i −1 i. and. if. i=0. if. i>0. if. i < 0,. g.. One type of conjugacy that works for the class of non-stationary dynamical systems is. topological conjugacy :. Denition 1.2.4. if. (hi : Mi →. We say that a topological conjugacy. Ni )i∈Z and (h−1 i. : Ni → Mi )i∈Z. h : M → N between f. and. are equicontinuous families (that is,. h. g. is. and. uniform uniform. h −1 are. uniformly conjugate.. uniformly continuous maps). In that case we will say that the families are. Since the composition of uniformly continuous functions is uniformly continuous, the class consisting of non-stationary dynamical systems becomes a category, where the objects are the nonstationary 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. hi : Mi → Ni for −1 (hi )i≤0 ) are equicontinuous and. homeomorphisms. i≥0. (for. i≤. hi+1 ◦ fi = gi ◦ hi : Mi → Ni+1 , That is,. (fi )i≥0. and. (gi )i≥0 ((fi )i≤0. and. (gi )i≤0 ). between. f. and. g. is a sequence of. 0) such that (hi )i≥0 and (h−1 i )i≥0 ((hi )i≤0 and. for every. i≥0. (for every. i ≤ −1).. 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..

(27) 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. norm on. and. X2. be compact Riemannian manifolds. For. and distr (·, ·) the metric induced by. Xr. φ : X1 → X2. and. ψ : X1 → X2 .. The. Hom(X1 , X2 ). d0. k · kr. r = 1, 2,. on. Xr .. let. k · kr. be the Riemannian. Consider two homeomorphisms. metric induced by dist(·, ·) on. = {h : X1 → X2 : h. is a homeomorphisms}. is given by. d0 (φ, ψ) = max dist2 (φ(x), ψ(x)) + max dist1 (φ−1 (y), ψ −1 (y)). x∈X1. If. φ. and. ψ. y∈X2. are dieomorphisms of class. 1. Di. C 1,. the. d1. metric on. (X1 , X2 ) = {φ : X1 → X2 : φ. C 1 -dieomorphism}. is a. is given by. −1 d1 (φ, ψ) = d0 (φ, ψ) + max kDφx − Dψx k2 + max kD(φ)−1 y − D(ψ)y k1 , x∈X1. where. Dφx. and. Dψx. y∈X2. are the derivatives of. 2 dieomorphisms of class C , the 2. Di. φ. ψ. and. x ∈ X1 ,. at. respectively. If. φ. and. ψ. are. d2 metric on. (X1 , X2 ) = {φ : X1 → X2 : φ. is a. C 2 -dieomorphism}. is given by. 2 −1 d2 (φ, ψ) = d1 (φ, ψ) + max kD2 φx − D2 ψx k2 + max kD2 (φ)−1 y − D (ψ)y k1 , x∈X1. where. D 2 φx. and. D2 ψx. Denition 1.3.1.. are the second derivatives of. Suppose that. is the metric induced by i. ii. iii.. y∈X2. k · ki .. Mi. For. φ. and. ψ. at. x ∈ X1 ,. respectively.. is a Riemannian manifold with Riemannian norm. m≥0. and. τ > 0,. set:. m Dm i = {φ : Mi → Mi+1 : φ is a C -dieomorphism}; m B m (φ, τ ) = {ψ ∈ Dm i : d (φ, ψ) < τ },. F m (M) = {f = (fi )i∈Z : fi ∈ Dm i. for. for each. φ ∈ Dm i ; i ∈ Z}.. Note that. F m (M ) =. +∞ Y i=−∞. Dm i .. k · ki. and. di.

(28) NON-STATIONARY DYNAMICAL SYSTEMS. 6. Denition 1.3.2 (Product Topology).. The. Y. U=. D. product topology on F m (M) is generated by the subsets m i. ×. i<−j where. Ui. is an open subset of. Dm i ,. for. j Y. [Ui ] ×. Y. i=−j. i>j. −j ≤ i ≤ j,. for some. Dm i ,. M), τprod ).. j ∈ N.. The space. F m (M ). endowed. m with the product topology will be denoted by (F (. Denition 1.3.3 (Uniform topology).. Given. f. = (fi )i∈Z. and. g = (gi )i∈Z in F m (M), take. m dm norm (f , g ) = sup(min{d (fi , gi ), 1}). i∈Z. uniform topology m F (M).. The. on. F m (M). is spanned by. dm norm (·, ·).. Let. τunif. be the uniform topology on. topology (or Whitney topology ): for each f ∈ M) and a sequence of positive numbers (i )i∈Z , a strong basic neighborhood of f is the set. We can also endow. F m (M). with the. C m -strong. F m(. B m (f , (i )i∈Z ) = {g ∈ F m (M) : gi ∈ B m (fi , i ). for all. i}.. Denition 1.3.4 (Strong Topology). The C m -strong topology on F m (M) is generated by the strong m m basic neighborhood of each f ∈ F (M). Thus, a subset A of F (M) is open if for all f ∈ A, there exists a strong basic neighborhood. F m (M). B m (f , (i )i∈Z ). of. f , such that B m (f , (i )i∈Z ) ⊆ A. The space. endowed with the strong topology will be denoted by. (F m (M), τstr ).. Unless stated otherwise, we are considering the strong topology on. τstr. For simplicity, the. will be omitted.. Denition 1.3.5.. We say that. basic neighborhood to. F m (M).. f . A subset A. f. ∈ F m (M) is structurally stable in F m (M) if there exists a strong. B m (f , (i )i∈Z ). of. f , such that each g. M) is structurally stable. m of F (. ∈ B m (f , (i )i∈Z ). if every element in. A. is uniformly conjugate. 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, stationary dynamical systems dened on. d the metric on both M and N.. f. = (fi )i∈Z. and. g. = (gi )i∈Z. will represent two non-. M and N, respectively. By simplicity, we will denote by. 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 ∈ Z there 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 −1 and (hi )i≤i0 ) are equicontinuous and hi+1 ◦ fi = gi ◦ hi : Mi → Ni+1 , for every i ≥ i0 (for every i ≤ i0 )..

(29) SOME PROPERTIES OF THE UNIFORM CONJUGACY. Denition 1.4.2. x∈X. and. ε > 0,. Let. φ:X →X. be a homeomorphism on a metric space. X. with metric. ρ.. 7. For. set:. 1.. V s (x, φ) = {y ∈ X : ρ(φn (x), φn (y)) → 0. when. 2.. Vεs (x, φ) = {y ∈ X : ρ(φn (x), φn (y)) < ε,. for all. 3.. V u (x, φ) = {y ∈ X : ρ(φn (x), φn (y)) → 0. when. 4.. Vεu (x, φ) = {y ∈ X : ρ(φn (x), φn (y)) < ε,. for all. n → +∞}:= n ≥ 0}:=. the. n → −∞}:= n ≤ 0}:=. at. x;. local stable set. for. φ. at. x;. unstable set. for. φ. at. x;. local stable set. for. φ. at. x.. the. the. stable set. φ. the. for. 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 (V s (x, f )) = V s (hi (x), g) and hi (V u (x, f )) = V u (hi (x), g). Proof. Let y ∈ V s (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 (f ni (x)), hn+i (f ni (y))) = d(g ni hi (x), g ni hi (y))). Thus. hi (y) ∈ V s (hi (x), g ).. Now, using the equicontinuity of. (h−1 i )i∈Z ,. we can prove that. s s h−1 i (V (hi (x), g )) ⊆ V (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 r such that hi (Vrr (x, f)) ⊆ Vδrr (hi (x), g) ⊆ hi (Vεrr (x, f)).. Proof. exists. We will prove the stable case. Since. δs > 0. x ∈ Mi .. Fix. Vδss (hi (x),. such that, for any. i ∈ Z,. We will show that if. g ), d(g. n h (x), i i. g. n (y)) i. if. y ∈. < δs. (h−1 i )i∈Z. v, w ∈ Ni Vδss (hi (x),. for all. n ≥ 0.. εs > 0, there −1 (hi (v), h−1 i (w)) < εs .. is an equicontinuous family, given. and. g). d(v, w) < δs , then d. then. s h−1 i (y) ∈ Vεs (x, f ).. Thus, for all. n≥0. Indeed, since. y ∈. we have. −1 n n n −1 n −1 n n −1 εs > d(h−1 n+i g i hi (x), hn+i g i (y)) = d(f i hi hi (x), f i hi (y)) = d(f i (x), f i hi (y)), and hence,. s h−1 i (y) ∈ Vεs (x, f ).. Therefore,. s s h−1 i (Vδs (hi (x), g )) ⊆ Vεs (x, f ). Since that. (hi )i∈Z. is an equicontinuous family, analogously we can prove that there exists. hi (Vss (x, f )) ⊆ Vδss (hi (x), g ),. s > 0 such.

(30) NON-STATIONARY DYNAMICAL SYSTEMS. 8. which proves the proposition. Take two homeomorphisms spaces. X1. and. respectively, to. X2 , φ. φ : X1 → X1. and. ψ : X2 → X2. fφ. respectively. We will denote by. and to. ψ.. It is clear that if. φ. ψ. and. fψ. and. dened on two compact metric. the constant families associated,. 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 is,. S1 = {z ∈ C : kzk = 1},. Rα (z) =. Rα1. and. 2παi z for e. Rα2. z∈. with. Rα : S1 → S1. α ∈ [0, 1],. that. α2 ∈ [R \ Q] ∩ (0, 1). then. the circle rotation by a number. S1 . It is well-known that if. α1 ∈ Q ∩ (0, 1). and. are not topologically conjugate. However:. Proposition 1.4.5. Given α1 , α2 ∈ [0, 1], the constant families associated to Rα1 and Rα2 , respectively, are uniformly conjugate. Proof.. It is sucient to prove that, for any. α ∈ [0, 1],. the family of homeomorphisms for. k∈Z. and. z∈. (hk :. S1. →. S1 ). Rα and 1 be the identity on S . Consider. the constant families associated to. 1 to the identity on S , respectively, are uniformly conjugate. Let k∈Z , where. I. hk (z) =. e. 2kπ(1−α)i z for all. k ∈ Z.. Thus,. S1 , Rα (hk+1 (z)) = e2παi e2(k+1)π(1−α)i z = e2kπ(1−α)i z = hk (I(z)),. i. e.,. (hk )k∈Z. is a conjugacy between. fR. α. and. f I . If z1 , z2 ∈ S1 and d(z1 , z2 ) ≤ 21 min{α, 1−α}, then. d(R(1−α) (z1 ), R(1−α) (z2 )) = d(z1 , z2 ). Consequently, if d(z1 , z2 ) ≤. 1 2. min{α, 1 − α},. for all. k ∈ Z,. we have. k k d(hk (z1 ), hk (z2 )) = d(R(1−α) (z1 ), R(1−α) (z2 )) = d(z1 , z2 ). This fact proves that. (hk )k∈Z. tinuous. Hence,. and. fR. α. fI. is equicontinuous. Analogously we can prove that. (h−1 k )k∈Z. is equicon-. 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. ρ(φ(x), φ(y)) = ρ(x, y). φ : X → X,. for all. on a metric space. X. with metric. ρ,. is called an. isometry. if. 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. that. Consider the family. (hk )k∈Z. (hk )k∈Z ,. where. hk (x) = φ−k (x). is a topological conjugacy between. is an isometry, the family. (hk )k∈Z. for every. f φ and f I , where I. x ∈ Mk = X × {k}. is the identity on. It is clear. X.. Since. φ. is equicontinuous.. It follows from Proposition 1.4.7 that all the constant families associated to any isometry on are uniformly conjugate. We will nish this chapter with the following proposition.. M.

(31) 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. by the family. g. are uniformly conjugate by. h˜ = (h˜ n )i∈Z :. h. = (hi )i∈Z ,. then. f˜ and g˜. are uniformly conjugate. i. fni−1. fn. −1. fn. fni+1 −1. gni−1. gn. −1. gn. gni+1 −1. i → · · · −−−−−→ Mni+1 Mni−1 −−−−→ · · · −−−i−→ Mni −−−−     hn ···  ··· yhni−1 yhni y i+1 i → · · · −−−−−→ Mni+1 Mni−1 −−−−→ · · · −−−i−→ Mni −−−−. It is clear that. h˜ = (h˜ n )i∈Z is equicontinuous. i.

(32) 10. NON-STATIONARY DYNAMICAL SYSTEMS.

(33) 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 of. X.. φ : X → X. on a compact topological space X via open covers. 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-autonomous dynamical systems, which generalizes the notion of entropy for single dynamical systems. They. (fi )i≥0. considered only sequences of type (possibly. +∞).. and the entropy for this sequence was a single number. 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 for. i → −∞. (see Remark 2.2.11).. First, the entropy of a non-stationary dynamical system of non-negative numbers that. (ai )i∈Z. fi. (ai )i∈Z ,. where each. ai. (fi )i∈Z. depends only on. fj. will be dened as a sequence for. j ≥ i.. Then we will see. 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 homeomorphism 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. M), the entropy depends continuously on each non-stationary dynamical system in F m (M). In m contrast, with the product topology on F (M), the entropy is discontinuous for any non-stationary m dynamical system. To prove that the entropy is continuous on F (M) with the uniform topology F m(. 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}, 11. for each. i ∈ Z,.

(34) 12. ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS. where. M. N. and. 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. (fi )i∈Z ,. system. 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 each. ai. depends only on. fj. for each. j ≥ i.. (ai )i∈Z , where. In Corollary 2.2.7 we will see that this sequence is. constant. We consider the following denitions: an. A = {Aλ }λ∈Λ , Mi = M × {i},. such that if. A. M=. S. open cover. λ Aλ . In this section,. is an open cover of. of notation, we will omit the sub index. M, i. then. of. Ai. A. M. of. and. B. Ai = A × {i}. for covers of. is a collection of open subsets of will denote open covers of. is an open partition of. Mi .. M.. M,. Since. By abuse. Mi .. Denition 2.1.1. Let N (A) be the number of sets in a nite subcover of A with smallest cardinality. The entropy of A is the number H(A) := log N (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 We say. B. is a. Wk. m=1 A. renement. m for a collection of open covers. A. of. if each element of. B. A1 , ..., Ak. of. M.. 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((fik )−1 (A)) for each i ∈ Z and k ≥ 0. iv. H(. Wn−1. f. k −1 k=0 ( i ) (A)). ≤ nH(A),. for each i ∈ Z and n ≥ 1.. v. The limit 1 Hi (f, A) = lim H n→+∞ n. n−1 _. (f. ! k −1 i ) (A). (2.1.1). k=0. exists and is nite, for each i ∈ Z.. Denition 2.1.3 (Topological entropy).. We dene the. entropy. H(f , A) = (Hi (f , A))i∈Z .. of. f relative to A as the sequence.

(35) ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS The. topological entropy of f. 13. is the sequence. H(f ) = (Hi (f ))i∈Z , where. Hi (f ) = sup{Hi (f , A) : A. From now on,. X. φ : X → X,. which we denote by. The above denition only makes sense when. cover of. M }.. will represent a compact metric space. We recall that the. of a homeomorphism. X.. is an open cover of. A. H(φ),. topological entropy. is dened considering open covers of. is an open cover of. M. instead of a general open. 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. Mi is a (n, ε)-spanning of Mi with respect f if for each x max d(f ji (x), f ji (y)) < ε, i. e.,. We say that a compact subset. ∈ Mi. there exists. y ∈ K. K ⊆. such that. 0≤j<n. [ n−1 \ Mi ⊆ (f ki )−1 (B(f ki (y), ε)), y∈K k=0. where. B(f ki (y), ε). Denote by. is the closed ball with center. f ki (y) ∈ Mi+k and radius ε.. r[n, i](ε, f ) the smallest cardinality of any (n, ε)-span of Mi. is compact, we have. r[n, i](ε, f ) < ∞. for each. i∈Z. and. n ≥ 1.. with respect. f . Since Mi. Set. 1 r[i](ε, f ) = lim sup log r[n, i](ε, f ). n→+∞ n. Denition 2.1.5.. The. ?-topological. H(f ) = (Hi (f ))i∈Z ,. entropy of f where. is the sequence. Hi (f ) = lim r[i](ε, f ) for each i ∈ Z. ε→0. 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 max d(f ji (x), f ji (y)) > ε, i. e., if for all 0≤j<n. x ∈ E,. the set. n−1 \ k=0 contains no other point of. E.. (f ki )−1 (B(f ki (x), ε)).

(36) ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS. 14. Denote by. to. f . Set. s[n, i](ε, f ). the largest cardinality of any. (n, ε)-separated. subset of. Mi. with respect. 1 s[i](ε, f ) = lim sup log s[n, i](ε, f ). n→+∞ n. 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 ) = lim s[i](ε, f ) ε→0. Consequently,. i ∈ Z.. 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),. Therefore,. for all. for all. i ∈ Z.. (2.1.2). 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. type,. non-stationary subshift of nite. 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. dynamical systems. measure-theoretic entropy for random. (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. formula. Lyapunov exponents. and the. Pesin's. 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 dynamical 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):.

(37) 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 The topological entropy for. n ∈ Z.. Hi (f ).. H(φ) of a single homeomorphism φ : X → X. satises. H(φn ) = |n|H(φ),. For non-stationary dynamical systems we have:. f) Proposition 2.2.2. Suppose f = (fi )i∈Z is an equicontinuous sequence. Fix n ≥ 1. Let ˜f ∈ F 1 (M fi = Mni and f˜i = fn(i+1)−1 ◦· · ·◦fni ; be the gathering obtained of f by the sequence (ni)i∈Z , that is, M f˜i =fn(i+1)−1 ◦···◦fni. f˜i−1 =fni−1 ◦···◦fn(i−1). · · · Mn(i−1) −−−−−−−−−−−−−−→ Mni −−−−−−−−−−−−−→ Mn(i+1) · · ·. Thus, for each i ∈ Z we have Proof.. For. i ∈ Z, x, y ∈ Mni max. and. m > 0,. d(f˜ i (x), f˜ i (y)) = k. 0≤k<m. Hi (˜f ) = nHin (f ).. k. we have. max. 0≤k<m. nk d(f nk ni (x), f ni (y)) ≤. max. 0≤j<nm. d(f jni (x), f jni (y)).. ε > 0, each (nm, ε)-span subset K of Mni with respect to f is a (m, ε)˜ . Consequently, we obtain r[m, ni](ε, f˜ ) ≤ r[nm, ni](ε, f ). respect to f. This fact proves that, for all span subset of. Mni. with. Hence,. Hi (f˜ ) ≤ nHin (f ).. On the other hand, since. f. is equicontinuous, we can prove that. is an equicontinuous collection of families. Consequently, given. n−1 (fni )i∈Z , (f 2ni )i∈Z , . . . , (f ni )i∈Z. ε > 0,. there exists. δ>0. such that. k+1 max {d(f k+1 nj (x), f nj (y)) : x, y ∈ Mnj , d(x, y) < δ} < ε.. 1≤k<n j∈Z Now, if that. K. is a. (m, δ)-span. of. Mni. with respect to. f˜ , then, for all x ∈ Mni , there exists y ∈ K such. max{d(x, y), d(f nni (x), f nni (y)), ..., d(f ni. (m−1)n. (x), f ni. (m−1)n. (y))} < δ.. Thus,. max {d(f kni (x), f kni (y))} < ε,. 0≤k<n. max {d(f kn(i+1) ◦ f nni (x), f kn(i+1) ◦ f nni (y))} < ε,. 0≤k<n. . . .. max {d(f kn(i+m−1) ◦ f ni. (m−1)n. 0≤k<n. (x), f kn(i+m−1) ◦ f ni. (m−1)n. (y))} < ε..

(38) ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS. 16. Consequently, we have. max {d(f kni (x), f kni (y))} < ε,. 0≤k<n. n+k max {d(f n+k ni (x), f ni (y))} < ε,. 0≤k<n. . . .. max {d(f ni. (m−1)n+k. 0≤k<n Therefore,. that is,. (x), f ni. (m−1)n+k. (y))} < ε.. max{d(f kni (x), f kni (y)) : k = 0, ..., mn − 1} < ε,. K. is a. r[nm, ni](ε, f ). (mn, ε)-span. of. Mni. and, therefore,. with respect to. f.. Consequently, we have. r[m, ni](ε, f˜ ) ≥. 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. Let. This follows directly from Denition 2.1.5.. S1 ⊆ R2 φ. morphisms. on. be the circle endowed with the Riemannian metric inherited from. S1. we have that. H(φ) = 0. R2 .. For homeo-. (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. (fj )j≥i all. Fix. i ∈ Z. It follows from Lemma 1.4.1 that there exists a uniform conjugacy (hj )j≥i between. and. j ≥ i,. if. (gj )j≥i .. Since. x, y ∈ Mj. (hj )j≥i. and. is equicontinuous, given. d(x, y). < δ,. then. ε>0. d(hj (x), hj (y)). there exists. < ε.. Let. K. δ>0. be a. such that, for. (m, δ)-span. of. Mi.

(39) PROPERTIES OF ENTROPY with respect to. 17. 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 > max. 0≤j<m. This fact proves that. d(hi+j ◦ f ji (x), hi+j ◦ f ji (y)) =. r[m, i](ε, f ) ≥ r[m, i](δ, g ).. max. 0≤j<m. d(g ji ◦ hi (x), g ji ◦ hi (y)).. Hence,. Hi (f ) ≥ Hi (g ). Since. (h−1 j )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 then. (fi )i≥i0. and. (gi )i≥i0. Hi0 (f ) = Hi0 (g ). Furthermore, entropy depends only on the future:. are uniformly conjugate. 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. It follows from Lemma 1.4.1 that we have. (gj )j≥i0 (fj )j≥0. Hi (f ) = Hi (g ) for all i ∈ Z.. are uniformly conjugate (take. and. (gj )j≥0. hj = Id. for each. j ≥ i0 ).. are uniformly conjugate. By Theorem 2.2.5. Corollary 2.2.7. For all i, j ∈ Z we have Hi (f ) = Hj (f ). Proof.. g. It is sucient to prove that. = (gj )j∈Z ,. where. Hi (f ). gj = Ij : Mj → Mj+1. Mi = M. (remember that. x, y ∈ Mi. and. n≥2. Hi+1 (f ). =. for each. Mi = M × {i}),. and. for all. j≤i. gj = fj. i ∈ Z.. Fix. i ∈ Z.. Take the family. is the identity, modulo the identication. for. j > i.. Thus. Hi (f ) = Hi (g ). For each. we have. max. 0≤j<n. d(g ji (x), g ji (y)) =. Using this fact we can prove that. max. 0≤j<n−1. d(g ji+1 (x), g ji+1 (y)).. 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.. 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 ). Remark 2.2.9.. We can consider the system. f. = (fi )i∈Z. as a homeomorphism.

(40) ENTROPY FOR NON-STATIONARY DYNAMICAL SYSTEMS. 18. 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. Consequently, the entropy for a non-stationary dynamical system on metrics on. this case, for. n>0. we have. and. f. as. d on M .. M is the same for equivalent. f −1 = (gi )i∈Z , where gi := fi−1 : Mi+1 → Mi for each i. In. (f −1 )0i := Ii+1 : Mi+1 → Mi+1. (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. Example 2.2.10. for. considering. M.. We can dene the inverse of. M. f. Let. I:M →M. be the identity on. with non-zero topological entropy. Let. M. and. fi : Mi → Mi+1. H(f ) 6= H(f −1 ).. φ:M →M. be a homeomorphism on. be the dieomorphisms dened as. fi = I. 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 entropy of. f. −1. f. depends only on the future, while the. depends only on the past.. Remark 2.2.11.. As a consequence of Example 2.2.10, we can also consider the entropy. H(−1) (f ). All the above results for H have analog versions for H(−1) .. which we denote by. H(f −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 for each. x ∈ Mi ,. h = (hi )i∈Z is a uniform conjugacy between f. g , then,. we have. hi (V s (x, f )) = V s (hi (x), g ) Let. and. M = S1 , pN. be the north pole and. and. pS. s is a homeomorphism with stable set V (pN , φ). hi (V u (x, f )) = V u (hi (x), g ).. be the south pole of. = M \ {pS }. S1 .. Suppose that. (see Figure 2.2.1). Let. f. φ:M →M and. g. be the.