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Universidade Federal da Bahia-UFBA

Instituto de Matem ´atica-IM

Programa de P ´os-Gradua¸c ˜ao em Matem ´atica-PGMAT

Tese de Doutorado

  Pesin’s Entropy Formula for C

  1

non-uniformly expanding maps.

  

Felipe Fonseca dos Santos

Salvador-BA

2017

  1 Pesin’s Entropy Formula for C non-uniformly expanding maps.

  

Felipe Fonseca dos Santos

  Tese de Doutorado apresentada ao Colegiado da Pós-Graduação em Matemática UFBA/UFAL como requisito parcial para obtenção do título de Doutor em Matemática.

  Orientador: Prof. Dr. Vitor D. Martins

  

de Araújo

Salvador-BA

  Agosto de 2017 Santos, Felipe Fonseca dos.

  

1

Pesin’s Entropy Formula for C non-uniformly expanding maps

  / Felipe Fonseca dos Santos. – Salvador, 2017. 93 f. : il.

  Orientador: Vitor Domingos Martins de Araújo.

  Tese (Doutorado - Doutorado em Matemática) – Universidade Federal da Bahia, Programa de Pós-graduação em Matemática, 2017.

  1. Fórmula de Entropia de Pesin 2. Estados de Equilíbrio.

3. Medidas SRB\física-fraca. 4. Expansão não uniforme 5.Teoria Ergódica. I. Araújo, Vitor Domingos Martins de. II. Título.

  CDD -

  1 C

Pesin’s Entropy Formula for non-uniformly expanding

maps.

  Tese de Doutorado submetida ao Colegiado de Pós-graduação em Ma- temática UFBA/UFAL como parte dos requisitos necessários à obtenção do título de Doutor em Matemática.

  

Banca examinadora:

Prof. Dr. Vitor D. Martins de Araújo (Orientador)

UFBA

  

Prof. Dr. Paulo César Rodrigues Pinto Varandas

UFBA

Prof. Dr. Augusto Armando de Castro Jr

UFBA

  

Profa. Dr. Maria José Pacífico

UFRJ

Profa. Dr. Vanessa Ribeiro Ramos

UFMA

  Salvador-BA Agosto de 2017

  Dedico este trabalho aos meus pais, meus irmãos, à Lara e à minha noiva Milena, por tudo que fizeram por mim para a realização deste sonho.

  

Agradecimentos

  Agradeço primeiramente à Deus por estar sempre comigo, iluminando a minha vida, por ter me capacitado e por estar sempre me protegendo e me dando força.

  À Milena, que tanto amo, companheira de todas as horas, obrigado pela sua confiança e compreensão em ceder grande parte do nosso tempo a esses anos de estudos. Seu amor é fundamental na minha vida e me faz sempre acreditar que posso ir mais longe.

  Ao meus pais Joel e Rosely, meus irmãos Willian e Liz, minha avó Terezinha e minha sobrinha Lara por fazerem de tudo para me ver feliz. O amor, carinho e apoio de Vocês é muito importante na minha vida.

  A todos os meus tios, tias, primos e amigos, por sempre torcerem e me apoiarem em todos os momentos e por compreenderem a minha ausência em muitas ocasiões.

  A toda à Família Borges, por todo apoio e carinho. Sou feliz por ter três maravilhosas Famílias. Ao Professor Vitor Araújo, que novamente me deu a honra e a satis- fação de sua valiosa orientação, pela sua amizade, incentivo, paciência, dedicação, por ter acreditado em mim e pelas grandes oportunidades de aprendizagem a que me proporcionou. Você é um exemplo para mim.

  Aos membros da banca, que muito me honraram com suas presenças,

  a Prof. Augusto Armado de Castro Jr, Prof . Maria José Pacífico Prof. a

  Paulo Cesar Pinto Varandas, Prof . Vanessa Ribeiro Ramos e Prof. Vitor Domingos Martins de Araújo agradeço pelos enriquecedores comentários e sugestões ao trabalho.

  À todos os meus Professores do curso de doutorado: Luciana Salgado, Manuel Stadlbauer, Paulo Varandas, Vilton Pinheiro, Vitor Araújo e Ter- tuliano Franco, pelos valiosos conhecimentos matemáticos que comparti- lharam comigo ao longo das disciplinas, seminários e conversas informais nesse período e por estarem sempre dispostos a ajudar. Vocês são inspira- ção para minha vida profissional.

  A minha turminha do doutorado: Darlan, Fabi, Jacq, Mari, Sara e Elaine. Aos meus maninhos: Andrêssa, Edvan e Junilson e todos os ami- gos e colegas da sala “282”, que ao longo desse tempo tive a felicidade de conhecer, obrigado a todos pelos conselhos, ensinamentos e pelos momen- tos de descontração e alegria durante estes anos. Mesmo correndo o risco de esquecer o nome de alguma pessoa que foi importante nesse período quero destacar os amigos: Adriana, Anderson, Alejandra, Diego, Diogo, Ed, Elen, Heides, Morro, Moa, Roberto, Cattai e Vinicius. Obrigado a todos Vocês!

  A todos os funcionários da pós-graduação do IM da UFBA por estarem sempre dispostos a me ajudar, em especial a Davilene, Mayara, Márcio, Kleber e Diogo.

  Agradeço também aos meus professores da UFRB e da UFBA (período do mestrado) por terem me incentivado e preparado para estar aqui. O apoio e amizade de vocês foram fundamentais.

  Por fim, agradeço a CAPES e a Fapesb pelo apoio financeiro.

  “Nele estão escondidos todos os tesouros da sabedoria e da ciência.” Colossenses 2:3.

  

Resumo

  Provamos a existência de estados de equilíbrio com propriedades espe- ciais para uma classe de homeomorfismos locais positivamente expansivos e potenciais contínuos, definidos em espaço métrico compacto. Além disso,

  1

  formulamos uma generalização de classe C da Fórmula de Entropia de Pesin: toda medida ergódica weak-SRB-like satisfaz a Fórmula de Entropia

  1 de Pesin para transformações de classe C não uniformemente expansoras.

  Mostramos que para transformação expansora fraca, tal que Leb-q.t.p x te- nha frequência positiva de tempos hiperbólicos, medidas weak-SRB-like, existem e satisfazem a Fórmula de Entropia de Pesin e são estados de equi- líbrio para o potencial ψ = − log | det D f |. Em particular, isso é válido para

  1

  qualquer transformação expansora de classe C e neste caso o conjunto de medidas de probabilidade invariantes que satisfazem a Fórmula de Entro-

  ∗

  pia de Pesin é o fecho convexo na topologia fraca das medidas ergódicas weak-SRB-like.

  Palavras-chave:

  Teoria Ergódica, expansão não uniforme, Medidas SRB\física- fraca, Estados de Equilíbrio e Fórmula de Entropia de Pesin.

  

Abstract

  We prove existence of equilibrium states with special properties for a class of distance expanding local homeomorphisms on compact metric

  1

  spaces and continuous potentials. Moreover, we formulate a C genera- lization of Pesin’s Entropy Formula: all ergodic weak-SRB-like measures

  1 satisfy Pesin’s Entropy Formula for C non-uniformly expanding maps.

  We show that for weak-expanding maps such that Leb-a.e x has positive frequency of hyperbolic times, then all the necessarily existing ergodic weak-SRB-like measures satisfy Pesin’s Entropy Formula and are equili- brium states for the potential ψ = − log | det D f |. In particular, this holds

  1

  for any C -expanding map and in this case the set of invariant probability

  ∗

  measures that satisfy Pesin’s Entropy Formula is the weak -closed convex hull of the ergodic weak-SRB-like measures.

  Keywords:

  Ergodic theory, non-uniform expansion, SRB\physical-like mea- sures, equilibrium states, Pesin’s entropy formula.

  x

  

Sumário

xiii

  

  1 . . . . . . . . . . . . . . . . .

  8

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  . . . . . . . . . . . . . . . 26

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  31

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  37

  . . 37

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  xii Sumário

  

  41

  . . . . . . . . . . . 41

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  . . . . . . . . . . . . . . . . . . . . . . . . . . 44

  . . . . . 44

  . . . . . . . . . . . . . . . . . . . . . . . . . . 45

  6.6 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . 49

  

  53

  . . . . . . . . . . . . . . . . . . . . . . . . . 53

  7.2 Proof of Theorem . . . . . . . . . . . . . . . . . . . . . . . . 60

  

  63

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  . . . . . . . . . 66

  

  69

  . . . . 69

  . . . . . . . . . . . . . . . . . . . . . . . . . . 70

  9.3 Proof of Corollary . . . . . . . . . . . . . . . . . . . . . . . . 71

  9.4 Proof of Corollary . . . . . . . . . . . . . . . . . . . . . . . 72

  

  74

  Lista de Figuras . . . . . . . . . . . . . . . . . . . . . . . . . .

  9

   . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

  . . . . . . . . . . . 56

  xiv

Lista de Figuras

  

Capítulo 1

Introdução e descrição dos

resultados

  Os estados de equilíbrio, um conceito originalmente da Mecânica Esta- tística, é uma classe especial de medidas de probabilidade. Na definição clássica, dadas uma transformação contínua T : X X definida em um espaço métrico compacto X e uma função contínua φ : X → R chamada de potencial, dizemos que uma medida de probabilidade T-invariante é φ-estado de equilíbrio (ou estado de equilíbrio para o potencial φ) se Z ( ) Z

  

P top (T, φ) = h (T) + φ dµ, onde P top (T, φ) = sup h (T) + φ dλ ,

µ λ λ∈M T T é o conjunto de todas as medidas de probabilidade T-invariantes.

  e M Nesta relação, P top (T, φ) é chamada pressão topológica e a igualdade acima do lado direito é conhecida por Princípio Variacional; veja por exemplo para as definições de entropia h (T) e pressão topológica. Postergamos as definições formais para os próximos capítulos.

  Dependendo do sistema dinâmico e potencial estudados (T, φ), os es- tados de equilíbrio podem ter propriedades que fornecem informação útil sobre o sistema. Sinai, Ruelle e Bowen foram os pioneiros da teoria dos estados de equilíbrio para sistemas dinâmicos uniformemente hiperbólicos. Eles estabeleceram uma importante conexão entre estados de equilíbrio e medidas que são caracterizadas por fornecer informações assintóticas sobre um conjunto de trajetórias, isto é, dada uma medida µ, esperamos que sua bacia n X −1 Z

  1 j

  

B (µ) = x φ(T x (X, R)

 →+∞  X : ) −−−−→ φdµ, ∀φ ∈ C n n j=

2 Capítulo 1. Introdução e descrição dos resultados

  seja grande do ponto de vista da medida de Lebesgue ou alguma outra medida de referência relevante. Essas medidas são chamadas medidas SRB em homenagem a Sinai, Ruelle e Bowen, e também medidas físicas devido à origem de muitos conceitos na Mecânica Estatística.

  Apesar do substancial progresso de vários autores em estender essa teoria, uma visão global ainda está longe de ser completa. Por exemplo, a estratégia básica utilizada por Sinai-Ruelle-Bowen foi (semi-)conjugar a dinâmica a um subshift de tipo finito, através de uma partição Markov, e usar as fortes propriedades topológicas e ergódicas das Cadeias de Mar- kov Topológicas para obter resultados para a dinâmica uniformemente hiperbólica original.

  No entanto, a existência de partições de Markov é conhecida apenas em alguns casos e, muitas vezes, tais partições podem não ser finitas. Além disso, existem transformações que não admitem estados de equilíbrio ou medidas físicas/SRB (como por exemplo, a transformação identidade).

  Em muitas situações com algum tipo de expansão, é possível garantir que os estados de equilíbrio sempre existam. No entanto, os estados de equilíbrio geralmente não são únicos e, pela forma como são obtidos, não fornecem informações adicionais para o estudo da dinâmica. No contexto das funções uniformemente expansoras, Ruelle mostrou em que existe um único estado de equilíbrio que é uma medida física/SRB se o potencial for Hölder contínuo e a dinâmica f for transitiva.

  Neste sentido, surgem algumas questões naturais:

  1. Quando um sistema dinâmico tem um estado de equilíbrio?

  2. Se tais medidas existem, que propriedades estatísticas essas proba- bilidades exibem em relação à medida de referência (possivelmente não-invariante)?

  3. Quando existe, o estado de equilíbrio é único? A existência de estados de equilíbrio é uma propriedade relativamente mais simples que muitas vezes pode ser estabelecida através de argumen- tos de compacidade. Entretanto, as propriedades estatísticas e a unicidade do estado de equilíbrio são geralmente mais sutis e requerem uma melhor compreensão da dinâmica. Em nosso contexto, não esperamos ter uma unicidade de estado de equilíbrio, porque consideramos:

  • sistemas dinâmicos com baixa regularidade: contínua ou de classe

1 C somente; e • apenas potenciais contínuos (com exceção do Teorema A).

  3 Do Formalismo Termodinâmico a resposta para a primeira questão é

  conhecida e afirmativa para todas as transformações expansivas (também chamadas transformações Ruelle expansiva), em um espaço métrico com- pacto X e para todo potencial contínuo; veja .

  Inspirado na definição de medida observável introduzida por Keller em a noção de medidas “SRB- like” ou pseudo-físicas (uma generalização da noção de medida física/SRB) e mostram que tais medidas sempre existem para toda transformação con- tínua definida em um espaço métrico compacto. Além disto, eles mostram que a definição de medida “SRB-like” é ótima em certo sentido, se o objetivo for descrever as estatísticas assintóticas de Lebesgue quase toda órbita do sistema. Mais precisamente, dado um ponto x X denotamos o conjunto X ∗  n j −1 1 w i

p ω(x) = (x) = δ µ .

  • T j n T (x)
  • µ ∈ M ; ∃n −−−−→ ∞ tal que σ j −−−−→ j →+∞ j →+∞ n j i= 

      Fixado uma medida de referência ν (não necessariamente a medida de T é ν-SRB-like se, Lebesgue) para o espaço métrico X, dizemos que µ ∈ M e só se, ν(A (µ)) > 0 para todo ε > 0, onde ε

      A ε

      (µ) = {x X; dist(pω(x), µ) < ε} é a bacia ε-fraca de atração de µ e dist é uma métrica que induz a topologia

      ∗ fraca no espaço das medidas de probabilidades Boreliana.

      Mostramos que, em nosso contexto, as medidas “ν-SRB-like” são par- ticularmente interessantes, porque podem ser vistas como medidas que surgem naturalmente como pontos de acumulação de medidas ν-SRB.

      Seja T Teorema A.

      : X X uma transformação aberta, expansiva e topologica-

      mente transitiva definida em um espaço métrico compacto X. Consideren ) n ≥1

    uma sequência de potenciais Hölder contínuas, (ν ) uma sequência de medi-

    n n

      ≥1

      ν

    das conformes associados a (T, φ n ) e n ) n uma sequência de medidas n -SRB.

      ≥1 Assuma que:

      φ φ na topologia da convergência uniforme;

      1. n j −−−−→ j →+∞ w

      ν ν na topologia fraca* de convergência de medidas;

      2. n j −−−−→ j →+∞ w µ µ também na topologia fraca*.

      3. n j −−−−→ j →+∞

    4 Capítulo 1. Introdução e descrição dos resultados

      

    Então ν é medida conforme para (T, φ) e µ é ν-SRB-like. Em particular µ é φ-estado

    de equilíbrio. Além disso, se T é topologicamente exata então (µ)) = 0

    ε

      ν(X\A para todo ε > 0. Relacionando agora as transformações que expandem distância e as medidas “ν-SRB-like”, obtemos uma resposta positiva para a segunda pergunta.

      Seja T Teorema B.

      : X X uma transformação aberta, expansiva e topolo-

      gicamente transitiva, definida em um espaço métrico compacto X e

      φ : X → R

      

    contínua. Então, para cada (necessariamente existente) medida φ-conforme ν = ν

    φ

    todas as (necessariamente existentes) medidas ν φ -SRB-like são φ-estado de equilí-

    brio.

      Esse resultado, em particular, estende o principal resultado obtido por Catsigeras-Henrich em , onde foram estudadas apenas as medidas

      1

      “SRB-like” com respeito à medida de Lebesgue para dinâmicas C expan- soras do círculo.

      Em , Catsigeras, Cerminara e Henrich estendem a noção de medidas SRB-like, as medidas “weak-SRB-like” ou weak pseudo-física. Diz-se que T é ν-weak-SRB-like se, e só se, uma medida µ ∈ M

      1 lim sup log ν(A ε,n (µ)) = 0 ∀ε > 0, n n

      →+∞

      onde A ε,n n (µ) = {x X : dist(σ (x), µ) < ε} é a bacia ε-fraca de µ em tempo n .

      Observamos que se µ é um φ-estado de equilíbrio e se “relaciona bem”

      ∗

      com a medida de referência ν então µ é combinação convexa fraca de medi- das ergódicas weak-SRB-like. Além disso, se µ é o único φ-estado de equi- líbrio então µ possui cota superior de grandes desvios, isto é, para qualquer

      ∗

      vizinhança fraca

      1 (espaço das medidas de probabilidade

      V de µ em M P n n (x) := δ 1 −1 j T (x)

      1 Boreliana), então a probabilidade ν({x X : σ j= ∈ M \V}) n decresce exponencialmente com n em uma taxa que depende do “tama-

      nho” de V.

      Corolário C. Sejam T

      : X X uma transformação aberta, expansiva e topolo-

      gicamente transitiva, definida em um espaço métrico compacto X,

      φ : X → R um

      

    potencial contínuo e ν uma medida φ-conforme. Se µ é um φ-estado de equilíbrio

    tal que h ν x de µ é uma medida

      (T, µ) < ∞ então quase toda componente ergódica µ

      

    de probabilidade ν-weak-SRB-like. Além disso, se µ é o único φ-estado de equilí-

      µ é uma medida ν-SRB tal que ν(B(µ)) = 1 e µ satisfaz a seguinte cota

      brio então

      5superior de grandes desvios: para toda vizinhança fraca

      V de µ temos

      1 lim sup n

      1

      log ν({x X : σ (x) ∈ M \V}) ≤ −I(V) n n

      →+∞ onde I r (V) = sup{r > 0 : K (φ) ⊂ V}.

      Considere agora M uma variedade Riemanniana compacta finita sem bordo. Na década de 1970, Pesin mostrou uma relação entre dois impor- tantes conceitos matemáticos: os expoentes de Lyapunov e a entropia para sistemas dinâmicos diferenciáveis.

      Mais precisamente, em , Pesin mostrou que se µ é uma medida

      2 1+α

    f -invariante para um difeomorfismo de classe C (ou C , α > 0) definido

      em uma variedade compacta que é absolutamente contínua com respeito + a medida de Lebesgue, então Z

      Σ

      h ( f ) = d µ, (1.0.1) µ

    • + onde Σ denota a soma dos expoentes de Lyapunov positivos em pontos regulares, contando suas multiplicidades. Dizemos que uma medida f - invariante µ satisfaz a Fórmula de entropia de Pesin se µ satisfaz a equação .

      Ledrappier e Young em , no mesmo contexto que Pesin, generaliza- ram a fórmula de entropia de Pesin, mostrando que uma medida invariante satisfaz a Fórmula de Entropia de Pesin se, e somente se, tiver medidas condicionais absolutamente contínuas em relação a medida de Lebesgue nas folhas instáveis.

      Podemos também citar, Liu, Qian e Zhu que obtiveram em generalizações da fórmula de entropia de Pesin para endomorfismos de

      2 classe C .

    1 Para funções de classe C , a ausência de distorção limitada impossibilita

      o uso de muitos métodos já conhecidos para obter a Fórmula de Entropia de

      2 1+α

      Pesin para dinâmicas de classe C (ou C , α > 0). Recentemente, Xueting Tian mostrou em a fórmula de entropia de Pesin para dinâmica entre

      1 1+α C e C .

      Entretanto, podemos encontrar diversos resultados já obtidos para sis-

      1

      temas dinâmicos de classe C :

    • Ali Tahzibi em mostrou a fórmula de entropia de Pesin para

      1

      difeomorfismos C genéricos que preserva área em superfícies de dimensão 2;

    6 Capítulo 1. Introdução e descrição dos resultados

    • Bessa-Varandas em mostraram que vale o resultado anterior para tempo contínuo (fluxos incompreensíveis) em variedades de dimensão 3;
      • 1

    • Hao Qiu em provou que se f é Anosov transitiva então C genericamente existe uma única medida SRB que satisfaz a fórmula de entropia de Pesin;
    • Em 2012, Sun e Tian exibem em condições para que uma medida invariante µ satisfaça a fórmula de entropia de Pesin para difeomor-

      1

      fismos C com decomposição dominada ao longo da órbita para µ quase todos os pontos. Além disso, entre outras coisas, eles esten- dem o resultado obtido em para qualquer dimensão;

    • Catsigeras e Enrich estabelecem relação entre as medidas “SRB-like” e

      1

      a fórmula de entropia de Pesin para funções C -expansoras no círculo em . Posteriormente, Catsigeras, Cerminara e Enrich estenderam

      1

      essa relação para difeomorfismos C com decomposição dominada em estendem o resultado para

      1

      difeomorfismos C Anosov para medidas “weak-SRB-like”;

    • Yang e Cao, mais recentemente, mostraram em , que se f é um di-

      1

      feomorfismo C que admite atrator Λ com decomposição dominada Λ

      T M = E

      ⊕ F tal que para qualquer medida suportada em Λ temos que todos os expoentes de Lyapunov ao longo de E são não positivos e todos os expoentes de Lyapunov ao longo de F são não negati- vos, então existe medida suportada em Λ que satisfaz a fórmula de entropia de Pesin. Apesar de muito progresso nessa direção, ainda há muitas questões em aberto. Nesse sentido, foi investigada a Fórmula de entropia de Pesin para

      1 difeomorfismos locais de classe C não uniformemente expansor.

      1 Teorema D. Seja f não unifor-

      : M M um difeomorfismo local de classe C

      

    memente expansor. Então toda medida de probabilidade expansora weak-SRB-like

    satisfaz a fórmula de entropia de Pesin. Além disso, todas as medidas de probabi-

    lidades ergódicas SRB-like são expansoras.

      Além disso, foram estudadas a existência de medidas ergódicas “weak- SRB-like” e algumas de suas propriedades para sistemas dinâmicos com alguma propriedade de expansão. Sabe-se que nem todas as dinâmicas admitem medidas ergódicas “weak-SRB-like”, como pode ser visto em Exemplo 5.4]. Porém, Catsigeras, Cerminara e Enrich mostram em

      7

      que medida ergódica “weak-SRB-like” sempre existe para difeomorfismos

    1 Anosov de classe C

      e, mais recentemente em , Catsigeras e Troubetzkoy mostram que, para funções contínuas C -genérica do intervalo, todas as medidas ergódicas são SRB-like.

      Relacionando os conceitos das medidas “weak-SRB-like”, Fórmula de Entropia de Pesin e estado de equilíbrio obtemos

      1 Corolário E. Seja f não unifor-

      : M M um difeomorfismo local de classe C

      

    memente expansor e ( f, ψ) = 0 então, toda medida de

    top

      ψ = − log | det D f |. Se P ψ-estado de equilíbrio e todas as (necessariamente

      probabilidade weak-SRB-like é

    existentes) medidas expansora ergódicas weak-SRB-like satisfazem a Fórmula de

    Entropia de Pesin.

      Ademais, supondo propriedades de expansão mais fortes obtemos:

      1 Corolário F. Seja f expansor

      : M M um difeomorfismo local de classe C

      fraco e não uniformemente expansor. Então,

      1. Todas as (necessariamente existentes) medidas de probabilidade weak-SRB- like são ψ-estados de equilíbrio e, em particular satisfazem a Fórmula de Entropia de Pesin;

      2. Existe medida de probabilidade ergódica weak-SRB-like;

      ψ < 0 não existe medida de probabilidade atômica weak-SRB-like;

      3. Se −1

      4. Se

      D = {x M : kD f (x) k = 1} é finito e ψ < 0 quase todas as ψ-estado de equilíbrio são medidas weak-

      componentes ergódicas de um SRB-like. Além disso, toda medida de probabilidade weak-SRB-like µ, suas

      µ

      componentes ergódicas x são medidas de probabilidades expansoras weak- SRB-like para µ-quase todo x no subconjunto expansor.

      Em particular, vemos que uma construção análoga ao resultado de existência de medida física/SRB e atômica para uma aplicação quadrática,

      1

      obtida por Keller em , não é possível entre transformações C uniforme- mente expansoras, embora genericamente tais medidas sejam singulares com respeito a qualquer forma de volume.

      

    Corolário G. Não existe medida atômica SRB para uma transformação expansora

    1 C de uma variedade compacta.

      1 Pelo trabalho de Ávila e Bochi em , é sabido que C -genericamente

      1

      as transformações de classe C de uma variedade compacta sem bordo de dimensão finita não admitem medida de probabilidade absolutamente

    8 Capítulo 1. Introdução e descrição dos resultados

      contínua em relação a medida de Lebesgue. Esse é um dos empecilhos que assegura que não é possível obter um análogo aos resultados obtidos por Pesin, Ledrappier e Young em de modo geral. Mas podemos nos perguntar se é possível estabelecer condições necessárias e suficientes para que um análogo à Fórmula de Entropia de Pesin seja satisfeita. Neste

      1 sentido, obtemos a seguinte caracterização para dinâmica C -expansora.

      1 Seja f . Então Corolário H.

      : M M uma função expansora de classe C

      

    uma medida de probabilidade f -invariante µ satisfaz a Fórmula de entropia de

    Pesin se, e somente se, suas componentes ergódicas µ são weak-SRB-like µ-q.t.p.

    x x

      ∈ M. Além disso, todas as medidas de probabilidade weak-SRB-like satisfazem

    a Fórmula de entropia de Pesin e, em particular, são ψ-estados de equilíbrio.

      µ weak-SRB-like,

      Adicionalmente, se existe uma única medida de probabilidade

    então µ é SRB, Leb(B(µ)) = 1, µ é o único ψ-estado de equilíbrio, é ergódico e

    possui cota superior de grandes desvios.

      Concluímos a introdução com a seguinte versão fraca de estabilidade

      1

      estatística para dinâmica C -expansora: os pontos de acumulação de toda sequência de medidas weak-SRB-like são combinações lineares convexas generalizadas de medidas ergódicas weak-SRB-like.

      Sejam uma sequência de funções expansoras de Corolário I. n n

      { f : M M} ≥1

      1

      1 classe C tal que f e f n

      → f na topologia C − : M M é uma função expansora de

      1

    classe C . Considere para cada n uma medida weak-SRB-like associada a

    n

      ≥ 1, µ

      

    f . Então todo ponto de acumulação da sequência (µ ) é um estado de equilíbrio

    n n n ≥1 para o potencial

      ψ = − log | det D f | (em particular, satisfaz a Fórmula de Entropia

    de Pesin) e quase todas as suas componentes ergódicas são medidas weak-SRB-like.

    1.1 Comentários e Questões futuras

      Nesta seção listamos algumas questões baseadas nos estudos realizados na presente tese e que serão objeto de trabalhos futuros.

      

    Questão 1.1.1. Nas mesmas hipóteses do Corolário será possível obter uma cota

    inferior de grandes desvios que tenha a mesma taxa encontrada no Corolário

    Talvez seja necessário considerar que a dinâmica é topologicamente misturadora.

      

    Questão 1.1.2. Será possível obter uma propriedade estatística para as medidas

    weak-SRB-like, como no caso do Corolário para transformação expansora fraca

    não uniformimente expansora?

    Questão 1.1.3. Nas hipóteses do Corolário é possível garantir que µ é uma

    medida weak-SRB-like?

    1.1. Comentários e Questões futuras

      9 Alguns exemplos que surgiram naturalmente durante o desenvolvi- mento deste trabalho também motivaram algumas questões.

      O primeiro exemplo é atribuído a Bowen, e pode ser encontrado em maior detalhe para nosso contexto no Exemplo 5.5 em .

      2 Exemplo 1.1.4. Considere um difeomorfismo f em uma bola de R com dois pontos

    de sela A e B, de modo que uma componente conexa da variedade global instável

    u W

      (A)\{A} é um arco que coincide com uma componente conexa da variedade s u

      global estável W s (B)\{B} e, inversamente, a componente conexa W (B)\{B} = W

      (A)\{A}. Seja f , de modo que exista uma fonte C U onde U é o conjunto u u aberto com bordo W (B).

      (A) ∪ W Figura 1.1: Olho de Bowen.

      Se os autovalores da derivada de f em A e B forem adequadamente escolhidos P n 1 −1 j

    conforme especificado em , então a sequência δ para todos

      (x) n

      { j= f } ≥0 n

      x

      ∈ U\{C} não é convergente. Neste caso existe pelo menos duas subsequências

    convergentes para diferentes combinações convexas dos deltas de Dirac δ e δ .

    A B Assim, como observado em , as medidas de probabilidade SRB-like são

      combinações convexas de δ e δ e formam um segmento no espaço A B

      M das me-

      

    didas de probabilidade. Este exemplo mostra que as medidas SRB-like não são

    necessariamente ergódicas.

      Além disso, os autovalores de D f nos pontos de sela A e B podem ser adequada- P n 1 −1 j

    mente modificados para obter a convergência da sequência δ como

    f (x) n ≥0

      { j= } n

      

    indicado no Lema (i) da página 457 em . Na verdade, fazendo uma pequena

    pertubação C de f fora de uma pequena vizinhança dos pontos de sela A e B temos

    que o

      ω-limite das órbitas em U\{C} ainda contém os pontos A e B, obtendo que

      

    a sequência converge para uma única medida µ = λδ A B para

    • todo x

      (1 − λ)δ

      ∈ U\{C}, com uma constante fixa 0 < λ < 1. Então, µ é a única medida

      

    SRB-like. Isso prova que o conjunto de medidas de probabilidade SRB-like não

    depende continuamente da função.

      Este exemplo motiva algumas questões:

    10 Capítulo 1. Introdução e descrição dos resultados

      

    Questão 1.1.5. Quando podemos afirmar que um sistema dinâmico ( f, φ) tem

    medida de probabilidade ergódica ν -SRB-like ou ν -weak-SRB-like? φ φ Pelo exemplo sabemos que o conjunto das medidas de Questão 1.1.6.

    probabilidade ν φ -SRB-like não depende, em geral, continuamente com a função.

    Será que adicionando a dinâmica propriedade de expansão forte (como nos casos do

    Corolário podemos ter variação contínua das medidas ν φ -SRB-like

    ou ν -weak-SRB-like? φ

    Questão 1.1.7. Existe dependência contínua entre as medidas de probabilidades

      ν ν φ? φ -SRB-like (ou φ -weak-SRB-like) e o potencial Uma resposta positiva nesta direção é dada no Teorema é ergódica.

      

    Questão 1.1.8. No caso de uma resposta positiva na Questão será que há

    estabilidade estocástica?

      O próximo exemplo é uma adaptação da transformação Intermitente (Manneville-Pomeau) em um homeomorfismo local do círculo.

      Exemplo 1.1.9. Considere I =

      [−1, 1] e a transformação ˆf : I I (veja a Figura

      dada por ( √

      2 x se x − 1 ≥ 0,

      ˆf(x) = √ 1 − 2 |x| caso contrário.

      1

      1

      : S

      Esta transformação induz um homeomorfismo local contínuo f

      → S

      1

      =

      através da identificação S

      I

      / ∼, onde −1 ∼ 1. Esta é uma transformação que só

      

    não é diferenciável no ponto 0, possui frequência positiva de tempos hiperbólicos

    para Lebesgue quase todos os pontos, como pode ser visto em seção 5].

      Este é um exemplo de transformação expansora fraca e não uniforme-

      1 mente expansora, mas não é um difeomorfismo local C em todo ponto.

      Este exemplo sugere que podemos pensar em uma generalização do Teorema substituindo a medida

      (X, R) de Lebesgue por uma medida expansora φ-conforme ν para φ ∈ C ν ν f , onde J f é o Jacobiano f com respeito a ν. e o potencial ψ por − log J

      Sejam T Questão 1.1.10.

      : X X um homeomorfismo local definido em um

      espaço métrico compacto X e

      φ : X → R um potencial contínuo. Se existe medida

      

    expansora φ-conforme ν então todas as (necessariamente existente) medidas ν-

      φ?

      SRB-like são estados de equilíbrio para o potencial

      1.2. Organização do trabalho y=f(x) 1

      11

      0.5 −1 −0.5 y=x 0.5 1 −0.5 −1 Figura 1.2: Gráfico de f .

      Pode ser necessário considerar hipóteses adicionais sobre os potenciais, como por exemplo que os potenciais contínuos sejam potenciais hiperbólicos (veja a definição e resultados sobre o potencial hiperbólico Hölder contínuo em ).

      Além disso, também podemos pensar em uma generalição do Teorema

      1 para o caso de difeomorfismo local de classe C afastado de conjunto

      crítico/singular com frequência positiva de tempos hiperbólicos (veja para definição de conjunto crítico/singular).

      1 Questão 1.1.11. Seja f afastado

      : M M um difeomorfismo local de classe C

      de um conjunto crítico/singular

      C com recorrência lenta a C e tal que para alguns 0 < σ < 1, b, δ > 0 e θ > 0 Leb −q.t.p. x tem frequência positiva ≥ θ de (σ, δ, b)-tempos hiperbólicos para f . Todas as medidas de probabilidade expansora

      weak-SRB-like satisfazem a Fórmula de Entropia de Pesin?

      1.2 Organização do trabalho

      A tese está dividida em nove capítulos, incluindo o capítulo introdutó- rio “Introdução e descrição dos resultados”. Nos capítulos denomina- dos respectivamente “Statement of the results” e “Preliminary definitions and results” serão introduzidos os principais resultados da tese, grande parte das definições básicas e notações necessárias ao longo do texto, além de alguns resultados auxiliares sobre medidas de probabilidades ν-SRB- like, ν -weak-SRB-like e entropia. φ

      No capítulo intitulado “Examples of application” apresentamos al- guns exemplos onde os principais resultados desse trabalho podem ser

    12 Capítulo 1. Introdução e descrição dos resultados aplicados.

      No Capítulo No Capítulo “Expanding maps on compact metric space” serão exi- bidas algumas propriedades das aplicações expansivas e a demonstração do Teorema

      No Capítulo “Entropy Formula” serão usadas as propriedades dos tempos hiperbólicos e das medidas weak-SRB-like para provar uma refor-

      1

      mulação da Fórmula de Entropia para difeomorfismo local de classe C não uniformemente expansor (o Teorema .

      No Capítulo “Ergodic weak-SRB-like measure” será estudado a exis- tência de medida ergódica weak-SRB-like, sua relação com a Fórmula de Entropia de Pesin e a prova do Corolário

      No Capítulo “Weak-Expanding non-uniformly maps” serão usados alguns resultados obtidos nos capítulos anteriores para provar os Corolá- rios

    Chapter 2 Statement of the results Let T : X → X be a continuous transformation defined on a compact metric space (X, d).

    2.1 Topological pressure

      The dynamical ball of center x X, radius δ > 0, and length n ≥ 1 is defined by j j

      B x , T y (x, n, δ) = {y X : d(T ) ≤ δ, 0 ≤ j n − 1}.

      Let ν be a Borel probability measure on X. We define

      1

      

    h (T, x) = lim lim sup log ν(B(x, n, δ)) and

    ν

      −

      δ→0 n n →+∞ h ν ν (T, x). (2.1.1)

      (T, µ) = µ − ess sup h Note that ν is not necessarily T-invariant, and if µ is ergodic T-invariant, µ µ (T, x) = h (T), the usual metric entropy of T we have for µ-a.e. x X, h with respect to µ.

      Let n be a natural number, ε > 0 and let K be a compact subset of X. A subset F of X is said to (n, ε) span K with respect to T if ∀x K there exists j j

      y x , T y

      ∈ F with d(T ) ≤ ε for all 0 ≤ j n − 1, that is, [

    K B (x, n, ε).

      ⊂ x

      ∈F

      If K is a compact subset of X. Let

      1

      h (T; K) = lim lim inf log N(n, ε, K), (2.1.2) n ε→0 →+∞ n

      14 Chapter 2. Statement of the results

      where N(n, ε, K) denote the smallest cardinality of any (n, ε)-spanning set for K with respect to T.

      Definition 1. The topological entropy of T is h top

      (T) = sup{h(T, K); K X}, where the supremum is taken over the collection of all compact subset of

      X .

      Let φ : X → R be a real continuous function that we call the potential. Given an open cover α for X we define the pressure P (φ, α) of φ with T respect to α by   X  1   n S φ(U)

      P (φ, α) := lim log inf e T n   n   →+∞ n U⊂α   U

      ∈U n n −1 −j

      = T (α), where the infimum is taken over all subcovers U of α ∨ P n j=

      −1 j S φ(x) := φ(T x ) and S n n n j= φ(U) := sup{S φ(x); x U}.

      

    Definition 2. The topological pressure P top (T, φ) of the potential φ with re-

      spect to the dynamics T is defined by      

      

    P top (T, φ) = lim sup P (φ, α)

        T δ→0 |α|≤δ

      where |α| denotes the diameter of the open cover α.

      For given n > 0 and ε > 0, a subset E X is called (n, ε)-separated if j j

      x x , T y ) > ε.

      , y E, x , y then there exists 0 ≤ j n − 1 such that d(T An alternative way of defining topological pressure is through the no- tion of (n, ε)-separated set.

      

    Definition 3. The topological pressure P (T, φ) of the potential φ with

    top

      respect to the dynamics T is defined by     X 1     S n φ(x)

      P top (T, φ) = lim lim sup log sup e     ε→0 n   n →+∞ xE

      where the supremum is taken over all maximal (n, ε)-separated sets E.

      We refer the reader to for more details and properties of the topo- logical pressure.

      2.2. Expanding maps

      15

      2.2 Expanding maps

      A continuous mapping T : X X is expanding (with respect to the metric d) if there exist constants λ > 1, η > 0 and n ≥ 1, such that for all

      x

      , y X n n if d(x, y) < 2η then d(T (x), T (2.2.1) (y)) ≥ λd(x, y). In the sequel we will always assume (without loss of generality, see chapter 3 in ) that n = 1, that is

      d

      (2.2.2) (x, y) < 2η =⇒ d(T(x), T(y)) ≥ λd(x, y). We refer the reader to for more details and properties of expand- ing map.

      2.3 Transfer operator T ,φ φ =

      We consider the Ruelle-Perron-Fröbenius transfer operator L L associated to T : X X and the continuous function (potential) φ : X → R as the linear operator defined on the space C (X, R) of continuous functions

      g

      : X → R by φ (g)(x) = e g (y). X φ(y) L T (y)=x

      The dual of the Ruelle-Perron-Fröbenius transfer operator is given by

      ∗

      1

      φ

    1 L : M → M

      ∗

      η η : C 7→ L (X, R) → R φ Z ψ φ ψd η.

      7→ L is the set of Borel probabilities in X.

      1

      where M

      

    2.4 Weak-SRB-like and SRB-like probability mea-

    sures

      For any point x X consider, X n −1

      1 j σ (x) = δ (2.4.1) n T (x)

      n

    j=

      where δ y is the Dirac delta probability measure supported in y X.

      16 Chapter 2. Statement of the results Definition 4. T the limit set with

      For each point x M, denote pω(x) ⊂ M initial state x, ( w )

    • →+∞ →+∞

      p ω(x) := (x) T j n µ .

      µ ∈ M ; ∃n −−−−→ ∞ s.t. σ j −−−−→ j j

      

    Definition 5. Fix a reference measure ν for the space X, we say that prob-

    T is ν-SRB (or ν-physical) if ν(B(µ)) > 0, where ability measure µ ∈ M B (µ) = x X; pω(x) = {µ} is the "ergodic basin" of µ. T and ε > 0. We will consider the following measurable sets

      Let µ ∈ M in X:

      A ε,n n (2.4.2)

      (µ) := {x X : dist(σ (x), µ) < ε};

      A ε

      (2.4.3) (µ) := {x X : dist(pω(x), µ) < ε}. We call A ε,n (µ) the ε-pseudo basin of µ up to time n, A ε (µ) the basin of ε-weak

      ∗

    statistical attraction that induces the weak

      1

      of µ and dist is a metric in M topology (see definition in ). Fix a reference probability measure ν for the space X. We say Definition 6. that a T-invariant probability measure µ is:

      1. ν-SRB-like (or ν-physical-like), if and only if ν(A ε (µ)) > 0 for all ε > 0; 2. ν-weak-SRB-like (or ν-weak-physical-like), if and only if

      1 lim sup log ν(A ε,n (µ)) = 0 ∀ε > 0. n →+∞ n When ν = Leb we say that µ is simply SRB-like (or weak-SRB-like).

      It is easy to see that every ν-SRB measure is also a ν-SRB-like Remark 2.4.1.

    measure. Moreover, the ν-SRB-like measures are particular case of ν-weak-SRB-

    like (see item B in ).

      ∗ T the set of ν-weak-SRB-like probability mea-

      We denote W (ν) ⊂ M T

      ∗ ∗ .

      sures. When ν = Leb we denote W (Leb) = W T T Definition 7.

      Given T : X X and an arbitrary continuous function φ : M → R, we say that a probability measure ν is conformal for T with

      ∗ ν = λν.

      respect to φ (or φ-conformal) if there exists λ > 0 so that L φ

      2.5. SRB-like measures as limits of SRB measures

      17

      

    2.5 SRB-like measures as limits of SRB measures

      The next result shows that we can see the ν-SRB-like measures as mea- sures that naturally arise as accumulation points of ν -SRB measures. n

      Theorem A. Let T

      : X X be an open expanding topologically transitive map

      

    of a compact metric space X, (φ ) a sequence of Hölder continuous potentials,

    n n ≥1

      (ν ) a sequence of conformal measures associated to the pair (T, φ ) and (µ ) n n n n n

      ≥1 ≥1 a sequence of ν -SRB measures. Assume that n 1. φ φ (in the topology of uniform convergence); n j −−−−→ j

      →+∞ w 2. ν ν (in the weak topology); n j −−−−→ j

      →+∞ w 3. µ µ (in the weak topology). n j −−−−→ j

      →+∞

    Then ν is a conformal measure for (T, φ) and µ is ν-SRB-like. In particular, µ is

    an equilibrium state for the potential φ. Moreover, if T is topologically exact then

    ε (µ)) = 0 for all ε > 0.

      ν(X \ A This is one of the motivations for the study of SRB-like measures as

      1 the natural extension of the notion of physical/SRB measure for C maps.

      Additional justification is given by the results of this work.

      It is worth noting that Hölder continuous potentials are only used in this work on the assumption of Theorem and all continuous potentials can be approximated by Lipschitz potentials (and all Lipschitz continuous potential are α-Hölder continuous for α ∈ (0, 1]) see Remark

      The next result extends, in particular, the main result obtained in Theorem 2.3] valid only for expanding circle maps.

      Theorem B. Let T

      : X X be an open expanding topologically transitive

      map of a compact metric space X and

      φ : X → R a continuous potential. For

      

    each (necessarily existing) conformal measure ν φ all the (necessarily existing)

    ν -SRB-like measures are equilibrium states for the potential φ. φ Definition 8.

      Let T : X X be a continuous map and φ : X → R a T continuous potential. Given r > 0 we consider the set in M Z r T µ top : h (T) + T is the set of all T-invariant Borel probability measures. K (φ) = {µ ∈ M φdµ ≥ P (T, φ) − r} where M

      18 Chapter 2. Statement of the results Corollary C. Let T

      : X X be an open expanding topologically transitive

      map of a compact metric space X,

      φ : X → R a continuous potential and ν a φ-conformal measure. If µ is a φ-equilibrium state such that h ν

      (T, µ) < ∞ then

      

    every ergodic component µ of µ is a ν-weak-SRB-like measure. Moreover, if µ is

    x

    the unique φ-equilibrium state then µ is ν-SRB, ν(B(µ)) = 1 and µ satisfies the

    following large deviation bound: for every weak neighborhood

      V of µ we have

      1 lim sup n

      1

      log ν({x X : σ (x) ∈ M \V}) ≤ −I(V) n n

      →+∞ where I r (V) = sup{r > 0 : K (φ) ⊂ V}.

    2.6 Non-uniformly expanding maps

      We denote by k · k a Riemannian norm on the compact m-dimensional boundaryless manifold M, m ≥ 1; by d(·, ·) the induced distance and by Leb a Riemannian volume form, which we call Lebesgue measure or volume and assume to be normalized: Leb(M) = 1. Note that Leb is not necessarily

      f -invariant.

    1 Recall that a C

    • map f : M M is uniformly expanding or just expanding if there is some λ > 1 such for some choice of a metric in M one has
    • x M kD f (x)vk > λkvk, for all x M and all v T \{0}.

        1

        local diffeomorphism In what follows we write always f : M M be C and ψ := − log | det D f |.

        Next corollary suggests that Theorem should be extended to weaker forms of expansion. Recall that a Borel set in a topological space is said to have total proba- bility if it has probability one for every f -invariant probability measure.

      1 Corollary 2.6.1. Let f local diffeomorphism and Y be a subset

        : M M be C

        with total probability. If n X −1

        1 j

        −1

        lim inf (x)) n log kD f ( f k < 0, ∀x Y,

        →+∞ n j=

      then all the (necessarily existing) SRB-like measures are equilibrium states for the

      potential ψ = − log | det D f |.

      2.6. Non-uniformly expanding maps

        19

      1 Proof. For a C

      • local diffeomorphism f : M M we have that Leb is ψ-conformal. Moreover, we know by Theorem 1.15 in that if n
      • X −1 1 j

          −1

          lim inf (x)) n →+∞ log kD f ( f k < 0

          n j=

          for x in a total probability subset, then f is uniformly expanding. Thus, by Theorem we concluded the proof.

        1 The Lyapunov exponents of a C local diffeomorphism f of a compact

          manifold M are defined by Oseledets Theorem which states that, for any

          f

        • invariant probability measure µ, for almost all points x M there is
        • x M = F

            1 2 κ(x) κ(x)+1

            κ(x) ≥ 1, a filtration T (x) ⊃ F (x) ⊃ · · · ⊃ F (x) ⊃ F (x) = {0}, and numbers λ

            1 (x) > λ 2 k i (x) = F i ( f (x)) and

            (x) > · · · > λ (x) such D f (x) · F

            1 n lim (x) i n log kD f (x)vk = λ

            →±∞ n i i+

            1 i (x) are called for all v F (x)\F (x) and 0 ≤ i ≤ κ(x). The numbers λ

          Lyapunov exponents of f at the point x. For more details on Lyapunov

          exponents and non-uniform hyperbolicity see .

          1 Definition 9. local diffeomorphism. We say that

            Let f : M M be C f satisfies the Pesin Entropy Formula if + µ ∈ M Z

            Σ

            

          h ( f ) = d µ,

          µ
          • + where Σ denotes the sum of the positive Lyapunov exponents at a regular point, counting multiplicities.

            Let 0 < σ < 1 we denote, n X  −1  −1  1  j

            H (σ) = x (x)) (2.6.1) M; lim sup log kD f ( f k < log σ n →+∞  n j= Definition 10.

            We say that f : M M is non-uniformly expanding if there exists σ ∈ (0, 1) such that Leb(H(σ)) = 1. A probability measure µ (not necessarily invariant) is expanding if there exists σ ∈ (0, 1) such that µ(H(σ)) = 1. Next result relates non-uniform expansion and expanding measures with the Entropy Formula.

            20 Chapter 2. Statement of the results Theorem D. Let f

            : M M be non-uniformly expanding. Every expanding

            

          weak-SRB-like probability measure satisfies Pesin’s Entropy Formula. In parti-

          cular, all ergodic SRB-like probability measures are expanding.

            Now we add a condition ensuring that there exists expanding ergodic weak-SRB-like probability measure and that all weak-SRB-like measures become equilibrium states.

            Corollary E. Let f top ( f, ψ) = 0,

            : M M be non-uniformly expanding. If P

            

          then all weak-SRB-like probability measures are ψ-equilibrium states and all the

          (necessarily existing) expanding ergodic weak-SRB-like measures satisfy Pesin’s

          Entropy Formula.

          2.7 Weak and non-uniformly expanding maps

            Now we strengthen the assumptions on non-uniform expansion to im- prove properties of weak-SRB-like measures.

            −1 Definition 11.

            k ≤ 1 for all We say that f is weak-expanding if, kD f (x)

            xM.

            We say that a probability measure is atomic if it is supported on a finite set. We denote supp(µ) the support of probability measure µ.

            Corollary F. Let f : M M be weak-expanding and non-uniformly expanding.

            Then, 1. all the (necessarily existing) weak-SRB-like probability measures are ψ-

          equilibrium states and, in particular, satisfy Pesin’s Entropy Formula;

          2. there exists some ergodic weak-SRB-like probability measure;

            ψ < 0 there is no atomic weak-SRB-like probability measure;

            3. if −1

            4. if

            D = {x M : kD f (x) k = 1} is finite and ψ < 0 almost all ergodic

            components of a ψ-equilibrium state are weak-SRB-like measures. Moreover all weak-SRB-like probability measures µ its ergodic components µ are x expanding weak-SRB-like probability measure for

            µ-a.e. x M\D. In particular, we see that an analogous result to the existence of an atomic physical measure for a quadratic map, as obtained by Keller in

            1

            , is not possible in the C expanding setting, although generically such measures must be singular with respect to any volume from.

          2.7. Weak and non-uniformly expanding maps

            21

            1 Corollary G. No atomic measure is a SRB measure for a C uniformly expanding map of a compact manifold.

            We now restate the previous results in the uniformly expanding setting.

            

          1

          Corollary H. Let f -expanding map. Then an f -invariant

            : M M be a C µ satisfies Pesin’s Entropy Formula if and only if its ergodic

            probability measure components µ are weak-SRB-like x

            µ-a.e. x M. Moreover, all the (necessarily

            

          existing) weak-SRB-like probability measures satisfy Pesin’s Entropy Formula, in

          particular, are ψ-equilibrium states. In addition, if µ is the unique weak-SRB-like

          probability measure then µ is SRB probability measure, Leb(B(µ)) = 1, is ergodic,

          µ is the unique ψ-equilibrium state and µ satisfies large deviation bound.

            Using this we get a weak statistical stability result in the uniformly

            1 ∗

          C -expanding setting: all weak accumulation points of weak-SRB-like

            measures are generalized convex linear combinations of ergodic weak- SRB-like measures, as follows.

            1 Corollary I. Let a sequence of C -expanding maps such that n n

            { f : M M} ≥1

            1

            1

          f -topology and f -expanding map. Let (µ ) a

          n n n

            → f in the C − : M M be a C ≥1

            

          sequence of weak-SRB-like measures associated f . Then all accumulation points of

          n

            (µ ) is an equilibrium state for the potential n n

            ≥1 ψ = − log | det D f | (in particular,

          satisfy Pesin’s Entropy Formula) and almost all ergodic components are weak-

          SRB-like measures.

            22 Chapter 2. Statement of the results

          Chapter 3 Examples of application Here we present some examples of applications. In the first section

            1

            we show the construction of a C uniformly expanding map with many different SRB-like measures which in particular satisfies the assumptions of Theorem

            In the second section we present a class of non uniformly expanding

          1 C local diffeomorphisms satisfying the assumptions of Theorem D. In the

            third section we exhibit examples of weak-expanding and non uniformly expanding transformations which are not uniformly expanding in the set- ting of Corollary

            

          3.1 Expanding map with several absolutely con-

          tinuous invariant probability measures

            1

            1

            1 We present the construction of a C -expanding map f : S that

            → S has several SRB-Like measures. Moreover, such measures are absolutely continuous with respect to Lebesgue measure (which in particular means

            1

            1 that f does not belong to a C generic subset of C (M, M), see ).

            

          Example 3.1.1. Following , we first adopt some notation for Cantor sets

          with positive Lebesgue measure. Let I be a closed interval and α > 0 numbers

          P n

            ∞

            α

            with n n= < |I|, where |E| denotes the one-dimensional Lebesgue measure of ′ ′ the subset E of I. Let a = a 1 a 2 . . . a denote a sequence of s and n 1 s of length

            = [a, b],

            n = n I = h i (a); we denote the empty sequence a = ∅ with n(a) = 0. Define Ia+b a+b α α

            ∗ ∗

          • 2

            = , aI I and I a recursively as follows.

            −

            2

            2

            2 ∅ ∗

            Let I and I a a 1 be the left and right intervals remaining when the interior of I α n (ak) a

          is removed from I a ; let I be the closed interval of length n (ak) and having the same

          a

            2

            24 Chapter 3. Examples of application center as I (k=0,1). ak T S

            =

            The Cantor set K is given by K I I a m= n (a)=m I .

            This is the standard construction of the Cantor set except that we allow our-

          selves some flexibility in the lengths of the removed intervals. The measure of K

          Iis Leb(K α > 0. I n

            ) = |I| − P n=

            Suppose that another interval J > 0 such that n PI is given together with β ∞ ∗ n= < |J|. One can then construct J a β β , J and K as above. Let us assume now n n a J that α n −−−→ γ ≥ 0. Following the construction of Bowen , we get g : I J n

            →∞

            1 ′

          a C orientation preserving homeomorphism so that g and

          I

            (x) = γ for all x K

            ′ g (x) > 1 for all x I. P

            ∞ More precisely, let us take J = > 0 with β < 2 n n β β β n+ 1 [−1, 1] and choose β n= h i n+ 1 P

            1 ∞

            .g. β = , 1 α = α <

            and e n 2 . Let I = and n . Then n

            → 1, β β β n n (n+100)

            2 2 n=

            =

            and γ = lim

            1 −

          2. We define a homeomorphism G : (−I) ∪ I → J by

            2 α ( n

          • + g (x) if x T

            ∈ I

            −n

            =

            G (x) = , where K G : K J K J J . n= (J) and G|

            −g(−x) if x ∈ (−I)

            3

            2 β β 3β 9β

            4

            4

            = =

            Consider now c 1 , c 2 ; f

            1

            − 2 − 3 , − β

            2 β β 4 : [−

            2 4 ] →

            ] given by [−1, −

            4 ! ! !

            3

            2

            β β β

          • 1 (x) = c

            f

            1 x +

          2 x +

            2 x + − c − 1

            2

            2

            2 β β β β β β

            3

            2

            and f 2 : [ , , 1] given by f 2 (x) = c 1 x c 2 x

            2 x 1. ] → [ − − −

            4

            2

            4

            2

            2

            2 Then, we have β β β β β β

            = = = =

            1. f , f and f

            1

            1

            1

            

          2

            2

            − −1, f − − β

            2 f ( h ( ) f ( h ( ) 1 )− f β0 β0 β0 β0 2

            4 1 β β 2 + +

            4

            4

            4

            2 1 )− f 4 + + 1 4

            = = = = =

            2. f lim 2, f 2, f lim +

            1

            2 h h

            

          2

            4

            1

            4 h h β →0 →0 −

            2.

            = 2 and f

            2

            2 h i β β β β β β 2 n

            = = =

            Consider now J 1 , , I 1 , and choose β > 0 with P n+ β β β β β ′ ′ ′

            −

            4 1 P P

            4

            8 4 n

            4 ∞ n+ 1 n+ 1 ∞ ∞ n+ 1 ′ ′ ′

            = = = n= n → 1. Let α n n= n n= β < and , then α < β β β 2 β β n β 2 n

            2

            2 2β n 1 1 n

            = = =

            − β −

            and γ = lim ′ lim 2.

            2

            2

            2

            2 4 α β n n+ 1 Similarly to the above construction, we obtain a homeomorphism G

            1

            1

            : (−I ) ∪

            I

            1 1 given by

            → J (

            g 1 (x) if x

            1

            ∈ I

            G 1 (x) =

            1 1 ) ,

            −g (−x) if x ∈ (−I

          • + 3.1. Expanding map T

            25 ∞ −n

            =

            where K G (J J 1 n= 1 ) is a Cantor set with positive Lebesgue measure, G 1 : K

            |

          1 J1

            1 K J and g 1 → J 1 : I

            1 1 is a C orientation preserving homeomorphism so that ′ ′ g J and g 1 .

            (x) = 2 for all x K 1 (x) > 1 for all x I

            1

            1 β β 2 β β 2 Similarly we obtain f : [0, ] and f , 1] such

            3

            4

            ] → [−1, − : [− , 0] → [ β β 2 β −β β

            8 2

            4

            8 + 2

            4 − that f

            3 3 ( , f ( ) = 2, f (0) = 2, f 4 ( ) = , f 4 (0) = 1,

            (0) = −1, f ) = − β 2

            8

            4

            3

            8

            3

            8

            4

          • +

            f ) = 2 and f (0) = 2

            (−

            4

            8

          4 Finally, define the function f

             : J J by (see Figure for its graph) (x)

            G if x f  β β ∈ (−I) ∪ I

            1 (x) if x ∈ − , − β β

            2

            4 f , 2

            (x) if x

            4

            2

          f (x) = ,

          G

            1

            1

            1 (x) if x ∈ (−I ) ∪ I β 2 f , 0  (x) if x ∈ −

            4 f

            8 β 2

          3 0,

            (x) if x

            8 Figure 3.1: Graph of f .

            Identifying

            −1 and 1 and making a linear change of coordinates we obtain a

            1

          1 C expanding map of the circle f : S .

            Consider Leb the normalized Lebesgue measure of the set K , this is, Leb (A) = K J J K J N

          • + =
          • J J

              )/ Leb(K

              and consider

              Leb(AK ) for all measurable A J. Denote Σ {0, 1}

            • + 2

              the homeomorphism h : K J J that associates each point x the sequence

              → Σ

            • + 2 ∈ K

              =

              a describing its location in the set K , this is, a is such that I J a J

              ∈ Σ ∩ K {x}. Let

            • + 2

              µ be the Bernoulli measure in Σ giving weight 1/2 to each digit.

              2

              26 Chapter 3. Examples of application Claim 3.1.2. µ = h Leb

              ∗ K J + We show that this relation holds for an algebra of subsets which generates the n

            Borel σ-algebra of Σ . For a = a a . . . a we have that µ([a]) = 1/2 . Moreover,

              1 2 n

              2 −1 −1

            h ([a]) = I , so Leb (h ([a])) = Leb (I ). By construction, at each

            a J K J K J a J

              ∩ KK

              

            step all remaining intervals in the construction of K have the same length. Thus,

            n J the n-th stage contains

              2 intervals I a (among them) all with the same Lebesgue

              measure. Since Leb is a probability measure, we have K J n

              1 2 Leb K (I a J K (I a J ) = . J JK ) = 1 =⇒ Leb ∩ K n

              2

              −1 We conclude that Leb (I ) = Leb (h ([a])) = µ([a]) for all a = K J a J K J

              ∩ K +

              a a . . . a , n

              1 2 n ≥ 1 proving the Claim.

              =

              Clearly h K

              ◦ f | J σ ◦ h where σ is the standard left shift σ : Σ . Since µ

              2

            is σ-invariant, then Leb is f -invariant. Consequently ( f, Leb ) is mixing,

            K K K J | J J

              ( f ) = log 2 = h ( f ) = h (σ) and Leb

              h top Leb µ K KJ J ≪ Leb.

              Hence lim σ (x) = Leb for Leb and n →+∞

            n K K J J

            J Ja.e, x K |K | > 0, so it follows

            that Leb K is a SRB-like measure. By Theorem Leb K is an equilibrium state

            J Jfor the potential ψ = − log | f |.

              Leb Leb Similarly K is also a SRB-like measure, but distinct form K . J1 J

              Note that this construction allows us to obtain countably many ergodic SRB-like measures by reapplying the construction to each removed subin- terval of the first Cantor set.

              Note also that, if we take a sequence of Hölder continuous potentials

              ′

              φ n n n a φ -conformal measure and a converging to ψ = − log | f |, choose ν

              ∗

              ν n -SRB measure µ n and weak accumulation points ν, µ as in Theorem since f is topologically exact then ν(A (µ)) = 1 for all ε > 0. Therefore ν is ε

              1 not Lebesgue measure on S .

            1 C

            3.2 Non uniformly expanding maps

              1 Next we present an example that is a robust (C open) class of non-

              1 uniformly expanding C local diffeomorphisms.

              This family of maps was introduced in Chapter 1]) and maps in this class exhibit non-uniform expansion Lebesgue almost everywhere but are not uniformly expanding.

              

            Let M be a compact manifold of dimensiom d

            Example 3.2.1.

              : M ≥ 1 and f

              1 is a C

              −expanding map. Let V M be some small compact domain, so that

              

            the restriction of f to V is injective. Let f be any map in a sufficiently small

            1 C -neighborhood so that:

              N of f

            3.3. Weak-expanding and non-uniformly expanding maps

              27 1. f is volume expanding everywhere: there exists σ 1 > 1 such that for every x

              1

              | det D f (x)| ≥ σ ∈ M

              2. f is expanding outside V: there exists σ < 1 such that −1 for every x

              kD f (x) k < σ ∈ M\V;

              3. f is not too contracting on V : there is some small δ > 0 such that −1 kD f (x) k < 1 + δ for every x V.

            1 Then every map f in such a C -neighborhood is non-uniformly ex-

              N of f

              

            panding (see a proof in subsection 2.1 in every expanding

            weak-SRB-like probability measure satisfies Pesin’s Entropy Formula, and every

            ergodic SRB-like measure is expanding.

              Such classes of maps can be obtained through deformation of a uniformly expanding map by isotopy inside some small region.

              

            3.3 Weak-expanding and non-uniformly expan-

            ding maps

              Consider α > 0 and the map T α α 1+α : [0, 1] → [0, 1] defined as follows ( x + 2 x if x ∈ [0, 1/2)

              

            T α (x) = .

            α 1+α x

              − 2 (1 − x) if x ∈ [1/2, 1]

              1 1+α

              This defines a family of C maps of the unit circle S := [0, 1]/ ∼ into itself, known as intermittent maps. These applications are expanding, except at a neutral fixed point, the unique fixed point is 0 and DT (0) = 1. The local α behavior near this neutral point is responsible for various phenomena. The above family of maps provides many interesting results in ergodic theory.

              If α ≥ 1, i.e. if the order of tangency at zero is high enough, then the Dirac mass at zero δ is the unique physical probability measure and so the Lyapunov exponent of Lebesgue almost all points vanishes (see ). This example shows that there exists systems which are weak-expanding but not non-uniformly expanding. Hence the assumption of weak expansion together with non-uniform expansion in Corollary is not superfluous.

              The following is an example of a map which is weak-expanding and non-uniformly expanding in the setting of Corollary

              28 Chapter 3. Examples of application Example 3.3.1. If, in the setting of the construction of the previous Example

            the region V of a point p of a periodic orbit with period k, and the deformation

            k

            weakens one of the eigenvalues of D f (p) in such a way that 1 becomes and

            k

            eigenvalue of D f (p), then f is an example of a weak expanding and non uniformly

              1 expanding C transformation. t More precisely, consider the function g (t) = (0) = 0.

              , 0 < t ≤ 1/2, g

              log(1/t) It is easy to see that

              ′ is stricly increasing;

            • g (t) > 0, 0 < t ≤ 1/2 and so g

              ′ is not of

            • g α-generalized bounded variation for any α ∈ (0, 1) (see for α

              ′ the definition of generalized bounded variation) and so g is not C for any 0 < α < 1.

              1 Setting g (t) = g (t)/g strictly increas-

              (1/2) we obtain g : [0, 1/2] → [0, 1] a C

              1+α ing function which is not C for any

              α ∈ (0, 1). Now we consider the analogous

              map to T α

              ( x + xg(x) if x ∈ [0, 1/2)

            T (x) = .

              x

              − (1 − x)g(1 − x) if x ∈ [1/2, 1]

              1 1+α

            This is now a C map of the circle into itself which is not C for any 0 < α < 1.

              =

              Moreover, letting f T satisfies

              × E where E(x) = 2x mod 1, we see that f

            items (1-3) in Example for V a small neighborhood of the fixed point (0, 0).

              1 ′

              Hence f is a C non-uniformly expanding map. Moreover, since T (0) = 1,

              1 we have that f is also a C weak expanding map with D = {(0, 0)}.

              Now we extend this construction to obtain a weak expanding and non

              1

              uniformly expanding C map so that D is non denumerable.

              Example 3.3.2. Let K n (x) =

              ⊂ I = [0, 1] be the middle third Cantor set and let β

              −n

            d ]), where d is the Euclidean distance on I. Note that β n is Lipschitz

              (x, K ∩ [0, 3

              ′

            and g is bounded and continuous but not of α-generalized bounded variation

            n

              ◦ β

              

            for any 0 < α < 1, where g was defined in Example Indeed, since

            kkn

              (3 ] for all k > n, then , 2 · 3 ) is a gap of K ∩ [0, 3

              1−k ′ ′ −k g (β(3 (β(3 ))

              /2)) − g k

              1

              ′

              =

              g

              2 · 3 k

              1−kk −1

              3 /2 − 3 2 · 3 k !

              1 2 · 3 =

              − 1 (1 − k) log 3 − log 2 (1 − k) log 3 − log 2 R x

              ′ is not bounded when k g R ր ∞. If h : I → R is given by h(x) = x + ◦

              1 −1 ′

              β n g n , where the integrals are with respect to Lebesgue measure on the ◦ β

            3.3. Weak-expanding and non-uniformly expanding maps

              29

              1

              1

              1

            real line, then h (0) = 0, h(1) = 2 and h induces a C map h : S whose

            α → S

              ′

            derivative is continuous but no C for any 0 < α < 1, such that h (x) = 1 for each

              −nx ] and h (x) > 1 otherwise.

              ∈ K ∩ [0, 3 R x

              (t+g ) n ◦β

              1

              1 We set now h (x) = x + R t 1 for t map of S ∈ [0, 1] which induces a C (t+g ) n ◦β

              ′

            into itself with continuous derivative and h > 1 for t > 0. We then define the

            t

              1

              1

            skew-product map f (x, y) = (E(x), h which is a weak

            sin πx × S

              (y)) for (x, y) ∈ S

              −n expanding map with ]).

              D = {0} × (K ∩ [0, 3

              For sufficiently big n it is easy to verify that f satisfies items (1-3) in Exam-

              1

            ple for V a small neighborhood of the fixed point (0, 0). Hence f is also a C

              1+α non-uniformly expanding map which is not a C map for any 0 < α < 1.

              30 Chapter 3. Examples of application

              

            Chapter 4

            Preliminary definitions and

            results

              The aim of this chapter is to fix notations, give definitions and state some facts which will be used throughout this work.

            4.1 Invariant and

              ν-SRB-like measures

              In this section, we revisit the definition and some results on the theory ν-SRB-like measures. Let X be a compact metric space and T : X X be continuous. Denote

            1 T the set of all

              M the set of all Borel probability measures on X and M

              ∗

            • + T

              1 fix the weak metric

              −invariant Borel probability measures on X. In M XZ Z 1 dist(µ, ν) := φ d φ d ν , (4.1.1) i i

              µ − i i is a countable family of continuous functions that is dense in i=

              2 } ≥0 where {φ the space C (X, [0, 1]).

              The following technical lemma will be used in the proofs of Propositions i i

              

            Lemma 4.1.1. For all ε > 0 there is δ > 0 such that if d(T (x), T (y)) < δ for all

            i = (x), σ (y)) < ε n n

              0, . . . , n − 1 then dist(σ Proof. k Given ε > 0 and a countable family of continuous functions F = P {φ : k ∈ N} that is dense in the space C (X, [0, 1]), fix N ≥ 1 such that

              ∞ −n n=N < ε/2. Let γ = ε/4 and consider F = {φ } the (N + 1)-first

              2 , . . . , φ N elements of F .

              32 Chapter 4. Preliminary definitions and results

              Note that V(σ ε (σ (x)), where n n (x), F, γ) ⊂ B ( Z Z )

              V (σ (x), F, γ) = φ d φ d σ (x) n i i n

              µ ∈ M; µ − < γ, ∀i = 0, . . . , N

              ∗ is a neighborhood in the weak topology. R R n i i n (x), F, γ) then φ d φ d σ (x) < γ for all

              In fact, given ν ∈ V(σ ν −

              i = 0, . . . , N and X N Z Z ∞ Z Z X

              1

              1 n i i n i i n dist(ν, σ (x)) = φ d φ d σ (x) φ d φ d σ (x) i i ν − ν −

            • i= i=N+

              2

              2

              1 X ∞ ∞ i X

              ε 1 ε ε 2 sup kφ k

            • =

              < 2γ + i i < < ε,

              2

              2

              2

              2

              2 ε (σ (x)). n i=N+ i=N

              1

              therefore ν ∈ B i i We will now show that there is δ > 0 such that if d(T (x), T (y)) < δ for n n n ε (σ (x)). all i = 0, . . . , n − 1 then σ (y) ∈ V(σ (x), F, γ) ⊂ B

              For each k = 0, . . . , N let δ(k) > 0 be a constant of uniform continuity of i i φ k

              (x), T (y)) < δ for all and set δ = min{δ(k); 0 ≤ k N} > 0. Thus, if d(T R

              i = n n k n (x), F, γ), since φ d σ (z) =

              0, . . . , n − 1 we conclude that σ (y) ∈ V

              1 P n −1 i n i= φ (T (z)), for all k = 0, . . . , N and k n n Z Z X −1 −1 X

              1 i i

              1 φ d σ φ d σ (x) (T (T γ = γ. k n k n k k (y) − ≤ |φ (y)) − φ (x))| <

              n n i= i=

              Next results ensure the existence of ν-SRB-like measure and conse-

              Proposition 4.1.2. Let T

              : X X be a continuous map of a compact metric

              space X. For each reference probability measure the T

              1

              ν there exist W (ν) ⊂ M

              ∗ unique minimal non-empty and weak compact set, such that p T (ν) for

              ω(x) ⊂ W T T .

              ν-a.e. x X. When ν = Leb we denote W (Leb) = W

              Proof. In the case that ν coincides with the Lebesgue measure this corre-

              sponds to Theorem 1.5 in , replacing Leb by ν.

              ∗

              Consider the family Υ of all the non-empty and weak compact sets

              A

              1 ⊂ M such that pω(x) ⊂ A for ν-a.e. x M. T

              The family Υ is not empty, since trivially M ∈ Υ. Define in Υ the partial order A

              1 2 if and only if A

              1 2 .

              AA

            4.2. Measure-theoretic entropy

              α

              33 α∈J

              Let, {A } ⊂ Υ is a chain if it is a totally ordered subset of Υ. Let us

              A α

              prove that A := ∩ α∈J belongs to Υ. For each fixed α ∈ J, and for each ε > 0 ε = define B α ε

              (α) = {x X; Pω(x) ⊂ A } and A {x X; Pω(x) ⊂ B (A)}, where

              B ε

              1

              (A) := {ν ∈ M ; dist(ν, A) < ε}. To conclude that A ∈ Υ, it is enough to ε prove that ν(A ) = 1 for all ε > 0.

              Claim 4.1.3. α ε

              For all (A).

              ε > 0 there exists α ∈ J such that AB If it did not exist then, by the property of finite intersections of compact α is totally ordered, we would deduce that the set

              } α∈I T sets, and since {A (A α ε

              \B (A)) , ∅ would be non-empty, contained in A, but disjoint with

              α∈J its open neighborhood B (A). ε

              We deduce that B (A). Since A (α)) = 1 ε α (α) ⊂ B ∈ Υ, we have that ν(B ε for all α ∈ I. Thus ν(A ) = 1 for all ε > 0, and therefore A ∈ Υ. We have proved that each chain in Υ has a minimal element in Υ.

              So, by Zorn’s Lemma there exist minimal elements in Υ, namely,

              ∗

              minimal non-empty and weak T 1 such that compact sets W (ν) ⊂ M

              p T

              ω(x) ⊂ W (ν) for ν almost all x X. T Finally, the minimal element W (ν) ⊂ Υ is unique since the intersection of two of them is also in Υ.

              

            Proposition 4.1.4. Given ν a reference probability measure, a probability measure

            is (ν).

            1 T

              µ ∈ M ν-SRB-like if and only if µ ∈ W Proof. See proof of Proposition 2.2 in . T T . For more details on SRB-like It is standard to check that W (ν) ⊂ M measures, see for more details on weak-SRB-like measures.

            4.2 Measure-theoretic entropy

              For any Borel measurable finite partition P of M, and for any (not necessarily invariant) probability ν it is defined X ν(P) log ν(P).

              H

              (P, ν) = − P ∈P The conditional entropy of partition P given the partition Q is the number X X

              ν(P Q) .

              H ν

              (P/Q) = − ν(P Q) log P ∈P Q ν(Q)

              ∈Q

              34 Chapter 4. Preliminary definitions and results Lemma 4.2.1. Given S

              ≥ 1 and ε > 0 there exists δ > 0 such that, for any finite

              

            partitions , . . . , P , . . . , ˜ P ) < δ for all

            1 S

              1 S i i

              P = {P } and ˜P = { ˜P } such that ν(P △ ˜P

              i = 1, . . . , S, then H ν ( ˜P/P) < ε. Proof. See Lemma 9.1.6 in q W q −1

              −j

              =

              f

              Denote P (P), where for any pair of finite partitions P and j= f then Q it is defined P ∨ Q = {P Q , ∅ : P ∈ P, Q ∈ Q}. If besides ν ∈ M

              1 q

            h H , ν).

            (P, ν) = lim (P q

              

            →+∞

            q

              Finally, the measure-theoretic entropy h ( f ) of an f -invariant measure ν ν is defined by h ν

              ( f ) = sup h(P, ν), where the sup is taken on all the Borel measurable finite partitions P of M. We define the diameter diam(P) of a finite partition P as the maximum diameter of its pieces. The following technical lemma will be used in the proof of the large deviation lemma that is essential for the proof of Theorem

              Lemma 4.2.2. Let f

              : M M be a measurable function. For any sequence of P n

              1 −1 j

            not necessarily invariant probabilites ν , let µ := ( f ) ν and µ be a weak

            n n n n j= ∗

              (µ

              accumulation point of n n ). Let P be a finite partition of M with S elements, with

              µ(∂P) = 0 = µ (∂P) for all n ≥ 1. Then for, for any ε > 0, there is a subsequence

              of integers n i

              ր ∞ such that ε

              1 n i

            • n
            • i

                H , ν h µ n i (P ) ≤ ( f ) ∀i ≥ 1.

                4 Proof. Fix integers q ≥ 1, and n q. Write n = aq + j where a, j are integer numbers such that 0 ≤ j q−1. Fix a (not necessarily invariant) probability

                ν. From the properties of the entropy function H of ν with respect to the partition P, we obtain n aq+j aq+q , ν)

                H

                (P , ν) = H(P , ν) ≤ H(P q

              _ _

              −1 a

              qiiq

                f f ), ν)

                ≤ H( (P), ν) + H( (P i= i= q

                1 X −1 a i q iq X H ) ν) + H , ( f ) ν) ∗ ∗

                ≤ (P, ( f (P i= i= X a q iq

                1 H , ( f ) ≤ q log S + (P ∗ ν) ∀q ≥ 1, N q. i=

                1

              4.2. Measure-theoretic entropy

                35 To obtain the inequality above recall that H(P, ν) ≤ log S for all ν ∈ M

                where S is the number of elements of the partition P. The inequality above

                −l

                holds also for f (P) instead of P, for any l ≥ 0, because it holds for any partition with exactly S pieces. Thus n q iq q l+iq X a a X

                −ll

              H ( f H ( f ), ( f ) ν) = q log S + H , ( f ) ν).

                ∗ ∗

                (P ), ν) ≤ q log S + (P (P i= i=

                1

                1 Adding the above inequalities for 0 ≤ l q − 1, we obtain: q q X −1 −1 a n X X 2 q l+iql H ( f log S + H , ( f ) ν). l= l= i= (P ), ν) ≤ q (P ∗

                1 Therefore, q aq+q X −1 −1 n X 2 q ll

                H ( f log S + H , ( f ) ν). (4.2.1) l= l= (P ), ν) ≤ q (P ∗

                On the other hand, for all 0 ≤ l q − 1, n n+l n X l −1

                −il

              H H ( f ), ν)

                (P , ν) ≤ H(P , ν) ≤ (P), ν) + H( f (P i= q

                

              l

              ), ν).

                ≤ q log S + H( f (P Therefore, adding the above inequalities for 0 ≤ l q − 1 and joining with the inequality , we obtain aq+q n X −1 2 q l

                

              qH log S + H , ( f ) ν).

                (P , ν) ≤ 2q (P ∗ l= Recall that n = aq + j with 0 ≤ j q − 1. So aq + q n + q and then n −1 aq+q n X −1 X 2 q l q l

                

              qH log S + H , ( f ) ν) + H , ( f ) ν)

                (P , ν) ≤ 2q (P ∗ (P ∗ n l= l=n X −1 2 q l log S + H , ( f ) ν). ≤ 3q (P ∗ l=

                36 Chapter 4. Preliminary definitions and results

                In the last inequality we have used that the number of non-empty pieces q q w is at most S . Now we fix a sequence n µ i n of P ր ∞ such that µ i −−−−→ n →+∞ put ν = ν and divide by n . Since H is convex we deduce: n i i n i −1

                2 X

                2 q 3q log S aq log S n q l

                1 i

                H , ν H , ( f ) ν ) + nn

              • n n n n i i i i

                n

                l=

                i −1

                (P i ) ≤ (P i

                2 X

                2

                3q log S aq log S

                1 q l

              • n i

                H , ( f ) ν ) +

                ≤ (P

                n n n i i i

              l=

                2

                3q log S q

              • n H , µ ).

                ≤ (P i

                n i n o

              36q log S

                (q) = max ,

                q

                Therefore, for all i i ε 1 ε n q i

                1 , ν , µ i

              • n
              • i w 12 q q

                  ) ≤

                  H n H n i (P (P ) ∀i i (q), ∀q ≥ 1.

                  Since µ µ and µ n n i −−−−→ (∂P) = µ(∂P) = 0 for all n ∈ N, then lim (P i i i H , µ ) = n q →+∞

                  →+∞ H . Thus there exists i f 1 > i

                  1

                  (P , µ) because µ ∈ M (q) such that for all i i

                  1 q q 1 ε H , µ H , µ) + . n (P i ) ≤ (P

                  12

                  q q

                1 Thus, for all i ≥ i

                  1 ε n q i

                  1 H , ν H , µ). n (P i ) ≤ (P

                • n
                • i 6 q 1 q

                    Moreover, by definition lim H q q (P , µ) = h(P, µ) then there exists q

                    →+∞

                    1 q ε

                    2 :=

                    , µ) ≤ h(P, µ) + q (P

                    N such that H . Thus taking i

                    12 for all q q

                    , i

                    1

                    max{q } we have 1 ε ε n i

                    H , ν h ( f ) + n µ 2 . (4.2.2)

                    (P i ) ≤ h(P, µ) + for all i i

                    n 4 ≤ i

                    4 This completes the proof using the subsequence (n ) as the sequence i i

                    ≥i 2 claimed in the statement of the lemma.

                    Lemma 4.2.3. Let a P ℓ ℓ P ℓ 1 , . . . , areal numbers and let p

                  a

                  k

                  1 , . . . , pnon-negative numbers

                    =

                    such that p k

                    1. Denote L = P e . Then p k (a k k k= k= k= ) ≤ log L.

                    1

                    1 e ak 1 − log p

                    = Moreover, the equality holds if, and only if, p for all k. k L Proof. See Lemma 10.4.4 in .

                    

                  Chapter 5

                  Continuous variation of SRB-like

                  measures

                    Here we prove Theorem showing that ν-SRB-like measures can be seen as measures that naturally arise as accumulation points of ν -SRB n measures.

                    

                  5.1 Measures with prescribed Jacobian. Confor-

                  mal measures

                    A measurable function J

                    Definition 12. ν T

                    : X → [0, +∞) is called the Jacobian of a map T : X X with respect to a measure ν if for every Borel set A X on which T is injective Z ν(T(A)) = J Td ν. A ν

                    Next result guarantees the existence of measures with prescribed Jaco- bian.

                    Theorem 5.1.1. Let T

                    : X X be a local homeomorphism of a compact metric

                    space X and let

                    φ : X → R be continuous. Then there exists a φ-conformal

                    ∗

                  probability measure ν = ν and a constant ν = λν. Moreover,

                  φ

                    λ > 0, such that L φ

                    −φ

                    λe ν.

                    the function J ν T = is the Jacobian for T with respect to the measure Proof. See Theorem 4.2.5 in .

                    38 Chapter 5. Continuous variation of SRB-like measures

                  5.2 SRB measures

                    We say that a continuous mapping T : X X is open, if open sets have open images. This is equivalent to saying that if f (x) = y and y y n −−−−→ n

                    →+∞ then there exist x x such that f (x ) = y for n large enough. n n n

                    −−−−→ n

                    →+∞ Definition 13.

                    A continuous mapping T : X X is called: (a) topologically exact if for all non-empty open set U X there exists N N = N (U) such that T U = X .

                    (b) topologically transitive if for all non-empty open sets U, V X there n exists n ≥ 0 such that T (U) ∩ V , ∅.

                    Topological transitiveness ensures conformal measures give positive mass to any open subset.

                    Proposition 5.2.1. Let T

                    : X X be an open expanding topologically transitive

                    map and φ is

                    φ : X → R be continuous. Then every conformal measure ν = ν

                    

                  positive on non-empty open sets. Moreover for every r > 0 there exists α = α(r) >

                  0 such that for every x X, ν(B(x, r)) ≥ α. Proof. See Proposition 4.2.7. in

                    For Hölder potentials it is known that there exists a unique ν-SRB probability measure.

                    Theorem 5.2.2. Let T

                    : X X an open expanding topologically transitive

                    

                  map, ν = ν a conformal measure associated to the a Hölder continuous function

                  φ φ ergodic invariant ν-SRB probability

                    φ : X → R. Then there exists a unique µ measure such that µ is the unique equilibrium state for T and φ. φ Proof. The proof follows the results presented in Chapter 4 of .

                    The following result shows that positively invariant sets with positive reference measure have mass uniformly bounded away from zero.

                    In the same setting of Theorem if G is an T-invariant set Theorem 5.2.3. such that ν(G) > 0, then there is a disk of radius δ/4 so that ν(∆\G) = 0.

                    Proof. See the proof of Lemma 5.3 in .

                    5.3. Limits of SRB-measures

                    39 Remark 5.2.4. It is easy to see that each continuous potential in a compact metric

                  space can be approximated in uniform convergence by Lipschitz continuous poten-

                  tials (in particular, all Lipschitz continuous potentials are α-Hölder continuous

                  for 0 < α ≤ 1).

                    In fact, let

                    A = { f : X → R; f is Lipschitz continuous} ⊂ C(X, R) and

                    observe that

                    A is an subalgebra of the algebra C(X, R), since A forms a vector

                    space over R and given f

                    , g ∈ A, x X then ( f ·g)(x) = f (xg(x) ∈ A. Moreover,

                    f

                    ≡ 1 belongs to A and A separates points, since given x, y X, x , y, we may

                    take f

                    : X → R given by f (z) = d(z, {x}), f is Lipschitz continuous (because, {x} is a closed set) therefore f ∈ A and f (x) , f (y). Thus, the Theorem of

                    Stone-Weierstrass ensures that A is dense in C(X, R).

                    5.3 Limits of SRB-measures Now we are ready to prove Theorem

                    ∗

                  Proof of Theorem Let λ (ν ) = λ ν (it is easy to

                  n φ n n n j −−−−→ λ, where L j j j j n j

                    →+∞ ∗

                    = see that λ > 0 since λ (ν )(1) for all n). n n L φ n

                    (X, R) we have Z Z Z Z Then for each ϕ ∈ C

                    ∗ ∗

                    ν ) = (ϕ) dν (ϕ) dν = ν) n φ n φ ϕ d(L j L j −−−−→ L ϕ d(L φ n j n j j φ w w ∗ ∗

                  →+∞

                  ∗ ∗

                    ν ν. Moreover, λ ν λν and by uniqueness of n n n thus, L j −−−−→ L j j −−−−→ φ φ n j j →+∞ j →+∞ the limit, it follows that

                    ∗ ∗ λν = lim λ ν = ν = ν. j j n n lim n j j L j L n j →+∞ →+∞ Thus, ν is a φ-confomal measure.

                    Let µ be the ν -SRB measure and let µ = lim µ . For each fixed ε > 0, n n n j j j j

                    →+∞

                    consider N = N(ε) such that ε ε dist(µ , µ ) < and dist(µ , µ) < j n n n j m j , m N.

                    4 4 ∀ Thus, A ε/4 (µ n ε/2 (µ n ) and A ε/2 (µ n ε j m j ) ⊂ A ) ⊂ A (µ) for all j, m N. ε/4 n n (µ ) we have dist(pω(x), µ ) < ε/4. Then,

                    In fact, for each x A j j ε dist(pω(x), µ ) + dist(µ , µ ) < n n n n m ) ≤ dist(pω(x), µ j j m

                    2

                    40 Chapter 5. Continuous variation of SRB-like measures ε/4 (µ ε/2 (µ n n m

                    for all j, m N. Therefore, A j ) ⊂ A ) for all j, m N. Analo- gously, A (µ (µ). ε/2 n ε j ) ⊂ A n j of radius δ

                    1 /4

                    By Theorem for each j ≥ 1 there exists a disk ∆ around x such that ν (∆ (µ )) = 0. Let x = lim x taking a subse- n n n ε/2 n n j j j \A j j j

                    →+∞

                    ∆ quence if necessary. By compactness, the sequence n accumulates on j j /8 a disc ˜∆ of radius δ

                    1

                    1

                    /4 around x. Let ∆ be the disk of radius 0 < s ≤ δ around x. Thus, there exists N . nN such that ∆ ⊂ ∆ j for all j N

                    ∆ = /8 such that ν ∂ (µ) 0 and note that for all

                    1 ε

                    Consider 0 < s ≤ δ \A ,

                    j

                    ≥ N ∆ ∆ ∆ , 0 = ν n n ε/2 (µ n ) n ε/2 (µ n ) n ε (µ) j j \A j ≥ ν j \A j ≥ ν j \A n j n j and A ε/2 (µ ε . since ∆ ⊂ ∆ ) ⊂ A (µ) for all j N

                    ∗

                    Thus, by weak convergence ν ∆ = ν ∆ = ε (µ) lim n ε (µ)

                    0. (5.3.1) \A j \A j

                    →+∞

                    Since ν is a conformal measure, it follows from ∆ that ν A (µ) > 0. As ε > 0 is arbitrary, we conclude that µ is ε

                    ≥ ν ν-SRB-like. Let P be a finite Borel partition of X with diameter not exceeding an expansive constant such that µ(∂P) = 0. Then P generates the Borel sigma- algebra for every Borel probability T-invariant in X (see Lemma 2.5.5 in η (T) is ) and by Kolmogorov-Sinai Theorem this implies that η 7→ h upper semi-continuous in µ. R R

                    Moreover, as φ µ φ dµ, µ is an equilibrium state for n d n n j j −−−−→ j j

                    →+∞

                    (T, φ n (T, ϕ) (see Theorem j ) (by Theorem and by continuity of ϕ 7→ P top

                    9.7 in ) it follows that Z Z ! =

                  h (T) + h (T) + φ d µ lim P (T, φ ) = P (T, φ).

                  µ µ n n top n top

                    φdµ ≥ lim j j j j j n j

                    →+∞ →+∞ This shows that µ is an equilibrium state for T with respect to φ.

                    Now we assume that T is topologically exact. We know that given ε ε > 0 there exists a disk ∆ such that ν(∆\A (µ)) = 0. Since ∆ ⊂ X is a N non-empty open set, there exists N > 0 such that X = T (∆). Moreover,

                    A (µ) = T(A (µ)), thus ε ε N N

                    0 = ν(T ε ε ε (µ)).

                    (∆\A (µ)) ≥ ν(T (∆)\A (µ)) = ν(X\A Since ε > 0 was arbitrary, this shows that ν(A (µ)) = 1 for all ε > 0. ε

                    

                  Chapter 6

                  Expanding maps on compact

                  metric spaces

                    Here we state the main results needed to obtain the proof of Theorem

                    We start by presenting some results about expanding maps that will be used throughout this chapter. We close the chapter with the proof of Theorem

                  6.1 Basic properties of expanding open maps In what follows X is a compact metric space.

                    Lemma 6.1.1. If T

                    : X X is a continuous open map, then for every ε > 0 there

                    exists δ > 0 such that T(B(x, ε)) ⊃ B(T(x), δ) for every x X. Proof. See Lemma 3.1.2 in .

                    Remark 6.1.2. If T

                    : X X is an expanding map, then by ,

                    

                  for all x is injective and therefore it has a local inverse

                  B (x,ε)

                    ∈ X, the restriction T| map on T (B(x, ε)).

                    If additionally T

                    : X X is an open map, then, in view of Lemma the

                    

                  domain of the inverse map contains the ball B (T(x), δ). So it makes sense to define

                  −1 the restriction of the inverse map, T x : B(T(x), δ) → B(x, ε) at each x X.

                    Lemma 6.1.3. Let T

                    : X X be an open expanding map. If x X and y, z

                    −1 −1 −1 −1 B (T(x), δ) then d(T (y), T d (y, z). In particular T x x (z)) ≤ λ x (B(T(x), δ)) ⊂

                    −1 B (x, λ

                    δ) ⊂ B(x, δ) and

                    −1

                    δ))

                    B

                    (6.1.1) (T(x), δ) ⊂ T(B(x, λ δ > 0 small enough.

                    for all

                    42 Chapter 6. Expanding maps on compact metric spaces Proof. See Lemma 3.1.4 from . Definition 14.

                    Let T : X X be an open expanding map. For every x X, j = j T (x). In view of Lemma every n ≥ 1 and every j = 0, 1, . . . , n − 1 write x the composition n

                    −1 −1 −1 T : B(T

                  xT x ◦ · · · ◦ T x (x), δ) → X

                  1 n

                  −1

                    −n is well-defined and will be denoted by T . x Lemma 6.1.4. Let T

                    : X X be an open expanding map. For every x X we

                    have:nn

                    1. T (A) = S T y (x) n y (A) for all A B(x, δ);T nnnn

                    2. d (T (y), T d (x), δ); x x (z)) ≤ λ (y, z) for all y, z B(T nnn

                  3. T (B(T r x (x), r)) ⊂ B(x, min{ε, λ }) for all r ≤ δ.

                    Proof. The lemma follow from Lemma

                    For more details on the proofs in this subsection, see subsection 3.1 in .

                  6.2 Conformal measures. Weak Gibbs property

                    Here we cite results relating φ-conformal measures and the topological pressure of φ. Next results says φ-conformal measures for expanding dynamics with continuous potentials are almost Gibbs measures; see for more details.

                    Proposition 6.2.1. Let T

                    : X X an open expanding topologically transitive

                    map, ν = ν a conformal measure associated the a continuous function φ

                    φ : X → R

                    −φ

                    λe ν. Given δ > 0, for

                    and J ν T = the Jacobian for T with respect to the measure all x

                    ∈ X and all n ≥ 1 there exists α(ε) > 0 such that ν((B(x, n, ε))) δ

                    

                  nδ n

                    α(ε)e ≤ ≤ e exp (S n

                    φ(x) − Pn) ε > 0 small enough, where P = log λ.

                    for all

                  6.2. Conformal measures. Weak Gibbs property

                    43 Proof.

                    Given δ > 0 there exist γ > 0 such that for all x, y X, with d(x, y) < γ we have |φ(x) − φ(y)| < δ. Fix 0 < ε < γ, x X and n ≥ 1 arbitrary. Then n n n Z ν(B(T x , ε)) = ν(T (B(x, n, ε))) = J T d ν. (6.2.1) B ν

                    (x,n,ε)

                    Hence, since it gives uniform weight to balls of fixed radius and the by uniform continuity of φ, we obtain n n n φ Z Z

                    −S n x , ε)) = J T d ν = λ e d ν ν

                    α(ε) ≤ ν(B(T B B nS n φ(x)+nδ (x,n,ε) (x,n,ε)

                    e

                    ≤ λ · ν(B(x, n, ε)), and n n n Z

                    −S n φ(x)−nδ x , ε)) = J T d e ν 1 ≥ ν(B(T ν ≥ λ · ν(B(x, n, ε)). B

                    (x,n,ε)

                    Hence ν(B(x, n, ε)) n δ

                    −nδ

                    α(ε)e ≤ ≤ e exp (S n

                    φ(x) − Pn) where P = log λ.

                    Lemma 6.2.2. Let T

                    : X X be an open expanding topologically transitive

                    map, let

                    φ : X → R be continuous and ν be a probability measure such that

                    ∗

                    (ν) = λν. Then P top L φ (T, φ) ≤ P = log λ.

                    

                  Proof. Given δ > 0 consider ε > 0 small enough as in Proposition

                  n

                    and n ≥ 1. Let EX be a maximal (n, ε)-separated set. Therefore n n {B(x, n, ε) : x E } covers X, {B(x, n, ε/2) : x E } is pairwise disjoint open sets. Thus by Proposition we have, n n X X (P+δ) (P+δ) S n φ(x) e e .

                    e

                    ≤ ν(B(x, n, ε/2)) · ≤ x x α(ε/2) α(ε/2)

                    ∈E nE n

                    Therefore, X

                    1 S n φ(x)

                    P top (T, φ) = lim lim sup log e

                    ≤ P + δ,

                    ε→0 n n →+∞ xE n

                    as δ > 0 can be taken small enough, we conclude that P top (T, φ) − P ≤ 0.

                    44 Chapter 6. Expanding maps on compact metric spaces

                    

                  6.3 Constructing an arbitrarily small initial par-

                  tition with negligible boundary

                    Let ν be a Borel probability measure on the compact metric space X. We say that T is ν-regular map or ν is regular for T if T

                    ∗ ν ≪ ν, that is, if E X is −1

                    0. It follows that if T admits a Jacobian with respect a T-regular measure.

                    = such that ν(E) = 0, then ν T (E)

                    Lemma 6.3.1. Let X be a compact metric space of dimension d

                    ≥ 1. If ν is a

                    

                  regular reference measure for T positive on non-empty open sets and δ > 0, then

                  there exists a finite partition

                    P of X with diam(P) < δ such that every atom

                    P ∈ P have non-empty interior and ν(∂P) = 0. Proof. Let δ > 0 and consider 0 < δ , δ

                    1 l

                    1

                    ≤ δ and B = {B( ˜x /8), l = 1, . . . , q} be a finite open cover of X by δ

                    1 /8- balls such that ν(∂B( ˜x , δ l 1 /8)) = 0 for all

                  l = 1, . . . , q. Note that such value of δ > 0 exists since the set of values of δ

                    

                  1

                    1

                    such that ν(∂B( ˜x , δ l

                    1

                    /8)) > 0 for some l ∈ {1, . . . , q} is denumerable, because ν is a finite measure and X is a compact metric space soon separable. From this we define a finite partition P of X as follows. We start by setting

                    δ δ

                    1

                    1

                    ˜x , , . . . , P ˜x ,

                    P 1 := B 1 k := B k 1 k )

                    \(P ∪ . . . ∪ P −1

                    8

                    8 for all k = 2, . . . , S where S q. ,

                    Note that if P has non-empty interior (since X is separable), k k ∅ then P δ δ 1 1 diameter smaller than 2 < and the boundary ∂P is a (finite) union of k

                    8

                    4

                    pieces of boundaries of balls with zero ν-measure. We define P by the elements P constructed above which are non-empty. k Note that since T is ν-regular the boundary of g(P) still has zero ν- n

                    , for any measure for every atom P ∈ P and every inverse branch g of T

                    n ≥ 1.

                    

                  6.4 Expansive maps and existence of equilibrium

                  states

                    A continuous transformation T : X X of a compact metric space X equipped with a metric ρ is (positively) expansive if and only if n n (x), T

                    ∃δ > 0[∀n ≥ 0 ρ(T (y)) ≤ δ] =⇒ x = y and the number δ above is called an expansive constant.

                    6.5. Large deviations

                    45 Theorem 6.4.1. If T T

                    : X X is positively expansive, then the function M ∋ µ (T) is upper semi-continuous and consequently each continuous potential µ 7→ h φ : X → R has an equilibrium state.

                    See Theorem 2.5.6 in .

                    Proof.

                    Theorem 6.4.2. Expanding property implies positively expansive property.

                    Proof. See Theorem 3.1.1 in .

                    Corollary 6.4.3. If T

                    : X X is expanding, then each continuous potential r T : φ : X → R has an equilibrium state. In particular, K (φ) = {µ ∈ M R

                    h µ (T) + top φdµ ≥ P (T, φ) − r} , ∅ for all r ≥ 0. Proof. The proof is immediate from Theorem

                    6.5 Large deviations

                    The statement of Theorem and in Proposition 6.1.11 in .

                    Proposition A. Let T

                    : X X be an open expanding topologically transitive

                    

                  map and let a φ-conformal measure, r > 0

                  φ

                    φ : X → R be continuous. Fix ν = ν

                    ∗

                  and consider the weak distance dist defined in . Then, for all 0 < ε < r,

                  there exists n n o ≥ 1 and κ > 0 such that

                    ν x n r . (6.5.1) ∈ X; dist(σ (x), K (φ)) ≥ ε < κ exp [n(ε − r)] ∀n n

                    

                  Proof. We know by Theorem that all the (neces-

                    sarily existing) conformal measures ν are positive on non-empty open sets

                    −φ and J λe is the Jacobian for T with respect to the measure ν. ν T =

                    Consider ν a conformal measure, fix r > 0 and let 0 < ε < r. For ε/6, fix ε whenever a constant γ > 0 of uniform continuity of φ, i.e., |φ(x) − φ(y)| <

                    6 d

                    (x, y) < γ. Consider 0 < ξ < γ and a partition P of X as in Lemma ξ . such that diam(P) <

                  4 Let us choose one interior point in each atom P ∈ P, and form the set

                    =

                    C , . . . , w

                    (where

                  1 S

                    l

                    {w } of representatives of the atoms of P = {P } 1≤lS S S =

                    S =

                    ∂P l #{P ∈ P : P , ∅}). Let d min{d(w, ∂P), w C } > 0 where ∂P = l=

                    1 is the boundary of P. r

                    Consider A := {µ ∈ M; dist(µ, K (φ)) ≥ ε}. Note that A is weak* , . . . , B compact, so it has a finite covering B

                    1 κ

                    for minimal cardinality κ ≥ 1,

                    46 Chapter 6. Expanding maps on compact metric spaces ε

                    = with open balls B i n ,i ⊂ M of radius . For any fixed n ≥ 1 write C {x S κ

                  3 S κ

                    = = =

                    

                  X ; σ n i n C n ,i , ˜C n ,i n i n ˜C n ,i ,

                    }, C

                    1 {x X; σ } and ˜C

                    1

                    (x) ∈ B i= (x) ∈ ˜B i=

                    2ε where ˜B are open balls concentric with B of radius for i = 1, . . . , κ. i i

                    3 Note that C n ,i n ,i n r n n .

                    ⊂ ˜C . Moreover, {x X; dist(σ (x), K (φ)) ≥ ε} ⊂ C ⊂ ˜C

                    Claim 6.5.1. For each > 0 such that ν(C i n ,i

                    1 ≤ i ≤ κ there exists n ) ≤ i . exp [n (ε − r)] for all n n

                    First, let us see that it is enough to prove the Claim to finish the = proof of the lemma. In fact, if Claim holds, put n max n .Then we i

                    1≤i≤κ

                    , as needed: obtain the following inequality for all n n X κ ν(C ν(C n n ,i ) ≤ ) ≤ κ exp [n (ε − r)]. i=

                    1 n ,i n

                    Let us prove Claim for x C let P ∈ P be the atom such that T (x) ∈ P

                    −n

                    and set Q = T of all such sets Q is finite since x (P). Then the family Q n both P and the number of inverse branches are finite. Moreover by the expression of J T in terms of φ ν Z Z −1   n  X  j

                    −n     n ,i ν ) = J T d ν = exp d ν.

                    ν(Q C  φ ◦ Tn log λ  T ) T ) n n     j=

                    (QC n ,i (QC n ,i We note that if ν(C n ,i ) = 0, then Claim becomes trivially proved.

                    1 , . . . , Q N n n ,i

                    Consider the finite family of atoms {Q } = {Q ∈ Q : ν(Q C ) > 0} which has N = N(n, i) elements for some N ≥ 1. N Note that ν(C ) = P ν(Q ). For each k = 1, . . . , N, consider n ,i k n ,i k= n 1 ∩ C

                    

                  x k k such that T (x k ) = w j for some j = 1, . . . , S (recall that w j are interior

                    ∈ Q k ,n for each points of each atom of the partition P, so there is only one j = j n x such that T (x ) = w ). k k k jQ n ξ

                    < ξ for all n > 0 (remember Lemma Since diam(P ) < diam(P) < j j

                    4

                  ε

                    (x k k , then |φ(T )) − φ(T (y))| < for all y Q and j = 0, . . . , n − 1.

                  6 Considering for each k, y k k n ,i then,

                    ∈ QC

                  6.5. Large deviations

                    N N n  

                    47 X X  Z    ε    X  −1 j

                    ν(C ,i ) = ν(Q ,i exp φ(T (y )) + d ν n k n kC ) ≤  − n log λ    k=  

                    6 N n

                    1 k=

                   

                  1 j=

                  T (Q ) n

                  kC n ,i

                  X 

                  X 

                  −1   j n ε ε     exp φ(T (x )) + (Q ))

                    ≤

                  •    − n log λ  · ν(TC k k n ,i k=  

                    6

                    6 N n

                    1 j=   X  X  −1       j ε exp φ(T (x k )) +

                    ≤  − n log λ     

                    3 k= j=

                    1 N n   X  X  −1

                    ε       j n log λ exp φ(T (x )) . k ≤ exp     3 −   k=

                  1 j=

                    Defining N n n     X 

                  X  

                  −1 −1 X              j j

                    1 L := exp φ(T (x )) , λ := exp φ(T (x ))         k k k k=     L N 1 j= j= λ = then P k 1 and by Lemma k=

                    1         X N n −1 N X  j         X log L = λ φ(T (x )) λ log λ .     −   k= k kk k    

                  1 j= k=

                  N N

                    1 Define the probability measures ν := P λ δ and µ := P λ σ (x ) n k x n k n k k=

                  R

                  1 k k= n

                    1 n n so that we may rewrite, log L = n

                    φdµ H , ν ). We fix a weak (P

                  • n n j accumulation point µ of (µ ) and take a subsequence n
                  • w

                    −−−−→ ∞ such j

                    →+∞

                    that µ n j −→ µ and

                    1

                    1 lim sup log ν(C ) = lim log ν(C ). (6.5.2) n ,i n ,i j j

                    →+∞ n n n j →+∞

                    We now make a small perturbation of the original partition so that the points x are still given by the image of the same n th inverse branch k kQ of T of an atom of the perturbed partition and the boundaries of the new partition also have negligible µ measure. d ξ

                    , does not depend on n) such that for Let 0 < η < min{ } (note that d

                    2

                    4

                    all l = 1, . . . , S and for each n ≥ 1 ξ ξ

                    = + + µ ∂B , η 0 = µ ∂B , η (6.5.3)

                    ˜x n ˜x l l

                    4

                    4

                    48 Chapter 6. Expanding maps on compact metric spaces

                    where the centers are the ones from the construction of the initial partition P in the proof of Lemma

                    Such value of η exists since the set of values of η > 0 such that some of the expressions in is positive for some l ∈ {1, . . . , S} and some n ≥ 1 is at most denumerable because the measures involved are probability measures. Thus we may take η > 0 satisfying arbitrarily close to zero.

                    We consider now the finite open cover l , ξ/4 + η) : l = 1, . . . , S (6.5.4) ˜C = Bx of X and construct the partition ˜P induced by ˜C by the same procedure as l before and following the some order of construction P, ( ˜x , l ∈ {1, . . . , S} are the same used in the construction of P in Lemma .

                    Note that d(w , ∂B( ˜x , ξ/4 + η)) > d /2 for all l = 1, . . . , S and l l − η > d

                    

                  w by construction. Therefore, each w is contained in some atom

                  l l

                    ∈ CC

                    P w l ∈ ˜P. Moreover there cannot be distinct wCP k l k l w , w such that w , by the choice of η. Thus, the number of atoms of P is less than or equal to the

                    number of atoms of ˜P. On the other hand, by construction the maximum number of elements ˜P is S = Card(P), because Card( ˜C) = S. In this way we conclude that the partition ˜P has the same number of atoms as P.

                    We have (1) diam( ˜P) < 2(ξ/4 + η) < ξ; n (2) µ(∂ ˜P) = 0 = µ (∂ ˜P) for all n ≥ 1; . (3) w ∈ int( ˜P(w)), ∀w C n n j j

                    , ν , ν ) by definition of the ν and by con- n n n Note that H( ˜P j ) = H(P j j struction of the ˜P as a perturbation of P. Using items (1) and (2) we get, by Lemma that there exists j > 0 such that

                    1 n n j j 1 ε

                    H , ν ) = H , ν (T) + . (6.5.5) n n µ

                    (P j ( ˜P j ) ≤ h , ∀j j

                    n n j j

                    3 ,

                    For the partition obtained above, we have that x Q and Q k k k k n ,i ∈ ˜ ∩QC

                    Q k n for k = 1, . . . , N(n, i), where this family is defined

                    ∅, where ˜ belongs to ˜Q n similarly to Q changing only the atoms of the original partition P by the atoms of the perturbed partition ˜P. k n n i k ,i . Then σ . As d(T (x ), T j j

                    Consider y QC (y) ∈ B (y)) ≤ diam(P) < n k n (x ), σ (y)) < ε/3 and as ξ, for all j = 0, . . . , n − 1 then, by Lemma dB i i concentrically, we conclude that σ n (x k i .

                    ⊂ ˜B ) ∈ ˜B

                    6.6. Proof of Theorem

                    49 Since the ball ˜B is convex and µ is a convex combination of the mea- i n

                    = sures σ (x ), (recall that P λ 1), we deduce that µ . Therefore, the n k k n i ∈ ˜B

                    ∗ ∗

                    weak n n belongs to the weak limit µ of any convergent subsequence of {µ }

                    2ε

                    closure ˜B . Since the ball ˜B has radius (φ). Then i i r we conclude that µ < K R

                    3

                    φdµ + h (T) < P µ top (T, φ) − r, and therefore

                    ε ε ν(C n log λ n log λ log L n ,i

                    ) ≤ exp · L = exp

                  • " #

                    3 − 3 − Z ε

                    1 n

                  • = n n exp n log λ + φdµ H , ν ) .

                    (P R R 3 − n φdµ φdµ (because φ is continuous and by

                    We know that n j −−−→ j

                    →∞ RR

                    weak convergence) and therefore, there exists j

                    1 > 0 such that φdµ n j

                    φdµ + ε/3 for all j > j 1 .

                    1 n j By there exists j > 0 such that H , ν . (P j ) ≤ h (T) + ε/3, ∀j j n µ n j

                    = Taking j

                    2 1 , j

                    2

                    max{j } we have, ∀j > j " Z # ε ε ε

                    ν(C n log λ + φdµ + h (T) + n ,i j µ j ) ≤ exp

                  • " #

                    3 − Z

                    3

                    3 = exp n φdµ + h µ (T) j h i h i ε − log λ +

                    n (T, φ) n j top j

                    ≤ exp ε − r − log λ + P ≤ exp ε − r where the last inequality follows from Lemma By we conclude that there exist n > 0 such that ν(C ,i n h

                    ) ≤ exp n ending the proof. ε − r i for all n n

                    6.6 Proof of Theorem r T . By the upper

                    Given r > 0, consider the (non-empty) set K (φ) ⊂ M semicontinuity of the metric entropy (see Theorem , we have that

                    ∗ r (φ) is closed, hence, weak is decreasing with r, r r

                    K compact. Since {K (φ)} (φ) = T (φ). r we have K K r >0 R By the Variational Principle h (T) + . µ top T φ dµ ≤ P (T, φ) for all µ ∈ M T (ν) of ν-SRB-

                    So, to prove Theorem we must prove that the set W n T r (φ) = like measures satisfy W (ν) ⊂ K (φ) for all r > 0, because K µ ∈ R

                    ∗ T ; h µ (T) + top (T, φ) r (φ) is weak compact, we have φdµ = P

                    M o. Since K

                    50 \ ε ε

                  Chapter 6. Expanding maps on compact metric spaces

                    n o r (φ) = (φ) = T r

                    K K r r µ ∈ M ε>0 (φ), where K ; dist(µ, K (φ)) ≤ ε

                    ∗

                    with the weak distance defined in . Therefore, it is enough to prove ε T (φ) for all 0 < ε < r/2 and for all r > 0. By Proposition that W (ν) ⊂ K r ε

                    ∗

                    (φ) is weak compact, it is enough to prove the following and since K r ε

                    (φ)

                    Lemma 6.6.1. The basin of attraction of s ε ε n o K r

                  W (φ)) := x (φ)

                  rX; pω(x) ⊂ K r

                    (K has full ν-measure.

                    Proof. By Proposition since 0 < ε < r, there exists n n o ε n (ε−r) ≥ 1 and κ > 0 such that ν x (φ) for any n > n . This implies n n oX; σ (x) < K r ≤ κe

                  • +ε

                    that P ν (φ) n= x n

                    1 ∈ X; σ (x) < K r < +∞. By the Borel-Cantelli Lemma it

                    follows that   + +    ε    \ [ ∞ ∞ n o =

                    ν x (φ) n 0.     =X; σ (x) < K r n r r 1 n=n n ε ε In other words, for ν-a.e. x X there exists n ≥ 1 such that σ (x) ∈ K r r

                    (φ) for all n n . Hence, pω(x) ⊂ K (φ) for ν-almost all the points x X, as required.

                    The proof of Theorem is complete.

                    Remark 6.6.2. If the set r r r (φ)

                    K (φ) is not closed, we may substitute K (φ) for K

                    in the proof of Theorem and by the same argument we conclude that T

                    W T (ν) ⊂ r (φ). Thus, in a more general context, where the Proposition is valid and r >0 K

                    K 1/n (φ) , ∅ for all n ≥ 1, we can say that ν-SRB-like measures are "almost φ-

                    

                  equilibrium states", since, given (ν) then µ = lim µ , µ (φ) for

                  T n n 1/n

                    µ ∈ W ∈ K n

                    →+∞ all n

                    ≥ 1. Therefore, we can find a sequence of T-invariant probability measures R

                    

                  1

                  so that h (T) + φdµ for all n µ n top n nP (T, φ) − ≥ 1 and µ → µ in the weak*

                  n

                  topology.

                    To obtain a φ-equilibrium state in the limit we need only assume that φ is uniformly approximated by continuous potentials, as follows.

                    Let T Corollary 6.6.3.

                    : X X be an open expanding topologically transitive

                    

                  map of a compact metric space X, (φ ) a sequence of continuous potentials,

                  n n ≥1

                    (ν ) a sequence of conformal measures associated to the (T, φ ) and µ a sequence n n n n

                    ≥1

                    ν

                    of n -SRB-like measures. Assume that

                  6.6. Proof of Theorem B

                    51 1. φ φ in the topology of uniform convergence; n j −−−−→ j →+∞ w

                    ∗ 2. µ µ in the weak topology. n j −−−−→ j →+∞ µ is an equilibrium state for the potential φ.

                    Then

                  Proof. Let µ be a ν -SRB-like measure and let µ = lim µ . Since any

                  n j n j j n j →+∞

                    finite Borel partition P of X with diameter not exceeding an expansive constant and satisfying µ(∂P) = 0 generates the Borel sigma-algebra for every Borel T-invariant probability measure in X (see Lemma 2.5.5 in ), η (T) is upper semi- then Kolmogorov-Sinai Theorem implies that η 7→ h continuous. R R

                    Moreover, as φ n d µ n φ dµ, µ n is an equilibrium state for j j j −−−−→ j

                    →+∞

                    (T, φ n (T, ϕ) (see Theorem j ) (by Theorem and by continuity of ϕ 7→ P top

                    9.7 in ) it follows that Z Z ! =

                  h µ (T) + h µ (T) + φ n d µ n lim P top (T, φ n ) = P top (T, φ).

                  j j j

                    φdµ ≥ lim j j n j

                    →+∞ →+∞ This shows that µ is an equilibrium state for T with respect to φ.

                    52 Chapter 6. Expanding maps on compact metric spaces

                  Chapter 7 Entropy Formula Here we state the main results needed to obtain the proof of Theorem D. Then we prove Theorem D in the last subsection.

                  7.1 Hyperbolic Times

                    The main technical tool used in the study of non-uniformly expanding maps is the notion of hyperbolic times, introduced in . We now outline some the properties of hyperbolic times.

                    Definition 15.

                    Given σ ∈ (0, 1), we say that h is a σ-hyperbolic time for a point x M if for all 1 ≤ k h, Y h −1 j k

                    −1

                    (x)) (7.1.1) j=h kD f ( f k ≤ σ

                    −k

                  Remark 7.1.1. Throughout this text the reader will find many quotes from works

                    2 1+α

                  where f is assumed to be of class C (or f (M, M), α > 0). But it is worth

                    ∈ C

                    

                  noting that these results cited are proven without using the bounded distortion

                  assumption, and therefore the proofs are easily adapted to our context (in general,

                  the proofs are the same).

                    Proposition 7.1.2. Given 0 < σ < 1, there exists δ 1 > 0 such that, whenever

                  h is a σ-hyperbolic time for a point x, the dynamical ball B(x, h, δ ) is mapped

                  h h

                    1 diffeomorphically by f onto the ball B ( f (x), δ

                  h h k /2 h h

                  1 ), with

                    −kk

                  d ( f (y), f (y), f (z))

                    (z)) ≤ σ · d( f

                    for every

                  1 ).

                    1 ≤ k h and y, z B(x, h, δ

                    54 Chapter 7. Entropy Formula Proof. See Lemma 5.2 in

                  Remark 7.1.3. For an open expanding and topologically transitive map T of a

                  compact metric space X every time is a hyperbolic time for every point x, that is,

                  every x satisfies the condition of Proposition

                    Definition 16.

                    We say that the frequency of σ-hyperbolic times for x M is positive, if there is some θ > 0 such that all sufficiently for large n ∈ N there < h < . . . < h

                    1 2 l

                    are l ≥ θn and integers 1 ≤ hn which are σ-hyperbolic times for x.

                    The following Theorem ensures existence of infinitely many hyperbolic times Lebesgue almost every point for non-uniformly expanding maps. A complete proof can be found in Ref. , Sec. 5.

                    1 Theorem 7.1.4. Let f non-uniformly expanding local dif-

                    : M M be a C

                    feomorphism. Then there are

                    σ ∈ (0, 1) and there exists θ = θ(σ) > 0 such

                    that

                    Leb-a.e. x M has infinitely many σ-hyperbolic times. Moreover if we 0 < h < h < h < . . . for the hyperbolic times of x then their asymptotic

                    write

                    1

                    2

                    3 frequency satisfies k

                    #{k ≥ 1 : hN} lim inf N ≥ θ for Leb −a.e.x M

                    →∞ N

                    The Lemma below shows that we can translate the density of hyperbolic times into the Lebesgue measure of the set of points which have a specific (large) hyperbolic time.

                    Lemma 7.1.5. Let B

                    ⊂ M, θ > 0 and g : M M be a local diffeomorphisms

                    such that g has density

                    > 2θ of hyperbolic times for every x B. Then, given any

                    probability measure

                    ν on B and any n ≥ 1, there exists h > n such that θ

                    ν({x B : h is a hyperbolic time of g for x}) >

                    2 See Lemma 3.3 in Proof. The next result is the flexible covering lemma with hyperbolic preballs which will enable us to approximate the Lebesgue measure of a given set through the measure of families of hyperbolic preballs.

                    Let a measurable set A Lemma 7.1.6.

                    ⊂ M, n ≥ 1 and ε > 0 be given with

                    

                  m (A) > 0. Let θ > 0 be a lower bound for the density of hyperbolic times for

                  Lebesgue almost every point. Then there are integers n < h < . . . < h for

                  1 k k = k i of subsets of M, i = 1, . . . k such that

                    (ε) ≥ 1 and families E

                  7.1. Hyperbolic Times

                    55 1.

                    1 is a finite pairwise disjoint family of subsets of M; k

                    E ∪ . . . ∪ E σ

                    2. h is a -hyperbolic time for every point in Q, for every element Q , i i

                    ∈ E

                    2 i = 1, . . . , k; 3. every Q is the preimage of some element P i h i ∈ E ∈ P under an inverse branch of f , i = 1, . . . , k; 4. there is an open set U

                    1 1 with k

                    ⊃ A containing the elements of E ∪ . . . E

                    m (U

                    1

                    \A) < ε; 5. m i i ) < ε. (A△ ∪ E Proof. See Lemma 3.5 in .

                    

                  Remark 7.1.7. This covering lemma is true replacing f by an open expanding and

                  topologically transitive map T of a compact metric space X; Leb by a φ-conformal

                  measure

                    ν for a continuous potential φ : M → R; recall Remark We use this covering lemma to prove the following.

                    Proposition B. Let f

                    : M M be a non-uniformly expanding map. For any

                    ∗

                    µ ∈ W f Z

                    h µ ( f ) + (7.1.2) ψdµ ≥ 0.

                    ∗ Proof. , consider δ 1 > 0 as in Proposition and as f :

                    Given µ ∈ W f

                    1 local diffeomorphism in particular f is a regular map.

                    M

                    → M is a C δ 1 Consider a partition P of M as in Proposition such that diam(P) <

                    4 and µ(∂P) = 0.

                    Since µ is f -invariant and P is a µ-mod0 partition such that µ(∂P) = 0, then the function λ 7→ h(P, λ) is upper semi-continuous in µ, that is, for each small enough τ > 0 we can find δ > 0 such that

                    2

                    (7.1.3)

                    2 if dist(µ, ˜µ) ≤ δ then h(P, ˜µ) ≤ h(P, µ) + τ.

                    Fix τ > 0 and 0 < δ < τ as above. Since µ is weak-SRB-like probability

                    2

                    measure, by definition, for any fixed value of 0 < ε < δ

                    2 /3 there exists a subsequence of integers n (µ)) > 0 for all l > 0. l ε,n

                    → +∞ such that Leb(A l Take δ

                    2 /3 > 0 and fix γ > 0 as in Lemma and γ 1 > 0 of uniform

                    /3 if d(x, y) < γ . We denote γ =

                    2

                    1

                    continuity of ψ, i.e., |ψ(x) − ψ(y)| < δ , γ

                    min{γ

                    1 }.

                    56 Chapter 7. Entropy Formula

                    Let us choose one interior point having density ≥ θ of σ-hyperbolic =

                    , . . . , w

                    1 S

                    times of f in each atom P ∈ P and form the set W {w } of

                    1≤ℓ≤S

                    representatives of the atoms of P = {P } , where S = #{P ∈ P : P , ∅}, S S =

                    ∂P and consider d ℓ is the } > 0, where ∂P = ℓ=1 min{d(w, ∂P), w W boundary of P.

                    We use now the Lemma for obtain the flexible covering of set =

                    A (µ) with hyperbolic preballs. Take positive integers l, m and β ε,n l m l

                    1 2 Leb(A (µ)) > 0 such that σ δ /4 < γ. Then there are integers n < ε,n n l l 1 l

                  n m < h takes the place of ε in

                  • l k l

                    1 2 < . . . < h

                    ≤ h with k = k (l) ≥ 1 (here β j of subsets of M, j = 1, . . . k so that Lemma and families E X k k X Leb(A (µ)) = Leb(A ) + Leb(A ) ε,n ε,n j ε,n j l l (µ) ∩ E l (µ)\E j= X h k 1 j= X

                    1

                    , hence Leb(A ε,n l

                    ≤ l (µ) ∩ Q) + β j= Q

                    1 ∈E j X h k

                  X

                  n l

                    Leb(A ε,n Leb(A ε,n (7.1.4) l (µ)) ≤ l (µ) ∩ Q),

                    n l

                    − 1 j= Q

                    1 S

                  ∈E j

                    ∈E jh j j is a family of all sets obtained as f (P) which intersect

                    = = j h ε,n j ε,n and A A where E E j l (µ) ∩ E Q l (µ) ∩ Q.

                    Figure 7.1: E (µ) in points for which h

                    A ε,n j l is a hyperbolic time, where P ∈ P. Analo- j+ 1 .

                    gously for E δ 1 h h j j , by Lemma f (Q) is diffeomorphism Q

                    As diam(P) < | : Q f j j is a -hyperbolic time for every point in

                    4 σ

                    2

                    ∀1 ≤ j k, where h for all Q ∈ E

                  7.1. Hyperbolic Times

                    Note that S h j ψ(y) = S n l ψ(y) + S h jn l ψ( f n l (y)) and since h jn l

                    57 Q

                    , for every element Q ∈ E j , j = 1, . . . , k and every Q ∈ E j is the preimage of some element P ∈ P under an inverse branch of f h j

                    , j = 1, . . . , k. Then Leb(Q A ε,n l

                    (µ)) = Z f h j

                    (QA ε,nl (µ))

                    | det D f

                    −h j

                    |d Leb = Z f h j

                    (QA ε,nl (µ)) e S h j ψ d Leb .

                  • m is a hyperbolic time for all y ∈ E j we have S h jn l

                    ψ( f n l (y)) ≤ 0 and so S h j ψ(y) ≤

                    ψ( f j

                    we can rewrite Leb(A ε,n l (µ)) ≤

                    n l n l

                    − 1 exp n l

                    2 δ

                    2

                    3

                    n l

                    log L(n l ) . (7.1.5) Note that since P x

                    ∈W nl

                    λ(x) = 1 then by Lemma log L(n l ) = X x

                    ∈W nl

                    λ(x) n l −1 X j=

                    1

                    x )

                    and λ(x) :=

                    − X x

                    ∈W nl

                    λ(x) log λ(x) . Defining the probability measures

                    ν n l := X x

                    ∈W nl

                    λ(xx and µ n l :=

                    1

                    

                  n

                  l

                  n l −1 X i=

                    ( f i )

                    ∗

                    (ν n l ) = X x

                    ∈W nl

                    λ(xn l (x),

                    1 L (n l ) e S nl ψ(x)

                    ∈W nl e S nl ψ(x)

                    S n l ψ(y) for all y ∈ E j

                    1 /4 < γ, for all j = 1, . . . , k, and by uniform

                    . Therefore, Leb(Q A ε,n l (µ)) ≤ Z f h j

                    (QA ε,nl (µ)) e S nl ψ

                  d

                    Leb ∀Q ∈ E jj = 1, . . . , k. For each Q ∈ E j such that Leb(QA ε,n l

                    (µ)) > 0, consider y QQA ε,n j

                    (µ) and let x QQ be such that f h j

                    (x Q ) ∈ W

                    (recall that elements of W are interior points of each atom of the partition P). We write W n l for the set of all points x Q for all Q ∈ E j such that Leb(Q A ε,n l (µ)) > 0 for all j = 1, . . . , k. From Proposition we know that max{diam( f l (Q)); Q ∈ E j

                    , l = 0, . . . n l

                    } < σ 1 2

                    (h jn l )

                    δ

                    1 /4 < σ m 2

                    δ

                    continuity of ψ, |ψ( f i (x Q

                    Setting L(n l ) := P x

                    )) − ψ( f i (y))| < δ

                    2

                    /3 for all y Q and for all

                    i = 0, . . . , n l

                    − 1. Altogether we get Leb(A ε,n l (µ) ∩ Q) ≤ Z f h j

                    (QA ε,nl (µ)) e S nl ψ d

                    Leb ≤ e S nl ψ(y Q

                    )+n l δ2 3

                    2n l δ2 3 + S nl ψ(x Q ) .

                    Thus, by we can write Leb(A ε,n l (µ)) ≤ n l n l −1

                    e 2δ2 3 n l P x

                    ∈W nl e S nl ψ(x) .

                    ≤ e

                  • 1

                    58 Chapter 7. Entropy Formula

                    we can rewrite Z n ψdµ , ν l log L(n l ) = n l n H n ). (7.1.6) l l

                  • n l l n l i i i li

                    (P w

                  • →+∞ →+∞

                    Fix (µ ) and take a subsequence n ˜µ −−−−→ ∞ such that µ −−−−→

                    and

                    1

                    1 i →+∞ lim log Leb(A ε,n (µ)) = lim sup log Leb(A ε,n (µ)). (7.1.7) li

                    n n l n i →+∞ We keep the notation n for simplicity in what follows. l n Q ε,n there exists y (µ) such that

                    Note that, for each x W lQ A l i i

                    x , y (x), f (y )) < γ for all i = 0, . . . , n Q Q l

                    ∈ Q, hence dist( f −1. By Lemma we have that dist(σ (x), σ (y )) < δ n n Q l l 2 /3, then by the triangular inequality δ

                    2

                    dist(σ (x), σ (y )) + dist(σ (y ), µ) <

                  • n n n Q n Q

                    ε < δ , l (x), µ) ≤ dist(σ l l l

                    2

                    3 because y Q ε,n (µ) and ε < δ l 2 (M, R), Z Z Z ZA /3. Thus, for any ϕ ∈ C X X =

                    ϕdµ ϕdµ λ(x) ϕdσ λ(x) ϕdµ n n ll (x) − x x XWW nl nl Z Z λ(x) ϕdσ ϕdµ < δ n 2 .

                    ≤ l (x) − x

                    ∈W nl

                    , and so

                    Z Z Z Z ψdµ ψdµ + δ and ψdµ + δ . (7.1.8) n l ≤ ψd ˜µ ≤

                  2 Therefore, dist(µ, ˜µ) ≤ δ

                    2

                    2 n still

                    We now make a small perturbation of P so that the points x W l belong to the same atom of the h j −th refinement the perturbed partition that intersect A ε,n j l (µ) for each Q ∈ E and j = 1, . . . , k and l ≥ 1. Now fix δ > 0 given by the statement of Lemma with the choice d δ 1 of S and ε = δ

                    2 , does not

                    /4. Let 0 < η < min{ } (by construction, d

                    2

                    4

                    depend on n l ) such that for all i = 1, . . . , S and for each l ≥ 1

                    δ δ δ

                    1

                    1

                    1

                  • = = = + +

                    µ ∂B ˜x , η ˜µ ∂B ˜x , η µ ∂B ˜x , η (7.1.9) i i n i l

                    4

                    4

                    4 and δ δ

                    1

                    1 i i

                    ˜µ B ˜x

                  • , η ˜x , < δ (7.1.10)

                    \B

                    4

                    4

                  7.1. Hyperbolic Times

                    59

                    where the centers are the ones from the construction in the proof of Propo- sition

                    Such value of η exists since the set of values of η > 0 such that some of the expressions in is positive for some i ∈ {1, . . . , S} and some l ≥ 1 is denumerable because the measures involved are probability measures. Moreover, we can also get because ˜µ is regular probability measure. Thus we may take η > 0 satisfying arbitrarily close to zero. i , δ /4 + η) : i = 1, . . . , S

                    1 We consider now the finite open cover ˜C = B( ˜x

                    as in and we analogously construction a new partition ˜P with the same number of elements of the original partition. i i Moreover, for P = {P ; 1 ≤ i S} and ˜P = { ˜P ; 1 ≤ i S}, we have by construction that ˜µ(P ) < δ for all i = 1, . . . S. Thus we have, the same i i

                    △ ˜P properties (1-3) in the proof of Claim replacing ξ by δ

                    1 /4 together

                    with n l (1) ˜µ(∂ ˜P) = 0 = µ (∂ ˜P) for all l ≥ 1; (2) ˜µ(P ) < δ for all i = 1, . . . S; and i i

                    △ ˜P h j (3) x Q Q j where ˜ Q j and f (x Q

                    ∈ ˜ ∩ Q for all Q ∈ E ∈ ˜E n n l l ) = w ∈ ˜P P.

                    , ν , ν ) by definition of the ν and by con- n n n Note that H( ˜P l ) = H(P l l struction of the ˜P as a perturbation of P. Following the proof of Lemma

                    ) there exists l ≥ 0 such that

                    1 n n l l 1 δ

                    2 H , ν ) = H , ν . n n

                    (P l ( ˜P l ) ≤ h( ˜P, ˜µ) + , ∀l l

                    n n l l

                    4 Since, h( ˜P, ˜µ) ≤ h(P, ˜µ) + H ˜µ ( ˜P/P) we have that by Lemma and δ 2 item (2) above, that H . Therefore,

                    ˜µ

                    ( ˜P/P) <

                    4

                    1 δ n l 2 δ

                    2 H , ν h , n

                  • n 2 ≤
                  • l

                      (P l ) ≤ h(P, ˜µ) + (P, µ) + τ, ∀l l

                      2 where the last inequality is valid by the choice of δ

                      2 in . Thus,

                      1 δ n

                      2

                      3

                    • l

                      H , ν ( f ) + . (7.1.11) n µ

                      (P l ) ≤ h(P, µ) + τ ≤ h τ, ∀l l

                      n l

                      2

                      2 < τ. and remember that δ

                      2

                      60 Chapter 7. Entropy Formula

                      Combining the assertions we have that

                      n δ l

                      1

                      2 ε,n l l l

                      Leb(A exp n 2 log L(n ) (µ)) ≤

                    • 3

                      n l n l

                      − 1 " !# Z

                      n l

                      2

                      1 n l l n n exp n τ + ψdµ H , ν )

                    • l (P l

                      n l l 3 n

                      − 1 " Z !#

                      n l

                      exp n 3τ + h ( f ) + ψdµ l µ

                      n l

                      − 1 Hence, Z

                      1 1 n l . log Leb(A ε,n log h µ ( f ) + l ψdµ + 3τ, ∀l l

                    • n n n l l l

                      (µ)) ≤

                      − 1

                      ∗

                      , we conclude that, Since µ ∈ W f Z

                      1

                      1 ψdµ+3τ. 0 = lim sup log Leb(A ε,n (µ)) = lim log Leb(A ε,n µ ( f )+ l l (µ)) ≤ h

                      →+∞ n l n n →+∞

                      As τ > 0 is arbitrary, the proof of the Lemma is complete.

                      

                    Remark 7.1.8. Proposition is true replacing f by an open expanding and

                    topologically transitive map T of a compact metric space X; Leb by a φ-conformal

                      ∗

                    measure (ν) = λν, for some λ > 0 and for a continuous potential

                      ν with L φ φ : X → R, since we used that Leb is ψ-conformal together with a covering

                      

                    lemma that clearly holds for expanding maps, recall Remark Thus, we have

                    Z

                    h ( f ) + (ν).

                    µ

                      φdµ − log λ ≥ 0, for all µ ∈ W T

                      

                    In particular, this shows together with the Lemma that P top (T, φ) = log λ

                    7.2 Proof of Theorem D

                      To prove Theorem first consider a weak-SRB-like measure µ. Since

                      ∗

                      µ ∈ W f is an expanding probability measure, there exists σ ∈ (0, 1) such

                      1 P n −1 j −1

                      that lim sup (x)) n log kD f ( f i= k ≤ log σ < 0 for µ-a.e x M.

                      →+∞ n

                    • + Thus, the Lyapunov exponents are non-negative. Hence the sum Σ (x) of the positive Lyapunov exponents of a µ−generic point x, counting multi-

                      1 n

                    • + plicities, is such that Σ (x) = lim
                    • R n →+∞ log | det D f n

                        (x)| (by the Multiplica- + Σ tive Ergodic Theorem) and

                        d µ =

                        R log | det D f|dµ = − R ψdµ by the standard Ergodic Theorem.

                      7.2. Proof of Theorem D

                        61

                      1 For C -systems, Ruelle’s Inequality [36] states that for any f -invariant

                        probability measure µ on the Borel σ-algebra of M, the corresponding R + Σ measure-theoretic entropy h µ ( f ) satisfies h µ R

                        d µ and consequently

                        ( f ) ≤

                        h µ ( f ) + ψdµ ≤ 0.

                        By definition, Pesin’s Entropy Formula holds if the latter difference is R

                        ∗

                        , by Proposition we have that h µ ( f )+ ψdµ = equal to zero. Since µ ∈ W f 0 which proves the first statement of Theorem

                        The next corollary concludes the proof of Theorem

                        Corollary 7.2.1. Let f

                        : M M be non-uniformly expanding. Then all the ergodic SRB-like probability measures are expanding probability measures.

                        Proof.

                        The assumptions on f ensure that there exists σ ∈ (0, 1) such that Leb(H(σ)) = 1. The proof uses a simple lemma.

                      1 Lemma 7.2.2. If f local diffeomorphism such that Leb(H(σ)) =

                        : M M is a C

                        −1

                      f

                      satisfies σ.

                        1 for some σ ∈ (0, 1), then each µ ∈ W R log k(D f) kdµ < log

                        1 −1 Proof.

                        Fix 0 < ε < − log σ small enough. Since that ϕ(x) := log kD f (x) k is

                        2 ∗

                        a continuous potential, from the definition of the weak topology in space

                        1 of probability measures, we deduce that there exists 0 < ε 1 < ε such

                        M R R that if dist(µ, ν) < ε

                        1 1 .

                        ϕdµ − ϕdν| < ε for all µ, ν ∈ M then | f ε ε , then Leb(A Let µ ∈ W 1 (µ)) > 0, take x A 1 (µ) ∩ H(σ) and consider

                        ν x x ) < ε

                        

                      1 . Thus,

                        ∈ pω(x) such that dist(µ, ν Z Z

                        −1 −1 x < ε, log k(D f ) kdµ − log k(D f ) kdν w and therefore there exists n k n (x) x and then Z Z ր ∞ so that σ k −→ ν −1 −1 n −1 X k x ε log k(D f ) kdµ ≤ log k(D f ) kdν

                      • 1 j

                        −1

                        = lim (x)) k log kD f ( f k + ε

                        →+∞

                      n k

                      j= n X −1

                        1 j

                        −1

                        (x)) ≤ lim sup log kD f ( f k + ε n n

                        →+∞ j=

                        √ < log σ + ε < log σ as stated.

                        62 Chapter 7. Entropy Formula

                        Going back to the proof of the Corollary, since µ is f -invariant and ergodic, then by the previous lemma and by the Ergodic Theorem n X −1 Z 1 j

                        −1 −1

                        lim (y)) n log kD f ( f k = log k(D f ) kdµ < log σ for µ − a.e y M.

                        →+∞ n j=

                        Therefore µ is an expanding measure. This finishes the proof of the coro- llary.

                        This completes the proof of Theorem

                      Chapter 8 Ergodic weak-SRB-like measure In this chapter, we prove Corollary E on existence of ergodic weak- SRB-like measures for non-uniformly expanding local diffeomorphisms

                        f : M M.

                      8.1 Ergodic expanding invariant measures

                        

                      1

                      Theorem 8.1.1. Let f local diffeomorphism. If µ is an ergodic

                        : M M be a C

                        expanding f -invariant probability measure, then Z

                        log(Leb(A (µ))) ε,n lim lim sup ψdµ + h ( f ). (8.1.1) µ +

                        ε→0 n n →+∞

                      Proof. Since µ is an ergodic probability measure, we have lim n σ n (x) = µ

                        →+∞

                        for µ-a.e. x M. So for µ-a.e. x M, there exists N(x) ≥ 1 such that dist(σ n (x), µ) < ε/4 ∀n N(x). Given ε > 0 and any natural value of N ≥ 1, define the set

                        B N n (8.1.2) := {x M : dist(σ (x), µ) < ε/4 ∀n N}.

                        Consider δ

                        1 h > 0 such that for each σ-hyperbolic time h ≥ 1 time for x, f ) maps B(x, h, δ B (x,h 1 ) diffeomorphically to the ball of radius δ 1 around h | 1 f (x).

                        Fix δ > 0 such that Lemma holds with ε/8 in the place of ε and fix ε if d(x, y) < ξ. ξ > 0 satisfying |ψ(x) − ψ(y)| <

                      4 Consider 0 < γ

                        1

                        &lt; min{ξ, δ, δ /2} and let N(n, γ, b) be the minimum number of points needed to (n, γ)-span a set of µ-measure b (see ). &lt; γ

                        Choose 0 &lt; γ

                        1 such that

                        1 ε lim inf γ &lt; γ , (8.1.3) µ

                        1 n log N(n, 4γ, 1/2) ≥ h ( f ) − →+∞ n 2 ∀

                        64 Chapter 8. Ergodic weak-SRB-like measure

                        and let 0 &lt; γ

                        2 1 be such that ( ≤ γ )! 1 ε

                        2 µ &gt; .

                        x log Leb(B(x, n, γ

                        2 Leb ( f, µ) +

                        ∈ M : lim sup − )) ≤ h n

                        n

                        4

                        3

                        →+∞

                        This is possible by definition of h ( f, µ) (see ). We have implicitly

                        

                      Leb

                        assumed that h Leb Leb ( f, µ) &lt; ∞ here. If h ( f, µ) = ∞, then there is nothing

                        1 to prove since h ( f ) &lt; h local diffeomorphism. µ top

                        ( f ) &lt; ∞ because f is a C Denote ( ) 1 ε log Leb(B(x, n, γ ( f, µ) + . (8.1.4)

                        A = x

                        2 Leb

                        ∈ M : lim sup − )) ≤ h n

                        n

                        4

                        →+∞

                        Note that B N N+ N

                        1

                        ⊂ B and µ(∪B ) = 1. So there exists N ≥ 1 such that

                        5

                        1

                        µ(B . If C then µ(C N N N N N ) ≥ := A B ) ≥ and for all x C and n N(x)

                        6

                        2

                        we have: (1) B(x, n, γ

                        2 (µ); ε,n

                        ) ⊂ A

                        −(h Leb ( f,µ)+ε/2)n

                        (2) Leb(B(x, n, γ 2 .

                        )) ≥ e Note that (2) immediately follows from . Moreover, (1) holds j j

                        (y), f because, given y B(x, n, γ) then d( f (x)) &lt; γ for all j = 0, . . . , n − 1. ε By Lemma we have dist(σ n (y), σ n (x)) &lt; N , by the

                        8 . Since x B

                        triangular inequality dist(σ (y), σ (x)) + dist(σ (x), µ) &lt; ε. n n n n ε,n (µ). (y), µ) ≤ dist(σ Therefore, y A

                        = For each n, let E E (2γ n n 2 ) be a maximal (n, 2γ 2 )-separated subset of points contained in C . Then S B (x, n, 4γ by maximality of E N

                        2 N n x ) ⊃ CE n

                        and so #E n , x , y then B(x, n, γ

                        2 n

                        2

                        ≥ N(n, 4γ , 1/2). Also, given x, y E ) ∩

                        B (y, n, γ

                        2

                        ) = ∅. Thus, X

                        1

                        1 lim inf log Leb(A ε,n log Leb(B(x, n, γ

                        2 )) n n (µ)) ≥ lim inf →+∞ →+∞ n n x

                        ∈E n

                        1

                        −(h Leb ( f,µ)+ε/2)n

                        log #E n

                        ≥ lim inf · e n →+∞

                        n

                        1 ε log N(n, 4γ .

                        2 Leb

                        ≥ lim inf , 1/2) − h ( f, µ) − n →+∞

                        n

                        2 Thus, by we have,

                        1 lim inf log Leb(A (8.1.5) ε,n µ Leb n (µ)) ≥ h ( f ) − h ( f, µ) − ε.

                        →+∞ n

                      8.1. Ergodic expanding invariant measures

                        65 Moreover, since µ is expanding, then there exist θ &gt; 0 and σ ∈ (0, 1)

                        such that µ-a.e. x M has positive frequency ≥ θ of σ-hyperbolic times h of f . Let ˜ M M , lim σ (x) = µ and f ) n B (x,h 2M be such that for all x ∈ ˜ | n

                        →+∞ h

                        maps B(x, h, γ ) diffeomorphically to the ball of radius γ around f (x), for

                        2 2 h a σ-hyperbolic time for x. j j j j ε

                        ) since d( f x , f y

                        4

                        2 Note that |ψ( f (y)) − ψ( f (z))| &lt; for all z B(y, h, γ ) ≤

                        γ

                        2 for all j = 0, . . . , h − 1.

                        Hence, since Lebesgue measure gives weight uniformly bounded away from zero to balls of fixed radius and the by uniform continuity of ψ, we obtain h h Z 0 &lt; α(γ x , γ )) =

                        2

                        2

                        ) ≤ Leb(B( f | det D f | d Leb B ) Z ψ ψ(x)+hε/4 (x,h2

                      SS

                      h h

                      )).

                        e d

                        2

                        ≤ Leb ≤ e · Leb(B(x, h, γ B )

                        (x,h2 Thus,

                        1 0 = lim inf log α(γ h 2 ) →+∞ h ε

                        1

                        1

                      • S h ψ(x) + lim inf log Leb(B(x, h, γ

                        2 ))

                        ≤ lim inf − h →+∞ h →+∞

                        h Z 4 h

                        ε

                        1 ψdσ (x) + lim sup log Leb(B(x, h, γ h 2 ))

                        ≤ − lim sup − h h 4 − h Z →+∞ →+∞ ε

                        = ψdµ + h Leb ( f, x).

                        − 4 − R ε Therefore, h Leb ψdµ + Leb

                        ( f, x) ≤ − for µ-a.e x M, hence h ( f, µ) ≤ R ε

                        

                      4

                        ψdµ + and by −

                      4 Z

                        1 lim lim sup log Leb(A ( f ) + ψdµ, ε,n µ (µ)) ≥ h

                      • + ε→0 n

                        n →+∞ which gives the desired estimate.

                        

                      1

                      Corollary 8.1.2. Let f local diffeomorphism. Every expanding

                        : M M be a C R µ such that ψdµ + h

                        ergodic f -invariant probability measure µ

                        ( f ) ≥ 0 is a weak-SRB-like probability measure.

                        

                      Proof. f be a expanding ergodic probability measure such that

                      R Let µ ∈ M

                        ψdµ + h µ ( f ) ≥ 0. By Theorem we have that Z log(Leb(A ε,n (µ))) lim lim sup ( f ) + µh ψdµ ≥ 0.

                      • + n

                        ε→0 n

                        →+∞

                        66 Chapter 8. Ergodic weak-SRB-like measure log(Leb(A ε,n (µ)))

                        Moreover, if ε &lt; ε then A (µ). So lim sup

                        1 2 ε ,n ε ,n 1 (µ) ⊂ A 2 n n →+∞ log(Leb(A ε,n (µ)))

                        is increasing with ε &gt; 0. Thus lim sup n ≥ 0 for all ε &gt; 0.

                        →+∞ n

                        But since Leb is a probability measure, we conclude that log(Leb(A (µ))) ε,n lim sup = 0 ∀ε &gt; 0. n n

                        →+∞ Thus, we deduce that µ is weak-SRB-like measure.

                        

                      Remark 8.1.3. Theorem is true replacing f by an open expanding and

                      topologically transitive map T of a compact metric space X; Leb by a φ-conformal

                        ∗

                      measure (ν) = λν for some λ &gt; 0, and for a continuous potential

                        ν, with L φ ν (T, µ) is finite. So we get φ : X → R with the extra assumption that h Z

                        1 lim lim sup log ν(A ( f ) + ε,n µ (µ)) ≥ h φdµ − log λ,

                      • + n

                        ε→0 n

                        →+∞

                      since we used that Leb is ψ-conformal together with general results from Ergodic

                      Theory. Analogously for Corollary

                        

                      1

                      Corollary 8.1.4. Let f local diffeomorphism. Every expanding

                        : M M be a C

                        

                      ergodic f -invariant probability measure that satisfies Pesin’s Entropy Formula is

                      a weak-SRB-like probability measure.

                      Proof. be an expanding ergodic probability measure such

                      f

                        Let µ ∈ M R P P + + that h ( f ) = d µ, where (x) is the sum of the positive Lyapunov µ

                      • + exponents of a µ-generic point x counting multiplicities. R R P µ =

                        d

                        ψdµ ≤ We deduce by the Multiplicative Ergodic that − R

                        h µ ( f ). Therefore, h µ ( f )+

                        ψdµ ≥ 0 and the corollary follows from Corollary

                      8.2 Ergodic expanding weak-SRB-like measures Now we are ready to prove Corollary E.

                        

                      Proof of Corollary that

                        all weak-SRB-like measures are equilibrium states for the potential ψ, that R

                        ∗ is, h µ ( f ) + .

                        ψdµ = 0 for all µ ∈ W f We know that there exists σ ∈ (0, 1) such that Leb(H(σ)) = 1. Given

                        √

                        −1 f , by Lemma we have that σ.

                        µ ∈ W kdµ ≤ log

                        R log k(D f)

                      8.2. Ergodic expanding weak-SRB-like measures

                        67 By the Ergodic Decomposition Theorem, there exists A M such that

                        √

                        −1 y y σ, where µ is an

                        µ(A) &gt; 0 and for all y A, R log k(D f) kdµ ≤ log ergodic component of µ.

                        Fix yA. By Birkhoff’s Ergodic Theorem, n X −1 Z

                        1 j

                        −1 −1

                        lim (y)) σ, y n log kD f ( f k = log k(D f ) kdµ ≤ log

                        →+∞ n j= for µ y y is an expanding probability measure.

                      • a.e. y M. Therefore µ R R

                        Since P top ( f, ψ) = 0 and h µ ( f ) + ψdµ = 0 we conclude that h µ ( f ) + y ψdµ = y 0 for all µ-a.e x M. In particular, by Corollary we con-

                        ∗

                        clude that there exist y showing that there exist yA such that µ ∈ W f expanding ergodic weak-SRB-like probability measure such that satisfies

                        Pesin’s Entropy Formula ending the proof of Corollary

                        68 Chapter 8. Ergodic weak-SRB-like measure

                        

                      Chapter 9

                      Weak-Expanding non-uniformly

                      expanding maps

                        In this chapter we reformulate Pesin’s Entropy Formula for a class of

                        1

                        weak-expanding and non-uniformly expanding maps with C regularity and prove Corollary

                        

                      9.1 Weak-SRB-like, equilibrium and expanding

                      measures

                        We divide the proof of Corollary into the next two corollaries below.

                        Corollary 9.1.1. Let f

                        : M M be weak-expanding and non-uniformly ex-

                        

                      panding. Then, all (necessarily existing) weak-SRB-like probability measures are

                        ψ-equilibrium states and, in particular, satisfy Pesin’s Entropy Formula. More- Proof.

                        Let f : M M be as in statement of Corollary then for every

                        1 n x M x n

                        ∈ M and all v T \{0} we have lim inf →+∞ log kD f (x) · vk ≥ 0. Thus, n the Lyapunov exponents for any given f −invariant probability measure µ R

                        1 n + +

                        Σ are non-negative. Hence Σ (x) = lim

                      n

                      d µ =

                        →+∞ log | det D f (x)| and n R log | det D f|dµ = − R ψdµ. R

                        1

                      • + Σ

                        For C -systems, Ruelle’s Inequality ensures h R µ d µ then ( f ) ≤

                        h ( f ) + µ top ( f, ψ) = 0 and R ψdµ ≤ 0. By Proposition we have that P

                        ∗ h µ ( f ) + .

                        ψdµ = 0 for all µ ∈ W f Therefore, all weak-SRB-like probability measures are ψ-equilibrium states and satisfy Pesin’s Entropy Formula. Moreover, by Corollary we conclude that there exist ergodic weak-SRB-like measures.

                        70 Chapter 9. Weak-Expanding non-uniformly expanding maps

                        Here we obtain a sufficient condition to guarantee that all ψ-equilibrium states are generalized convex combinations of weak-SRB-like measures.

                        Corollary 9.1.2. Let f

                        : M M be weak-expanding and non-uniformly expand- ψ &lt; 0 there is no atomic weak-SRB-like probability measure. Moreover,

                        ing. If −1 if

                        D = {x M; kD f (x) k = 1} is finite and ψ &lt; 0 then almost all ergodic ψ-equilibrium state are weak-SRB-like measures and all weak-

                        components of a

                      SRB-like probability measures µ have ergodic components µ which are expanding

                      x weak-SRB-like probability measure for R µ-a.e. x M\D.

                        

                      Proof. Note that if ψ &lt; 0 then . On the other hand,

                      f

                        ψdµ &lt; 0 for all µ ∈ M f is an atomic invariant probability measure, then h ( f ) = 0. There- µ if µ ∈ M fore µ does not satisfy Pesin’s Entropy Formula and by Corollary we conclude that there is no atomic weak-SRB-like probability measure. R R

                        Since ψdµ &lt; 0 and h , then a ψ- µ f ( f ) ≤ − ψdµ for all µ ∈ M R R

                        = equilibrium state satisfies h ( f ) + ψdµ = 0 and so h ( f ) + ψdµ 0, µ µ x x µ-a.e. x by the Ergodic Decomposition Theorem and h µ ( f ) &gt; 0. x Hence µ is non-atomic and thus expanding because supp(µ x x

                        ) D so

                        −1 x x x &lt; 0 and µ is ergodic, for µ-a.e x. Such µ also satisfies

                        R log k(D f) kdµ the Entropy Formula. Therefore Corollary ensures that µ is weak- x SRB-like for µ-a.e. x M.

                        ∗ −1

                        , then Consider µ ∈ W R log k(D f) kdµ &lt; 0, otherwise, we have f supp(µ) ⊂ D and we see that this is not possible. R Because µ is a ψ-equilibrium state, then h ( f ) + ψdµ = 0. More- R µ

                        −1 −1

                        over 0 &gt; R log k(D f) kdµ = log k(D f ) kdµ, thus by the Ergodic M

                        \D

                        Decomposition Theorem we conclude that for µ-a.e x M\D we have R

                        −1

                        ψdµ = &lt; 0.

                        h µ ( f )+ x 0 (remember that P top ( f, ψ) = 0) and x x

                        R log k(D f) kdµ Therefore, µ-a.e x M\D have expanding ergodic components that are ψ-equilibrium states. By Corollary we deduce that µ is an weak- x SRB-like probability measure for µ-a.e x M\D and finishes the proof.

                        Putting Corollaries

                      9.2 Expanding Case

                        The Corollary and allows rewriting

                        1

                        all the results from

                        9.3. Proof of Corollary

                        71 Proof of Corollary The assumptions on f ensure that all f -invariant pro-

                        bability measures µ are expanding. Moreover, by Proposition and Ru- elle’s Inequality we conclude that P top ( f, ψ) = 0 and every weak-SRB-like probability measures are ψ-equilibrium states and satisfy Entropy For- mula. f satisfies Entropy Formula, then

                        Then, on the one hand, if µ ∈ M R

                        

                      h ( f ) + ψdµ = 0. By Ergodic Decomposition Theorem we have that

                      µ R

                        =

                        

                      h ( f ) + ψdµ top ( f, ψ) = 0. By Corollary

                      µ x x 0 for all µ-a.e x M, because P

                        we have that µ x is weak-SRB-like probability measure for µ-a.e x M. f x is such that its ergodic components µ are On the other hand, if µ ∈ M R

                        = µ x ( f )+ ψdµ weak-SRB-like probability measures for µ-a.e. x M, then h x 0 for µ-a.e x M. Thus, by the Ergodic Decomposition Theorem we have that Z Z Z ! Z

                        ψdµ µ(x) = µ(x) = h x d h µ ( f ) d µ ( f ). − ψdµ = − x Assume now that µ is the unique weak-SRB-like probability measure.

                        By item G of Theorem 1 in we have that µ is SRB and Leb(B(µ)) = 1.

                        See the proof of Corollary As observed in , the SRB-like condition is a sufficient but not neces- sary condition for a measure µ to be an equilibrium state for the potential

                        ψ, because it may exist a non-ergodic invariant measure µ &lt;&lt; Leb that is neither SRB nor SRB-like (see ). In such a case µ satisfies Pesin’s Entropy Formula, as stated the following lemma.

                        

                      1

                        µ be a non-ergodic

                        Corollary 9.2.1. Let f -expanding map. Let

                        : M M be a C

                        

                      f -invariant probability such that µ &lt;&lt; Leb. Then µ satisfies Pesin’s Entropy

                      Formula. Proof. See Corollary 2.6 in .

                        9.3 Proof of Corollary

                        1 Proof of Corollary Let f f in the C -topology, where f n n

                        −−−−→ , f : M n →+∞

                      1 M are C

                        n a

                      • expanding maps for all n ≥ 1. For each n ≥ 1 consider µ

                        ∗

                        weak-SRB-like measures associated to f and let µ be a weak limit point: n µ = lim µ j n .

                        →+∞ j

                        72 Chapter 9. Weak-Expanding non-uniformly expanding maps n

                        Fix P a generating partition for every f j for all j ≥ 1 and such that µ(∂P) = 0.

                        1

                        1 This is possible, since f is C -expanding and f -topology n n jf in C 1 and f is also C -expanding.

                        By Kolmogorov-Sinai Theorem this implies that h ( f ) and µ n n n ) = h(P, µ

                        h µ

                        ( f ) = h(P, µ), that is,

                        1 k k

                        1

                        

                      h ( f ) = inf H , µ ) and h ( f ) = inf H , µ)

                      µ n n µ n j j (P n j (P

                      k k

                      j

                      ≥1 ≥1

                      k k k

                        , µ ) converge n Since µ gives zero measure to the boundary of P then H(P n j k j to H(P , µ) as j → ∞. Furthermore, for every ε &gt; 0 there is n ≥ 1 such that

                        1 n

                        1 n , µ

                      h µ ( f n H n H µ ( f ) + 2ε.

                      j n j , µ) + ε ≤ h n j ) ≤ (P j ) ≤ (P

                        n n R

                        = By Corollary h µ ( f ) + ψ d µ is weak- n j n j n j n j n j 0 for all j ≥ 0, since µ

                        = SRB-like probability measure and ψ n n j − log | det D f j |.

                        Since ψ n j R R → ψ in the topology of uniform converge, we have that R ψ d µ ψdµ. By Ruelle’s inequality, h ν ( f ) + n j n j

                        → ψdν ≤ 0 for any

                        f -invariant probability measure ν on the Borel σ-algebra of M. Thus, Z Z !

                        = µ µ n n n ( f ) + h ( f ) + ψ d µ 0. 0 ≥ h ψdµ ≥ lim sup n n

                        →+∞

                        This shows that µ satisfies Pesin’s Entropy Formula, is a ψ-equilibrium state since P top ( f, ψ) = 0 and by Corollary its ergodic components µ are x weak-SRB-like probability measures for µ-a.e x M.

                      9.4 Proof of Corollary C Finally we prove Corollary C.

                        

                      Proof of Corollary

                      T

                        Theorem Let µ ∈ M such that Z Z Z !

                        h (T) + φdµ = P top (T, φ) = h (T) + φdµ d µ(x) µ µ x x R R

                        = we also have h (T) + φdµ (T, φ) and so h (T) + φdµ P (T, φ) µ x top µ x top xP x for µ-a.e x. Now from Theorem since h ν (T, µ x ) &lt; ∞

                      9.4. Proof of Corollary C

                        73

                        for µ-a.e. x X we get Z

                        1 lim lim sup log ν(A (µ (T) + φdµ ε,n x µ x )) ≥ h x − log λ = 0

                      • + ε→0 n

                        n →+∞ ∗

                        and then we conclude that µ (ν) following the same argument in the x ∈ W T proof of Corollary

                        Assume now that µ is the unique φ-equilibrium state. By Proposition follows that µ is ν-SRB and ν(B(µ)) = 1. is

                        1 r r

                        Let now V be a small neighborhood of µ in M . Since {K (φ)} r &gt;0 r (φ) we have that there exists decreasing with r and {µ} = K (φ) = ∩ K

                        ∗ r r r (φ) is weak

                        &gt; 0 such that K (φ) ⊂ V for all 0 &lt; r &lt; r . Since K r (φ) = compact (by upper semicontinuity of the metric entropy) we have K T ε ε n

                        ∗ ε&gt;0 K r (φ), where K r µ ∈ M ; dist(µ, K (φ)) ≤ ε (φ) = T r o with the weak ε distance defined in . Consider 0 &lt; ε &lt; r

                        , such that K r (φ) ⊂ V. By Proposition there exists n

                        ≥ 1 and κ = κ(ε, r) &gt; 0 such that ε n n

                        1

                        1

                        ν({x X : σ (x) ∈ M \V}) ≤ ν({x X : σ (x) ∈ M \K r (φ)}) n (ε−r) = n r ,

                        ν({x X; dist(σ (x), K (φ)) ≥ ε}) &lt; κe

                        1

                        . Thus lim sup n

                        1

                        for all n n log ν({x X : σ (x) ∈ M \V}) &lt; ε−r. As n n

                        →+∞

                        1

                        ε &gt; 0 can be taken arbitrary small, we conclude that lim sup n n log ν({x

                        →+∞

                        =

                        X : σ n 1 and r

                        I r (x) ∈ M \V}) &lt; −r for all 0 &lt; r &lt; r (V) := sup{r &gt;

                        0; K (φ) ⊂ V}. Therefore

                        1 lim sup n

                        1 log ν({x X : σ (x) ∈ M \V}) ≤ −I(V). n n →+∞ n

                        1 This shows that the probability ν({x X : σ (x) ∈ M \V}) decreases exponentially fast with n at a rate that depends on the “size” of V.

                        The assumptions on µ are the same as in Corollary with ν = Leb and φ = ψ, so the upper large deviations statement of this corollary follows with the same proof.

                        74 Chapter 9. Weak-Expanding non-uniformly expanding maps

                        

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