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Doctoral Thesis
DOI
https://doi.org/10.11606/T.55.2022.tde-18112022-191726
Document
Author
Full name
Richard Javier Cubas Becerra
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2022
Supervisor
Committee
Tahzibi, Ali (President)
Carvalho, André Salles de
Gelfert, Katrin Grit
Silva, Tiago Pereira da
Title in Portuguese
Sistemas de medidas Margulis e medidas de máxima entropia para difeomorfismos parcialmente hiperbólicos com folheação central compacta
Keywords in Portuguese
AB-sistema
Difeomorfismo parcialmente hiperbólico
Entropia topológica
Expoente de Lyapunov
Medida de máxima entropia
Sistema de Marguli
su-folha
Abstract in Portuguese
Este trabalho trata sobre a construção e caracterização das medidas de máxima entropia para certos sistemas parcialmente hiperbólicos usando o conceito de medidas Margulis. Consideramos a classe dos difeomorfismos C2 parcialmente hiperbólicos com folheação central uniformemente compacta de dimensão um, sobre uma variedade fechada M, denotada por PHC2c=1(M). Para sistemas f ∈ PHC2c=1(M), supondo que a dinâmica induzida no espaço das folhas centrais é topologicamente transitiva, construímos uma família de medidas ao longo da folheação instável chamadas medidas Margulis e exibimos a sua relação com a desintegração ao longo da folheação instável de medidas de máxima entropia. Usando esta caracterização, quando a folheação instável é dinamicamente minimal provamos que ou f possui uma única medida de máxima entropia a qual é provada ter expoente central zero, ou f possui exatamente duas medidas de máxima entropia ergódicas, as quais são hiperbólicas e com expoente central de sinal oposto. Também estudamos a natureza do suporte das medidas de máxima entropia com expoente central zero para difeomorfismos f ∈ PHC2c=1(M) que são infra-AB-sistemas, e provamos que toda medida de máxima entropia com expoente central zero possui uma sub-variedade compacta e periódica, tangente aos fibrados estável e instável, a qual chamamos de su-folha. Ainda neste contexto, quando o sistema f é topologicamente transitiva, mostramos que f possui no máximo duas medidas de máxima entropia com expoente central nulo. Além disso, para o caso f ∈ PHC2c=1(T3 ) mostramos finitude de medidas de máxima entropia ergódicas usando algumas hipóteses adicionais e aplicando os resultados obtidos.
Title in English
Margulis systems of masures and measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center foliation
Keywords in English
AB-system
Lyapunov Exponent
Margulis system
Maximal entropy measures
Partially hyperbolic diffeomorphism
su-leaf
Topological entropy
Abstract in English
This work is the construction and characterization of maximal entropy measures for certain partially hyperbolic systems using the concept of Margulis measures. We consider the class of C2 partially hyperbolic diffeomorphisms with one-dimensional center bundle and uniformly compact center foliation on a closed manifold M, denoted by PHC2c=1(M). For systems f ∈ PHC2c=1(M), assuming that the dynamics induced in the space of the central leaves is topologically transitive, we construct a family of measures along the unstable foliation called Margulis measures, and characterize their relationship with the disintegration along the unstable foliation of maximal entropy measures. using this characterization, we prove that when the unstable foliation is dynamically minimal there is a dichotomy: either f has a unique entropy-maximizing measure, which is proved to have zero center exponent; or the system f has exactly two ergodic maximal measures, which are hyperbolic and have a central exponent of opposite sign. We also study the nature of the support of measures of maximal entropy with vanishing center exponent for diffeomorphisms f ∈ PHC2c=1(M) infra-AB-systems, and we prove that every measure of maximal entropy with vanishing center exponent has a periodic submanifold tangent to the stable and unstable bundles, which we call su-leaf. Still in this context, when the system f is topologically transitive, we show that f has at most two ergodic maximal entropy measures with vanishing center exponent. Furthermore, for the case M = T3, if f ∈ PHC2c=1 (T3) we show finiteness of the ergodic maximal entropy measures using some additional assumptions and applying the previous results.
 
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Publishing Date
2022-11-21
 
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