Margulis systems of masures and measures of maximal entropy for partially hyperbolic diffeomorphisms with compact center foliation

Author(s)

Richard Javier Cubas Becerra

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2022-10-21

Examining board members

Ali Tahzibi; André Salles de Carvalho; Katrin Grit Gelfert; Tiago Pereira da Silva

• Advisor: Ali Tahzibi

Abstract

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. (AU)

• FAPESP's process: 18/04076-5 - Margulis measures and partially hyperbolic systems
• Grantee: Richard Javier Cubas Becerra
• Support Opportunities: Scholarships in Brazil - Doctorate