Partially hyperbolic diffeomorphisims with a finite number of measures of maximal entropy

TL;DR

This paper proves that under certain conditions, partially hyperbolic diffeomorphisms have only finitely many ergodic measures of maximal entropy, with applications to specific classes and stability properties.

Contribution

It establishes finiteness of ergodic measures of maximal entropy for a broad class of partially hyperbolic diffeomorphisms with dominated center bundles and Lyapunov bounds.

Findings

Finiteness of measures of maximal entropy for certain partially hyperbolic systems.

Finiteness results for derived classes on $ ext{T}^4$ and skew products.

Upper semicontinuity of the number of maximal entropy measures.

Abstract

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute value of the Lyapunov exponents. As applications we prove finiteness for a class derived from Anosov partially hyperbolic diffeomorphisms defined on $\mathbb{T}^4$ and that in a class of skew product over partially hyperbolic diffeomorphisms there exists a $C^1$ open and $C^r$ dense set of diffeomorphisms with a finite number of ergodic measures of maximal entropy. We also study the upper semicontinuity of the number of measures of maximal entropy with respect to the diffeomorphism.