Measures of maximal entropy for non-uniformly hyperbolic maps

TL;DR

This paper investigates the uniqueness and finiteness of maximal entropy measures in non-uniformly hyperbolic dynamical systems, including billiards and Viana maps, extending understanding of measure-theoretic properties in complex systems.

Contribution

It proves that each homoclinic class of an ergodic hyperbolic measure has at most one maximal entropy measure and applies this to various classes of non-uniformly hyperbolic maps.

Findings

Uniqueness of maximal entropy measures in certain hyperbolic systems

Finiteness results for measures in dispersing billiards and Viana maps

Extension of measure-theoretic entropy concepts to non-invertible maps

Abstract

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study the finiteness/uniqueness of such measures in several different settings: finite horizon dispersing billiards, codimension one partially hyperbolic endomorphisms with ``large'' entropy, robustly non-uniformly hyperbolic volume-preserving endomorphisms as in Andersson-Carrasco-Saghin (2025), and Viana maps (1997).