Measures of maximal entropy for non-uniformly hyperbolic maps

Abstract

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that the homoclinic class of each 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).