Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms

Abstract

Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>\lambda_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$ case studied in \cite{BCS22}, answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of $f$ admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the $C^\infty$ case.