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

TL;DR

This paper studies how the number and nature of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms change under small perturbations, revealing conditions for upper semicontinuity and local constancy.

Contribution

It extends the understanding of ergodic measures of maximal entropy to $C^r$ surface diffeomorphisms, establishing upper semicontinuity and criteria for local constancy.

Findings

Number of ergodic measures of maximal entropy is upper semicontinuous at $f$.

Local constancy of the number occurs iff ergodic measures admit ergodic continuations.

Accumulation points of measures are ergodic under certain conditions.

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.