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, $h_{\mathrm{top}}(f)>\hat{I}{»}_{\min }(f)/r$). 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.

35pages, 4Figures. Compared with the previous version, this one provides an equivalent characterization of the continuity of the number of ergodic MMEs and constructs examples showing that discontinuity can indeed occur. Most of these ideas originate from Jérôme Buzzi, and thus we completed the writing of the paper in collaboration