Uniqueness of the measure of maximal entropy for the standard map

TL;DR

This paper proves the uniqueness and Bernoulli property of the measure of maximal entropy for the standard map at large parameters, along with results on equidistribution, Hausdorff dimension, and robustness of these properties.

Contribution

It establishes the uniqueness and Bernoulli nature of the measure of maximal entropy for large parameters in the standard map, with new estimates on dimension and support.

Findings

Unique measure of maximal entropy for large parameters

The m.m.e. is Bernoulli and supports equidistribution of periodic points

Support of the m.m.e. has Hausdorff dimension 2 for generic large parameters

Abstract

In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from zero equidistribute with respect to the m.m.e. We prove some estimates regarding the Hausdorff dimension of the m.m.e. and about the density of the support of the measure on the manifold. For a generic large parameter, we prove that the support of the m.m.e. has Hausdorff dimension $2$. We also obtain the $C^2$-robustness of several of these properties.