Uniqueness of the measure of maximal entropy for the standard map

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. Contents 1 Introduction 2 Preliminaries 3 Estimates on invariant manifolds for measures with large exponents 4 Homoclinic relation for measures with large exponents 5 Proof of Theorem A 6 Growth and equidistribution of periodic points: proof of Theorem B 7 Further properties of the m.m.e.: proof of Theorems C and D Bibliography * D.O. was supported by the projects ANR BEKAM : ANR-15-CE40-0001 and ERC project NUHGD.

Equilibrium states for natural extensions of non-uniformly expanding local homeomorphisms

“…Buzzi, Crovisier, and Sarig [3] then proved uniqueness of maximal entropy measures for transitive C ∞ surface diffeomorphisms with positive topological entropy. Adapting these techniques Obata [13] recently proved for sufficiently large parameters the standard map has a unique measure of maximal entropy.…”