Properties of the maximal entropy measure and geometry of H\'enon attractors

TL;DR

This paper studies the properties of the maximal entropy measure for strongly regular Hénon-like diffeomorphisms, establishing its uniqueness, mixing properties, and statistical behavior, thus advancing understanding of complex dynamical systems.

Contribution

It proves the existence and uniqueness of the maximal entropy measure, its Bernoulli property, exponential mixing, and statistical limit theorems for a class of Hénon-like maps.

Findings

Existence and uniqueness of maximal entropy measure.

Maximal entropy measure is Bernoulli and equidistributed on periodic points.

Maximal entropy measure is exponentially mixing and satisfies the CLT.

Abstract

We consider an abundant class of non-uniformly hyperbolic $C^2$-H\'enon like diffeomorphisms called strongly regular and which corresponds to Benedicks-Carleson parameters. We prove the existence of $m>0$ such that for any such diffeomorphism $f$, every invariant probability measure of $f$ has a Lyapunov exponent greater than $m$, answering a question of L. Carleson. Moreover, we show the existence and uniqueness of a measure of maximal entropy, this answers a question of M. Lyubich and Y. Pesin. We also prove that the maximal entropy measure is equi-distributed on the periodic points and is finitarily Bernoulli, which gives an answer to a question of J.P. Thouvenot. Finally, we show that the maximal entropy measure is exponentially mixing and satisfies the central limit Theorem. The proof is based on a new construction of Young tower for which the first return time coincides with the symbolic return time, and whose orbit is conjugated to a strongly positive recurrent Markov shift.