# Properties of the maximal entropy measure and geometry of Hénon attractors

### Pierre Berger

Université Paris 13, Villetaneuse, France

## Abstract

We consider an abundant class of non-uniformly hyperbolic $C_{2}$-Hénon 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, which answers a question of M. Lyubich and Y. Pesin. We also prove that the maximal entropy measure is equidistributed 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 a Young tower for which the first return time coincides with the symbolic return time, and whose orbit is conjugate to a strongly positive recurrent Markov shift.

## Cite this article

Pierre Berger, Properties of the maximal entropy measure and geometry of Hénon attractors. J. Eur. Math. Soc. 21 (2019), no. 8, pp. 2233–2299

DOI 10.4171/JEMS/884