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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.

Properties of the maximal entropy measure and geometry of Hénon attractors