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We consider an abundant class of non-uniformly hyperbolic -Hénon like diffeomorphisms called strongly regular and which corresponds to Benedicks–Carleson parameters. We prove the existence of such that for any such diffeomorphism , every invariant probability measure of has a Lyapunov exponent greater than , 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. 2233–2299DOI 10.4171/JEMS/884