  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.
