Abundance of one dimensional non uniformly hyperbolic attractors for surface dynamics

We present a (new) proof of the existence of a non uniformly
hyperbolic attractor for a positive set of parameters $a$ in the
family of endomorphisms:
\[(x,y)\mapsto (x2+a+2y,0)+B(x,y)\]
where $B$ is any fixed  $C^2$ small function. For $B=0$, this is the
Jackoson theorem. For $B=b.(0,x)$, we get the Benedicts-Carleson
theorem for the Henon map.

The proof is done thanks to analytical and probabilistic tools of
(B-C)  in the geometric and combinatorial formalism of Yoccoz puzzles
generalised in a very algebraic way (pseudo-semi-group).  These
theorems are notably generalised to the  $C^2$-case and to the
endomorphisms. The theorem is an answer to  question of
Pesin-Yurchenko réaction-diffusion EDPs in applied mathematics.


The article is avaible on arxiv.