Abundance of one dimensional non uniformly hyperbolic attractors for surface endomorphisms

For every C-small perturbation B, we prove that the map (x, y) 7→ (x +a+2y, 0)+B(x, y) preserves a physical, SRB probability, for a Lebesgue positive set of parameters a. When the perturbation B is zero, this is the Jackobson theorem; when the perturbation is a small constant times (0, 1), this is the celebrated Benedicks-Carleson theorem. In particular, a new proof of the last theorems is given, by basically mixing analytical ideas of Benedicks-Carleson and the combinatorial formalism of Yoccoz puzzle with new geometrical and algebraic ingredients. These proofs are enlarged to the C-topology. Also the dynamics can be an endomorphism. Our aim is to prove the existence of a non-uniformly hyperbolic attractor for a large set of parameters a ∈ R, for the following family of maps: fa,B : (x, y) 7→ (x + a+ 2y, 0) +B(x, y), where B is a fixed C2-map of R2 close to 0. We denote by b an upper bound of the uniform C2-norm of B and of the Jacobian of fa,B. For B fixed, we prove that for a large set ΩB of parameters a, the dynamics fa,B is strongly regular. This ought to imply a lot of properties. Belong those, a simple consequence is the following theorem. Theorem 0.1. For any t ∈ (0, 1), there exist a0 greater but close to −2 and b > 0 such that for any B with C2-norm less than b, there exists a subset ΩB ⊂ [−2, a0] of relative measure greater than t satisfying the following property. For any a ∈ ΩB, there exists an union of unstable manifolds A which support a unique SRB measure which is ergodic and whose basin has positive Lebesgue measure. This answers to a question of Pesin-Yurchenko for reaction-diffusion PDEs in applied mathematics [PY04]. 1 ar X iv :0 90 3. 14 73 v1 [ m at h. D S] 9 M ar 2 00 9