Abundance of one dimensional non uniformly hyperbolic attractors for surface endomorphisms

Details Abundance of one dimensional non uniformly hyperbolic attractors for surface endomorphisms Abundance of one dimensional non uniformly hyperbolic attractors for surface endomorphisms

2009-03-09

arxiv.org/abs/0903.1473v2

Mathematics

Dynamical Systems

To clarify all the details and simply the proof, I made many changes to the latter version, however the proof follows the same

Scientific paper

For every $C^2$-small perturbation $B$, we prove that the map $(x,y)\mapsto (x^2+a+y,0)+B(x,y,a)$ preserves a physical, SRB probability, for a positive Lebesgue measure set of parameters $a$. When the perturbation $B$ is zero, this is the Jakobson 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 Theorem is given, by basically mixing analytical ideas of Benedicks-Carleson and the combinatorial formalism of Yoccoz puzzle with new geometrical and arithmetic ingredients. These proofs are enlarged to the $C^2$-topology. Also the dynamics can be an endomorphism.

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