Abundance of non-uniformly hyperbolic Hénon like endomorphisms

Abstract

For every $C^2$-small function $B$, we prove that the map $(x,y)\mapsto (x^2+a,0)+B(x,y,a)$ leaves invariant a physical, SRB probability measure, for a set of parameters $a$ of positive Lebesgue measure. When the perturbation $B$ is zero, this is the Jakobson Theorem; when the perturbation is a small constant times $(0,x)$, this is the celebrated Benedicks-Carleson Theorem. In particular, a new proof of the last theorem is given, based on devellopment of the combinatorial formalism of the Yoccoz puzzles. By adding new geometrical and combinatorial ingredients, and restructuring classic analytical ideas, we are able to carry out our proof in the $C^2$-topology, even when the underlying dynamics are given by endomorphisms.

Proof completely rewritten, with special emphasis on the parameter selection, so that it is now induction free

References in corpus (1)

• Persistence of laminations

Cited by in corpus (4)

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