Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models
Abstract
We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called H\'enon maps. We show that Smale's Axiom A property is -dense among the systems in this family, despite the existence of -open subsets (closely related to the so-called Newhouse phenomena) where Smale's Axiom A is violated. In particular, this provides some evidence towards Smale's conjecture that Axiom A is a -dense property among surface diffeomorphisms. The basic tools in the proof of this result are: 1) a recent theorem of Moreira saying that stable intersections of dynamical Cantor sets (one of the main obstructions to Axiom A property for surface diffeomorphisms) can be destroyed by -perturbations; 2) the good geometry of the dynamical critical set (in the sense of Rodriguez-Hertz and Pujals) thanks to the particular form of Benedicks-Carleson toy models.
Keywords
Cite
@article{arxiv.1004.5453,
title = {Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models},
author = {Carlos Matheus and Carlos Gustavo Moreira and Enrique Pujals},
journal= {arXiv preprint arXiv:1004.5453},
year = {2014}
}
Comments
20 pages, 2 figures. Final version based on the referee's comments. To appear in Annales Scientifiques de l'Ecole Normale Superieure