Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

TL;DR

This paper demonstrates that Smale's Axiom A property is dense in a family of planar systems related to Hénon maps, despite the presence of Newhouse phenomena, supporting Smale's conjecture for surface diffeomorphisms.

Contribution

It proves the $C^1$-density of Axiom A in Benedicks-Carleson models, using recent results on dynamical Cantor sets and geometric properties of the models.

Findings

Axiom A is $C^1$-dense among the systems studied.

Existence of $C^2$-open sets where Axiom A fails due to Newhouse phenomena.

Destruction of stable intersections of Cantor sets by $C^1$-perturbations.

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 $C^1$-dense among the systems in this family, despite the existence of $C^2$-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 $C^1$-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 $C^1$-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.