Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

Matheus, Carlos; Moreira, Carlos G.; Pujals, Enrique R.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 46 (2013) no. 6, pp. 857-878.

Nous considérons une famille de systèmes introduite en 1991 par Benedicks et Carleson comme un modèle jouet pour la dynamique des applications d’Hénon. Nous montrons que l’axiome A de Smale est une propriété $C^1$-dense parmi les systèmes dans cette famille, même si nous trouvons aussi des ensembles $C^2$-ouverts (liés au phénomène de Newhouse) où l’axiome A de Smale n’est pas satisfait. En particulier, notre résultat soutient la conjecture de Smale selon laquelle l’axiome A est une propriété $C^1$-dense parmi les difféomorphismes de surfaces. Les outils utilisés dans la preuve de notre résultat sont  : (1) un théorème récent de Moreira qui dit que les intersections stables des ensembles de Cantor dynamiques (une des obstructions majeures à l’axiome A pour les difféomorphismes de surfaces) peuvent être enlevées par des perturbations $C^1$-petites  ; (2) la bonne géométrie de l’ensemble de points critiques dynamiques (au sens de Rodriguez-Hertz et Pujals) due à la forme particulière des modèles jouets de Benedicks-Carleson.

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 Hénon 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.

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