Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 6, pp. 857-878.

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.

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.

DOI: 10.24033/asens.2204
Classification: 37D40,  37D20
Keywords: axiom a, Newhouse phenomena, Benedicks-Carleson toy models, hénon maps, dynamical critical points, stable intersections of dynamical Cantor sets, two-dimensional dynamical systems
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     title = {Axiom~A versus {Newhouse} phenomena for {Benedicks-Carleson} toy models},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {857--878},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
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Matheus, Carlos; Moreira, Carlos G.; Pujals, Enrique R. Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 6, pp. 857-878. doi : 10.24033/asens.2204. http://www.numdam.org/articles/10.24033/asens.2204/

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