Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models
[Axiome A versus phénomène de Newhouse pour les modèles jouets de Benedicks-Carleson]
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.

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
Mot clés : axiome A, phénomène de Newhouse, modèles jouets de Benedicks-Carleson, applications d'Hénon, points critiques dynamiques, intersections stables des ensembles de Cantor dynamiques, systèmes dynamiques en dimension deux
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     title = {Axiom~A versus {Newhouse} phenomena for {Benedicks-Carleson} toy models},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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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, Série 4, Tome 46 (2013) no. 6, pp. 857-878. doi : 10.24033/asens.2204. http://archive.numdam.org/articles/10.24033/asens.2204/

[1] F. Abdenur, C. Bonatti, S. Crovisier & L. J. Díaz, Generic diffeomorphisms on compact surfaces, Fund. Math. 187 (2005), 127-159. | MR

[2] D. V. Anosov, Extension of zero-dimensional hyperbolic sets to locally maximal ones, Sb. Math. 201 (2010), 935-946. | MR

[3] M. Benedicks & L. Carleson, The dynamics of the Hénon map, Ann. of Math. 133 (1991), 73-169. | MR

[4] M. V. Jakobson, Smooth mappings of the circle into itself, Mat. Sb. (N.S.) 85 (127) (1971), 163-188. | MR

[5] R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys. 100 (1985), 495-524. | MR

[6] R. Mañé, Ergodic theory and differentiable dynamics, Ergebnisse Math. Grenzg. (3) 8, Springer, 1987. | MR

[7] W. De Melo & S. Van Strien, One-dimensional dynamics, Ergebnisse Math. Grenzg. (3) 25, Springer, 1993. | MR

[8] C. G. Moreira, There are no C 1 -stable intersections of regular Cantor sets, Acta Math. 206 (2011), 311-323. | MR

[9] S. E. Newhouse, Nondensity of axiom A(a) on S 2 , in Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., 1970, 191-202. | MR

[10] S. E. Newhouse, Hyperbolic limit sets, Trans. Amer. Math. Soc. 167 (1972), 125-150. | MR | Zbl

[11] S. E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9-18. | MR | Zbl

[12] S. E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. I.H.É.S. 50 (1979), 101-151. | EuDML | Numdam | MR | Zbl

[13] J. Palis & F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Math. 35, Cambridge Univ. Press, 1993, Fractal dimensions and infinitely many attractors. | MR | Zbl

[14] V. A. Pliss, On a conjecture of Smale, Differencialʼnye Uravnenija 8 (1972), 268-282. | MR

[15] E. R. Pujals & F. Rodriguez Hertz, Critical points for surface diffeomorphisms, J. Mod. Dyn. 1 (2007), 615-648. | MR | Zbl

[16] E. R. Pujals & M. Sambarino, Homoclinic tangencies and hyperbolicity for surface diffeomorphisms, Ann. of Math. 151 (2000), 961-1023. | EuDML | MR | Zbl

[17] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. | MR | Zbl

[18] R. Ures, Abundance of hyperbolicity in the C 1 topology, Ann. Sci. École Norm. Sup. 28 (1995), 747-760. | EuDML | Numdam | MR | Zbl

[19] R. Ures, On the approximation of Hénon-like attractors by homoclinic tangencies, Ergodic Theory Dynam. Systems 15 (1995), 1223-1229. | MR | Zbl

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