Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models

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

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 6, page 857-878
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Matheus, Carlos, Moreira, Carlos G., and Pujals, Enrique R.. "Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models." Annales scientifiques de l'École Normale Supérieure 46.6 (2013): 857-878. <http://eudml.org/doc/272176>.

@article{Matheus2013,
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 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.},
author = {Matheus, Carlos, Moreira, Carlos G., Pujals, Enrique R.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {axiom a; Newhouse phenomena; Benedicks-Carleson toy models; hénon maps; dynamical critical points; stable intersections of dynamical Cantor sets; two-dimensional dynamical systems},
language = {eng},
number = {6},
pages = {857-878},
publisher = {Société mathématique de France},
title = {Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models},
url = {http://eudml.org/doc/272176},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Matheus, Carlos
AU - Moreira, Carlos G.
AU - Pujals, Enrique R.
TI - Axiom A versus Newhouse phenomena for Benedicks-Carleson toy models
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 6
SP - 857
EP - 878
AB - 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.
LA - eng
KW - axiom a; Newhouse phenomena; Benedicks-Carleson toy models; hénon maps; dynamical critical points; stable intersections of dynamical Cantor sets; two-dimensional dynamical systems
UR - http://eudml.org/doc/272176
ER -

References

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