Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale

Moreira, Carlos Gustavo ; Yoccoz, Jean-Christophe

Annales scientifiques de l'École Normale Supérieure, Tome 43 (2010), p. 1-68 / Harvested from Numdam

Résumé

Soit $F_0$ un difféomorphisme d’une surface possédant deux fers à cheval $\Lambda , \Lambda ^{\prime }$ tels que $W^s \Lambda $ et $W^u \Lambda ^{\prime }$ aient en un point $q$ une tangence quadratique isolée. Nous montrons que, si la somme des dimensions transverses de $W^s \Lambda $ et $W^u \Lambda ^{\prime }$ est strictement plus grande que 1, les difféomorphismes voisins de $F_0$ tels que $W^s \Lambda $ et $W^u \Lambda ^{\prime }$ soient stablement tangents au voisinage de $q$ forment une partie de densité inférieure strictement positive en $F_0$.

Abstract

Let $F_0$ be a surface diffeomorphism with two horseshoes $\Lambda , \Lambda ^{\prime }$ such that $W^s \Lambda $ and $W^u \Lambda ^{\prime }$ have a quadratic tangency at a point $q$. We show that, if the sum of the transverse dimension of $W^s \Lambda $ and $W^u \Lambda ^{\prime }$ is larger than one, the set of diffeomorphisms close to $F_0$ such that $W^s \Lambda $ and $W^u \Lambda ^{\prime }$ have a stable tangency near $q$ has positive density at $F_0$.

Détail

Publié le : 2010-01-01

MR 2583264 | Zbl 1200.37020

DOI : https://doi.org/10.24033/asens.2115
Classification:  37D05,  37D20,  37E30
Mots clés: bifurcation homocline, tangence homocline, fer à cheval, dimension fractale

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