Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale

Moreira, Carlos Gustavo; Yoccoz, Jean-Christophe

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 1, pp. 1-68.

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

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

Classification : 37D05, 37D20, 37E30
Mot clés : bifurcation homocline, tangence homocline, fer à cheval, dimension fractale
Keywords: homoclinic bifurcation, homoclinic tangency, horseshoe, fractal dimension

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