On the hyperbolicity of $ {\mathrm{C}}^{1}$-generic homoclinic classes

Xiaodong Wang

doi:10.1016/j.crma.2015.07.017

Dynamical systems

On the hyperbolicity of

C^{1}

-generic homoclinic classes
[Sur l'hyperbolicité des classes homoclines

C^{1}

-génériques]

Présenté par : Étienne Ghys

Xiaodong Wang ^1,²

¹ School of Mathematical Sciences, Peking University, Beijing, 100871, China
² Laboratoire de mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay, France

Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1047-1051.

Résumé
Abstract

Des travaux de Liao, Mañé, Franks, Aoki et Hayashi ont caractérisé le manque d'hyperbolicité des difféomorphismes par l'existence d'orbites périodiques faibles. Dans cette note, nous annonçons un résultat qui peut être considéré comme une version locale de ces travaux : pour les difféomorphismes $C^{1}$ -génériques, une classe homocline, ou bien est hyperbolique, ou bien contient une suite d'orbites périodiques qui ont un exposant de Lyapunov arbitrairement proche de 0.

The works of Liao, Mañé, Franks, Aoki, and Hayashi characterized a lack of hyperbolicity for diffeomorphisms by the existence of weak periodic orbits. In this note, we announce a result that can be seen as a local version of these works: for $C^{1}$ -generic diffeomorphisms, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0.

Reçu le : 2014-12-08
Accepté le : 2015-07-10
Publié le : 2015-10-21

DOI : 10.1016/j.crma.2015.07.017

Affiliations des auteurs :

Xiaodong Wang ^{1, 2}

¹ School of Mathematical Sciences, Peking University, Beijing, 100871, China
² Laboratoire de mathématiques d'Orsay, Université Paris-Sud 11, 91405 Orsay, France

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Xiaodong Wang. On the hyperbolicity of $ {\mathrm{C}}^{1}$-generic homoclinic classes. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1047-1051. doi : 10.1016/j.crma.2015.07.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.07.017/

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