Dynamical systems
On the hyperbolicity of C1-generic homoclinic classes
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 1047-1051.

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 C1-generic diffeomorphisms, a homoclinic class either is hyperbolic or contains a sequence of periodic orbits that have a Lyapunov exponent arbitrarily close to 0.

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 C1-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.

Published online:
DOI: 10.1016/j.crma.2015.07.017
Wang, Xiaodong 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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Wang, Xiaodong. 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. http://www.numdam.org/articles/10.1016/j.crma.2015.07.017/

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