On the hyperbolicity of homoclinic classes

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December 2009, 25(4): 1143-1162. doi: 10.3934/dcds.2009.25.1143

On the hyperbolicity of homoclinic classes

Christian Bonatti ^1, , Shaobo Gan ^2, and Dawei Yang ^3,

1.	Institut de Mathématiques de Bourgogne, Université de Bourgogne, Dijon 21004, France
2.	School of Mathematical Science, Peking University, Beijing 100871
3.	School of Mathematical Sciences, Peking University, Beijing 100871, China

Received October 2008 Revised May 2009 Published September 2009

Abstract
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We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class $H(p)$ admits a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where $E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all periodic points homoclinically related with $p$ are uniformly $E^u$-expanding at the period, then $H(p)$ is hyperbolic. We also give some consequences of this result.

Keywords: hyperbolic time, shadowing lemma., homoclinic class.

Mathematics Subject Classification: Primary: 37D20, 37D30; Secondary: 37D2.

Citation: Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143

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