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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

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 and Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143
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