# American Institute of Mathematical Sciences

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

## On the hyperbolicity of homoclinic classes

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