# American Institute of Mathematical Sciences

August  2010, 27(3): 1133-1145. doi: 10.3934/dcds.2010.27.1133

## Hyperbolicity of $C^1$-stably expansive homoclinic classes

 1 Department of Mathematics, Chungnam National University, Daejeon, 305-764 2 Department of Mathematics, Mokwon University, Daejeon, 302-729, South Korea

Received  November 2008 Revised  February 2010 Published  March 2010

Let $f$ be a diffeomorphism of a compact $C^\infty$ manifold, and let $p$ be a hyperbolic periodic point of $f$. In this paper we introduce the notion of $C^1$-stable expansivity for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $\mathcal {R}(f)$ of $f$ is $C^1$-stably expansive if and only if $f$ satisfies both Axiom A and no-cycle condition, $(ii)$ the homoclinic class $H_f(p)$ of $f$ associated to $p$ is $C^1$-stably expansive if and only if $H_f(p)$ is hyperbolic, and $(iii)$ $C^1$-generically, the homoclinic class $H_f(p)$ is $C^1$-stably expansive if and only if $H_f(p)$ is $C^1$-persistently expansive.
Citation: Keonhee Lee, Manseob Lee. Hyperbolicity of $C^1$-stably expansive homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1133-1145. doi: 10.3934/dcds.2010.27.1133
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