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2008 International Workshop on Dynamical System and Realted Topics
(2008 / vol.10 / no.2)
Hyperbolicity of $C^1$-stably expansive homoclinic classes
Keonhee Lee, Manseob Lee
Pages. 37-43   



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 $C^1$-robustly expansive and the $H_f(p)$-germ of $f$
is expansive.



1. Introduction



homoclinic class, $C^1$-stably expansive, $C^1$- persistently expansive, germ expansive, shadowing, chain recurrent, chain component, hyperbolic, Axiom A



37B20, 37C29, 37C50, 37D20, 37