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2008 International Workshop on Dynamical System and Realted Topics
(2008 / vol.10 / no.2)
Hyperbolicity of $C^1$-stably shadowing homoclinic classes
Khosro Tajbakhsh, Keonhee Lee
Pages. 79-82   



Let $f$ be a diffeomorphism on a closed manifold
$M$, and let $pin M$ be a hyperbolic periodic point of $f$. Denote $H_{f}(p)$
the homoclinic class of $f$ associated to $p$. We say that $H_{f}(p)$ is $C^1$-stably
shadowing if $H_{f}(p)$ is locally maximal (in $Usubseteq M$) and there is
a $C^1$-neighborhood $mathcal{U}(f)$ of $f$ such that for any $gin mathcal{U}(f)$, $g$ has
the shadowing property on $Lambda_{g}$, where $Lambda_g;=cap_{n in }g^n(U)$
and which is called the {em continuation} of $H_{f}(p)= cap_{n in } f^n(U)$.
We show in this paper that $H_{f}(p)$ is $C^1$-stably shadowing if and only if $H_{f}(p)$ is hyperbolic.



homoclinic class, $C^1$-stably shadowing, $C^1$- persistently shadowing, , shadowing, hyperbolic, Axiom A



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