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2008 International Workshop on Dynamical System and Realted Topics
(2008 / vol.10 / no.2)
$C^1$-stably weak shadowing chain components are partially hyperbolic
Kazuhiro Sakai
Pages. 71--78   



Let $f$ be a diffeomorphism of a closed $C^infty$ three-dimensional manifold.
In this paper, we introduce the notion of $C^1$-stably weak shadowing for a closed $f$-invariant set, and prove that $C^1$-generically, for an aperiodic chain component $C_f$ of $f$ isolated in the chain recurrent set, if $f_{|C_f}$ is $C^1$-stably weak shadowing, then there are a $C^1$-neighborhood ${mathcal U}(f)$ of $f$ and an open and dense subset ${mathcal V}$ of ${mathcal U}(f)$ such that for any $g in {mathcal V}$, there is a chain component (of $g$ nearby $C_f$) which is partially hyperbolic.



1. Introduction
2. Statement of Results