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September  2008, 22(3): 683-697. doi: 10.3934/dcds.2008.22.683

$C^1$-stable shadowing diffeomorphisms

Keonhee Lee 1, , Kazumine Moriyasu 2, and  Kazuhiro Sakai 3,

1. 

Department of Mathematics, Chungnum National University, Daejeon 305-764, South Korea
2. 

Department of Mathematics, Tokushima University, Tokushima 770-8502, Japan
3. 

Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505

Received  July 2007 Revised  March 2008 Published  August 2008

Let $f$ be a diffeomorphism of a closed $C^\infty$ manifold. In this paper, we define the notion of the $C^1$-stable shadowing property for a closed $f$-invariant set, and prove that $(i)$ the chain recurrent set $R(f)$ of $f$ has the $C^1$-stable shadowing property if and only if $f$ satisfies both Axiom A and the no-cycle condition, and $(ii)$ for the chain component $C_f(p)$ of $f$ containing a hyperbolic periodic point $p$, $C_f(p)$ has the $C^1$-stable shadowing property if and only if $C_f(p)$ is the hyperbolic homoclinic class of $p$.
Keywords: hyperbolic set, $C^1$-stable shadowing property, chain component, Axiom A., chain recurrent set, shadowing, pseudo-orbit, homoclinic class.
Mathematics Subject Classification: Primary: 37B20, 37C29, 37C50, 37D20, 37D30; Secondary: 37C7.
Citation: Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683
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