# American Institue of Mathematical Sciences

2010, 27(1): 205-216. doi: 10.3934/dcds.2010.27.205

## $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings

 1 LMAM, School of Mathematical Sciences, Peking University, Beijing 100871 2 Department of Mathematics, Utsunomiya University, Utsunomiya 321-8505 3 School of Mathematic Sciences, Peking University, Beijing, 100871

Received  June 2009 Revised  November 2009 Published  February 2010

Let $f$ be a diffeomorphism of a closed $n$-dimensional $C^\infty$ manifold, and $p$ be a hyperbolic saddle periodic point of $f$. In this paper, we introduce the notion of $C^1$-stably weakly shadowing for a closed $f$-invariant set, and prove that for the homoclinic class $H_f(p)$ of $p$, if $f_{|H_f(p)}$ is $C^1$-stably weakly shadowing, then $H_f(p)$ admits a dominated splitting. Especially, on a 3-dimensional manifold, the splitting on $H_f(p)$ is partially hyperbolic, and if in addition, $f$ is far from homoclinic tangency, then $H_f(p)$ is strongly partially hyperbolic.
Citation: Shaobo Gan, Kazuhiro Sakai, Lan Wen. $C^1$ -stably weakly shadowing homoclinic classes admit dominated splittings. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 205-216. doi: 10.3934/dcds.2010.27.205
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