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

Shaobo Gan; Kazuhiro Sakai; Lan Wen

doi:10.3934/dcds.2010.27.205

Article Contents

2010, Volume 27, Issue 1: 205-216. Doi: 10.3934/dcds.2010.27.205

This issue Previous Article Front propagation for a two-dimensional periodic monostable lattice dynamical system Next Article Upper bounds for the number of limit cycles of some planar polynomial differential systems

$C^1$ -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: March 2010

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 37B20, 37C29, 37C50, 37D20, 37D30.

Citation:

Article Metrics

HTML views() PDF downloads(95) Cited by(0)

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

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

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

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content