On the hyperbolicity of homoclinic classes
type: Item
instance of: scholarly article
title: On the hyperbolicity of homoclinic classes
zbMATH Open document ID: 1200.37023
author: C. Bonatti
author: Shaobo Gan
author: Da-Wei Yang
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-12-16T00:00:00Z
review text: The authors study the hyperbolicity of a diffeomorphism \(f\) on a compact manifold. More precisely, they seek to investigate the hyperbolicity of an invariant compact set by using the hyperbolicity of the periodic points in this set. In general, the hyperbolicity of periodic orbits is not enough to conclude the hyperbolicity of the set. The main result states that if \(p\) is a hyperbolic periodic point and its homoclinic class \(H(p)\) admits a (homogeneous) dominated splitting \(T_{H(p)}M=E \oplus F\), where \(E\) is contracting and \(\text{dim}(E)=\text{ind}(p)\). If, moreover, all points homoclinically related with \(p\) are uniformly \(F\)-expanding, then \(F\) is uniformly expanding on \(H(p)\). In particular, \(H(p)\) is hyperbolic. The paper contains an introduction to the results and the proofs; the authors also derive and describe some consequences of their main result.
reviewed by: Peter Giesl
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37D25
zbMATH DE Number: 5648551
zbMATH Keywords: homoclinic class
zbMATH Keywords: hyperbolic time
zbMATH Keywords: shadowing lemma
MaRDI profile type: Publication
full work available at URL: dcds.2009.25.1143
OpenAlex ID: W1994920597
DOI: 10.3934/DCDS.2009.25.1143
description: scientific article; zbMATH DE number 5648551