On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions
type: Item
instance of: scholarly article
title: On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions
zbMATH Open document ID: 1111.37008
author: Enrique R. Pujals
published in: Discrete and Continuous Dynamical Systems
publication date: 2007-02-05T00:00:00Z
review text: Hyperbolic systems constitute an ample and relevant class of dynamical systems (DS), but not a generic one: it is known that there are open sets in the space of DS which are nonhyperbolic. This was first shown by Abraham and Smale and by Shub in the early 70s by exhibiting open sets of diffeomorphisms in three dimensions with hyperbolic periodic points of different indices inside a transitive set. When periodic points of different indices are linked through the intersection (these are necessarily not all transversal) of their stable and unstable manifolds, one speaks of a heterodimensional cycle. Later on, Newhouse showed that hyperbolicity is also non dense within \(C^r\) (with \(r\geq 2\)) diffeomorphisms of compact surfaces. The underlying mechanism here is that of homoclinic tangency, i.e., nontransversal intersection of the stable and unstable manifolds of a periodic point. In the 80s, Palis conjectured that the mechanisms mentioned above are rather common in the complement of hyperbolic systems within DS. His conjectures are: (1) Every \(C^r\) diffeomorphism of a compact manifold \(M\) can be \(C^r\) approximated either by a hyperbolic one, or by one with a heterodimensional cycle, or by one exhibiting a homoclinic tangency. (2) When \(M\) is a two-dimensional compact manifold, every \(C^r\) diffeomorphism of \(M\) can be \(C^r\) approximated either by a hyperbolic one or by one exhibiting a homoclinic tangency. The present paper goes in the direction of proving the Palis conjecture in the \(C^1\) topology for an attracting homoclinic class of a three-dimensional \(C^2\) diffeomorphism. Given a periodic point \(p\), one defines the homoclinic class \(H_p\) associated to \(p\) as the closure of the transversal intersections of the stable and unstable manifolds of \(p\). Given \(H_p\), we say it is attracting if there exists an open set \(U\) such that \(H_p \subset U\) and \(H_p = \cap_{n>0} f^n (U)\). The present paper proves that given a maximally invariant a…
reviewed by: Giuseppe Gaeta
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37G25
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37D30
zbMATH DE Number: 5121856
zbMATH Keywords: nonhyperbolicity
zbMATH Keywords: Palis conjecture
MaRDI profile type: Publication
full work available at URL: dcds.2006.16.179
OpenAlex ID: W2079277084
DOI: 10.3934/DCDS.2006.16.179
description: scientific article; zbMATH DE number 5121856