On maximal transitive sets of generic diffeomorphisms
type: Item
instance of: scholarly article
title: On maximal transitive sets of generic diffeomorphisms
zbMATH Open document ID: 1032.37011
author: C. Bonatti
author: L. J. Díaz
published in: Publications Mathématiques
publication date: 2003-09-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: Locally generic \(C^1\)-diffeomorphisms \(f\) of \(3\)-manifolds with maximal transitive Cantor sets without periodic points are constructed. The locally generic diffeomorphisms constructed also exhibit strongly pathological features generalizing the Newhouse phenomenon (coexistence of infinitely many sinks or sources). These features are the following: coexistence of infinitely many nontrivial (hyperbolic and nonhyperbolic) attractors and repellors, and coexistence of infinitely many nontrivial (nonhyperbolic) homoclinic classes. These phenomena are proved to be associated to the existence of a homoclinic class \(H(P,f)\) (\(P\) is a periodic point of \(f\)) with the following properties: -- in a \(C^1\)-robust way, the homoclinic class \(H(P,f)\) does not admit any dominated splitting, -- there is a periodic point \(P'\) homoclinically related to \(P\) such that the Jacobians of \(P'\) and \(P\) are greater than and less than one, respectively. The paper's main result is the following: Theorem. Given any compact \(3\)-manifold \(M\) there is a locally residual subset \(\mathcal F(M)\) of \(\text{Diff}^1(M)\) of diffeomorphisms \(f\) having maximal transitive Cantor sets \(\Lambda _f\) without periodic points. In particular, the maximal transitive Cantor sets \(\Lambda _f\) are not homoclinic classes.
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37E30
zbMATH DE Number: 1986171
zbMATH Keywords: locally generic diffeomorphisms
zbMATH Keywords: wild homoclinic classes
zbMATH Keywords: maximal transitive invariant sets
zbMATH Keywords: robustly transitive sets
zbMATH Keywords: Newhouse phenomenon
reviewed by: Eugene Ershov
MaRDI profile type: Publication
OpenAlex ID: W2068316139
DOI: 10.1007/S10240-003-0008-0
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description: scientific article; zbMATH DE number 1986171