Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows.
type: Item
instance of: scholarly article
title: Homoclinic bifurcations and uniform hyperbolicity for three-dimensional flows.
zbMATH Open document ID: 1045.37006
DOI: 10.1016/S0294-1449(03)00016-7
author: Aubin Arroyo
author: F. Rodriguez Hertz
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2003-09-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78598
review text: Let \(M\) be a compact three-dimensional manifold (without boundary), and let \(X\) be a \(C^{1}\)-vector field on \(M\). It is proved that \(X\) can be approximated in the \(C^{1}\)-topology by a \(C^{1}\)-vector field~\(Y\) on~\(M\) showing one of the following properties: (1) \(Y\) is uniformly hyperbolic and satisfies the no-cycles condition; (2) \(Y\) has a homoclinic tangency; (3) \(Y\) has a singular cycle. This solves a conjecture of J.~Palis in this special case. For diffeomorphisms of a surface this conjecture has been solved by \textit{E. R. Pujals} and \textit{M. Sambarino} [Ann. Math. (2) 151, 961--1023 (2000; Zbl 0959.37040)]. Although the proofs are technical the paper is well written. It is well organized and the exposition is clear.
reviewed by: Peter Raith
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37D20
Mathematics Subject Classification ID: 37D30
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37G20
Mathematics Subject Classification ID: 37E30
zbMATH DE Number: 1975935
zbMATH Keywords: \(C^1\)-vector field
zbMATH Keywords: three-dimensional manifold
zbMATH Keywords: \(C^1\)-topology
zbMATH Keywords: uniformly hyperbolic
zbMATH Keywords: homoclinic tangency
zbMATH Keywords: singular cycle
zbMATH Keywords: Poincaré flow
zbMATH Keywords: dominated splitting
MaRDI profile type: Publication
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OpenAlex ID: W1985639436
description: scientific article; zbMATH DE number 1975935