Nonsingular vector fields in \({\mathcal G}^1(M^3)\) satisfy Axiom A and no cycle: A new proof of Liao's theorem
type: Item
instance of: scholarly article
title: Nonsingular vector fields in \({\mathcal G}^1(M^3)\) satisfy Axiom A and no cycle: A new proof of Liao's theorem
zbMATH Open document ID: 0965.37030
author: Hiroyoshi Toyoshiba
published in: Hokkaido Mathematical Journal
publication date: 2000-11-22T00:00:00Z
review text: Let \(M\) be a compact 3-dimensional smooth manifold without boundary. Let \({\mathcal G}^1(M)\) be the set of \(C^1\) vectorfields on \(M\) which have a neighborhood \({\mathcal U}\) such that any \(Y \in {\mathcal U}\) has all its periodic orbits and singularities hyperbolic. The author proves that if \(X \in {\mathcal U}\) has no singularities, then it satisfies Axiom A and no-cycle property. The main tool in the proof was Mañé's Ergodic Closing Lemma. The author noted that the same result was proved in [\textit{S. T. Liao}, Obstruction sets. II, Acta Sci. Natur. Univ. Pekin. 2, 1-36 (1981)], by using unfamiliar techniques.
Mathematics Subject Classification ID: 37D20
zbMATH DE Number: 1448528
zbMATH Keywords: axiom A
zbMATH Keywords: hyperbolic orbits
reviewed by: Jerzy Ombach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.14492/hokmj/1350912956
OpenAlex ID: W2054287899
Wikidata QID: Q114007357
DOI: 10.14492/HOKMJ/1350912956
description: scientific article; zbMATH DE number 1448528