Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-02-02T02:22:36.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Bistable vector fields are axiom A

Published online by Cambridge University Press:  17 April 2009

Mike Hurley
Mike Hurley
Affiliation:
Department of MathematicsCase Western Reserve UniversityCleveland OH 44106-7058, United States of America
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Recently L. Wen showed that if a C1 vector field (on a smooth compact manifold without boundary) is both structurally stable and topologically stable then it will satisfy Axiom A. The purpose of this note is to indicate how results from an earlier paper can be used to simplify somewhat Wen's argument.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 51 , Issue 1 , February 1995 , pp. 83 - 86
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Abraham, R. and Marsden, J.E., Foundations of mechanics (Benjamin-Cummings, Reading MA, 1978).Google Scholar
[2]Chi, D.P., Choi, S.K. and Park, J.S., ‘Structurally stable flows’, Bull. Austral. Math. Soc. 45 (1992), 79–90.CrossRefGoogle Scholar
[3]Conley, C., Isolated invariant sets and the Morse index, C.B.M.S. publication 38 (American Mathematical Society, Providence, RI, 1978).CrossRefGoogle Scholar
[4]Hurley, M., ‘Consequences of topological stability’, J. Differential Equations 54 (1984), 60–72.CrossRefGoogle Scholar
[5]Hurley, M., ‘Combined structural and topological stability are equivalent to Axiom A and the strong transversality condition’, Ergodic Theory Dynamical Systems 4 (1984), 81–88.CrossRefGoogle Scholar
[6]Hurley, M., ‘Fixed points of topologically stable flows’, Trans. Amer. Math. Soc. 294 (1986), 625–633.CrossRefGoogle Scholar
[7]Liao, S., ‘Obstruction sets (II)’, Acta Sci. Natur. Univ. Pekinensis 2 (1981), 1–36.Google Scholar
[8]Mañé, R., ‘An ergodic closing lemma’, Ann. of Math. 116 (1982), 503–540.CrossRefGoogle Scholar
[9]Mañé, R., ‘A proof of the C1 stability conjecture’, Publ. I.H.E.S. 66 (1987), 161–210.Google Scholar
[10]Markus, L., Lectures in differentiable dynamics, C.B.M.S. publication 3 (American Mathematical Society, Providence, RI, 1980).Google Scholar
[11]Palis, J. Jr and Melo, W. de, Geometric theory of dynamical systems (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[12]Wen, L., ‘Combined two stabilities imply Axiom A for vector fields’, Bull. Austral. Math. Soc. 48 (1993), 23–30.CrossRefGoogle Scholar