Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T10:09:05.871Z Has data issue: false hasContentIssue false
  • English
  • Français

The chain recurrent set, attractors, and explosions

Published online by Cambridge University Press:  19 September 2008

Louis Block  and
John E. Franke
Louis Block
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, USA
John E. Franke
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695, USA
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.

Charles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 3 , September 1985 , pp. 321 - 327
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Auslander, J. & Katznelson, Y.. Maps of the circle without periodic points. Israel J. Math. 32, (1979), 375–381.CrossRefGoogle Scholar
[2]Block, L. & Franke, J.. The chain recurrent set for maps of the interval. Proc. Amer. Math. Soc. 87, (1983), 723–727.CrossRefGoogle Scholar
[3]Bowen, R.. ω-Limit sets for Axiom A diffeomorphisms. J. Differential Equations 18, (1975), 333–339.CrossRefGoogle Scholar
[4]Conley, C.. Isolated invariant sets and the Morse index. CBMS Regional Conf. Ser. in Math.,no. 38,Amer. Math. Soc:Providence, R.I.,1976.Google Scholar
[5]Coven, E. M., Madden, J. & Nitecki, Z.. A note on generic properties of continuous maps. In Ergodic Theory and Dynamical Systems II. Birkhauser: Boston, 1982.Google Scholar
[6]Denjoy, A.. Sur les courbes definies par les equations differentielles a la surface du tore. J. de Math Pures et Appliquees 11 (ser 9) (1932), 333–375.Google Scholar
[7]Dowker, Y. & Friedlander, F.. On limit sets in dynamical systems. Proc. London Math. Soc. 4 (1954), 168–176.CrossRefGoogle Scholar
[8]Newhouse, S.. Lectures on Dynamical Systems. In Dynamical Systems, Progress in Math. Vol. 8. Birkhauser, 1978, pp. 1–114.Google Scholar
[9]Nitecki, Z.. Differentiable Dynamics, an Introduction to the Orbit Structure of Diffeomorphisms. M.I.T. Press: Cambridge, Massachusetts, 1971.Google Scholar
[10]Nitecki, Z.. Explosions in completely unstable flows; I: preventing explosions. Trans. Amer. Math. Soc. 245, (1978), 43–61.Google Scholar
[11]Nitecki, Z. and Shub, M.. Filtrations, decompositions, and explosions. Amer. J. Math. 97 (1976), 1029–1047.CrossRefGoogle Scholar
[12]Sawada, K.. On the iterations of diffeomorphisms without C°-Ω-explosions: an example. Proc. Amer. Math. Soc. 79, (1980), 110–112.Google Scholar
[13]Shub, M.. Stabilité-globale des systemes dynamiques. Asterisque 56, Societe Mathematique de France, Paris, 1978.Google Scholar
[14]Shub, M. & Smale, S.. Beyond hyperbolicity. Annals of Math. 96, (1972), 578–591.CrossRefGoogle Scholar
[15]Williams, R. F.. The zeta function of an attractor. In Michigan State meeting on Algebraic Topology. Prindle, Weber & Smith, pp. 155–161.Google Scholar
[16]Williams, R. F.. The structure of attractors. Proc. International Congress Math. 2, (1970), 947–951.Google Scholar