Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T19:57:52.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Chain recurrence and attraction in non-compact spaces

Published online by Cambridge University Press:  19 September 2008

Mike Hurley
Mike Hurley
Affiliation:
Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio 44106–7058, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In the study of a dynamical system f: X → X generated by a continuous map f on a compact metric space X, the chain recurrent set is an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ of f. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 11 , Issue 4 , December 1991 , pp. 709 - 729
Copyright
Copyright © Cambridge University Press 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[C]Conley, C.. Isolated Invariant Sets and the Morse Index. CBMS Publication 38. Amer. Math. Soc., Providence, 1978.CrossRefGoogle Scholar
[E]Easton, R.. Isolating blocks and epsilon chains for maps. Physica D 39 (1989), 95–110.CrossRefGoogle Scholar
[F1]Franks, J.. Recurrence and fixed points of surface homeomorphisms, Ergod. Th. & Dynam. Sys. 8* (1988), 99–107.Google Scholar
[F2]Franks, J.. A variation on the Poicaré-Birkhoff theorem. Hamiltonian Dynamical Systems. Vol. 81 of Contemporary Mathematics, Amer. Math. Soc., Providence, 1988, pp. 111–117.CrossRefGoogle Scholar
[G]Garay, B. M.. Uniform persistence and chain recurrence. J. Math. Anal. Appl. 139 (1989), 372–381.CrossRefGoogle Scholar
[H1]Hurley, M.. Attractors: persistance, and density of their basins. Trans. Amer. Math. Soc. 269 (1982), 247–271.CrossRefGoogle Scholar
[H2]Hurley, M.. Bifurcation and chain recurrence. Ergod. Th. & Dynam. Sys. 3 (1983), 231–240.CrossRefGoogle Scholar
[H3]Hurley, M.. Generic properties of attractors in Euclidean spaces. Proc. Berkeley-Ames Conf. on Nonlinear Problems in Control and Fluid Dynamics. Hunt, L., Martin, C., eds., Math. Sci. Press, Brookline Mass., 1984, pp. 283–298.Google Scholar
[H4]Hurley, M.. Attractors in cellular automata. Ergod. Th. & Dynam. Sys. 10 (1990), 131–140.CrossRefGoogle Scholar
[M]McGehee, R.. Some metric properties of attractors with applications to computer simulations of dynamical systems. Preprint. University of Minnesota, 1988.Google Scholar
[N]Norton, D.. A metric approach to the Conley decomposition theorem. PhD Thesis. University of Minnesota, 1989.Google Scholar
[R]Robinson, C.. Stability theorems and hyperbolicity in dynamical systems. Rocky Mm. J. Math. 7 (1977), 425–437.Google Scholar