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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
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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

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References

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