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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
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Abstract

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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

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