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The topology of attractors

Published online by Cambridge University Press:  14 October 2010

Judy A. Kennedy
Judy A. Kennedy
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, De 19716, USA, (e-mail: [email protected])
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Abstract

We prove for a large class of compact metric spaces, including those manifolds of dimension at least two, Hilbert cube manifolds, and homogeneous Menger manifolds, that ‘most’ self-homeomorphisms (in the sense of residual set of homeomorphisms) have certain properties. Specifically, if F: X → X is one of these homeomorphisms, then F admits

• a dense, open wandering set;

• a nowhere dense chain recurrent set;

• an infinite collection of attractors (and repellers), each of which has nonempty interior and cannot be reduced to a ‘smallest’ attractor (or ‘largest’ repeller); and an uncountable collection of pairwise disjoint quasi-attractors.

We also discuss the topology of the boundaries of attractors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1311 - 1322
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[AHK] Akin, E., Hurley, M. and Kennedy, J.. The generic homeomorphism is complicated, but not chaotic. SubmittedGoogle Scholar
[Be] Bestvina, M.. Characterizing k-dimensional universal Menger continua. Mem. Amer. Math. Soc. (380) 71 (1988), 1.Google Scholar
[B] Bing, R. H.. Concerning hereditarily indecomposable continua. Pacific J. Math. 1 (1951), 43.CrossRefGoogle Scholar
[HI] Hurley, M., Generic homeomorphisms have no smallest attractors. Proc. Amer. Math. Soc. 123 (1995), 1227–1280.CrossRefGoogle Scholar
[H2] Hurley, M., Properties of attractors of generic homeomorphisms. Ergod. Th. & Dynam. Sys. pp 1297–1310 of this issue.CrossRefGoogle Scholar