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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Abstract $ \omega $-limit sets, chain recurrent sets, and basic sets for flows


Authors: John E. Franke and James F. Selgrade
Journal: Proc. Amer. Math. Soc. 60 (1976), 309-316
MSC: Primary 58F20; Secondary 58F10
MathSciNet review: 0423423
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Abstract: An abstract $ \omega $-limit set for a flow is an invariant set which is conjugate to the $ \omega $-limit set of a point. This paper shows that an abstract $ \omega $-limit set is precisely a connected, chain recurrent set. In fact, an abstract $ \omega $-limit set which is a subset of a hyperbolic invariant set is the $ \omega $-limit set of a nearby heteroclinic point. This leads to the result that a basic set is a hyperbolic, compact, invariant set which is chain recurrent, connected, and has local product structure.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0423423-X
PII: S 0002-9939(1976)0423423-X
Keywords: Flows, abstract $ \omega $-limit set, chain recurrent, invariant set, hyperbolic, Axiom A, basic set
Article copyright: © Copyright 1976 American Mathematical Society