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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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Keywords: Flows, abstract <IMG WIDTH="19" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\omega$">-limit set, chain recurrent, invariant set, hyperbolic, Axiom A, basic set
Article copyright: © Copyright 1976 American Mathematical Society