Abstract 𝜔-limit sets, chain recurrent sets, and basic sets for flows

Franke, John E.; Selgrade, James F.

doi:10.1090/S0002-9939-1976-0423423-X

Abstract $\omega$-limit sets, chain recurrent sets, and basic sets for flows
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by John E. Franke and James F. Selgrade PDF

Proc. Amer. Math. Soc. 60 (1976), 309-316 Request permission

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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