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

1976; American Mathematical Society; Volume: 60; Issue: 1 Linguagem: Inglês

10.1090/s0002-9939-1976-0423423-x

1088-6826

John E. Franke, James F. Selgrade,

Chaos control and synchronization

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.