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

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.