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

Abstract

o-LIMIT SETS, CHAIN RECURRENT SETS, AND BASIC SETS FOR FLOWS JOHN E. FRANKE AND JAMES F. SELGRADE ABSTRACT. An abstract w-limit set for a flow is an invariant set which is conjugate to the w-limit set of a point. This paper shows that an abstract wlimit set is precisely a connected, chain recurrent set. In fact, an abstract'wlimit set which is a subset of a hyperbolic invariant set is the w-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. An abstract w-limit set for a flow is an invariant set which is conjugate to the w-limit set of a point. This paper shows that an abstract wlimit set is precisely a connected, chain recurrent set. In fact, an abstract'wlimit set which is a subset of a hyperbolic invariant set is the w-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.