Isotopy Stable Dynamics Relative to Compact Invariant Sets

Abstract

Let f be an orientation-preserving homeomorphism of a compact orientable manifold. Sufficient conditions are given for the persistence of a collection of periodic points under isotopy of f relative to a compact invariant set A. Two main applications are described. In the first, A is the closure of a single discrete orbit of f, and f has a Smale horseshoe, all of whose periodic orbits persist; in the second, A is a minimal invariant Cantor set obtained as the limit of a sequence of nested periodic orbits, all of which are shown to persist under isotopy relative to A. 1991 Mathematics Subject Classification: 58F20, 58F15.