Isotopy Stable Dynamics Relative to Compact Invariant Sets

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