Isotopy Stable Dynamics Relative to Compact Invariant Sets

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.