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.