Bifurcation and chain recurrence

Abstract

Abstract We show that there is a residual subset of the set of C 1 diffeomorphisms on any compact manifold at which the map is continuous. As this number is apt to be infinite, we prove a localized version, which allows one to conclude that if f is in this residual set and X is an isolated chain component for f , then (i) there is a neighbourhood U of X which isolates it from the rest of the chain recurrent set of f , and (ii) all g sufficiently C 1 close to f have precisely one chain component in U , and these chain components approach X as g approaches f . (ii) is interpreted as a generic non-bifurcation result for this type of invariant set.