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Bifurcation and chain recurrence

Abstract We show that there is a residual subset of the set of C1 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 C1 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.