Chaotic extensions of continuous maps on compact manifolds

Abstract

Let X be a compact connected Lipschitz manifold, with or without boundary. We show that for every continuous map f : X → X and every nowhere dense closed subset K of X, there is a topologically transitive continuous map g : X → X having a dense set of periodic points in X such that g|K = f |K. Combined with a deep theorem of Sullivan, our result extends to all compact connected topological manifolds of dimension ≠.