Uniform Convergence and Transitive Subsets

Let (X, d) be a metric space and a sequence of continuous maps f n: X → X that converges uniformly to a map f. We investigate the transitive subsets of f n whether they can be inherited by f or not. We give sufficient conditions such that the limit map f has a transitive subset. In particular, we show the transitive subsets of f n that can be inherited by f if f n converges uniformly strongly to f. © 2012 Lei Liu et al.