Uniform convergence and transitive subsets (Q764598)

Summary: Let \((X, d)\) be a metric space and a sequence of continuous maps \(f_n : X \rightarrow X\) 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 \(f\).