Devaney’s chaos on uniform limit maps

Let (X, d) be a compact metric space and fn : X → X a sequence of continuous maps such that (fn) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all fn mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.

► The transitivity may not been inherited even if the sequence functions mixing. ► The sensitivity may not been inherited even if the iterates of sequence have some uniform convergence. ► Some equivalence conditions for the transitivity and sensitivity for uniform limit function are given. ► A non-transitive sequence may converge uniformly to a transitive map.