Equicontinuity of maps on a dendrite with finite branch points

Abstract

Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by ω(x, f) and P(f) the ω-limit set of x under f and the set of periodic points of f, respectively. Write Ω(x, f) = {y| there exist a sequence of points xk ∈ T and a sequence of positive integers n1 < n2 < ··· such that limk→∞xk = x and \({\lim _{k \to \infty }}{f^{{n_k}}}\left( {{x_k}} \right) = y\)}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) ω(x, f) = Ω(x, f) for any x ∈ T. (3) ∩n=1∞fn(T) = P(f), and ω(x, f) is a periodic orbit for every x ∈ T and map h: x → ω(x, f) (x ∈ T) is continuous. (4) Ω(x, f) is a periodic orbit for any x ∈ T.