Equicontinuity of Maps on a Dendrite with Finite Branch Points

Let (T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote by omega(x, f) and P(f) the omega-limit set of x under f and the set of periodic points of f, respectively. Write Omega(x, f) = {y| there exist a sequence of points x(k) is an element of T and a sequence of positive integers n(1) < n(2) < ... such that lim(k ->infinity) x(k) = x and lim(k ->infinity) f(nk)(x(k)) = y}. In this paper, we show that the following statements are equivalent: (1) f is equicontinuous. (2) omega(x, f) = Omega(x, f) for any x is an element of T. (3) boolean AND(alpha)(n-1) f(n) (T) = P(f), and omega(x, f) is a periodic orbit for every x is an element of T and map h: x -> omega(x, f) (x is an element of T) is continuous. (4) Omega(x, f) is a periodic orbit for any x is an element of T.