Topological entropy of a graph map

Let G be a graph and f: G . G be a continuous map. Denote by h(f), P(f),AP(f),R(f) and .(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the .-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent: (1) h(f) > 0. (2) There exists an x . G such that .(x, f) . P(f) . Ø and .(x, f) is an infinite set. (3) There exists an x . G such that .(x, f) contains two minimal sets. (4) There exist x, y . G such that .(x, f) . .(y, f) is an uncountable set and .(y, f) . .(x, f) . Ø. (5) There exist an x . G and a closed subset A . .(x, f) with f(A) . A such that .(x, f) . A is an uncountable set. (6) R(f) . AP(f) . Ø. (7) f| P ( f ) is not pointwise equicontinuous.