Entropy, Periodicity, and Graphs with Zero Euler Characteristic

Let G be a graph which contains exactly one simple closed curve. We prove that a continuous map f: G G has zero topological entropy if and only if there exist at most k [(Edg(G)*End(G)* 3)/2] different odd numbers n 1, . . . ,n k such that Per(f) is contained in ki=1j=0ni2j, where Edg(G) is the number of edges of G and End(G) is the number of end points of G.