The ω-limit set of a graph map

AbstractLet G be a graph and f:G→G be continuous. Denote by P(f), P(f)¯, ω(f) and Ω(f) the set of periodic points, the closure of the set of periodic points, ω-limit set and non-wandering set of f, respectively. In this paper we show that: (1) v∈ω(f) if and only if v∈P(f) or there exists an open arc L=(v,w) contained in some edge of G such that every open arc U=(v,c)⊂L contains at least 2 points of some trajectory; (2) v∈ω(f) if and only if every open neighborhood of v contains at least r+1 points of some trajectory, where r is the valence of v; (3) ω(f)=⋂n=0∞fn(Ω(f)); (4) if x∈ω(f)−P(f)¯, then x has an infinite orbit