The ω-limit set of a graph map

Let 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 ∞ f n ( Ω ( f ) ) ; (4) if x ∈ ω ( f ) − P ( f ) ¯ , then x has an infinite orbit.