Graph maps whose periodic points form a closed set

A continuous map f from a graph G to itself is called a graph map. Denote by P ( f ) , R ( f ) , ω ( f ) , Ω ( f ) and CR ( f ) the sets of periodic points, recurrent points, ω-limit points, non-wandering points and chain recurrent points of f respectively. It is well known that P ( f ) ⊂ R ( f ) ⊂ ω ( f ) ⊂ Ω ( f ) ⊂ CR ( f ) . Block and Franke (1983) [5] proved that if f : I → I is an interval map and P ( f ) is a closed set, then CR ( f ) = P ( f ) . In this paper we show that if f : G → G is a graph map and P ( f ) is a closed set, then ω ( f ) = R ( f ) . We also give an example to show that, for general graph maps f with P ( f ) being a closed set, the conclusion ω ( f ) = R ( f ) cannot be strengthened to Ω ( f ) = R ( f ) or ω ( f ) = P ( f ) .