The Full Periodicity Kernel for a Class of Graph Maps

Let X be a graph and let C be a class of X-maps (that is, of continuous maps from X into itself). A map f ∈ C is said to have full periodicity if Per (f) = N (here, Per (f) denotes the set of periods of all periodic points of f and N the set of positive integers). The set K ⊆ N is a full periodicity kernel of C if it satisfies the following two conditions: (i) If f ∈ C and K ⊆ Per (f) then f has full periodicity and (ii) if S ⊂ N is a set such that, for every f ∈ C, S ⊆ Per (f) implies Per (f) = N, then K ⊆ S. In this paper we show the existence and characterize the full periodicity kernel of the class of continuous maps from a graph with zero Euler characteristic to itself having all branching points fixed.