Periodic behavior on trees

Abstract

We characterize the set of periods for tree maps. More precisely, we prove that the set of periods of any tree map $f:T \to T$ is the union of finitely many initial segments of Baldwin's orderings $_p{\geq}$ and a finite set $\mathcal{F}$. The possible values of p and explicit upper bounds for the size of $\mathcal{F}$ are given in terms of the combinatorial properties of the tree T. Conversely, given any set $\mathcal{A}$ which is a union of finitely many initial segments of Baldwin's orderings $_p{\geq}$ with p of the above type and a finite set, we prove that there exists a tree map whose set of periods is $\mathcal{A}$.