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Periodic behavior on trees

Published online by Cambridge University Press:  08 June 2005

Ll. ALSEDÀ
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain (e-mail: alseda@mat.uab.es)
D. JUHER
Affiliation:
Departament d'Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain (e-mail: juher@ima.udg.es)
P. MUMBRÚ
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain (e-mail: mumbru@mat.ub.es)

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 $\mathcal{A}$.

Type
Research Article
Copyright
2005 Cambridge University Press

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