# American Institute of Mathematical Sciences

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July  2008, 20(3): 511-541. doi: 10.3934/dcds.2008.20.511

## Minimal dynamics for tree maps

 1 Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain 2 Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain 3 n/a, Spain

Received  December 2006 Revised  September 2007 Published  December 2007

We prove that, given a tree pattern $\mathcal{P}$, the set of periods of a minimal representative $f: T\rightarrow T$ of $\mathcal{P}$ is contained in the set of periods of any other representative. This statement is an immediate corollary of the following stronger result: there is a period-preserving injection from the set of periodic points of $f$ into that of any other representative of $\mathcal{P}$. We prove this result by extending the main theorem of [6] to negative cycles.
Citation: Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511
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