Minimal dynamics for tree maps

Lluís Alsedà; David Juher; Pere Mumbrú

doi:10.3934/dcds.2008.20.511

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2008, Volume 20, Issue 3: 511-541. Doi: 10.3934/dcds.2008.20.511

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Minimal dynamics for tree maps

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

Received: November 30, 2006

Revised: August 31, 2007

Published: August 2008

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Abstract

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.

Keywords:

tree maps,
minimal dynamics.

Mathematics Subject Classification: Primary: 37E25.

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