Minimal dynamics for tree maps
type: Item
instance of: scholarly article
title: Minimal dynamics for tree maps
zbMATH Open document ID: 1146.37024
author: Lluís Alsedà
author: David Juher
author: Pere Mumbrú
published in: Discrete and Continuous Dynamical Systems
publication date: 2008-06-18T00:00:00Z
review text: The authors of the present paper aim at studying the forcing problem for tree maps. They give a positive answer to the period forcing problem by showing that, given a tree pattern \({\mathcal P}\) of the tree \(T\), there is a period-preserving injection from the set of periodic points of a minimal representative \(f: T\rightarrow T\) of \(\mathcal P\) into that of any other representative of \(\mathcal P\). This makes them to complete the characterization of the set of periods for tree maps obtained in [\textit{Ll. Alsedà, D. Juher} and \textit{P. Mumbrú}, Ergodic Theory Dyn. Syst. 25, No. 5, 1373-1400 (2005; Zbl 1077.37031)]. As a corollary of this important result, it is shown that the set of periods of a minimal representative \(f\) is contained in the set of periods of any other representative. The method and the tools used in the proof of the main theorem come from the theory of patterns and minimal dynamics for graph maps developed in [\textit{Ll. Alsedà, F. Gautero, J. Guaschi, J. Los, F. Mañosas} and \textit{P. Mumbrú}, Proc. Lond. Math. Soc., III. Ser. 91, No. 2, 414--442 (2005; Zbl 1082.37041)]. Though the proof of the main theorem is very skillful, the detailed introduction of the notions and the well organization of the materials make the paper more readable to readers.
Mathematics Subject Classification ID: 37E25
Mathematics Subject Classification ID: 37E15
zbMATH DE Number: 5290020
zbMATH Keywords: forcing problem
zbMATH Keywords: periodic points
zbMATH Keywords: periods
zbMATH Keywords: minimal representative
reviewed by: Enhui Shi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5290020