Minimal dynamics for tree maps
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.
scientific article; zbMATH DE number 5290020