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Transactions of the American Mathematical Society

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On the minimum positive entropy for cycles on trees


Authors: Lluís Alsedà, David Juher and Francesc Mañosas
Journal: Trans. Amer. Math. Soc. 369 (2017), 187-221
MSC (2010): Primary 37E15, 37E25
DOI: https://doi.org/10.1090/tran6677
Published electronically: February 24, 2016
MathSciNet review: 3557772
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Abstract: Consider, for any $n\in \mathbb {N}$, the set $\operatorname {Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\operatorname {Irr}_n\subsetneq \operatorname {Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\operatorname {Pos}_n$ and $\operatorname {Irr}_n$. Let $\lambda _n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty )$. We explicitly construct an irreducible $n$-periodic tree pattern $\mathcal {Q}_n$ whose entropy is $\log (\lambda _n)$. For $n=m^k$, where $m$ is a prime, we prove that this entropy is minimum in the set $\operatorname {Pos}_n$. Since the pattern $\mathcal {Q}_n$ is irreducible, $\mathcal {Q}_n$ also minimizes the entropy in the family $\operatorname {Irr}_n$.


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Additional Information

Lluís Alsedà
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
MR Author ID: 212847
Email: alseda@mat.uab.cat

David Juher
Affiliation: Departament d’Informàtica i Matemàtica Aplicada, Universitat de Girona, Lluís Santaló s/n, 17071 Girona, Spain
MR Author ID: 680107
Email: david.juher@udg.edu

Francesc Mañosas
Affiliation: Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain
MR Author ID: 254986
Email: manyosas@mat.uab.cat

Keywords: Tree maps, periodic patterns, topological entropy
Received by editor(s): April 25, 2014
Received by editor(s) in revised form: December 16, 2014
Published electronically: February 24, 2016
Additional Notes: The authors have been partially supported by MINECO grant numbers MTM2008-01486 and MTM2011-26995-C02-01
Article copyright: © Copyright 2016 American Mathematical Society