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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
Email: alseda@mat.uab.cat

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

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

DOI: https://doi.org/10.1090/tran6677
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

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