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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the minimum positive entropy for cycles on trees
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by Lluís Alsedà, David Juher and Francesc Mañosas PDF
Trans. Amer. Math. Soc. 369 (2017), 187-221 Request permission

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
  • 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
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 187-221
  • MSC (2010): Primary 37E15, 37E25
  • DOI: https://doi.org/10.1090/tran6677
  • MathSciNet review: 3557772