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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Bowen’s entropy-conjugacy conjecture is true up to finite index
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by Mike Boyle, Jérôme Buzzi and Kevin McGoff PDF
Proc. Amer. Math. Soc. 143 (2015), 2991-2997 Request permission

Abstract:

For a topological dynamical system $(X,f)$, consisting of a continuous map $f : X \to X$, and a (not necessarily compact) set $Z \subset X$, Bowen (1973), defined a dimension-like version of entropy, $h_X(f,Z)$. In the same work, he introduced a notion of entropy-conjugacy for pairs of invertible compact systems: the systems $(X,f)$ and $(Y,g)$ are entropy-conjugate if there exist invariant Borel sets $X’ \subset X$ and $Y’ \subset Y$ such that $h_X(f,X\setminus X’) < h_X(f,X)$, $h_Y(g,Y \setminus Y’) < h_Y(g,Y)$, and $(X’,f|_{X’})$ is topologically conjugate to $(Y’,g|_{Y’})$. Bowen conjectured that two mixing shifts of finite type are entropy-conjugate if they have the same entropy. We prove that two mixing shifts of finite type with equal entropy and left ideal class are entropy-conjugate. Consequently, in every entropy class Bowen’s conjecture is true up to finite index.
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Additional Information
  • Mike Boyle
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 207061
  • ORCID: 0000-0003-0050-0542
  • Email: mmb@math.umd.edu
  • Jérôme Buzzi
  • Affiliation: Laboratoire de Mathématiques d’Orsay - Université Paris-Sud, 91400 Orsay, France
  • Email: jerome.buzzi@math.u-psud.fr
  • Kevin McGoff
  • Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708
  • Email: mcgoff@math.duke.edu
  • Received by editor(s): October 11, 2013
  • Received by editor(s) in revised form: February 19, 2014
  • Published electronically: February 6, 2015
  • Communicated by: Yingfei Yi
  • © Copyright 2015 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 2991-2997
  • MSC (2010): Primary 37A35; Secondary 37B10, 37C45
  • DOI: https://doi.org/10.1090/S0002-9939-2015-12491-4
  • MathSciNet review: 3336623