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Bowen's entropy-conjugacy conjecture is true up to finite index


Authors: Mike Boyle, Jérôme Buzzi and Kevin McGoff
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
Published electronically: February 6, 2015
MathSciNet review: 3336623
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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\vert _{X'})$ is topologically conjugate to $ (Y',g\vert _{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
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

DOI: https://doi.org/10.1090/S0002-9939-2015-12491-4
Keywords: Symbolic dynamics, subshifts of finite type, topological entropy, dimensional entropy, entropy-conjugacy, topological conjugacy, left ideal class, resolving maps
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
Article copyright: © Copyright 2015 American Mathematical Society

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