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

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- by Mike Boyle, Jérôme Buzzi and Kevin McGoff
- Proc. Amer. Math. Soc.
**143**(2015), 2991-2997 - DOI: https://doi.org/10.1090/S0002-9939-2015-12491-4
- Published electronically: February 6, 2015
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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 $(X’,f|_{X’})$ 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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## Bibliographic 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