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On ${\bar d}$-approachability, entropy density and $\mathscr {B}$-free shifts

Part of: Topological dynamics Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  15 February 2022

JAKUB KONIECZNY Open the ORCID record for JAKUB KONIECZNY [Opens in a new window] ,
MICHAL KUPSA Open the ORCID record for MICHAL KUPSA [Opens in a new window]  and
DOMINIK KWIETNIAK Open the ORCID record for DOMINIK KWIETNIAK [Opens in a new window]
JAKUB KONIECZNY*
Affiliation:
Camille Jordan Institute, Claude Bernard University Lyon 1, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
MICHAL KUPSA
Affiliation:
The Czech Academy of Sciences, Institute of Information Theory and Automation, CZ-18208 Prague 8, Czech Republic (e-mail: kupsa@utia.cas.cz)
DOMINIK KWIETNIAK
Affiliation:
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Łojasiewicza 6, 30-348 Kraków, Poland (e-mail: dominik.kwietniak@uj.edu.pl)
*
e-mail: jakub.konieczny@gmail.com
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Abstract

We study approximation schemes for shift spaces over a finite alphabet using (pseudo)metrics connected to Ornstein’s ${\bar d}$ metric. This leads to a class of shift spaces we call ${\bar d}$ -approachable. A shift space is ${\bar d}$ -approachable when its canonical sequence of Markov approximations converges to it also in the ${\bar d}$ sense. We give a topological characterization of chain-mixing ${\bar d}$ -approachable shift spaces. As an application we provide a new criterion for entropy density of ergodic measures. Entropy density of a shift space means that every invariant measure $\mu $ of such a shift space is the weak $^*$ limit of a sequence $\mu _n$ of ergodic measures with the corresponding sequence of entropies $h(\mu _n)$ converging to $h(\mu )$ . We prove ergodic measures are entropy-dense for every shift space that can be approximated in the ${\bar d}$ pseudometric by a sequence of transitive sofic shifts. This criterion can be applied to many examples that were beyond the reach of previously known techniques including hereditary $\mathscr {B}$ -free shifts and some minimal or proximal systems. The class of symbolic dynamical systems covered by our results includes also shift spaces where entropy density was established previously using the (almost) specification property.

Keywords

specification propertytopological entropyshift spacePoulsen simplexBesicovitch pseudometric

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.)
Secondary: 37A35: Entropy and other invariants, isomorphism, classification 37B10: Symbolic dynamics 37B40: Topological entropy 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 3 , March 2023 , pp. 943 - 970
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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