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

Published online by Cambridge University Press:  15 February 2022

Camille Jordan Institute, Claude Bernard University Lyon 1, 43 Boulevard du 11 novembre 1918, 69622Villeurbanne Cedex, France
The Czech Academy of Sciences, Institute of Information Theory and Automation, CZ-18208Prague 8, Czech Republic (e-mail:
Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Łojasiewicza 6, 30-348Kraków, Poland (e-mail:


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.

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© The Author(s), 2022. Published by Cambridge University Press

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