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

TL;DR

This paper introduces the concept of $ar{d}$-approachability in shift spaces, providing new criteria for entropy density of ergodic measures, and applies these results to classes like hereditary $ extbf{B}$-free shifts and minimal systems.

Contribution

It develops a topological characterization of $ar{d}$-approachable shift spaces and establishes entropy density criteria applicable to broader classes of systems.

Findings

Ergodic measures are entropy-dense in $ar{d}$-approachable shift spaces.

New criterion for entropy density applies to hereditary $ extbf{B}$-free shifts.

Includes shift spaces previously inaccessible to existing techniques.

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 $\bar{d}$-approachable when its canonical sequence of Markov approximations converges to it also in the $\bar{d}$ sense. We give a topological characterisation 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 out of 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.