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Mathematics > Dynamical Systems
Title: On $\bar{d}$-approachability, entropy density and $\mathscr{B}$-free shifts
(Submitted on 29 Jul 2021 (v1), last revised 4 Jan 2022 (this version, v3))
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.
Submission history
From: Jakub Konieczny [view email][v1] Thu, 29 Jul 2021 20:12:28 GMT (34kb)
[v2] Thu, 5 Aug 2021 23:36:25 GMT (34kb)
[v3] Tue, 4 Jan 2022 11:34:50 GMT (34kb)
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