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

We study approximation schemes for shift spaces over a finite alphabet using (pseudo)metrics connected to Ornstein’s d̄ metric. This leads to a class of shift spaces we call d̄-approachable. A shift space d̄-approachable when its canonical sequence of Markov approximations converges to it also in the d̄ sense. We give a topological characterisation of chain mixing d̄-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 μ of such a shift space is the weak limit of a sequence μn of ergodic measures with the corresponding sequence of entropies h(μn) converging to h(μ). We prove ergodic measures are entropy-dense for every shift space that can be approximated in the d̄ 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 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. We study approximation schemes for shift spaces over a finite alphabet A (subshifts of the full shift A ∞ over A ). For every shift space X there exists a canonically defined sequence of shifts of finite type (topological Markov approximations) converging to X in a natural topology on the space of all subshifts of A . This fact, however, is of little practical use, because dynamical properties usually do not convey from a sequence of shift spaces to its limit. Here, we consider another, stronger topology on the powerset of A ∞ which is induced by the d̄-pseudometric on A ∞ or one of its relatives. The pseudometric d̄ is given for x = (xj)j=0, y = (yj) ∞ j=0 ∈ A ∞ by d̄(x, y) = lim sup n→∞ 1 n |{0 ≤ j < n : xj 6= yj}|. It is an analog of Ornstein’s metric d̄M on A -valued stationary stochastic processes. Since d̄ is bounded, it induces a Hausdorff pseudometric d̄ on the space of all nonempty subsets of A . We examine the existence of an approximating sequence in the d̄ sense and study its consequences. In particular, we study shift spaces we call d̄-approachable, which are d̄ -limits of their own topological Markov approximations. We provide a topological characterisation of chain mixing d̄-approachable shift spaces using the d̄-shadowing property. This can be considered as an analogue for Friedman and Ornstein’s characterisation of Bernoulli processes as totally ergodic d̄M-limits of their own Markov approximations [23]. We prove that many specification properties imply chain mixing and d̄-approachability. It follows that all shift spaces with the almost specification property (this class includes all β-shifts and all mixing sofic shifts, in particular all mixing shifts of finite type) are d̄-approachable. In a forthcoming paper [29], we also construct minimal and Date: August 9, 2021. 2010 Mathematics Subject Classification. 37B05 (primary) 37A35, 37B10, 37B40, 37D20 (secondary).