Minimal and proximal examples of $\bar{d}$-stable and $\bar{d}$-approachable shift spaces

We study shift spaces over a finite alphabet that can be approximated by mixing shifts of finite type in the sense of (pseudo)metrics connected to Ornstein's $\bar{d}$ metric ($\bar{d}$-approachable shift spaces). The class of $\bar{d}$-approachable shifts can be considered as a topological analog of measure-theoretical Bernoulli systems. The notion of $\bar{d}$-approachability together with a closely connected notion of $\bar{d}$-shadowing were introduced by Konieczny, Kupsa, and Kwietniak [in \emph{Ergodic Theory and Dynamical Systems}, vol. \textbf{43} (2023), issue 3, pp. 943--970]. These notions were developed with the aim to significantly generalize specification properties. Indeed, many popular variants of the specification property, including the classic one and almost/weak specification property ensure $\bar{d}$-approachability and $\bar{d}$-shadowing. Here, we study further properties and connections between $\bar{d}$-shadowing and $\bar{d}$-approachability. We prove that $\bar{d}$-shadowing implies $\bar{d}$-stability (a notion recently introduced by Tim Austin). We show that for surjective shift spaces with the $\bar{d}$-shadowing property the Hausdorff pseudodistance $\bar{d}^H$ between shift spaces induced by $\bar{d}$ is the same as the Hausdorff distance between their simiplices of invariant measures with respect to the Hausdorff distance induced by the Ornstein's metric $\bar{d}$ between measures. We prove that without $\bar{d}$-shadowing this need not to be true (it is known that the former distance always bounds the latter). We provide examples illustrating these results including minimal examples and proximal examples of shift spaces with the $\bar{d}$-shadowing property. The existence of such shift spaces was announced in our earlier paper [op. cit.]. It shows that $\bar{d}$-shadowing indeed generalises the specification property.