  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.
