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However, I do not know of any examples of expansive maps without specification for which the inclusion $\mathcal{V} \subset \mathcal{W}$ is known. Does anybody know of such an example?

Edit: Of course if $f$ is uniquely ergodic then $\mathcal{W}$ contains all potential functions. The most obvious examples of uniquely ergodic systems, irrational circle rotations (or rather, their symbolic counterparts, which are expansive) possess a weak version of the specification property, but I don't know if this weak specification holds for every uniquely ergodic system.

What I'd really like to know is if there is an expansive map that is not uniquely ergodic and does not have the specification property for which $\mathcal{V} \subset \mathcal{W}$. I'd also be interested in knowing whether unique ergodicity implies weak specification.

(By "weak specification" I mean that orbits can be consecutively shadowed with uniformly bounded gaps, as in the usual specification property, but that we do not require the shadowing orbit to be periodic, and we allow the length of the gaps to vary.)

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# Unique equilibrium states for systems without specification

Let $X$ be a compact metric space and let $f\colon X\to X$ be a continuous expansive map. Let $\mathcal{V}$ denote the space of Hölder continuous potential functions $\phi\colon X\to \mathbb{R}$, and let $\mathcal{W}$ denote the set of potential functions (not necessarily Hölder continuous) that have a unique equilibrium state.

It is well known that if $f$ satisfies the specification property, then every $\phi\in \mathcal{V}$ has a unique equilibrium state, and hence $\mathcal{V} \subset \mathcal{W}$ (Rufus Bowen, Some systems with unique equilibrium states, Math. Systems Theory 8 (1974/75), no. 3, 193–202).

There are many systems without the specification property for which something is known about the set of potentials $\mathcal{W}$. For example, intrinsic ergodicity (existence of a unique measure of maximal entropy), which is equivalent to the statement that $\mathcal{W}$ contains the constant functions, has been studied in a number of recent works (Buzzi-Fisher, Bufetov-Gurevich, Climenhaga-Thompson), and there are stronger results for particular examples, such as $\beta$-shifts, for which $\mathcal{W}$ is known to contain the space of Lipschitz functions (Peter Walters, Equilibrium states for $\beta$-transformations and related transformations, Math. Z. 159 (1978), no. 1, 65–88).

However, I do not know of any examples of maps without specification for which the inclusion $\mathcal{V} \subset \mathcal{W}$ is known. Does anybody know of such an example?

(Note that it should be not too difficult to adapt the answers to this question to obtain a system for which $\mathcal{W}$ contains the constant functions, but does not contain all of $\mathcal{V}$.)