On the finite-dimensional marginals of shift-invariant measures

Let $\Sigma$ be a finite alphabet, $\Omega=\Sigma^{\mathbb{Z}^{d}}$ equipped with the shift action, and $\mathcal{I}$ the simplex of shift-invariant measures on $\Omega$. We study the relation between the restriction $\mathcal{I}_n$ of $\mathcal{I}$ to the finite cubes $\{-n,...,n\}^d\subset\mathbb{Z}^d$, and the polytope of "locally invariant" measures $\mathcal{I}_n^{loc}$. We are especially interested in the geometry of the convex set $\mathcal{I}_n$ which turns out to be strikingly different when $d=1$ and when $d\geq 2$. A major role is played by shifts of finite type which are naturally identified with faces of $\mathcal{I}_n$, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of $\mathcal{I}_n$, although in dimension $d\geq 2$ there are also extreme points which arise in other ways. We show that $\mathcal{I}_n=\mathcal{I}_n^{loc}$ when $d=1$, but in higher dimension they differ for $n$ large enough. We also show that while in dimension one $\mathcal{I}_n$ are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of $\mathcal{I}_n$ for all large enough $n$.