On the finite-dimensional marginals of shift-invariant measures

Let i be a finite alphabet, i=i ^ℤd equipped with the shift action, and A the simplex of shift-invariant measures on i. We study the relation between the restrictionxn of to the finite cubes {nn} ^dǎš,ℤ ^d, and the polytope of locally invariant' measures ^loc n. We are especially interested in the geometry of the convex set n, which turns out to be strikingly different when d=1 and when dǎ2. A major role is played by shifts of finite type which are naturally identified with faces of n, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of n, although in dimension dǎ2 there are also extreme points which arise in other ways. We show that n = ^loc n when d=1 , but in higher dimensions they differ for n large enough. We also show that while in dimension one n are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of n for all large enough n.