On the finite-dimensional marginals of shift-invariant measures

Abstract

AbstractLet Σ be a finite alphabet, Ω=Σℤdequipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐnof ℐ to the finite cubes {−n,…,n}d⊂ℤd, and the polytope of ‘locally invariant’ measures ℐlocn. We are especially interested in the geometry of the convex set ℐn, which turns out to be strikingly different whend=1 and whend≥2 . A major role is played by shifts of finite type which are naturally identified with faces of ℐn, and uniquely ergodic shifts of finite type, whose unique invariant measure gives rise to extreme points of ℐn, although in dimensiond≥2 there are also extreme points which arise in other ways. We show that ℐn=ℐlocnwhend=1 , but in higher dimensions they differ fornlarge enough. We also show that while in dimension one ℐnare polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐnfor all large enoughn.