TY - JOUR

T1 - On the finite-dimensional marginals of shift-invariant measures

AU - Chazottes, J. R.

AU - Gambaudo, J. M.

AU - Hochman, M.

AU - Ugalde, E.

PY - 2012/10

Y1 - 2012/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866041272&partnerID=8YFLogxK

U2 - 10.1017/S0143385711000526

DO - 10.1017/S0143385711000526

M3 - Article

AN - SCOPUS:84866041272

SN - 0143-3857

VL - 32

SP - 1485

EP - 1500

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

IS - 5

ER -