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On the finite-dimensional marginals of shift-invariant measures

Published online by Cambridge University Press:  08 November 2011

J.-R. CHAZOTTES ,
J.-M. GAMBAUDO ,
M. HOCHMAN  and
E. UGALDE
J.-R. CHAZOTTES
Affiliation:
CPHT, CNRS-École polytechnique, 91128 Palaiseau Cedex, France (email: jeanrene@cpht.polytechnique.fr)
J.-M. GAMBAUDO
Affiliation:
Laboratoire J. A. Dieudonné, CNRS-Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France (email: gambaudo@unice.fr)
M. HOCHMAN
Affiliation:
Department of Mathematics, Fine Hall, Washington Rd., Princeton, NJ 08544, USA (email: hochman@math.princeton.edu)
E. UGALDE
Affiliation:
Instituto de Física, Universidad Autónoma de San Luis Potosí, Av. Manuel Nava #6, Zona Universitaria, San Luis Potosí, S.L.P., 78290, México (email: ugalde@ifisica.uaslp.mx)
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Abstract

Let Σ be a finite alphabet, Ω=Σℤd equipped with the shift action, and ℐ the simplex of shift-invariant measures on Ω. We study the relation between the restriction ℐn of 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 when d=1 and when d≥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 dimension d≥2 there are also extreme points which arise in other ways. We show that ℐn =ℐlocn when d=1 , but in higher dimensions they differ for n large enough. We also show that while in dimension one ℐn are polytopes with rational extreme points, in higher dimensions every computable convex set occurs as a rational image of a face of ℐn for all large enough n.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1485 - 1500
Copyright
Copyright © Cambridge University Press 2011

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