Realization of a Choquet simplex as the set of invariant probability measures of a tiling system

Abstract

In this paper we show that, for every Choquet simplex K and for every d > 1, there exists a Z(d)-Toeplitz system whose set of invariant probability measures is affine homeomorphic to K. Then, we conclude that K may be realized as the set of invariant probability measures of a tiling system (Omega(T), R-d).