Abstract
In this paper we show that, for every Choquet simplex $K$ and for every $d>1$, there exists a ${\mathbb Z}^d$-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 $(\Omega_T,{\mathbb R}^d)$.
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