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Realization of a Choquet simplex as the set of invariant probability measures of a tiling system. (English) Zbl 1138.37009

The main result of this paper shows that for every Choquet simplex \(K\) and every \(d>1\), there exists a \(\mathbb{Z}^d\)-Toeplitz system whose set of invariant probability measures is affine homeomorphic to \(K\). The author makes use of a result of T. Downarowicz [in: S. Kolyada (ed.) et. al., Algebraic and topological dynamics, Proceedings of the conference, Bonn, Germany 2004, Contemp. Math. 385, 7–37 (2005; Zbl 1096.37002)] which says there is a dyadic \(\mathbb{Z}\)-Toeplitz flow whose set of invariant probability measures is affine homeomorphic to \(K\). The clopen Kakutani-Rohlin partition on this Toeplitz flow, as shown in [R. H. Herman, I. F. Putnam, and C. F. Skau, Int. J. Math. 3, No. 6, 827–864 (1992; Zbl 0786.46053)] and [I. F. Putnam, Pac. J. Math. 136, No. 2, 329–353 (1989; Zbl 0631.46068)], is modified to produce the wanted \(\mathbb{Z}^d\)-Toeplitz system.
In the final section, the author relates Toeplitz systems to tiling systems in order to conclude that every Choquet simplex can be realized as the set of invariant probability measures of a tiling system.

MSC:

37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37A35 Entropy and other invariants, isomorphism, classification in ergodic theory
54H20 Topological dynamics (MSC2010)
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