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Beta-expansions, natural extensions and multiple tilings associated with Pisot units. (English summary) Trans. Amer. Math. Soc. 364 (2012), no. 5, 2281–2318.1088-6850 Let β> 1, A be a finite subset of R and X be the disjoint union of non-empty bounded subsets Xa (a ∈ A) of R. Define T: X → X by T x = βx − a for a ∈ Xa. Then for each x ∈ X one has x = ∑∞ k=1 bk(x)β−k, where bk(x) = a if T k−1 (x) ∈ Xa. This generalizes the well-studied transformation T x = βx + α for fixed β> 1 and 0 ≤ α < 1 (Example 2.10). In particular, for α = 0 one has the classical β-transformation, which gives the greedy β-expansion of x. Following W. Thurston [“Groups, tilings and finite state automata”, Summer 1989 AMS Colloquium Lectures, Res. Rep., Geometry Computing Group, Minneapolis, MN, 1989], the authors consider the case when β is a Pisot unit of degree d> 1 and A ⊂ Z[β], and extend many earlier results to this more general setting. As a representative of one of many results (Theorem 4.10), they construct a multiple aperiodic tiling of a hyperplane of Rd. Reviewed by Christopher Smyt

Year: 2013

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