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ISSN 1088-6850(e) ISSN 0002-9947(p)

Beta-expansions, natural extensions and multiple tilings associated with Pisot units

Authors: Charlene Kalle and Wolfgang Steiner
Journal: Trans. Amer. Math. Soc. 364 (2012), 2281-2318
MSC (2010): Primary 11A63, 11R06, 28A80, 28D05, 37B10, 52C22, 52C23
Posted: January 6, 2012
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Abstract: From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $\beta$ and the greedy $\beta$ -transformation. In this paper, we consider different transformations generating expansions in base $\beta$ , including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $\beta$ -expansions. Remarkably, the symmetric $\beta$ -transformation does not satisfy this condition when $\beta$ is the smallest Pisot number or the Tribonacci number. This means that the Pisot conjecture on tilings cannot be extended to the symmetric $\beta$ -transformation.

Closely related to these (multiple) tilings are natural extensions of the transformations, which have many nice properties: they are invariant under the Lebesgue measure; under certain conditions, they provide Markov partitions of the torus; they characterize the numbers with purely periodic expansion, and they allow determining any digit in an expansion without knowing the other digits.

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