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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
Published electronically: January 6, 2012
MathSciNet review: 2888207
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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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Additional Information

Charlene Kalle
Affiliation: Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA Utrecht, The Netherlands
Address at time of publication: Institute of Mathematics, Leiden University, Postbus 9512, 2300RA, Leiden, The Netherlands

Wolfgang Steiner
Affiliation: LIAFA, CNRS, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France

Received by editor(s): July 31, 2009
Received by editor(s) in revised form: January 26, 2010
Published electronically: January 6, 2012
Additional Notes: The first author was partly supported by the EU FP6 Marie Curie Research Training Network CODY (MRTN 2006 035651).
The second author was supported by the French Agence Nationale de la Recherche, grant ANR–06–JCJC–0073 “DyCoNum”.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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