| Publication type: |
 |
Article in peer-reviewed journal |
|
| Subject: |
|
|
|
| Title: |
|
Beta-expansions, natural extensions and multiple tilings associated with Pisot units |
|
| Author(s): |
|
Charlene Kalle 1, Wolfgang Steiner ( ) 2 |
|
| Laboratory: |
|
|
|
| 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. |
|
| Fulltext language: |
|
English |
|
|
| Journal: |
|
| Transactions of the American Mathematical Society (Trans. Am. Math. Soc.) |
| Publisher |
American Mathematical Society |
| ISSN |
0002-9947 |
|
|
| Audience: |
|
international |
|
| Publication date: |
|
2012-01-06 |
|
| Volume: |
|
364 |
|
| Issue: |
|
5 |
|
| Page, identifiant, ...: |
|
2281-2318 |
|
|
| Classification: |
|
11A63, 11R06, 28A80, 28D05, 37B10, 52C22, 52C23 |
|
|
| ANR Project: |
|
| Project Id |
ANR-06-JCJC-0073 |
| Year |
2006 |
| Project acronyme |
DyCoNum |
| Project title |
Etudes diophantiennes dynamiques et combinatoires de différentes numérations |
| Intitule |
Programme "Jeunes chercheuses et jeunes chercheurs |
| Acronyme |
JCJC |
|
|
Export this paper
