Beta-expansions, natural extensions and multiple tilings

Charlene Kalle (1) , Wolfgang Steiner (2)

1 Department of Mathematics
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

Since 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 $\beta$. 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 the $k$-th digit in an expansion without knowing the previous digits.

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

Charlene Kalle, Wolfgang Steiner. Beta-expansions, natural extensions and multiple tilings. 2009. ⟨hal-00404226v1⟩