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

Abstract

International audienceFrom 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

Multiple tilings associated to d-Bonacci beta-expansions

“…x ∈ X S \ supp µ S have zero measure and excluding them allows us to use Theorem 4.10 of Kalle and Steiner [KS12] (see Remark 4.12 therein). As ψ is a conjugacy (Lemma 1), we know that supp µ B = ψ(supp µ S ).…”

“…If we drop the "greedy" hypothesis, things are getting more interesting. C. Kalle and W. Steiner [KS12] showed that the symmetric β-expansions for two particular cubic Pisot numbers β induce a double tiling -i.e., a multiple tiling such that almost every point of the tiled space lies in exactly two tiles. More generally, they proved that every "well-behaving" β-transformation with a Pisot unit β induces a multiple tiling.…”

“…Since − h = d − 1 − h , we get that L d−h = −L h , therefore x ∈ L h .Proof of Theorem 1. The collection of tiles T = { R(x) : x ∈ Z[β] ∩ X S } is a multiple tiling by Theorem 4.10 of[KS12]. By Lemma 8, the degree is at least d − 1 since all points of Φ(Z[β]) lie in at least that many tiles.…”

Intermediate $\beta$-shifts of finite type

“…An equivalent condition, in terms of the upper and lower kneading invariants, for the finite type property of greedy and the lazy β-shifts is given in [26] and reads as follows. 1,4,5,15,16]). For (β, α) ∈ ∆, the spaces Ω ± β,α are completely determined by upper and lower kneading invariants of Ω β,α ; indeed, we have…”

Substitutions, Rauzy fractals and tilings

“…In other words, a multiple tiling is a union of tiles (γ,i)∈Γ K i + γ that covers the full space H with possible overlaps in such a way that almost every point belongs to exactly p tiles. This is illustrated in Figure 5.3 for p = 2 with an example obtained in the framework of symmetric beta-expansions taken from (Kalle and Steiner 2009). If p = 1, then the multiple tiling is called a tiling.…”