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Beta-expansions, natural extensions and multiple tilings associated with Pisot units. (English) Zbl 1295.11010
The present article deals with generalized $$\beta$$-expansions. In particular, the authors construct natural extensions and show multiple tilings associated with them.
Let $$T: X\to X$$ be defined by $$Tx=\beta x-a$$ for $$x\in X_a$$ and $$a\in A$$, where $$A$$ is a finite subset of $$\mathbb{R}$$. This generalizes several kinds of transformations like the classical greedy $$\beta$$-transformation, linear modulo $$1$$ transformations, minimal weight expansions and symmetric $$\beta$$-transformations.
In the first part they provide several properties of these transformation like admissibility, ordering, continuity. They also characterize when the shift space is sofic.
The second part deals with natural extensions. They start by characterizing those points which are eventually and purely periodic. Then they construct the natural extension and consider its domain. Finally they provide natural extensions for the different examples originating from the first part.
In the third part they consider multiple tilings. The authors provide a sufficient criterion for the presence of a multiple tiling of a hyperplane and of the torus. At the end of this part they construct such a multiple tiling.

##### MSC:
 11A67 Other number representations 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 28A80 Fractals 28D05 Measure-preserving transformations 37B10 Symbolic dynamics 52C23 Quasicrystals and aperiodic tilings in discrete geometry
##### Keywords:
beta-expansion; natural extension; multiple tiling; Pisot unit
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