Multiple tilings associated to the symmetric β-expansions

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Date(s) - 24/11/2015
11 h 00 min - 12 h 00 min


It is a well-known fact that when β is a d-Bonacci number, the Rauzy fractals arising from the greedy (Rényi) β-transformation tile the contracting hyperplane. Recently, it was shown that all Pisot units satisfy this (the so-called Pisot conjecture for β-numeration; proved by M. Barge). However, the Rauzy fractals arising from the symmetric Tribonacci transformation form a double tiling, i.e., almost every point of the hyperplane lies in exactly 2 tiles. This means that the Pisot conjecture for beta-numeration is not true for symmetric transformations.

We show that for the d-Bonacci numbers, the degree of the multiple tiling (MT) is d-1. We can also determine which tiles form each layer of the multiple tiling. This relates to toral automorphisms; in the case when the MT is a tiling, the natural extension of the transformation is a toral automorphism.

For general Pisot unit bases 1<β<2, we show show how to compute the degree of the MT from the degree of the MT of a different transformation. We also show a necessary condition for the MT to be a tiling.
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