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Tilings for Pisot beta numeration

Abstract : For a (non-unit) Pisot number $\beta$, several collections of tiles are associated with $\beta$-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the $\beta$-transformation and a Euclidean one made of integral beta-tiles. We show that all these collections (except possibly the periodic translation of the central tile) are tilings if one of them is a tiling or, equivalently, the weak finiteness property (W) holds. We also obtain new results on rational numbers with purely periodic $\beta$-expansions; in particular, we calculate $\gamma(\beta)$ for all quadratic $\beta$ with $\beta^2 = a \beta + b$, $\gcd(a,b) = 1$.
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Contributor : Wolfgang Steiner <>
Submitted on : Friday, October 4, 2013 - 3:52:09 PM
Last modification on : Saturday, March 28, 2020 - 2:14:02 AM
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Milton Minervino, Wolfgang Steiner. Tilings for Pisot beta numeration. Indagationes Mathematicae, Elsevier, 2014, 25 (4), pp.745-773. ⟨10.1016/j.indag.2014.04.008⟩. ⟨hal-00869984⟩

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