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1 Montan Universität Leoben 2 LIAFA - Laboratoire d-informatique Algorithmique : Fondements et Applications

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 $\beta^2 = a \beta + b$, $\gcda,b = 1$.





Author: Milton Minervino - Wolfgang Steiner -

Source: https://hal.archives-ouvertes.fr/



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