Tilings for Pisot beta numeration

Milton Minervino (1) , Wolfgang Steiner (2)

1 MUL - Montanuniversität Leoben
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

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 $\beta^2 = a \beta + b$, $\gcd(a,b) = 1$.

Domaines: Théorie des nombres [math.NT]; Systèmes dynamiques [math.DS]

Milton Minervino, Wolfgang Steiner. Tilings for Pisot beta numeration. Indagationes Mathematicae, 2014, 25 (4), pp.745-773. ⟨10.1016/j.indag.2014.04.008⟩. ⟨hal-00869984⟩