Tilings for Pisot beta numeration

Abstract

For a (non-unit) Pisot number β, several collections of tiles are associated with β-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the β-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 β-expansions; in particular, we calculate γ(β) for all quadratic β with β2=aβ+b, gcd(a,b)=1.