Tilings for Pisot beta numeration

For a (non-unit) Pisot number ββ, several collections of tiles are associated with ββ-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the ββ-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 ββ-expansions; in particular, we calculate γ(β)γ(β) for all quadratic ββ with β2=aβ+bβ2=aβ+b, gcd(a,b)=1gcd(a,b)=1.