Tilings for Pisot beta numeration

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) = alpha beta + b, gcd(a, b) = 1. (C) 2014 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.