2013/10/04 by Milton Minervino, Minervino, Milton, Wolfgang Steiner +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.1310.1277
arxiv created 2013/10/04 · arxiv updated 2013/10/07
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 = a β+ b, gcd(a,b) = 1.