Multiple points of tilings associated with Pisot numeration systems

This paper deals with a kind of aperiodic tilings associated with Pisot numeration systems, originally due to W.P. Thurston, in the formulation of S. Akiyama. We treat tilings whose generating Pisot units ββ are cubic and not totally real. Each such tiling gives a numeration system on the complex plane; we can express each complex number zz in the following form:z=ckα-k+ck-1α-k+1+⋯+c1α-1+c0+c-1α1+c-2α2+⋯,z=ckα-k+ck-1α-k+1+⋯+c1α-1+c0+c-1α1+c-2α2+⋯,where αα is a conjugate of ββ, and c-mc-m+1⋯ck-1ckc-mc-m+1⋯ck-1ck is the ββ-expansion of some real number for any integer mm. We determine the set of complex numbers which have three or more representations. This is equivalent to determining the triple points of the tiling, which is shown to be a collection of model sets (or cut-and-project sets). We also determine the set of complex numbers with eventually periodic representations.