Multiple points of tilings associated with Pisot numeration systems

摘要

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 β are cubic and not totally real. Each such tiling gives a numeration system on the complex plane; we can express each complex number z in the following form: Z = c{sub}kα{sup}(-k) + c{sub}(k-1) α{sup}(-k+1) + … + c{sub}1α{sup}(-1) + c{sub}0 + c{sub}(-1) α{sup}1 + c{sub}(-2)α{sup}2 + …where a is a conjugate of β, and c{sub}(-m)c{sub}(-m+1) ... c{sub}(k-1)c{sub}k is the β-expansion of some real number for any integer m. 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-prqject sets). We also determine the set of complex numbers with eventually periodic representations.