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Self affine tiling and Pisot numeration system

A Pisot number is an algebraic integer greater than 1 whose conjugates other than itself have modulus smaller than 1. We have a particular interest in the case when β > 1 is a Pisot number. Surprisingly, one can find many similar phenomena with binary or decimal expansion. See [1]. We use a term ’Pisot numeration system’ to call this method to represent real numbers in a power series in Pisot number base. In this paper, we will prove fundamental properties of tilings generated by Pisot numeration system. Let Fin(β) be a set of all finite greedy expansion in base β. Consider two properties