SUBSTITUTIONS. ATOMIC SURFACES, AND PERIODIC BETA EXPANSIONS

We study the pure periodicity of β-expansions where β is a Pisot number satisfing the following two conditions: the β-expansion of 1 is equal to k 1 k 2...k d−11, k i ≥ 0, and the minimal polynomial of β is given by x d − k 1 x d−1 − ... − k d−l x − 1. From the substitution associated with the Pisot number β, a domain with a fractal boundary, called atomic surface, is constructed. The essential point of the proof is to define a natural extension of the β-transformation on a d-dimensional product space which consists of the unit interval and the atomic surface.