The aim of this paper is to characterize the formal power series which have purely periodic β-expansions in Pisot or Salem unit base under some condition. Furthermore, we will prove that if β is a quadratic Pisot unit base, then every rational f in the unit disk has a purely periodic β-expansion and discuss their periods.

Abstract. In [6], it is proved that the lengths of periods occurring in the β-expansion of a rational series r noted by P er β (r) depend only on the denominator of the reduced form of r for quadratic Pisot unit series. In this paper, we will show first that every rational r in the unit disk has strictly periodic β-expansion for Pisot or Salem unit basis under some condition. Second, for this basis, if r = P Q is written in reduced form with |P | < |Q|, we will generalize the curious property "P er β (