Beta-expansions for cubic pisot numbers

Abstract

Real numbers can be represented in an arbitrary base β > 1 using the transformation Tβ: x → βx (mod 1) of the unit interval; any real number x ∈ [0, 1] is then expanded into dβ(x) = (xi)i≥1 where (formula presented). The closure of the set of the expansions of real numbers of [0, 1] is a subshift of (formula presented), called the beta-shift. This dynamical system is characterized by the beta-expansion of 1; in particular, it is of finite type if and only if dβ(1) is finite; β is then called a simple beta-number. We first compute the beta-expansion of 1 for any cubic Pisot number. Next we show that cubic simple beta-numbers are Pisot numbers.