Beta-Expansions for Cubic Pisot Numbers

Real numbers can be represented in an arbitrary base 脽 1 using the transformation T脽 : x 驴 脽x (mod 1) of the unit interval; any real number x 驴 [0, 1] is then expanded into d脽(x) = (xi)i驴1 where xi = 驴脽T脽i (x)驴.The closure of the set of the expansions of real numbers of [0, 1] is a subshift of {a 驴 N | a N, 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 d脽(1) is finite; 脽 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.