Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each x ∈ Iβ := [0, m-1/β-1] can be represented in the form, where εk ∈ {0, 1, ..., m-1} for all k (a β-expansion of x). It is known that a. e. x ∈ Iβ has a continuum of distinct β-expansions. In this paper we prove that if β is a Pisot number, then for a. e. x this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by β. When β <1+√5/2, we show that the set of β-expansions grows exponentially for every internal x. © 2010 Springer-Verlag.