Let $\beta>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{ m-1}{ \beta-1}]$ can be represented in the form \[ x=\sum_{ k=1}^\infty \epsilon_k\beta^{ -k}, \] where $\epsilon_k\in\{ 0,1,...,m-1\}$ for all $k$ (a $\beta$-expansion of $x$). It is known that a.e. $x\in I_\beta$ has a continuum of distinct $\beta$-expansions. In this paper we prove that if $\beta$ 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 $\beta$. When $\beta<\frac{ 1+\sqrt5}2$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.
Categories: math.NT math.DS