Growth rate for beta-expansions

Let $β>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{β-1}]$ can be represented in the form $x=\sum_{k=1}^\infty ε_kβ^{-k},$ where $ε_k\in\{0,1,...,m-1\}$ for all $k$ (a $β$-expansion of $x$). It is known that a.e. $x\in I_β$ 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 $β<\frac{1+\sqrt5}2$, we show that the set of $β$-expansions grows exponentially for every internal $x$.

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