Growth rate for beta-expansions

Abstract

Let $\beta >1$ 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 ${\epsilon }_k\in \{0,1,...,m-1\}$ 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 $\beta <\frac{1+\sqrt{5}}{2}$, we show that the set of $\beta$-expansions grows exponentially for every internal $x$.

21 pages, 2 figures

References in corpus (1)

• Combinatorics of linear iterated function systems with overlaps