Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x\in I_\beta:=[0,\frac{m-1}{\beta-1}]}$$\end{document} can be represented in the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=\sum_{k=1}^\infty \epsilon_k\beta^{-k},$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\epsilon_k\in\{0,1,\ldots,m-1\}}$$\end{document} for all k (a β-expansion of x). It is known that a.e. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x\in I_\beta}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\beta < \frac{1+\sqrt5}2}$$\end{document}, we show that the set of β-expansions grows exponentially for every internal x.