# Measures of maximal entropy for random $\beta$-expansions

### Karma Dajani

Universiteit Utrecht, Netherlands### Martijn de Vries

Vrije Universiteit, Amsterdam, Netherlands

## Abstract

Let $\beta >1$ be a non-integer. We consider $\beta$-expansions of the form $\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]$. We show that $K_{\beta}$ has a unique mixing measure $\nu_{\beta}$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure $\nu_{\beta}$ the digits $(d_i)_{i \geq 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of $\beta$-expansions.