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

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 $(d_i{)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on $\{0,1{\}}^{\mathbb{N}}\times \left[0,\lfloor \beta \rfloor /(\beta -1)\right]$. 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 $(d_i{)}_{i\ge 1}$ 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.