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 \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.