JournalsjemsVol. 7 , No. 1DOI 10.4171/jems/21

Measures of maximal entropy for random β\beta-expansions

  • Karma Dajani

    Universiteit Utrecht, Netherlands
  • Martijn de Vries

    Vrije Universiteit, Amsterdam, Netherlands
Measures of maximal entropy for random $\beta$-expansions cover

Abstract

Let β>1\beta >1 be a non-integer. We consider β\beta-expansions of the form i=1diβi\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}, where the digits (di)i1(d_i)_{i \geq 1} are generated by means of a Borel map KβK_{\beta} defined on {0,1}N×[0,β/(β1)]\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]. We show that KβK_{\beta} has a unique mixing measure νβ\nu_{\beta} of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure νβ\nu_{\beta} the digits (di)i1(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 β\beta-expansions.