Let be a non-integer. We consider -expansions of the form , where the digits are generated by means of a Borel map defined on . We show that has a unique mixing measure of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure the digits 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 -expansions.
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Karma Dajani, Martijn de Vries, Measures of maximal entropy for random -expansions. J. Eur. Math. Soc. 7 (2005), no. 1, pp. 51–68DOI 10.4171/JEMS/21