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 existence and uniqueness of an absolutely continuous -invariant probability measure w.r.t. , where is the Bernoulli measure on with parameter and is the normalized Lebesgue measure on . Furthermore, this measure is of the form , where is equivalent with . We establish the fact that the measure of maximal entropy and are mutually singular. In case has a finite greedy expansion with positive coefficients, the measure is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].
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Karma Dajani, Martijn de Vries, Invariant densities for random <em>β</em>-expansions. J. Eur. Math. Soc. 9 (2007), no. 1, pp. 157–176DOI 10.4171/JEMS/76