Measures of Maximal Entropy for Random Β-expansions

Let β > 1 be a non-integer. We consider β-expansions of the form ∑∞ i=1 di βi , where the digits (di)i≥1 are generated by means of a random map Kβ defined on {0, 1}N× [0, bβc/(β − 1)]. We show that Kβ has a unique measure νβ of maximal entropy log(1 + bβc). Under this measure, the digits (di)i≥1 form a uniform Bernoulli process, and the projection of this measure on the second coordinate is an infinite convolution of Bernoulli measures. 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-expansions.