Invariant densities for random $\beta$-expansions

Source abstract

Let \beta >1 be a non-integer. We consider 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 existence and uniqueness of an absolutely continuous K_{\beta} -invariant probability measure w.r.t. m_p \otimes \lambda , where m_p is the Bernoulli measure on \{0,1\}^{\N} with parameter p ( 0 < p < 1) and \lambda is the normalized Lebesgue measure on [0 ,\lfloor \beta \rfloor /(\beta -1)] . Furthermore, this measure is of the form m_p \otimes \mu_{\beta,p} , where \mu_{\beta,p} is equivalent with \lambda . We establish the fact that the measure of maximal entropy and m_p \otimes \lambda are mutually singular. In case 1 has a finite greedy expansion with positive coefficients, the measure m_p \otimes \mu_{\beta,p} is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].