Invariant densities for random $β$-expansions

Let $β>1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i β^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_β$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor β\rfloor /(β-1)]$. We show existence and uniqueness of an absolutely continuous $K_β$-invariant probability measure w.r.t. $m_p \otimes λ$, where $m_p$ is the Bernoulli measure on $\{0,1\}^{\N}$ with parameter $p$ $(0 < p < 1)$ and $λ$ is the normalized Lebesgue measure on $[0 ,\lfloor β\rfloor /(β-1)]$. Furthermore, this measure is of the form $m_p \otimes μ_{β,p}$, where $μ_{β,p}$ is equivalent with $λ$. We establish the fact that the measure of maximal entropy and $m_p \otimes λ$ are mutually singular. In case the number 1 has a finite greedy expansion with positive coefficients, the measure $m_p \otimes μ_{β,p}$ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].