Invariant densities for random $\beta$-expansions

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 $(d_i{)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on $\{0,1{\}}^{\mathbb{N}}\times \left[0,\lfloor \beta \rfloor /(\beta -1)\right]$. 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 $\{0,1{\}}^{\mathbb{N}}$ 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].