# Invariant densities for random $β$-expansions

### Karma Dajani

Universiteit Utrecht, Netherlands### Martijn de Vries

Vrije Universiteit, Amsterdam, Netherlands

## Abstract

Let $β>1$ be a non-integer. We consider expansions of the form $∑_{i=1}β_{i}d_{i} $, where the digits $(d_{i})_{i≥1}$ are generated by means of a Borel map $K_{β}$ defined on ${0,1}_{N}×[0,⌊β⌋/(β−1)]$. We show existence and uniqueness of an absolutely continuous $K_{β}$-invariant probability measure w.r.t. $m_{p}⊗λ$, 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 $[0,⌊β⌋/(β−1)]$. Furthermore, this measure is of the form $m_{p}⊗μ_{β,p}$, where $μ_{β,p}$ is equivalent with $λ$. We establish the fact that the measure of maximal entropy and $m_{p}⊗λ$ are mutually singular. In case $1$ has a finite greedy expansion with positive coefficients, the measure $m_{p}⊗μ_{β,p}$ is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].

## Cite this article

Karma Dajani, Martijn de Vries, Invariant densities for random $β$-expansions. J. Eur. Math. Soc. 9 (2007), no. 1, pp. 157–176

DOI 10.4171/JEMS/76