On the Invariant Density of the Random β-transformation

We construct a Lebesgue measure preserving natural extension of a skew product system related to the random β-transformation Kβ. This allows us to give a formula for the density of the absolutely continuous invariant probability measure of Kβ, answering a question of Dajani and de Vries, and also to evaluate some estimates on the typical branching rate of the set of β-expansions of a real number