The natural extension of the random beta-transformation

We construct a geometrico-symbolic version of the natural extension of the random β-transformation introduced by Dajani and Kraaikamp. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random β transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky about the greedy transformation