The natural extension of the random beta-transformation

Abstract

Abstract We construct a geometrico-symbolic version of the natural extension of the random $\beta $ -transformation introduced by Dajani and Kraaikamp [Random $\beta $ -expansions. Ergod. Th. & Dynam. Sys.23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $ -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 [ $\beta $ -automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar.24 (1973), 273–278] about the greedy transformation.