On the Invariant Density of the Random Beta-Transformation

TL;DR

This paper constructs a natural extension of the random beta-transformation to derive its invariant measure density and analyze the branching rate of beta-expansions, addressing previous open questions.

Contribution

It provides a formula for the invariant measure density of the random beta-transformation and evaluates estimates on the branching rate of beta-expansions.

Findings

Derived a formula for the density of the invariant measure

Constructed a Lebesgue measure preserving natural extension

Evaluated estimates on the branching rate of beta-expansions

Abstract

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