Invariant densities for random continued fractions

TL;DR

This paper investigates random continued fraction expansions generated by Gauss and Rényi maps, demonstrating the existence of a unique smooth invariant measure for the associated random dynamical system.

Contribution

It establishes the existence and uniqueness of an absolutely continuous invariant measure with smooth density for the combined random continued fraction system.

Findings

Existence of a unique invariant measure with smooth density.

The invariant measure is absolutely continuous.

The system exhibits stable statistical properties.

Abstract

We continue the study of random continued fraction expansions, generated by random application of the Gauss and the R\'enyi backward continued fraction maps. We show that this random dynamical system admits a unique absolutely continuous invariant measure with smooth density.