A Gauss-Kuzmin-L\'evy theorem for R\'enyi-type continued fractions

TL;DR

This paper extends the Gauss-Kuzmin-Lévy theorem to a generalized Réný-type continued fraction transformation, analyzing the asymptotic distribution behavior of the associated continued fraction expansion.

Contribution

It introduces a new interval map generalizing the Réný transformation and proves a Gauss-Kuzmin-Lévy-type theorem for its continued fraction expansion.

Findings

Established asymptotic distribution behavior for the generalized continued fractions

Applied Sz"usz's method to prove the theorem

Extended classical results to a broader class of transformations

Abstract

We consider an interval map which is a generalization of the R\'enyi transformation. For the continued fraction expansion arising from this transformation, we prove a result concerning the asymptotic behavior of the distribution functions of this map. More exactly, we use Sz\"usz's method to prove a Gauss-Kuzmin-L\'evy-type theorem.