On a Gauss-Kuzmin Type Problem for a Family of Continued Fractions

TL;DR

This paper investigates a family of continued fraction expansions, analyzes their ergodic properties, and solves a Gauss-Kuzmin type problem to understand the distribution of their partial quotients.

Contribution

It introduces a new family of continued fractions, derives their Perron-Frobenius operator, and solves a Gauss-Kuzmin problem for this specific expansion.

Findings

Derived the Perron-Frobenius operator for the expansion

Established ergodic properties of the associated system

Solved a variant of the Gauss-Kuzmin problem

Abstract

We study a family of continued fraction expansion of reals from the unit interval. The Perron-Frobenius operator of the transformation which generates this expansion under the invariant measure of this transformation is given. Using the ergodic behavior of homogeneous random system with complete connections associated with this expansion we solve a variant of Gauss-Kuzmin problem for this continued fraction expansion.