Ergodic Properties Of $\theta$-Expansions And A Gauss-Kuzmin-Type Problem

TL;DR

This paper investigates the ergodic properties of a generalized continued fraction expansion called $ heta$-expansions, analyzing its invariant measure and solving a Gauss-Kuzmin-type problem using advanced operator methods.

Contribution

It extends the understanding of $ heta$-expansions by deriving their invariant measures and solving the associated Gauss-Kuzmin problem with a novel application of the Rockett and Sz"usz method.

Findings

Derived the invariant measure for $ heta$-expansions.

Solved the Gauss-Kuzmin-type problem for this expansion.

Provided new insights into the ergodic properties of generalized continued fractions.

Abstract

A generalization of the regular continued fractions was given by Chakraborty and Rao \cite{CR-2003}. For the transformation which generates this expansion and its invariant measure, the Perron-Frobenius operator is given and studied. For this expansion, we apply the method of Rockett and Sz\"usz \cite{RS-1992} and obtained the solution of its Gauss-Kuzmin-type problem.