A gauss-kuzmin theorem and related questions for $\theta$-expansions

TL;DR

This paper investigates the statistical properties of $ heta$-expansions using ergodic theory, providing an infinite-order-chain representation and solving a Gauss-Kuzmin type problem for these expansions.

Contribution

It introduces a new infinite-order-chain representation for $ heta$-expansions and addresses a related Gauss-Kuzmin problem using ergodic theory.

Findings

Established an infinite-order-chain representation of incomplete quotients.

Solved a variant of the Gauss-Kuzmin problem for $ heta$-expansions.

Demonstrated ergodic behavior of a related homogeneous random system.

Abstract

Using the natural extension for $\theta$-expansions, we give an infinite-order-chain representation of the sequence of the incomplete quotients of these expansions. Together with the ergodic behavior of a certain homogeneous random system with complete connections, this allows us to solve a variant of Gauss-Kuzmin problem for the above fraction expansion.