A two-dimensional Gauss-Kuzmin theorem for $N$-continued fraction expansions

TL;DR

This paper establishes a two-dimensional Gauss-Kuzmin theorem for N-continued fraction expansions, analyzing the convergence rate of the distribution function to its limit using advanced dynamical systems and operator theory.

Contribution

It introduces a new two-dimensional Gauss-Kuzmin theorem for N-continued fractions and derives explicit bounds on the convergence rate using the transition operator.

Findings

Proves a two-dimensional Gauss-Kuzmin theorem for N-continued fractions.

Provides explicit bounds for the convergence rate of the distribution function.

Analyzes the natural extension of the measure-dynamical system for these expansions.

Abstract

A two-dimensional Gauss-Kuzmin theorem for $N$-continued fraction expansions is shown. More exactly, we obtain a Gauss-Kuzmin theorem related to the natural extension of the measure-dynamical system corresponding to these expansions. Then, using characteristic properties of the transition operator associated with the random system with complete connections underlying $N$-continued fractions on the Banach space of complex-valued functions of bounded variation we derive explicit lower and upper bounds for the convergence rate of the distribution function to its limit.