Dependence with complete connections and the Gauss-Kuzmin theorem for N-continued fractions

TL;DR

This paper generalizes the Gauss transformation to a family of interval maps, solving the Gauss-Kuzmin problem for N-continued fractions using the theory of random systems with complete connections.

Contribution

It introduces a new family of interval maps for N-continued fractions and applies advanced probabilistic theory to solve the Gauss-Kuzmin problem for these expansions.

Findings

Solved the Gauss-Kuzmin-type problem for N-continued fractions.

Extended the theory of random systems with complete connections to new interval maps.

Provided a framework for analyzing convergence properties of N-continued fractions.

Abstract

We consider a family $\{T_N:N \geq 1 \}$ of interval maps as generalizations of the Gauss transformation. For the continued fraction expansion arising from $T_N$, we solve its Gauss-Kuzmin-type problem by applying the theory of random systems with complete connections by Iosifescu.