Random Systems with Complete Connections and the Gauss Problem for the Regular Continued Fractions

TL;DR

This paper explores how random systems with complete connections can be used to solve the Gauss problem related to regular continued fractions, employing ergodic theory to establish a Gauss-Kuzmin type theorem.

Contribution

It demonstrates the application of ergodic properties of homogeneous random systems with complete connections to address the Gauss problem for continued fractions.

Findings

Solved the Gauss problem using random systems with complete connections.

Established a Gauss-Kuzmin type theorem for regular continued fractions.

Highlighted the role of ergodic behavior in these systems.

Abstract

This paper present the important role that random system with complete connections played in solving the Gauss problem associated to the regular continued fractions. Hence, using the ergodic behavior of homogeneous random system with complete connections, we will solve a Gauss - Kuzmin type theorem.