On an invariant Möbius measure and the Gauss-Kuzmin face distribution

On Invariant Möbius Measure and Gauss - Kuzmin Face Distribution

Consider an n-dimensional real vector space with lattice of integer points in it. The boundary of the convex hull of all integer points contained inside one of the n-dimensional invariant cones for a hyperbolic n-dimensional linear operator without multiple eigenvalues is called a sail in the sense of Klein. The set of all sails of such n-dimensional operator is called (n−1)-dimensional continu...

On a Gauss-kuzmin Type Problem for Piecewise Fractional Linear Maps with Explicit Invariant Measure

A random system with complete connections associated with a piecewise fractional linear map with explicit invariant measure is defined and its ergodic behaviour is investigated. This allows us to obtain a variant of Gauss-Kuzmin type problem for the above linear map.

The Gauss-kuzmin-wirsing Operator

This paper presents a review of the Gauss-Kuzmin-Wirsing (GKW) operator. The GKW operator is the transfer operator of the Gauss map, and thus has connections to the theory of continued fractions – specifically, it is the shift operator for continued fractions. The operator appears to have a reasonably smooth, well-behaved structure, however, no closed-form analytic solutions are known, and thes...

On a Gauss-Kuzmin-Type Problem For a Generalized Gauss-Kuzmin Operator

A Gauss-Kuzmin-Lévy theorem for a certain continued fraction

We consider an interval map which is a generalization of the well-known Gauss transformation. In particular, we prove a result concerning the asymptotic behavior of the distribution functions of this map. 1. Introduction. In 1800, Gauss studied the following problem. In modern notation, it reads as follows. Write x ∈ [0, 1) as a regular continued fraction