On an Invariant Möbius Measure and the Gauss–Kuzmin Face Distribution

We study Möbius measures of the manifold of $n$-dimensional continued fractions in the sense of Klein. By definition any Möbius measure is invariant under the natural action of the group of projective transformations $\mathrm{PGL}(n+1)$ and is an integral of some form of the maximal dimension. It turns out that all Möbius measures are proportional, and the corresponding forms are written explicitly in some special coordinates. The formulae obtained allow one to compare approximately the relative frequencies of the $n$-dimensional faces of given integer-affine types for $n$-dimensional continued fractions. In this paper we make numerical calculations of some relative frequencies in the case of $n=2$.