On Gauss-Kuzmin Statistics and the Transfer Operator for a Multidimensional Continued Fraction Algorithm: the Triangle Map

TL;DR

This paper derives Gauss-Kuzmin statistics for the triangle map, a multidimensional continued fraction algorithm, by analyzing the transfer operator's eigenfunctions and spectral properties in specialized function spaces.

Contribution

It introduces a novel approach to studying the transfer operator for the triangle map using Banach and Hilbert space frameworks, revealing its nuclear operator structure.

Findings

Gauss-Kuzmin statistics explicitly derived for the triangle map

Transfer operator acts as a nuclear operator of trace class zero

Spectral analysis provides insights into the map's statistical properties

Abstract

The Gauss-Kuzmin statistics for the triangle map (a type of multidimensional continued fraction algorithm) are derived by examining the leading eigenfunction of the triangle map's transfer operator. The technical difficulty is finding the appropriate Banach space of functions. We also show that, by thinking of the triangle map's transfer operator as acting on a one-dimensional family of Hilbert spaces, the transfer can be thought of as a family of nuclear operators of trace class zero.