Functional analysis behind a Family of Multidimensional Continued Fractions: Part II

TL;DR

This paper advances the functional analysis of transfer operators for multidimensional continued fractions, identifying eigenfunctions, proving nuclearity, and deriving associated Gauss-Kuzmin distributions for specific triangle partition maps.

Contribution

It extends previous work by analyzing transfer operators on Banach spaces, establishing their nuclearity, and deriving distribution results for multidimensional continued fractions.

Findings

Eigenfunctions of transfer operators identified for certain triangle maps

Transfer operators shown to be nuclear of trace class zero

Gauss-Kuzmin distributions derived for specific maps

Abstract

This paper is a direct continuation of "Functional analysis behind a Family of Multidimensional Continued Fractions: Part I," in which we started the exploration of the functional analysis behind the transfer operators for triangle partition maps, a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This allows us now to find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces. We proceed to prove that the transfer operators, viewed as acting on one-dimensional families of Hilbert spaces, associated with select triangle partition maps are nuclear of trace class zero. We finish by deriving Gauss-Kuzmin distributions associated with select triangle partition maps.