On some symmetric multidimensional continued fraction algorithms

TL;DR

This paper explicitly computes the density of the invariant measure for certain symmetric multidimensional continued fraction algorithms, using a method that reveals properties of their invariant domains, including potential fractal boundaries.

Contribution

It applies a method to compute invariant measure densities for the Reverse, Brun, and Cassaigne algorithms, providing new insights into their invariant domains.

Findings

Explicit density computation for the Reverse algorithm

Application of the method to Brun and Cassaigne algorithms

Observation of fractal boundaries in invariant domains

Abstract

We compute explicitly the density of the invariant measure for the Reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear that it is of positive measure.