On some symmetric multidimensional continued fraction algorithms
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.
Keywords
Cite
@article{arxiv.1508.07814,
title = {On some symmetric multidimensional continued fraction algorithms},
author = {Pierre Arnoux and Sébastien Labbé},
journal= {arXiv preprint arXiv:1508.07814},
year = {2018}
}
Comments
Version 1: 22 pages, 12 figures. Version 2: new section on Cassaigne algorithm, 25 pages, 15 figures. Version 3: corrections during review process