Absorbing sets of homogeneous subtractive algorithms

TL;DR

This paper proves the existence of absorbing sets in certain homogeneous multidimensional continued fraction algorithms and shows their nonergodic nature, challenging previous conjectures and analyzing other algorithms including ergodic ones.

Contribution

It confirms Schweiger's conjecture on absorbing sets and demonstrates nonergodicity of these algorithms, providing new insights into their dynamical properties.

Findings

Existence of absorbing sets in specific algorithms

Nonergodicity of the renormalizations

Analysis of ergodic and nonergodic algorithms

Abstract

We consider homogeneous multidimensional continued fraction algorithms, in particular a family of maps which was introduced by F. Schweiger. We prove his conjecture regarding the existence of an absorbing set for those maps. We also establish that their renormalisations are nonergodic which disproves another conjecture due to Schweiger. Other homogeneous algorithms are also analysed including ones which are ergodic.