Classification of Quaternionic Projective Transformations by Equicontinuity Regions

TL;DR

This paper characterizes the regions of equicontinuity for cyclic subgroups of quaternionic projective transformations, linking these regions to the dynamical types of the generators such as elliptic, parabolic, or loxodromic.

Contribution

It provides a classification of equicontinuity regions based on the dynamical type of elements in quaternionic projective linear groups, offering an analytic perspective.

Findings

Elliptic groups act equicontinuously on all of quaternionic projective space.

Equicontinuity regions for parabolic, loxodromic, and loxoparabolic elements are explicitly described.

Regions depend solely on the Jordan form of the generator.

Abstract

We describe the equicontinuity regions of cyclic subgroups of the quaternionic projective linear group $\mathrm{PSL}(n+1,\mathbb{H})$. We show that these regions depend solely on the dynamical type of the generator $g$, i.e. whether $g$ is elliptic, parabolic, loxodromic or loxoparabolic. This yields an analytic interpretation of the dynamical classification of the elements. In particular, elliptic cyclic groups act equicontinuously on all of the quaternionic projective space, while for the parabolic, loxodromic and loxoparabolic elements the equicontinuity region is determined by explicit quaternionic projective subspaces arising from the generator's Jordan form.