Classification of Quaternionic Projective Transformations by Equicontinuity Regions
Sandipan Dutta, Krishnendu Gongopadhyay, Rahul Mondal
TL;DR
The paper classifies the equicontinuity regions for cyclic subgroups of $PSL(n{+}1,H)$ acting on quaternionic projective space, with the region $Eq(G)$ determined solely by the dynamical type of the generator: elliptic, parabolic, loxodromic, or loxoparabolic. Using quaternionic Jordan form and pseudo-projective limits, the authors express $Eq(G)$ as the complement of the kernels of limit maps, and provide explicit subspace descriptions in each case. The elliptic case yields full equicontinuity on $P^n_H$, while parabolic, loxodromic, and loxoparabolic cases give precise subspace complements described by the generator's Jordan blocks. This analytic classification extends complex projective dynamics to the quaternionic setting and lays groundwork for further study of global quaternionic projective dynamics and invariants such as limit sets.
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.