Classification of Quaternionic Projective Transformations by Equicontinuity Regions

We describe the equicontinuity regions of cyclic subgroups of the quaternionic projective linear group $\mathrm{P}\mathrm{S}\mathrm{L}(n+1,\mathbb{H})$. 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.