On the classification of quaternionic Möbius transformations

Abstract

In this paper we consider quaternionic Möbius transformations preserving the unit ball in the quaternions $\bh$. In other words, maps of the form $g(z)=(az+b)(cz+d)^{-1}$ where $a$, $b$, $c$ and $d$ all lie in $\bh$ with the property that $|g(z)|<1$ for all $|z|<1$. We give an explicit expression for the fixed points of $g$ in terms of $a$, $b$, $c$ and $d$ and we use this to classify quaternionic Möbius transformations into six categories determined by their dynamics.