On the classification of quaternionic Mobius transformations

In this paper we consider quaternionic Mobius transformations preserving the unit ball in the quaternions H. In other words, maps of the form g(z) = (az + b)(cz + d)(-1) where a, b, c and d all lie in H 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 Mobius transformations into six categories determined by their dynamics.