Mathematical Proceedings of the Cambridge Philosophical Society

On the classification of quaternionic Möbius transformations

a1 Institute of Mathematics and Software, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China. e-mail:
a2 Department of Mathematical Sciences, University of Durham, Durham DH1 3LE. e-mail:
a3 Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P. R. China. e-mail:

Article author query
cao w   [Google Scholar] 
parker jr   [Google Scholar] 
wang x   [Google Scholar] 


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.

(Received April 7 2003)
(Revised July 28 2003)