Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T08:53:58.322Z Has data issue: false hasContentIssue false

On the classification of quaternionic Möbius transformations

Published online by Cambridge University Press:  07 September 2004

WENSHENG CAO
Affiliation:
Institute of Mathematics and Software, Hunan University of Science and Technology, Xiangtan, Hunan 411201, P. R. China. e-mail: cwsxtpu@263.net
JOHN R. PARKER
Affiliation:
Department of Mathematical Sciences, University of Durham, Durham DH1 3LE. e-mail: j.r.parker@durham.ac.uk
XIANTAO WANG
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, P. R. China. e-mail: xtwang@mail.hunu.edu.cn

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.

Type
Research Article
Copyright
2004 Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)