Multipliers and Classification of <i>m</i>-Möbius Transformations

Abstract

It is known that any m -M &#246; bius transformation is an ordinary M &#246; bius transformation in every one of its variables when the other variables do not take the values a and 1/ a , where a is a parameter defining the respective m -M &#246; bius transformation. For ordinary M &#246; bius transformations having distinct fixed points, the multiplier associated with one of these points completely characterizes the nature of that transformation, i.e. it tells us if it is elliptic, hyperbolic or loxodromic. The purpose of this paper is to show that fixed points exist also for m -M &#246; bius transformations and multipliers associated with them can be computed as well. As in the classical case, the values of those multipliers describe completely the nature of the transformations. The method we used was that of a thorough study of the coefficients of the variables involved, with which occasion we discovered surprising symmetries. These were the results allowing us to prove the main theorem regarding the fixed points of a m -M &#246; bius transformation, which is the key to further developments. Finally we were able to illustrate the geometric aspects of these transformations, making the whole theory as intuitive as possible. It was as opening a window into a space of several complex variables. This allows us to prove that if a bi-M &#246; bius transformation is elliptic or hyperbolic in z 1 at a point z 2 it will remain the same on a circle or line passing through z 2 . This property remains true when we switch z 1 and z 2 . The main theorem, dealing with the fixed points of an arbitrary m -M &#246; bius transformation made possible the extension of this result to these transformations.