Some Geometric Properties of the m -Möbius Transformations

Möbius transformations, which are one-to-one mappings of  onto  have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations m f mapping m  onto  . Even for the simplest entity, the pre-image by m f of a unique point, there is no way of visualization. Pre-images by m f of figures from  are like ghost figures in m  . This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most im-portant achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in m  are well known and vector calculus in m  is familiar, yet the pre-image by m f of a vector from  is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in m  . Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points


Introduction
The bi-Möbius transformations are functions : f × →  An easy computation shows that: When we checked for the fixed points of ( ) 4 f z as a Möbius transformation in 4 z we got the equation: It is expected similar properties to be true for any m-Möbius transformation, yet for higher values of m, the computation becomes too tedious.
We notice that the coefficients of k s , which are polynomials in ω become more and more complicated as k increases. Yet, an interesting pattern should be noticed, namely that in every m f the coefficient of k s at the numerator is the same as the coefficient of m k s − at the denominator. Also, if we compare the coefficients from m f and from 2 m f + we find another surprising pattern, which will be studied in detail in Section 3.
More generally, if

Images of Circles
The geometric properties of the m-Möbius transformations concern the way these mappings transform figures from each one of the planes (z k ) into figures situated in the (w)-plane. As Möbius transformations, they are obviously conformal mappings, hence they will preserve the angles of those figures, except at singular points. They will transform circles (including straight lines, which can be considered circles of infinite radius) into circles. However, there are details which need to be examined.
A circle C centered at 0 w and of radius r in the (w)-plane has the equation: It is convenient to write this equation under the form: where , a β ∈  and b ∈  . Indeed, we obtain (5) when we replace 0 Then, when 0 α = , this equation becomes that of a straight line. We will continue to call it circle (of infinite radius, or centered at infinity). Theorem 1. The circle (5) is the image by ( ) fixed, for j k ≠ of circles: from the (z k )-planes, 1, 2, , k m =  . More exactly, for every k, fixing , ( ) Then: hence, (5) becomes: Obviously, different values of , j z j k ≠ will determine different circles (6)  The concept of pre-image by m f can be useful in order to describe this change. , , , m z z z  belongs to W, then so does any point obtained by a permutation of these coordinates.
Let us deal for simplicity with the case of ( ) 2 1 2 , f z z given by (2). We can chose . Solving for 2 z the equation: , z z W ∈ . We notice that 2 z , as a function of 1 z is a Möbius transformation and therefore a bijective mapping of  onto  , which means that the pre-image by 2 f of a single is a subset of 2  in one-to-one correspondence with  .
Given 1 z in the ( 1 z )-plane there is a unique 2 z in the ( 2 z )-plane, namely that given by (8) such that ( ) The theorem states that the mapping The general case can be treated similarly. We choose ( ) is an arbitrary permutation of ( ) 1 2 , , , m z z z  . The formula (9) represents a Möbius transformation in w which maps a circle C of equation (5)  Let us deal now with the symmetry with respect to a circle (see [4], page 80) of Equation (5). As we have seen, that equation represents a proper circle when 0 α ≠ or a line when 0 α = . When that line is the real axis, we say that the points w and w are symmetric with respect to it. Yet, for any line it is known what symmetric points with respect to that line mean, namely z and z * are symmetric points with respect to the line L if and only if L is the bisecting normal of the segment determined by z and z * . This concept can be extended to the case when 0 α ≠ . In that case the Equation (5) is the center of the circle and r is its radius. As shown in [4], page 81 by using the tool of cross ratios, w and w * are symmetric with respect to this circle if and only if: The symmetry principle states that if a Möbius transformation carries a circle 1 C into a circle 2 C , then it transforms any pair of symmetric points with respect to 1 C into a pair of symmetric points with respect to 2 C . Here circle means proper circle or line. This principle can be extended to m-Möbius transformations in the following way.
Theorem 3 (The Main Theorem). Let w and w * be symmetric points with respect to the circle (5) and let W and W * be the pre-images by m f of w and respectively w * . Then the projection onto the ( k z )-plane of any section of W and W * obtained by keeping fixed, j k ≠ . The projection of the respective section of C is k C and that of the sections of W and W * is k z and k z * . The symmetric of k z with respect to k C .
The theorem states that a m-Möbius transformation carries symmetric points with respect to C into symmetric points with respect to C. There is a one-to-one mapping of W onto W * assigning to every ( ) with reflection in C . The second affirmation is true since the pre-image of the intersection of two sets is equal to the intersection of the pre-images of those sets.
Next, m f moves * L into the reflection of L in C which passes through 0 w , hence the pre-image of 0 w is included in * L . Finally, if a proper circle 1 C passes through 0 w , then 1 C * should pass through ∞ , which is the reflection in C of 0 w . Thus 1 C * is a line not passing through 0 w and then Theorem 5. Given a circle ( ) for every j k ≠ there are infinitely many circles j C in every ( j z )-plane such that k C and j C have the same image by m f when all the other variables are kept fixed.
Proof: Suppose that a circle C of Equation (5) is given in the (w)-plane. We are looking for a circle (6)   The projection onto the ( k z )-plane of the sections as previously defined of the pre-image by m f of a triangle is a curvilinear triangle having the same angles as the original one. Some of these triangles can be infinite, in the sense that one side is an arc of a circle and the other two are half-lines. An analogous situation appears for the pre-image of an arbitrary polygon.

Uniqueness of m-Möbius Transformations
It is known that there is a unique Möbius transformation in the plane moving three distinct points into other three distinct points. In what follows, we will study similar properties of m-Möbius transformations.
This theorem is a particular case of the following: