## Möbius transformations

The group of Möbius transformations, , gets a natural topology from the group (which is an open subset of ). More precisely, a sequence of transformations converges to a transformation if and only if there exists matrices, and , representing and respectively, such that , , and .

Prop 1.6   A non-identity Möbius transformation has at least one and at most two fixed points on the Riemann sphere.

Möbius transformations can be classified in four different, mutually exclusive types as follows:
(i) the identity;
(ii) parabolic if it has only one fixed point. In that case the transformation is conjugate to ;
(iii) elliptic if it has two fixed points and it is conjugate to a transformation of the form , where ;
(iv) loxodromic if it is conjugate to a transformation of the form , where . In the particular case of real, positive and not equal to 0 or , the transformation is called hyperbolic (we include hyperbolic transformations in the loxodromic case).

This classification can also be made with the square of the trace the matrices representing Möbius transformations. Let be a Möbius transformation and a matrix representing it; denote by the number . We have then the following way of classifying :
(i) if and only if is the identity or parabolic;
(i) is real and in the interval if and only if is elliptic;
(iii) is real and greater than if and only if is hyperbolic;
(iv) if not in the real interval if and only if is loxodromic but not hyperbolic.

Prop 1.7   A Möbius transformation keeps a disc invariant if and only if .

Prop 1.8   Let be the collection of lines and circles in the complex plane. Let be an element of and a Möbius transformation. Then is also an element of .

Pablo Ares Gastesi 2005-08-31