Möbius transformations

The group of Möbius transformations, $ \mathrm{PGL}(2,{\mathbb{C}})$, gets a natural topology from the group $ \mathrm{GL}(2,{\mathbb{C}})$ (which is an open subset of $ {\mathbb{C}}^4$). More precisely, a sequence of transformations $ \{A_n\}$ converges to a transformation $ A_0$ if and only if there exists matrices, $ (\begin{smallmatrix}a_n & b_n  c_n & d_n\end{smallmatrix})$ and $ (\begin{smallmatrix}a_0 & b_0  c_0 & d_0\end{smallmatrix})$, representing $ A_n$ and $ A_0$ respectively, such that $ a_n \to a_0$, $ b_n\to b_0$, $ c_n \to c_0$ and $ d_n \to d_0$.

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 $ z \mapsto z + 1$;
(iii) elliptic if it has two fixed points and it is conjugate to a transformation of the form $ z \mapsto \lambda z$, where $ \vert\lambda\vert = 1$;
(iv) loxodromic if it is conjugate to a transformation of the form $ z \mapsto \lambda z$, where $ \vert\lambda\vert \neq 0,1$. In the particular case of $ \lambda$ real, positive and not equal to 0 or $ 1$, 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 $ M$ be a Möbius transformation and $ (\begin{smallmatrix}a & b c & d\end{smallmatrix})$ a matrix representing it; denote by $ tr^2(M)$ the number $ (a+d)^2$. We have then the following way of classifying $ M$:
(i) $ tr^2(M) = 4$ if and only if $ M$ is the identity or parabolic;
(i) $ tr^2(M)$ is real and in the interval $ [0,4)$ if and only if $ M$ is elliptic;
(iii) $ tr^2(M)$ is real and greater than $ 4$ if and only if $ M$ is hyperbolic;
(iv) $ tr^2(M)$ if not in the real interval $ [0,\infty)$ if and only if $ M$ is loxodromic but not hyperbolic.

Prop 1.7   A Möbius transformation $ M$ keeps a disc invariant if and only if $ tr^2(M) \geq 0$.

Prop 1.8   Let $ \mathfrak{C}$ be the collection of lines and circles in the complex plane. Let $ C$ be an element of $ \mathfrak{C}$ and $ M$ a Möbius transformation. Then $ M(C)$ is also an element of $ \mathfrak{C}$.

Pablo Ares Gastesi 2005-08-31