##

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