Editing Möbius transformation (section)

== Fixed points ==
Every non-identity Möbius transformation has two [[fixed point (mathematics)|fixed points]] <math>\gamma_1, \gamma_2</math> on the Riemann sphere. The fixed points are counted here with [[Multiplicity (mathematics)|multiplicity]]; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity.

=== Determining the fixed points ===
The fixed points of the transformation
<math display="block">f(z) = \frac{az + b}{cz + d}</math>
are obtained by solving the fixed point equation {{nowrap|1=''f''(''γ'') = ''γ''}}. For {{nowrap|''c'' ≠ 0}}, this has two roots obtained by expanding this equation to
<math display="block">c \gamma^2 - (a - d) \gamma - b = 0 \ ,</math>
and applying the [[quadratic formula]]. The roots are
<math display="block">\gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a - d)^2 + 4bc}}{2c} = \frac{(a - d) \pm \sqrt{\Delta}}{2c}</math>
with discriminant
<math display="block"> \Delta = (\operatorname{tr}\mathfrak{H})^2 - 4\det\mathfrak{H} = (a+d)^2 - 4(ad-bc),</math>
where the matrix
<math display="block">\mathfrak{H} = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}</math>
represents the transformation.
Parabolic transforms have coincidental fixed points due to zero discriminant. For ''c'' nonzero and nonzero discriminant the transform is elliptic or hyperbolic.

When {{nowrap|1=''c'' = 0}}, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When {{nowrap|''a'' ≠ ''d''}} the second fixed point is finite and is given by
<math display="block">\gamma = -\frac{b}{a-d}.</math>

In this case the transformation will be a simple transformation composed of [[Translation (geometry)|translation]]s, [[rotation (mathematics)|rotation]]s, and [[dilation (metric space)|dilation]]s:
<math display="block">z \mapsto \alpha z + \beta.</math>

If {{nowrap|1=''c'' = 0}} and {{nowrap|1=''a'' = ''d''}}, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:
<math display="block">z \mapsto z + \beta.</math>

=== Topological proof ===
Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the [[Euler characteristic]] of the sphere being 2:
<math display="block"> \chi(\hat{\Complex}) = 2.</math>

Firstly, the [[projective linear group]] {{nowrap|PGL(2, ''K'')}} is [[n-transitive|sharply 3-transitive]]&nbsp;– for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially [[dimension counting]], as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity.

Next, one can see by identifying the  Möbius group with <math>\mathrm{PGL}(2,\Complex)</math> that any Möbius function is homotopic to the identity. Indeed, any member of the [[general linear group]] can be reduced to the identity map by [[Gaussian elimination|Gauss-Jordan elimination]], this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The [[Lefschetz–Hopf theorem]] states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the [[Lefschetz number]] of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.

By contrast, the projective linear group of the real projective line, {{nowrap|PGL(2, '''R''')}} need not fix any points&nbsp;– for example <math>(1+x) / (1-x)</math> has no (real) fixed points: as a complex transformation it fixes ±''i''<ref group="note">Geometrically this map is the [[stereographic projection]] of a rotation by 90° around ±''i'' with period 4, which takes <math>0 \mapsto 1 \mapsto \infty \mapsto -1 \mapsto 0.</math></ref>&nbsp;– while the map 2''x'' fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.

=== Normal form ===
Möbius transformations are also sometimes written in terms of their fixed points in so-called '''normal form'''. We first treat the non-parabolic case, for which there are two distinct fixed points.

''Non-parabolic case'':

Every non-parabolic transformation is [[conjugacy class|conjugate]] to a dilation/rotation, i.e., a transformation of the form
<math display="block">z \mapsto k z </math>
{{nowrap|(''k'' ∈ '''C''')}} with fixed points at 0 and ∞. To see this define a map
<math display="block">g(z) = \frac{z - \gamma_1}{z - \gamma_2}</math>
which sends the points (''γ''<sub>1</sub>, ''γ''<sub>2</sub>) to (0, ∞). Here we assume that ''γ''<sub>1</sub> and ''γ''<sub>2</sub> are distinct and finite. If one of them is already at infinity then ''g'' can be modified so as to fix infinity and send the other point to 0.

If ''f'' has distinct fixed points (''γ''<sub>1</sub>, ''γ''<sub>2</sub>) then the transformation <math>gfg^{-1}</math> has fixed points at 0 and ∞ and is therefore a dilation: <math>gfg^{-1}(z) = kz</math>. The fixed point equation for the transformation ''f'' can then be written
<math display="block">\frac{f(z)-\gamma_1}{f(z)-\gamma_2} = k \frac{z-\gamma_1}{z-\gamma_2}.</math>

Solving for ''f'' gives (in matrix form):
<math display="block">\mathfrak{H}(k; \gamma_1, \gamma_2) =
\begin{pmatrix}
\gamma_1 - k\gamma_2 & (k - 1) \gamma_1\gamma_2 \\
1 - k                &  k\gamma_1 - \gamma_2
\end{pmatrix}</math>
or, if one of the fixed points is at infinity:
<math display="block">\mathfrak{H}(k; \gamma, \infty) =
\begin{pmatrix}
k & (1 - k) \gamma \\
0 &  1
\end{pmatrix}.</math>

From the above expressions one can calculate the derivatives of ''f'' at the fixed points:
<math display="block">f'(\gamma_1) = k </math> and <math display="block"> f'(\gamma_2) = 1/k.</math>

Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (''k'') of ''f'' as the '''characteristic constant''' of ''f''. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:
<math display="block">\mathfrak{H}(k; \gamma_1, \gamma_2) = \mathfrak{H}(1/k; \gamma_2, \gamma_1).</math>

For loxodromic transformations, whenever {{nowrap|{{abs|''k''}} > 1}}, one says that ''γ''<sub>1</sub> is the '''repulsive''' fixed point, and ''γ''<sub>2</sub> is the '''attractive''' fixed point. For {{nowrap|{{abs|''k''}} < 1}}, the roles are reversed.

''Parabolic case'':

In the parabolic case there is only one fixed point ''γ''. The transformation sending that point to ∞ is
<math display="block">g(z) = \frac{1}{z - \gamma}</math>
or the identity if ''γ'' is already at infinity. The transformation <math>gfg^{-1}</math> fixes infinity and is therefore a translation:
<math display="block">gfg^{-1}(z) = z + \beta\,.</math>

Here, ''β'' is called the '''translation length'''. The fixed point formula for a parabolic transformation is then
<math display="block">\frac{1}{f(z)-\gamma} = \frac{1}{z-\gamma} + \beta.</math>

Solving for ''f'' (in matrix form) gives
<math display="block">\mathfrak{H}(\beta; \gamma) =
\begin{pmatrix}
1+\gamma\beta    & - \beta \gamma^2  \\
\beta            &   1-\gamma \beta
\end{pmatrix}</math>Note that
<math>\det\mathfrak{H}(\beta;\gamma)=|\mathfrak{H}(\beta;\gamma) |
=\det
\begin{pmatrix}
1+\gamma\beta & -\beta\gamma^2\\
\beta & 1-\gamma\beta
\end{pmatrix}
=1-\gamma^2\beta^2+\gamma^2\beta^2=1

</math>

If {{nowrap|1=''γ'' = ∞}}:
<math display="block">\mathfrak{H}(\beta; \infty) =
\begin{pmatrix}
1 & \beta  \\
0 & 1
\end{pmatrix}</math>

Note that ''β'' is ''not'' the characteristic constant of ''f'', which is always 1 for a parabolic transformation. From the above expressions one can calculate:
<math display="block">f'(\gamma) = 1.</math>