Classification of Mobius transforms

The parabolic transformation has only one fixed point. From (28), p. 153: $\infty$ is the sole fixed point if and only if $M(z)$ is a translation, $M(z) = z + b$. On p. 169, we find

If $M(z)=\frac{az+b}{cz+d}$ is normalized, then we know from (27), p. 152, that it is parabolic if and only if $(a+d) = \pm 2$, in which case the single fixed point is $\xi = (a-d)/2c$. That is all we need to define this class.

Rewrite the second $M$ as $M(z) = \frac{z+b}{0 \cdot z + 1}$. Then a = 1, c = 0, and d = 1. Notice that b does not figure into (27). Substitute these normalized coefficients into (27),

$\hspace{5em}\xi = \frac{(1-1) \pm \sqrt{(\pm 2)^2 - 4}}{2 \cdot 0} = \frac{0}{0} = ?$

For $M(z) = 2z + 3$, [using $\xi = (a-d)/2c$ with normalized coefficients], where $N = \frac{1}{\sqrt{ad-bc}}$ [with the given coefficients],

$\hspace{3em}N = \frac{1}{\sqrt{2}}$.

$\hspace{3em}M(z) = \frac{2N-N}{0} = \frac{\frac{1}{\sqrt{2}}}{0} = \infty$

It's probably worth noting that the decomposition (3) on page 124 also breaks down when $c=0$ and the new decomposition of the resulting similarity transformation $Az+B$ is:

(i) $z\mapsto Az$ which is an expansion and a rotation

(ii)$z\mapsto z+B$ which is a translation

Vasco

I agree that Vasco has clarified the construction. It also helps to clarify other things, at least for me. [Archetypal] elliptic, hyperbolic, and parabolic transformations all have a fixed point at infinity, but only the parabolic lacks a second fixed point at 0. (27) is given as a formula that applies to any [normalized] Möbius transformation, except that in the parabolic case, the equation reduces to $\xi = \frac{a-d}{2c}$, which implies that $c \neq 0$ for at least some normalized parabolic Möbius transformations. I think one might form the impression that this is the case for all [normalized] parabolic Möbius transformations. Subsection 5 takes up Möbius transformations where $c = 0$, which entails at least one fixed point at infinity. The four subtypes are projected onto the Riemann sphere. The parabolic transformation is one of these. At this point, one might form the impression that the parabolic transformation entails $c = 0$. Then a careful reading of (28) and the prior paragraph shows that the parabolic type is the pure translation $M(z) = (z+b)$, where $c = 0$. The subsection ends with the statement that each Möbius transformation [not just the ones where $c = 0$] is equivalent, in a certain sense, to one (and only one) of the four types … [for which $c = 0$]. With that statement, one would like to know more about what is implied by equivalent, in a certain sense. Perhaps is is a reference to the list (44) on p. 170 and the list on p. 180.

This is a rather complicated construction. Perhaps it leaps off the page to some, but it took me a few readings and Vasco’s clarification to see it. It will take more work to sort out the connections to list (44).

Gary

I do not disagree, but I think this point is where confusion easily arises. In $\S\ 5, \S\S\ 5$ Fixed Points at Infinity, the idea of $\widetilde{M}$ as a Möbius transformation in the new plane created by $F$ and illustrated in [29] has not yet been introduced. Even so, in $\S\S 5$, we are in the realm of $c = 0$, fixed points 0 and $\infty$, and Möbius transformations labeled with unvarnished $M(z)$. These transformations are all similarity transformations and they provide the “simplest, archetypal” examples of elliptic, hyperbolic, loxodromic, and parabolic Möbius transformations. The framework defining the relation of these archetypal examples to ordinary Möbius transformations does not appear until $\S VII$ Visualization and Classification, $\S\S 1$ The Main Idea, which presents $F(z) = \frac{z-\xi_+}{z-\xi_-}$

Gary

Replying to reply 9
Subsection 4 Fixed points of chapter 3 is dealing with the general Mobius transformations where both of the fixed points lie in the finite plane, and this means that $c$ must be non-zero (see the first sentence of subsection 5).

When, at the end of subsection 4, Needham talks about the exceptional case when the two fixed points coalesce into one fixed point, he is still talking about a quadratic, with two repeated roots that lie in the finite plane, and so where $c$ is non-zero.

The last sentence of subsection 4 is just saying that in the cases where there is only one fixed point the transformation is called parabolic, regardless of the value of $c$.

We can't use the formula he gives there for the repeated root when $c=0$ because it has been derived using a quadratic equation, assuming that $c$ is non-zero.

Subsection 5 deals specifically with the cases where $c=0$ i.e. when $M(z)=az/d+b/d=Az+B$, and shows that when $M$ is a translation ($A=1$) then it only has one fixed point, and so by the definition at the end of subsection 4 referred to above, it is parabolic.

Vasco