(Exer 3.15) Using the cross ratio, with different choices of $z_k$, find two different Möbius transformations that transform $C[1 + i, 1]$ onto the real axis plus $\infty$. In each case, find the image of the center of the circle.
Note: $C[i+1,1] = \{|z-(i+1)|=1\}$ is the circle with centre (1,1) and radius 1
(Part I) I'm going to try $g=[z,(2,1),(1,2),(0,1)], -g=[z,(2,1),(1,0),(0,1)]$. Please point out and explain any errors.
I observe that that $\pm g$ map $C[i+1,1] \to \mathbb R$. By plugging in $z=(1+i)+e^{i \phi}$, we can show that $\pm g((1+i)+e^{i \phi}) \in \mathbb R$. Then we extend $\pm g$ to be defined for the singularity $z=z_3=i=(0,1)$.
Finally, the images are (WA: plus, minus)
$$\pm g(1+i) = \pm i$$
Did I go wrong anywhere, and why?
(Part II) Can I actually pick any 3 points on $C[i+1,1]$ to form a required cross-ratio?
By inputting in the 3 points used to make up a cross ratio, into the cross ratio, it seems that the cross ratio will output $0,1,\infty$, soooo it looks like all cross ratios will map 3 distinct points on a circle to the real line.
$\therefore,$ although the question asks for 2 Möbius transformations, there are uncountably $\infty$ly many Möbius transformations/cross-ratios that work, so it seems?
So how will we compute the images $[i+1,z_1,z_2,z_3]$? Is it simply that with no particular pattern?
Best Answer
For Part I, you are almost there. The instructions ask you to try
which you tried to do using $\ g=[z,(2,1),(1,2),(0,1)], \ -g=[z,(2,1),(1,0),(0,1)] \ $ which produces $\ g(z) = (-i z -1 +2i)/(z-i). \ $ Even though $\ -g(z) \ $ is a Möbius transformation, you did not get it using a different choice of $\ z_i$. Notice that, by the characteristic property of cross ratio, we immediately know that $\ g(2+i) = 0, \ g(1+2i) = 1, \ g(i) = \infty. \ $ and also $\ g(1) = -1, \ g(1+i) = i. \ $ It follows that the Möbius transformation $\ h=[z,(2,1),(1,0),(0,1)] \ $ is the same as $\ -g. \ $ Now $\ g \ $ maps the center of the circle to $\ i \ $, and $\ -g = h \ $ maps it to $\ -i. \ $
For part II, you can pick any three distinct points on the circle and each such choice will produce different Möbius transformations which share the common property that they map the circle one-to-one onto the real line. Depending on the order of the three points, the interior of the circle, including the center, will get mapped one-to-one onto either the upper or lower half plane.
You can think of all Möbius transformations as being conjugate to two cases depending on the number of fixed points. Case 1 with one fixed point is a parabolic transformation $\ f(z) = z + a \ $ for some complex constant $\ a $ and the fixed point is at infinity. Case 2 with two fixed points is $\ f(z) = b\ z \ $ for some complex constant $\ b $ and the two fixed points are zero and infinity.
There is much more information about all this in the Wikipedia article Möbius transformation.