Does there exist any Möbius transformation that preserves upper unit disc?

Question

I know how to find Möbius transformations that preserve unit disc.Can I link my question with that and how? Or maybe Möbius transformations that fix upper unit disc don't exist(exept identity)?Why?

Answer 1

Let $T$ be a Möbius transformations which maps the upper half of the unit disk into itself. Möbius transformations preserve angles, so that $\{ -1, 1 \}$ must be mapped to $\{ -1, 1 \}$. There are two possibilities:

1. $T(1) = 1$ and $T(-1) = -1$. Then $T$ maps the segment $[-1,1]$ onto itself, so that $T(r) = 0$ for some $r \in (0, 1)$. A simple calculation shows that $$ T(z) = \frac{z-r}{1-rz} \, . $$ In this case, $T$ is the restriction of an automorphism of the unit disk.

2. $T(1) = -1$ and $T(-1) = 1$. Then $T$ maps the upper half of the unit circle onto the segment $[-1, 1]$, so that $T(\lambda) = 0$ for some $\lambda$ with $|\lambda|=1$ and $\operatorname{Im} \lambda > 0$. Now one gets $$ T(z) = \frac{z-\lambda}{\lambda z - 1} \, . $$ In this case, $T$ is the restriction of a conformal mapping of the unit disk to the upper halfplane.