Due: April 30th, 2020

Math 185 Assignment 12

$$ \newcommand{\res}{\operatorname{res}} \newcommand{\aut}{\operatorname{aut}} \newcommand{\H}{\mathbb{H}} \newcommand{\mat}{\textsf{Mat}_{2\times2}(\C)} $$

  1. Automorphisms of the disc

    1. Let $a, b \in \D$ be distinct, and suppose that $f, g \in \aut(\D)$ are such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that $f = g$.
    2. Let $a, b \in \D$ (not necessarily distinct), and suppose that $f, g \in \aut(\D)$ are such that $f(a) = g(a)$ and $f'(b) = g'(b)$. Show that $f = g$.

    3. Must every automorphism of $\D$ have a fixed point? That is, given $f \in \aut(\D)$, is there necessarily some $z \in \D$ so that $f(z) = z$?

  2. Injective maps and poles

    Suppose that $f : \Omega \to \C$ is injective with a singularity at $s$.

    1. Show that if the singularity is removable, then the holomorphic extension of $f$ across the singularity remains injective. (Hint: the Open Mapping Theorem may be useful here.)
    2. Show that if the singularity is not removable, then it is a pole of order $1$, and that $f$ can have at most one pole.

  3. Möbius transformations

    Recall that given \[M = \begin{bmatrix}a&b\\c&d\end{bmatrix} \in \mat\] such that $\det(M) = ad-bc \neq 0$, we define the corresponding Möbius transformation \begin{align*} f_M : \C_\infty &\longrightarrow \C_\infty \\ z &\longmapsto \frac{az+b}{cz+d}. \end{align*} Recall also that $\det(M)\neq 0$ if and only if $M$ is invertible. Throughout this entire problem, all matrices are assumed to be invertible.

    1. Verify that for $A, B \in \mat$, \[f_A\circ f_B = f_{AB}.\]
    2. Show that for any $M \in \mat$, $f_M$ is a bijection.

    Since $f_M$ is holomorphic on its domain and its singularity is a pole, this tells us $f_M \in \aut(\C_\infty)$.

    1. Describe all $M$ for which $f_M(0) = 0$ and $f_M(\infty) = \infty$.
    2. Suppose that $\varphi \in \aut(\C_\infty)$ is such that $\varphi(0) = 0$ and $\varphi(\infty) = \infty$. Show that for some $\lambda \in \C^\times$, $\varphi(z) = \lambda z$. (Hint: show $\varphi(1/z)$ has a simple pole at zero; remove it.)
    3. Conclude that $\aut(\C_\infty) = \set{f_M \mid M \in \mat}$.

    We can conclude that any automorphism of $\C_\infty$ is completely determined by its pole, its zero, and a scale. Alternatively, given three input points and three output points, there is precisely one automorphism of $\C_\infty$ which takes each input point to its corresponding output.