2015-26.md

course	course_year	question_number	tags	title	year
Geometry	IB	26	IB 2015 Geometry	Paper 3, Section II, F	2015

Let $T: \mathbb{C}{\infty} \rightarrow \mathbb{C}{\infty}$ be a Möbius transformation on the Riemann sphere $\mathbb{C}_{\infty}$.

(i) Show that $T$ has either one or two fixed points.

(ii) Show that if $T$ is a Möbius transformation corresponding to (under stereographic projection) a rotation of $S^{2}$ through some fixed non-zero angle, then $T$ has two fixed points, $z_{1}, z_{2}$, with $z_{2}=-1 / \bar{z}_{1}$.

(iii) Suppose $T$ has two fixed points $z_{1}, z_{2}$ with $z_{2}=-1 / \bar{z}{1}$. Show that either $T$ corresponds to a rotation as in (ii), or one of the fixed points, say $z{1}$, is attractive, i.e. $T^{n} z \rightarrow z_{1}$ as $n \rightarrow \infty$ for any $z \neq z_{2}$.