3.II.12A

Describe geometrically the stereographic projection map $\pi$ from the unit sphere $S^{2}$ to the extended complex plane $\mathbf{C}_{\infty}=\mathbf{C} \cup\{\infty\}$, positioned equatorially, and find a formula for $\pi$.

Show that any Möbius transformation $T \neq 1$ on $\mathbf{C}_{\infty}$ has one or two fixed points. Show that the Möbius transformation corresponding (under the stereographic projection map) to a rotation of $S^{2}$ through a non-zero angle has exactly two fixed points $z_{1}$ and $z_{2}$, where $z_{2}=-1 / \bar{z}_{1}$. If now $T$ is a Möbius transformation with two fixed points $z_{1}$ and $z_{2}$ satisfying $z_{2}=-1 / \bar{z}_{1}$, prove that either $T$ corresponds to a rotation of $S^{2}$, or one of the fixed points, say $z_{1}$, is an attractive fixed point, i.e. for $z \neq z_{2}, T^{n} z \rightarrow z_{1}$ as $n \rightarrow \infty$.

[You may assume the fact that any rotation of $S^{2}$ corresponds to some Möbius transformation of $\mathbf{C}_{\infty}$ under the stereographic projection map.]

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