# Mobius Transformations and Stereographic Projections

1. Jan 20, 2010

### Mathmos6

1. The problem statement, all variables and given/known data

Hi all - I've been battering away at this for an hour or so, and was hoping someone else could lend a hand!

Q: Show that any Mobius transformation T not equal to 1 on $\mathbb{C}_{\infinity}$ has 1 or 2 fixed points. (Done) Show that the Mobius transformation corresponding (under the stereographic projection map) to a rotation of S^2 through a nonzero angle has exactly 2 fixed points $z_1$ and $z_2$, where $z_2=\frac{-1}{z_1^*}$. If now T is a Mobius Transformation with 2 fixed points $z_1$ and $z_2$ satisfying $z_2=\frac{-1}{z_1^*}$, 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 not equal to $z_2$, $T^nz_1 \to z_1$ as $n \to \infty$).

Now I believe I've shown that the Mobius transformation corresponding to a rotation is mapped to mobius transformations with 2 fixed points, but I'm unsure as to how to show that $z_2=\frac{-1}{z_1^*}$, and I'm extra extra unsure how to show the later point about attractive fixed points! Please do reply, the more you can help me the better, and I certainly do need it! Many thanks, Mathmos6

2. Jan 21, 2010

### Mathmos6

No worries, I got it sorted on my own anyways.

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