Mobius Transformations and Stereographic Projections

[Mathmos6]

Homework Statement

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

 

[Mathmos6]

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