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Homework Help: Mobius Transformations and Stereographic Projections

  1. Jan 20, 2010 #1
    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 [itex]\mathbb{C}_{\infinity}[/itex] 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 [itex]z_1[/itex] and [itex]z_2[/itex], where [itex]z_2=\frac{-1}{z_1^*}[/itex]. If now T is a Mobius Transformation with 2 fixed points [itex]z_1[/itex] and [itex]z_2[/itex] satisfying [itex]z_2=\frac{-1}{z_1^*}[/itex], prove that either T corresponds to a rotation of S^2, or one of the fixed points, say [itex]z_1[/itex], is an attractive fixed point (i.e. for z not equal to [itex]z_2[/itex], [itex]T^nz_1 \to z_1[/itex] as [itex]n \to \infty[/itex]).

    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 [itex]z_2=\frac{-1}{z_1^*}[/itex], 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. jcsd
  3. Jan 21, 2010 #2
    No worries, I got it sorted on my own anyways.
     
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