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Given a non-parabolic transformation which is also an orientation preserving isometry in the hyperbolic upper half plane union the boundary, if I know the two fixed points and they are two different irreducible fractions on the boundary, how can I find the corresponding Möbius transformation?

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    I haven't thought about this stuff in a while, but if I recall correctly this only nails down that you have an elliptic Mobius transformation (of which the standard example takes the form $z\mapsto az$ for $a>0$). Check out Alan Beardon's "Geometry of Discrete Groups".2011-02-13
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    Small correction - they are hyperbolic, not elliptic, transformations. Beardon's book is indeed a great reference.2011-02-13

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