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By an analytic isomorphism of $\Omega$, I mean an analytic function $\Omega \to \Omega$, with an analytic inverse.

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    Can you extend the function to the whole sphere by continuity?2012-10-22
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    @MarianoSuárez-Alvarez Thanks. You mean I should look at analytic isomorphisms of the sphere (that is Mobius transformations), that preserve the set $\{p_1,\ldots,p_n \}$.2012-10-22

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