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Let $f$ be a linear fractional transformation of the unit disc in itself, fixing points 1 and -1. Can i conclude that $f$ fixes the real axis?

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Hint: You should know precisely what all the automorphisms of the disk are: up to rotation they are just $\displaystyle \frac{z-a}{1-\bar{a}z}$ where $a\in\mathbb{D}$. Figure out what $a$ and the rotation can be for this to be true, and then see if it must map $\mathbb{R}$ to itself.

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Yes, of course, thank you! –  Alex Youcis Dec 9 '12 at 7:55
    
i found that $a$ must be real –  Federica Maggioni Dec 9 '12 at 8:01
    
i found $a$ must be real even if $f$ fixes only 1, or only -1. So can i generalize saying: a fractional linear map of the unit disc, fixing 1, fixes the whole real axis? –  Federica Maggioni Dec 9 '12 at 8:23
    
@FedericaMaggioni What about the possible rotation? –  Alex Youcis Dec 9 '12 at 8:24
    
a rotation of $2n\pi$, $n$ integer? –  Federica Maggioni Dec 9 '12 at 8:25

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