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?
2026-03-31 21:07:19.1774991239
linear fractional transformation with two fixed point on the unit circle
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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.