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?
linear fractional transformation with two fixed point on the unit circle
1
$\begingroup$
complex-analysis
conformal-geometry
1 Answers
1
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.
-
0a rotation of $2n\pi$, $n$ integer? – 2012-12-09