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I have the following problem.

I need to find all linear fractional transformations $f$ such that $f(\{|z-2|=3\})=\{|z-2|=3\}$

I thought that maybe one could use that a LFT maps symmetric points with respect to $\{|z-2|=3\}$ to symmetric points with respect to $\{|z-2|=3\}$. But I'm a bit confused since I need to find all of them not only one. Could someone give me a hint?

Thanks for your help.

user1294729
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  • I know little about the topic but one method that seems possible is considering the maps circle -> $\mathbb{H}$ -> $\mathbb{H}$ -> circle, described here – Gareth Ma Mar 13 '22 at 20:38
  • the one I find easy to remember: the upper half plane is mapped to itself by $\frac{az+b}{cz+d}$ where $a,b,c,d$ are real and $ad-bc > 0.$ There are simple maps between the half plane and the standard unit disc $|z| < 1.$ Finally simple maps between your disc and the standard. – Will Jagy Mar 13 '22 at 20:38

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