If I have a Mobius map $f: \mathbb{D} \rightarrow \mathbb{D}$ that has no fixed points in the disk. Is it true that every orbit of $f$ escapes any compact set $K \subset \mathbb{D}$?
If so, I'm a little stuck on seeing how.
Also, a small side question, if $f$ does have a fixed point, must $f$ be a rotation? If $0$ is the fixed point, then I understand that it is a rotation, but I am not sure for an arbitrary fixed point.
Any help would be greatly appreciated.