For context I am working through Sullivan's proof for the No-Wandering domain theorem.
My question is, can you restrict that Measurable Riemann Mapping Theorem to functions that are not defined on the entire complex plane. For example for a simply connected domain $U$, if $f:U \rightarrow U$ fixes three points and has conformal distortion $\mu$, is this function unique?
On the wikepedia page, https://en.wikipedia.org/wiki/Measurable_Riemann_mapping_theorem, they say a similar result is true on replacing $\mathbb{C}$ with $\mathbb{D}$, but I can't seem to figure out what that result is...
Any help would be greatly appreciated.