I fell a bit behind on the continuation of the class field theory series because I was setting up a new laptop. Before I resume that, I want to talk about something very weird that I learned today.
Let be a set that omits at least two points. If holomorphic and is such that at one , then is the identity.
This is a striking rigidity phenomenon!
But how do we prove it? The idea is to consider the sequence of iterates . Suppose for simplicity . Then in a neighborhood of , we can write , where the are omitted higher terms. If is not identically the identity, then .
So, similarly, by direct computation, in some neighborhood of , we have . Similarly, if we define for notational convenience, we have
But the are all holomorphic maps into . Since omits at least two points, the family is normal by Montel’s theorem and consequently has a subsequence that converges uniformly on compact sets.
Thus the derivatives converge, which is impossible unless .
Huh? I didn’t exactly see that coming. If is the unit disk, then at least it looks familiar. A holomorphic map of the unit disk into itself sending zero to zero must satisfy , and if equality holds is a rotation. So perhaps this result should be thought of as a generalization of Schwarz’s lemma? (Nevertheless, the use of Montel’s theorem is quite a sledgehammer to prove something as elementary as Schwarz.)
I should say where I got this from: Krant’z Complex Analysis: The Geometric Viewpoint. Krantz didn’t prove exactly this, but the argument is the same. Either this is standard fare that I missed when learning basic complex analysis, or I’m turning Climbing Mount Bourbaki into a comedy routine.