(08/07/2022, 10:53 AM)bo198214 Wrote: Though I wonder whether for any non-entire function there can be two fixed points in a domain D such that \(f^{\circ n}\) is holomorphic on D for all \(n\in\mathbb{N}\) (nothing to do with non-integer iterations).
Maybe you have a proof for this - as a first step. (I mean if there is such domain then D then the function is already entire)
Okay, so I'm good for 3 cases. These are the same three cases Milnor calls:
\(D\) is Hyperbolic; which means it's a simply connected domain biholomorphic to the unit disk.
\(D\) is Euclidean; which means it's a simply connected domain isomorphic to \(\mathbb{C}\)
\(D\) is spherical; which means it's a simply connected domain isomorphic to \(\widehat{\mathbb{C}}\)
Unfortunately I don't know much advanced shit outside these options.
But I do know, That usually it will behaved like the spherical case. But it will be on more complex Riemann surfaces than the Riemann sphere. So you can do dynamics on very exotic Riemann spheres, and technically it relates to iterations on \(\arcsin\) or whatever. But at that point, you better pull out your Riemann surface calculus tools. Because shit gets crazy.
That's the limit of my knowledge. I know a lot about the first three cases though
