Oberseminar: Josias Reppekus (G.15.25)
I want to present a technique of identifying Fatou components of holomorphic maps using properties of plurisubharmonic functions with some examples of successful applications and some examples where one could hope to apply similar methods.
A Fatou component of a holomorphic self-map F of a complex manifold is a maximal connected open set of stable dynamics.
The hardest part in identifying a Fatou component is often proving maximality.
In 2019, Arosio, Benini, Fornaess, Peters identified a candidate component with dynamics tending to infinity by constructing a plurisubharmonic function encoding the dynamics of F on a hypothetical bigger containing component and reaching a contradiction by finding an open set inside of a polar set.
I will present their example and two examples with convergence to a finite point where I expect a similar argument should work, if one can find the correct plurisubharmonic function.
