Teichm\"uller spaces and normal forms associated to wandering domains

TL;DR

This paper investigates the structure of Teichmüller spaces linked to wandering domains of entire functions, revealing their infinite dimensionality under certain conditions and establishing normal forms for the dynamics.

Contribution

It demonstrates that a discrete grand orbit relation causes the Teichmüller space to be infinite dimensional and develops normal forms for dynamics on wandering domains.

Findings

Discrete grand orbit relation implies infinite dimensional Teichmüller space.

Normal forms provide global linearising coordinates in discrete cases.

Power-type dynamics are characterized between annuli in the indiscrete case.

Abstract

We study the dynamical Teichm\"uller space ${\mathcal T}(U,f)$ associated to a wandering domain $U$ of an entire function $f$. We show that a discrete grand orbit relation in $U$ forces ${\mathcal T}(U,f)$ to be infinite dimensional, thereby answering a question of Fagella--Henriksen. We further describe the geometry of these spaces by developing normal forms for the dynamics on wandering domains, yielding global linearising coordinates in the discrete case and power-type dynamics between annuli in the indiscrete case.