Iterates of Meromorphic Functions on Escaping Fatou Components

TL;DR

This paper extends the understanding of escaping Fatou components for meromorphic functions, analyzing the behavior of iterates, and providing examples of wandering domains with various escape rates.

Contribution

It generalizes results from entire to meromorphic functions with infinitely many poles and investigates fast escaping components using Nevanlinna theory.

Findings

Ratio of iterates' modulus can be bounded in certain escaping components

Multiply-connected wandering domains can be part of the fast escaping set

Examples of wandering domains with arbitrary escape rates

Abstract

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component may be bounded even if the orbit of the component contains an infinite modulus annulus sequence and this case cannot happen when the maximal modulus of the meromorphic function is large enough. Therefore, we extend the related results for entire functions to ones for meromorphic functions with infinitely many poles. And we investigate the fast escaping Fatou components of meromorphic functions defined in [44] in terms of the Nevanlinna characteristic instead of the maximal modulus in [12] and show that the multiply-connected wandering domain is a part of the fast escaping set under a growth condition of the maximal modulus. Finally we give examples of wandering domains escaping at arbitrary fast rate and slow rate.