# Fatou components and singularities of meromorphic functions

## TL;DR

This paper investigates the geometric and dynamical properties of meromorphic functions, focusing on the behavior of their Baker and wandering domains and the influence of the postsingular set on these domains.

## Contribution

It provides new results on the positioning of postsingular points relative to Baker and wandering domains, answering open questions and establishing conditions that exclude wandering domains.

## Key findings

- Bounded diameter wandering domains imply postsingular points approach their boundaries.
- If postsingular set is away from Julia set, wandering domains contain large disks.
- Certain meromorphic maps with many poles cannot have wandering domains.

## Abstract

We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljevi\'c-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that ${\rm dist}(p_n,\partial U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\mathbb C$) then any sequence of iterates of wandering domains must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.