Omitted values and dynamics of meromorphic functions

TL;DR

This paper investigates the dynamics of transcendental meromorphic functions with omitted values, classifying their Fatou components, analyzing Julia set connectivity, and proving the non-existence of certain types of domains.

Contribution

It provides a comprehensive classification of Fatou components and Julia set properties for meromorphic functions with omitted values, including new results on invariant domains and connectivity.

Findings

Julia set is not totally disconnected unless all omitted values are in one Fatou component

Non-existence of Baker wandering domains and invariant Herman rings

Existence of eventual connectivity of wandering domains

Abstract

Let $M$ be the class of all transcendental meromorphic functions $f: \mathbb{C} \to \mathbb{C} \bigcup\{\infty\}$ with at least two poles or one pole that is not an omitted value, and $M_o =\{f \in M:f{has at least one omitted value}\}$. Some dynamical issues of the functions in $M_o$ are addressed in this article. A complete classification in terms of forward orbits of all the multiply connected Fatou components is made. As a corollary, it follows that the Julia set is not totally disconnected unless all the omitted values are contained in a single Fatou component. Non-existence of both Baker wandering domains and invariant Herman rings are proved. Eventual connectivity of each wandering domain is proved to exist. For functions with exactly one pole, we show that Herman rings of period two also do not exist. A necessary and sufficient condition for the existence of a dense subset of singleton buried components in the Julia set is established for functions with two omitted values. The conjecture that a meromorphic function has at most two completely invariant Fatou components is confirmed for all $f \in M_o$ except in the case when $f$ has a single omitted value, no critical value and is of infinite order. Some relevant examples are discussed.