On the iterations and the argument distribution of meromorphic functions

TL;DR

This paper investigates the relationship between the argument distribution, Julia set, and Fatou set of meromorphic functions, establishing conditions for singular directions and their implications for the structure of Fatou components.

Contribution

It provides new insights into the existence of singular directions in meromorphic functions and links these directions to the structure of Fatou sets and annuli.

Findings

Existence of singular directions under certain growth conditions

Absence of singular directions implies no annuli in Fatou set

Connection between Fatou set structure and argument distribution

Abstract

This paper consists of tow parts. One is to study the existence of a point $a$ in the intersection of Julia set and escaping set such that $\arg z=\theta$ is a singular direction if $\theta$ is a limit point of $\{\arg f^n(a)\}$ under some growth condition of a meromorphic function. The other is to study the connection between the Fatou set and singular direction. We prove that the absent of singular direction deduces the non-existence of annuli in the Fatou set.