Dynamics of meromorphic functions outside a countable set of essential singularities

TL;DR

This paper studies the dynamics of a class of meromorphic functions outside a countable set of essential singularities, exploring Fatou and Julia sets, escaping sets, and the behavior of escaping hairs.

Contribution

It extends the understanding of meromorphic function dynamics by analyzing the role of essential singularities and introducing new results on escaping hairs and wandering singularities.

Findings

Analysis of Fatou and Julia sets for class K functions

Extension of dynamics to singularities via escaping hairs

Example of wandering singular end point in escaping hair

Abstract

We consider a class of functions, denoted by K in this paper, which are meromorphic outside a compact and countable set B(f), investigated by A. Bolsch in his thesis in 1997. The set B(f) is the closure of isolated essential singularities. We review main definitions and properties of the Fatou and Julia sets of functions in class K. It is studied the role of B(f) in this context. Following Eremenko it is defined escaping sets and we prove some results related to them. For instance, the dynamics of a function is extended to its singularities using escaping hairs. We give an example of an escaping hair with a wandering singular end point, where the hair is contained in a wandering domain of f in K.