Dynamics of Nearly Abelian Transcendental Semigroup

TL;DR

This paper introduces the concept of nearly abelian transcendental semigroups, extending previous rational semigroup results, and explores their properties, including function classes and relations between postsingular sets.

Contribution

It extends the notion of nearly abelian semigroups from rational to transcendental functions and characterizes their properties and function classes.

Findings

Defined nearly abelian transcendental semigroups.

Extended rational semigroup results to transcendental case.

Established relations between postsingular sets of composed functions.

Abstract

The notion of nearly abelian rational semigroup was introduced by Hinkannen and Martin. In this paper, we have introduced the notion of nearly abelian transcendental semigroup. We have extended the results of nearly abelian rational semigroups to the best possible class of nearly abelian transcendental semigroups. We have given a class of functions which nearly permute with a given transcendental entire function. In addition, a relation between the postsingular set of composition of two entire functions with that of individual functions is obtained.