Semigroups of transcendental entire functions and their dynamics

TL;DR

This paper explores the dynamics of semigroups of transcendental entire functions, extending classical iteration results, analyzing Julia set connectivity, and studying properties of finitely generated and abelian semigroups.

Contribution

It extends Fatou-Julia theory to transcendental semigroups and investigates their connectivity, structure, and limit functions, providing new insights into their complex dynamics.

Findings

Conditions for Julia set connectivity in transcendental semigroups

Extension of iteration results to semigroup context

Analysis of limit functions on invariant Fatou components

Abstract

We study the dynamics of an arbitrary semigroup of transcendental entire functions using Fatou-Julia theory. Several results of the dynamics associated with iteration of a transcendental entire function have been extended to transcendental semigroups. We provide some conditions for connectivity of the Julia set of the transcendental semigroups. We also study finitely generated transcendental semigroups, abelian transcendental semigroups and limit functions of transcendental semigroups on its invariant Fatou components.