A Study of Fatou Set, Julia set and Escaping Set in Conjugate Transcendental Semigroup

TL;DR

This paper explores the properties of Fatou, Julia, and escaping sets in conjugate transcendental semigroups, establishing their invariance under conjugation and characterizing nearly abelian semigroups.

Contribution

It introduces the concept of conjugate semigroups via commutators and proves invariance of key sets and structural properties of nearly abelian semigroups.

Findings

Invariance of escaping, Julia, and Fatou sets under conjugation.

Characterization of nearly abelian semigroups.

Representation of semigroup elements as compositions involving commutators.

Abstract

We define commutator of a transcendental semigroup, and on the basis of this concept, we define conjugate semigroup. We prove that the conjugate semigroup is nearly abelian if and only if the given semigroup is nearly abelian. We also prove that image of each of escaping set, Julia set and Fatou set under commutator (affine complex conjugating maps) is equal to respectively escaping set, Julia set and Fatou set of conjugate semigroup. Finally, we prove that every element of the nearly abelian semigroup $ S $ can be written as the composition of an element from the set generated by the set of commutators $ \Phi(S) $ and the composition of the certain powers of its generators.