Completely Invariant Escaping Set of Transcendental Semigroup

TL;DR

This paper investigates the properties and structure of the completely invariant escaping set K(S) in transcendental semigroups, extending understanding beyond abelian cases and exploring its relation to the general escaping set I(S).

Contribution

It introduces a new perspective on completely invariant escaping sets in non-abelian transcendental semigroups and analyzes their properties and relationships with I(S).

Findings

K(S) can be characterized even when S is non-abelian.

Relations between K(S) and I(S) are established.

Properties of K(S) differ from those in abelian semigroups.

Abstract

For a non-trivial transcendental semigroup, escaping set I(S) is in general S-forward invariant and it is S-completely invariant if semigroup S is abelian. In the contrary of this result, we investigate completely invariant escaping set K(S) in different way even if semigroup S is not abelian and we discuss some properties and structure of such type of escaping set. Also, we establish some relations between completely invariant escaping set K(S) and the general escaping set I(S).