Fatou Set, Julia Set and Escaping Set in Holomorphic Subsemigroup Dynamics

TL;DR

This paper explores the relationships between Fatou, Julia, and escaping sets in transcendental semigroups and their subsemigroups, introducing new concepts like fundamental sets to analyze their properties.

Contribution

It introduces the concepts of partial fundamental set and fundamental set for transcendental semigroups and proves the non-emptiness of their Fatou and escaping sets.

Findings

Fatou and escaping sets of transcendental semigroups are non-empty.

Definitions of fundamental sets help analyze semigroup dynamics.

Relationships between semigroup and subsemigroup sets are partially characterized.

Abstract

We investigate to what extent Fatou set, Julia set and escaping set of transcendental semigroup is respectively equal to the Fatou set, Julia set and escaping set of its subsemigroup. We define partial fundamental set and fundamental set of transcendental semigroup and on the basis of this set, we prove that Fatou set and escaping set of transcendental semigroup $ S $ are non-empty.