Invariant Properties of Fatou Set, Julia Set and Escaping Set of Holomorphic Semigroup

TL;DR

This paper investigates the invariant properties of Fatou, Julia, and escaping sets in holomorphic semigroups, revealing conditions under which these sets exhibit invariance similar to classical complex dynamics.

Contribution

It establishes new invariance properties of escaping, Fatou, and Julia sets in holomorphic semigroups, especially for abelian cases, linking semigroup dynamics to classical complex dynamics.

Findings

Escaping set of transcendental semigroup is S-forward invariant

Fatou, Julia, and escaping sets are S-completely invariant in abelian semigroups

Holomorphic semigroup dynamics can mirror classical complex dynamics under certain conditions

Abstract

In this paper, we prove that escaping set of transcendental semigroup is S-forward invariant. We also prove that if holomorphic semigroup is abelian, then Fatou set, Julia set and escaping set are S-completely invariant. We see certain cases and conditions that the holomorphic semigroup dynamics exhibits same dynamical behavior just like the classical complex dynamics. Frequently, we also examine certain amount of connection and contrast between classical complex dynamics and holomorphic semigroup dynamics.