On Dynamics of the Bungee set and the Filled Julia set of a Transcendental Semigroup

TL;DR

This work extends Fatou–Julia dynamics to transcendental semigroups by introducing the bungee set $BU(H)$ and the filled Julia set $K(H)$. It defines $BU(H)$ and $K(H)$ for a semigroup $H=[h_1,h_2,\dots]$, proves nonemptiness and intersections with the Julia set $J(H)$, and establishes that in abelian cases $K(H)=\bigcap_{h\in H} K(h)$ and $I(H)=\bigcap_{h\in H} I(h)$. The authors show $BU(H)=\bigcup_{h\in H} BU(h)$, provide invariance and conjugacy results, and obtain a plane partition into $BU(H)$, $K(H)$, and $I(H)$ under appropriate conditions, with $J(H)=\partial BU(H)$ when oscillatory wandering domains are absent. These results generalize single-function dynamics to semigroups, yielding a structured framework for understanding the combined action of families of transcendental entire functions and their collective Fatou–Julia sets.

Abstract

We have introduced the notion of the bungee set and the filled Julia set of a transcendental semigroup using Fatou-Julia theory. Numerous results of the bungee set of a single transcendental entire function have been generalized to a transcendental semigroup. For a transcendental semigroup having no oscillatory wandering domain, we provide some conditions for the containment of the bungee set inside the Julia set. The filled Julia set has also been explored in the context of a transcendental semigroup, and some of its properties are discussed. We have also explored some new features of the escaping set of a transcendental semigroup. The bungee set of a conjugate semigroup and an abelian transcendental semigroup has also been investigated.

Theorems & Definitions (60)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 3.1
• proof
• Example 3.2
• proof
• Definition 4.1
• Remark 4.2
• Lemma 4.3