On Bungee Set of Composition of Transcendental Entire Functions

TL;DR

This paper introduces a new, simplified definition of the Bungee set for transcendental entire functions and explores its properties, especially under composition and permutation of such functions.

Contribution

It provides an alternative, more accessible definition of the Bungee set and investigates its characteristics in composite and permutable transcendental entire functions.

Findings

New easy-to-use definition of Bungee set.

Properties of Bungee sets under composition.

Behavior of Bungee sets in permutable functions.

Abstract

Let $f$ be a transcendental entire function. For $n \in \mathbb{N},$ let $ f^{n}$ denote the $n^{th}$ iterate of $f$. Let $ I(f) = \{z \in \mathbb{C} : f^n \rightarrow \infty $ as $ n \rightarrow \infty \} $ and $ K(f) = \{z: \textrm{ there exists } R > 0 \textrm{ such that } | f^n(z) | \leq R \textrm{ for } n \geq 0 \}. $ Then the set $ \mathbb{C}\ \setminus (I(f) \cup K(f)) $ denoted by $ BU(f) $ is called Bungee set of $f$. In this paper we give an alternate definition for $ BU(f)$ which is very easy to work with, and we illustrate it by proving some properties of Bungee sets of composite transcendental entire functions and also of Bungee sets of permutable transcendental entire functions.