Results on Dynamics of Bungee set of Composite Entire Functions in the Eremenko-Lyubich Class

TL;DR

This paper explores the complex dynamics of composite entire functions within the Eremenko-Lyubich class, focusing on the relationships between bungee sets, escaping sets, and filled-in Julia sets, especially for permutable functions.

Contribution

It establishes new relationships between the dynamics of composite entire functions and their component functions, particularly regarding bungee and Julia sets.

Findings

Union of bungee sets of two functions contains the bungee set of their composition

Filled-in Julia set of composite functions contains the Julia sets of component functions

Results are supported by several illustrative examples

Abstract

In this paper, we have discussed the dynamics of composite entire functions in terms of relationship between bungee set, escaping set and filled-in Julia set. We have established some relation between the dynamics of composition of entire functions and the functions taken for composition. We have shown that the union of the bungee set of two entire functions contains the bungee set of the composite function. In addition, it is shown that the filled-in Julia set of composite entire functions contains the filled-in Julia set of functions used for the composition. The results have been illustrated with several examples. We have mostly dealt with permutable(commuting) functions.