On escaping sets of some families of entire functions and dynamics of composite entire functions

TL;DR

This paper investigates the escaping sets of two specific families of entire functions and explores the dynamics of their compositions, providing conditions under which two transcendental entire functions share the same escaping set.

Contribution

It introduces new results on the structure of escaping sets for these families and establishes criteria for equality of escaping sets in composite entire functions.

Findings

Characterization of escaping sets for the families  and '

Conditions for equality of escaping sets of two entire functions

Analysis of the dynamics of composite entire functions

Abstract

We consider two families of functions $\mathcal{F}=\{f_{{\la},{\xi}}(z)= e^{-z+\la}+\xi: \la,\,\xi\in\C, \RE{\la}<0, \RE\xi\geq 1\}$ and $\mathcal{F}'=\{f_{{\mu},{\ze}}(z)= e^{z+\mu}+\ze: \mu,\,\ze\in\C, \RE{\mu}<0, \RE\ze\leq-1\}$ and investigate the escaping sets of members of the family $\mathcal F$ and $\mathcal F'.$ We also consider the dynamics of composite entire functions and provide conditions for equality of escaping sets of two transcendental entire functions.