Results on escaping set of an entire function and its composition

TL;DR

This paper explores the relationships between escaping sets of permutable entire functions and their compositions, providing new examples where Eremenko's conjecture holds and analyzing the dynamics of a specific transcendental entire function.

Contribution

It establishes key relationships between escaping sets of permutable entire functions and their compositions, and offers new families of functions satisfying Eremenko's conjecture.

Findings

Established relationships between escaping sets of permutable entire functions and their compositions.

Provided examples of transcendental entire functions satisfying Eremenko's conjecture.

Analyzed the dynamics of the function f(z)=z+1+e^{-z}.

Abstract

Given two permutable entire functions $f$ and $g,$ we establish vital relationship between escaping sets of entire functions $f, g$ and their composition. We provide some families of transcendental entire functions for which Eremenko's conjecture holds. In addition, we investigate the dynamical properties of the mapping $f(z)=z+1+e^{-z}.$