On dynamics of composite entire functions and singularities

TL;DR

This paper investigates the dynamical behavior of transcendental entire functions and their compositions, focusing on conditions for their Fatou sets to coincide and exploring relationships between their singular values.

Contribution

It provides new conditions under which the Fatou sets of a transcendental entire function and its composition with another such function are the same, and relates their singular values.

Findings

Identifies conditions for Fatou set equality in compositions

Establishes relationships between singular values of functions and their compositions

Advances understanding of the dynamics of transcendental entire functions

Abstract

We consider the dynamical properties of transcendental entire functions and their compositions. We give several conditions under which Fatou set of a transcendental entire function $f$ coincide with that of $f\circ g,$ where $g$ is another transcendental entire function. We also prove some result giving relationship between singular values of transcendental entire functions and their compositions.