Sum of the exponential and a polynomial: Singular values and Baker wandering domains

TL;DR

This paper analyzes the singular values of functions combining exponential and polynomial parts, revealing their complex preimage structures and critical value properties, and investigates the existence of Baker wandering domains in their iteration.

Contribution

It characterizes the singular values and preimage domains of functions of the form exponential plus polynomial, and examines the presence of Baker wandering domains in their iteration.

Findings

Preimages of neighborhoods of infinity are infinitely connected domains.

Functions have infinitely many critical values with no finite asymptotic value.

Certain examples do not have Baker wandering domains.

Abstract

This article studies the singular values of entire functions of the form $E^k (z)+P(z)$ where $E^k$ denotes the $k-$times composition of $e^z$ with itself and $P$ is any non-constant polynomial. It is proved that the full preimage of each neighborhood of $\infty$ is an infinitely connected domain without having any unbounded boundary component. Following the literature, the point at $\infty$ is called a Baker omitted value for the function in such a situation. More importantly, there are infinitely many critical values and no finite asymptotic value and in fact, the set of all critical values is found to be unbounded for these functions. We also investigate the iteration of three examples of entire functions with Baker omitted value and prove that these do not have any Baker wandering domain. There is a conjecture stating that the number of completely invariant domains of a transcendental entire function is at most one. How some of these maps are right candidates to work upon in view of this conjecture is demonstrated.