Sum of the exponential and a polynomial: Singular values and 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.