Slow-growing counterexamples to the strong Eremenko conjecture

TL;DR

This paper constructs slow-growing counterexamples to the strong Eremenko conjecture for transcendental entire functions in class , demonstrating functions with specific growth rates that challenge previous assumptions about the structure of their escaping sets.

Contribution

It adapts tract constructions to produce new counterexamples with controlled slow growth, extending understanding of the Eremenko conjecture's limitations.

Findings

Counterexamples with slow growth rates are constructed.

Functions exhibit  class properties with specific logarithmic growth bounds.

The results include functions with growth rates like (rac{1}{(\u00a3 ext{log log } t)^\u03b1}) for  .

Abstract

Let $f\colon\mathbb{C}\to\mathbb{C}$ be a transcendental entire function. In 1989, Eremenko asked the following question concerning the set $I(f)$ of points that tend to infinity under iteration: can every point of $I(f)$ be joined to $\infty$ by a curve in $I(f)$? This is known as the \emph{strong Eremenko conjecture} and was disproved in 2011 by Rottenfu{\ss}er, R\"uckert, Rempe and Schleicher by the construction of a counterexample. The function has relatively small infinite order: it can be chosen such that $\log \log \,\lvert f(z)\rvert = (\log \lvert z\rvert)^{1+o(1)}$ as $f(z)\to \infty$. Moreover, $f$ belongs to the \emph{Eremenko--Lyubich class $\mathcal{B}$}. When a function belongs to this class, we can study the function via a \textit{logarithmic change of coordinates}. In this frame of coordinates, we are able to study the function via the \textit{tracts} that arise which are Jordan domains with unbounded real part. The key feature of the tracts in the counterexample of Rottenfu{\ss }er et al is that of large \textit{wiggling} sections. In this article we adapt the tracts used by Benitez and Rempe in order to deduce the existence of counterexample functions $f \in \mathcal{B}$ satisfying certain growth properties. We consider how slowly such an $f$ may grow. Suppose that $\Theta\colon [t_0,\infty)\to [0,\infty)$ is a function such that $\Theta(t) \to 0$ and \[ (\log t)^{\beta \Theta(\log t)}\Theta(t) \to \infty \quad\text{ as $t\to \infty$} \] for some $0<\beta<1$, along with a certain regularity assumption. Then there exists a counterexample $f\in\mathcal{B}$ as above such that \[ \log \log \lvert f(z) \rvert = O\bigl( (\log \lvert z \rvert)^{1 + \Theta(\log \lvert z \rvert)}\bigr) \quad\text{as $f(z) \to\infty$}. \] The hypotheses are satisfied, in particular, for $\Theta(t) = 1/(\log \log t)^{\alpha}$, for any $\alpha>0$.