Eremenko's conjecture for functions with real zeros: the role of the minimum modulus

TL;DR

This paper investigates conditions under which the escaping set of certain transcendental entire functions forms a connected spider's web, supporting Eremenko's conjecture, and provides new examples illustrating these conditions.

Contribution

It establishes that iterated minimum modulus growth implies the escaping set is a spider's web, confirming Eremenko's conjecture for these functions, and introduces new function families with this property.

Findings

Escaping set forms a spider's web under minimum modulus growth

Eremenko's conjecture holds for these functions

New examples of functions satisfying or not satisfying the condition

Abstract

We show that for many families of transcendental entire functions $f$ the property that $m^n(r)\to\infty$ as $n\to \infty$, for some $r>0$, where $m(r)=\min\{|f(z)|:|z|=r\}$, implies that the escaping set $I(f)$ of $f$ has the structure of a spider's web. In particular, in this situation $I(f)$ is connected, so Eremenko's conjecture holds. We also give new examples of families of functions for which this iterated minimum modulus condition holds and new families for which it does not hold.