From pathological to paradigmatic: A retrospective on Eremenko and Lyubich's entire functions

TL;DR

The paper surveys how Eremenko and Lyubich used a refined Runge–type approximation to realize entire functions with prescribed transcendental dynamics, marking a shift from rational dynamics to rich, pathbreaking phenomena. It details five landmark constructions—univalent wandering and Baker domains, an oscillating wandering domain, and Julia sets of positive area supporting invariant line fields—demonstrating both methodological power and conceptual novelty. The approach provided foundational techniques that spurred extensive subsequent work on wandering domains, boundary control, and the geometry of Julia and escaping sets, including counterexamples to long-standing conjectures. Overall, the work reframes pathological dynamics as central to transcendental dynamics, with lasting impact on theory, methods, and the landscape of questions in the field.

Abstract

This article surveys the impact of Eremenko and Lyubich's paper ''Examples of entire functions with pathological dynamics'', published in 1987 in the Journal of the LMS. Through a clever extension and use of classical approximation theorems, the authors constructed examples exhibiting behaviours previously unseen in holomorphic dynamics. Their work laid foundational techniques and posed questions that have since guided a good part of the development of transcendental dynamics.

Figures (3)

• Figure 1: Left: Schematic of Example 2 (univalent wandering domain): disjoint vertical strips iterate to $+\infty$; the separating vertical lines belong to an attracting basin containing the unit disc. Right: Schematic of Example 3 (univalent Baker domain): dynamics modelled by $z\mapsto 2z$ on a right half-plane bounded by a line $L$ that maps to a basin of attraction.
• Figure 2: The original figure in EL_87 illustrating the construction of an oscillating wandering domain containing $\mathscr{D}_0$. For $m\geq 0$, the model map translates every ball $B_m$ to the right so that $a_m$ is sent to $a_{m+1}$. After $m$ iterates, $\mathscr{D}_m$ maps inside $Q_m$, which in turn is mapped inside $\mathscr{D}_{m+1}$, creating an oscillating pattern. Sets and errors are carefully chosen so that the resulting approximation map satisfies the same inclusions, and hence $a_0=0$ is a limit point for every orbit in $\mathscr{D}_0$. Hence each $a_m$ and $\infty$ occur as limit functions.
• Figure 3: Schematic of the construction of Example 4, a Julia set of positive measure which is not the entire complex plane.

Theorems & Definitions (1)

• Lemma 2.1