Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture

TL;DR

This paper investigates the topological structure of Julia sets of disjoint-type entire functions, revealing complex continua like arc-like and pseudo-arc structures, and addresses Eremenko's conjecture about escaping sets.

Contribution

It provides a detailed topological classification of Julia continua for disjoint-type entire functions and constructs examples realizing diverse continua, including solutions to longstanding questions.

Findings

Julia continua have terminal points and span zero; some are arc-like.

Constructed entire functions with Julia sets containing complex continua like pseudo-arc.

Demonstrated Julia sets can have non-uniformly escaping points, addressing Eremenko's conjecture.

Abstract

A hyperbolic transcendental entire function with connected Fatou set is said to be "of disjoint type". It is known that a disjoint-type function provides a model for the dynamics near infinity of all maps in the same parameter space; hence a good understanding of these functions has implications in wider generality. Our goal is to study the topological properties of the Julia sets of entire functions of disjoint type. In particular, we give a detailed description of the topology of their connected components. More precisely, consider a "Julia continuum" C of such a function, i.e. the closure in the Riemann sphere of a component of the Julia set. We show that infinity is a terminal point of C, and that C has span zero in the sense of Lelek; under a mild geometric assumption on the function C is arc-like. (Whether every span zero continuum is also arc-like was a famous question in continuum theory, only recently resolved in the negative.) Conversely, we construct a single disjoint-type entire function with the remarkable property that each arc-like continuum with at least one terminal point is realised as a Julia continuum. The class of arc-like continua with terminal points is uncountable. It includes, in particular, the sin(1/x)-curve, the Knaster buckethandle and the pseudo-arc, so these can all occur as Julia continua of a disjoint-type entire function. We give similar descriptions of the possible topology of Julia continua that contain periodic points or points with bounded orbits, and answer a question of Bara\'nski and Karpi\'nska by showing that Julia continua need not contain points that are accessible from the Fatou set. Furthermore, we construct an entire function whose Julia set has connected components on which the iterates tend to infinity pointwise, but not uniformly. This is related to a famous conjecture of Eremenko concerning escaping sets of entire functions.