Arc-like continua, Julia sets of entire functions, and 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 Barański and Karpińska 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.

Figures (10)

• Figure 1: Some examples of arc-like continua; terminal points are marked by grey circles. (The numbers of terminal points in these continua are two, three, one and zero, respectively.)
• Figure 2: Two embeddings of the $\sin(1/x)$-continuum that are not ambiently homeomorphic
• Figure 3: A logarithmic tract containing points whose imaginary parts are further apart than $2\pi$. (The translate $T+2\pi i$ is shown in light grey to demonstrate that $T$ is indeed disjoint from its $2\pi i\mathbb{Z}$-translates.) Subfigure \ref{['subfig:Iz']} illustrates the definition of the segment $I_z$, and shows that we may have $\operatorname{sep}_T(a,b,z)=0$ even though $\operatorname{Re} a < \operatorname{Re} z < \operatorname{Re} b$. The configuration in \ref{['subfig:sep']} demonstrates that the number $\operatorname{sep}_T(a,b;z)$ can decrease under perturbation of $z$ (it will change from $2$ to $0$ if we move the point $z$ slightly to the right).
• Figure 4: The tract $T$.
• Figure 5: Construction of the domain $U_k$. The shaded intervals at the top of the figure are $[\omega_k^2,\omega_k^3]$ (in the domain of $g_k$) and $I_k^3$ (in the range).

Theorems & Definitions (161)

• Definition 1.1: Hyperbolicity and disjoint type
• Definition 1.2: Bounded slope strahlen
• Theorem 1.3: Topology of bounded-slope Julia continua
• Theorem 1.4: Topology of Julia continua
• Theorem 1.5: Pseudo-arcs in Julia sets
• Theorem 1.6: Non-uniform escape to infinity
• Definition 2.1: Terminal points; span zero; arc-like continua
• Definition 2.2: Irreducibility
• Theorem 2.3: Non-escaping and accessible points
• Definition 2.4: Periodic and bounded-address Julia continua