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


Preprint available here and also at http://pcwww.liv.ac.uk/~lrempe/Papers/arclike_9.pdf. Comments are welcome!

A sonnet

(From my talk at the conference "Perspectives of Modern Complex Analysis" in Będlewo, July 2014, celebrating Alex Eremenko's 60th birthday)


Let $f$ a transcendental function be
whose sing'lar points - which we assume are bounded,
as Eremenko-Lyubich once demanded -
immediately to a fixed point flee.

Within its Julia set we then may see
maximal shapes connected and unbounded
which - not being arcs - leave us confounded:
What might their topolog'cal structure be?

My friends, mayhap we ought to light some candles
for those who hoped that only simple sets escape.
Continua in Julia sets that may arise
we can describe and find that they comprise
not only pseudo-arcs and bucket handles,
but any other kind of arc-like shape.

Extended Summary


This article is concerned with studying the topology of Julia sets for transcendental entire functions that are of disjoint type, i.e. hyperbolic functions in the Eremenko-Lyubich class B that have connected Fatou set. These functions play a key role in understanding the dynamics of general functions in the class B.

In previous work, joint with Rottenfußer, Rückert and Schleicher [RRRS], it was shown that there exists a disjoint-type function whose Julia set contains no arcs. This article expands on this result by giving an almost complete description of the possible topology of components of Julia sets of disjoint-type entire functions. If C is a component of such a Julia set, let us call its one-point compactification (i.e., the union of C with the point at infinity) a Julia continuum of the function f.

In particular, we prove the following strengthening of the previously mentioned theorem from [RRRS].

Theorem. There exists a disjoint-type entire function for which every Julia continuum is a pseudo-arc.

Recall that the pseudo-arc is a well-known hereditarily indecomposable continuum. In particular, the Julia set in question contains no arc. This result arises from a much more general investigation of Julia continua. In particular, we prove the following result, which uses some standard terminology from continuum theory. (We refer to the article for definitions.)

Theorem. Every Julia continuum C of a disjoint-type entire function has span zero, and infinity is a terminal point of C. If the function additionally has bounded slope in the sense of [RRRS], then C is arc-like.
Conversely, there exists a bounded-slope, disjoint-type entire function f with the following property. If X is an arc-like continuum containing a terminal point x_0, then there exists a Julia continuum C of f that is homeomorphic to X, with x_0 corresponding to the point at infinity.

(We remark that any continuum of span zero that is not arc-like is a counterexample to an old and famous question of Lelek, which remained open for 40 years. Recently, a counterexample was constructed by Hoehn; it seems plausible that such examples could also be constructed as Julia continua.)

There are similar results concerning Julia continua at bounded external addresses (i.e., those whose forward iterates all contain a point at some uniformly bounded distance from the origin), and invariant Julia continua. In each case, we obtain a complete topological characterization of those continua of the given type that can arise for bounded-slope, disjoint-type entire functions (and the classes of continua are different for each of the three classes of Julia continua).

The construction also gives rise to a number of examples with interesting properties.

Theorem. There exist examples of Julia continua C of disjoint-type entire functions with each of the following properties:
  1. The set of non-escaping points in C is empty, but the iterates of f do not tend to infinity uniformly on C;
  2. The number of non-escaping points in C is finite but greater than one;
  3. The set of non-escaping points in C is dense;
  4. The set of non-escaping points in C is a Cantor set;
  5. The Julia continuum C contains no finite point that is accessible from the Fatou set of f.

(The final statement answers a question of Barański and Karpińska.)

The first statement in the preceding theorem is also interesting, because it shows that there may be escaping points that cannot be connected to infinity by a connected set on which the iterates tend to infinity uniformly. This has some connections with a famous conjecture of Eremenko. We study the question of such 'uniform escape', and it turns out that there is quite a close connection between this property and the topology of Julia continua. In addition, we can strengthen the above example further to obtain the following.

Theorem. There is an example of a Julia continuum C of a disjoint-type entire function such that C is homeomorphic to an arc, every finite point of C tends to infinity under iteration, but there is no non-degenerate connected set A containing the finite endpoint such that the iterates of f tend to infinity uniformly on A.

[RRRS] Rottenfußer, Rückert, Rempe & Schleicher: Dynamic rays of bounded-type entire functions, Ann. of Math. 173 (2011), no. 1, 77-125.