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Abstract

We show that there exists a hyperbolic entire function of finite order of growth such that the hyperbolic dimension—that is, the Hausdorff dimension of the set of points in the Julia set of whose orbit is bounded—is equal to two. This is in contrast to the rational case, where the Julia set of a hyperbolic map must have Hausdorff dimension less than two, and to the case of all known explicit hyperbolic entire functions.

In order to obtain this example, we prove a general result on constructing entire functions in the Eremenko-Lyubich class with prescribed behavior near infinity, using Cauchy integrals. This result significantly increases the class of functions that were previously known to be approximable in this manner.

Furthermore, we show that the approximating functions are quasiconformally conjugate to their original models, which simplifies the construction of dynamical counterexamples. We also give some further applications of our results to transcendental dynamics.

Hyperbolic entire functions with full hyperbolic dimension]Hyperbolic entire functions
with full hyperbolic dimension
and approximation by Eremenko-Lyubich functions 1 \subjclass[2010]30D05, 30E10, 37F10 (Primary); 30D10, 30D15, 37F35 (Secondary)

1 Introduction

The Hausdorff dimension of the Julia set of a rational function has been extensively studied. A related quantity, the hyperbolic dimension , was introduced by Shishikura [Sh] as the supremum over the Hausdorff dimensions of hyperbolic subsets of . (Here a hyperbolic set is a compact, forward invariant subset of such that sufficiently high iterates of are expanding when restricted to .)

Clearly the hyperbolic dimension is a lower bound for . If the rational function is hyperbolic, then by definition is a hyperbolic set itself, and hence

(1.1)

It is natural to ask whether (1.1) holds more generally. In other words, how prevalent is expanding dynamics in the Julia set of a rational function?

Question 1.1.

Is there a rational function such that ?

The relation (1.1) is known to hold in a vast number of cases, including all non-recurrent rational functions. The tantalizing possibility of an example of a rational function where (1.1) fails was first suggested by results of Avila and Lyubich [AL] on Feigenbaum quadratic polynomials with periodic combinatorics. They show that, if such a map exists whose Julia set has positive area (and hence dimension ), then its hyperbolic dimension would need to be strictly less than two. Since this article was submitted for publication, Avila and Lyubich have announced a proof that such Feigenbaum Julia sets of positive area do indeed exist, answering Question 1.1 in the positive. It remains open whether there is a rational function such that .

The results of Avila and Lyubich resonate strongly with the iteration theory of transcendental entire functions. Here, in stark contrast to the rational case, even hyperbolic functions (see Definition 1.2) frequently satisfy and . Stallard [S2] was the first to construct hyperbolic examples with (in slightly different terminology), while Urbański and Zdunik [UZ] proved that this situation occurs for hyperbolic exponential maps , where by a result of McMullen [McM]. This suggests that a systematic understanding of the measurable dynamics of transcendental entire functions is not only interesting in its own right, but can also help to shed further light on phenomena such as those discovered by Avila and Lyubich.

A class of hyperbolic entire functions that has received particular attention in recent years is given by those of finite order and disjoint type:

Definition 1.2.

A transcendental entire function is hyperbolic if there exists a compact set with such that the restriction

is a covering map. An entire function is said to be of disjoint type if it is hyperbolic and the Fatou set is connected, or equivalently if the set can be chosen to be connected.

An entire function has finite order if, setting , we have

The topology of the Julia set of a hyperbolic entire function of finite order is completely understood [Ba1, BJR, R2, RS]. Moreover, the Hausdorff dimension of is equal to two in this case [Ba2], and the hyperbolic dimension is greater than one [BKZ]. More precisely, suppose that is of disjoint type and finite order. Then:

  • The Julia set is a disjoint uncountable union of curves to , each consisting of a finite endpoint and a ray connecting this endpoint to infinity. In fact, is ambiently homeomorphic to a straight brush in the sense of [AO] (i.e., a certain universal plane topological object).

  • The set of endpoints in has Hausdorff dimension equal to two.

  • The union of rays in (without endpoints) has Hausdorff dimension equal to one.

Furthermore, for a large class of hyperbolic entire functions of finite order, including all hyperbolic maps in the exponential family , the trigonometric family and many others, the measurable dynamics is described in detail by the results of [MU1, MU2]. In particular, these maps satisfy . This suggests the following problem.

Question 1.3.

Let be a hyperbolic transcendental entire function of finite order. Is it always the case that ?

In this article, we give a negative answer.

Theorem 1.4 ((Hyperbolic functions with full hyperbolic dimension)).

There exists a transcendental entire function of disjoint type and finite order such that . Furthermore, can be chosen such that has positive measure.

Approximation

In order to obtain the desired counterexample, we shall first construct a suitable “model function” with the desired behavior, and then approximate this map by an entire function. The idea of using approximation to construct interesting examples in complex dynamics was introduced by Eremenko and Lyubich [EL1], who used Arakelyan’s theorem [G, Satz IV.2.3], an important result of approximation theory. Given a closed set , this theorem states that any function , defined and continuous on and holomorphic on its interior, can be uniformly approximated by entire functions if and only if satisfies certain simple topological conditions.

Arakelyan’s theorem, while powerful, has the drawback that there is little we can say about the behavior of the approximating function outside of the set . In particular, we have no control over the set of critical and asymptotic values of , which prevents us from being able to restrict the global function-theoretic or dynamical properties. Indeed, in order to obtain hyperbolic examples, we will at least need to be sure that the approximating function belongs to the Eremenko-Lyubich class

which was introduced in [EL2]. A related problem, which we mention here for completeness although it is not treated in this article, is approximation by functions in the Speiser class

Leaving aside dynamics for a moment, let us discuss the question of the function-theoretic behavior that these maps can exhibit. As it turns out, this will be the main problem to deal with when constructing hyperbolic examples.

If , then for sufficiently large , every component of is simply connected and mapped by as a universal covering. These components are called the tracts of (over ). If and is a tract of as above, then we can define a branch of on , and is a conformal isomorphism (where denotes the right half plane). Conversely, it is natural to ask which universal coverings can be approximated by entire functions in the class .

Question 1.5.

Suppose that is a Jordan domain whose boundary passes through infinity and that is a conformal isomorphism with . Under which conditions on does there exist an entire function (resp. ) such that

Arakelyan’s theorem implies that such a function always exists if we drop the requirement that .

A well-known way of building functions with a given tract is to use Cauchy integrals; see e.g. [PS, Part III, problem 158]. Although this method is rather old, and has had many applications over the years, it does not seem to have been treated systematically in the classical literature. The only theorem of a general nature that we are aware of was recently stated in [RS], following a construction from a paper by Eremenko and Gol’dberg [GE]. The result in [RS, Proposition 7.1] states that approximation is always possible when is the restriction of a conformal isomorphism , where is a sector, .

In [RS], this general theorem is used to construct a counterexample to the so-called strong Eremenko conjecture. However, the requirement that extends to a conformal isomorphism onto a sector of opening angle greater than is rather strong. It prevents, for instance, the construction of functions of lower order , as well as of tracts such as the one depicted in [RRS, Figure 1].

In this note, we present a considerable strenghtening of [RS, Proposition 7.1], which states that approximation is always possible if extends to a conformal isomorphism whose domain is only “slightly” larger than the half plane . It is convenient to first introduce the following definition.

Definition 1.6 ((Model functions)).

A model function is a conformal isomorphism

where

  • is an unbounded simply-connected domain;

  • is a simply-connected domain with (recall that denotes the right half plane);

  • if is a sequence with in , then in .

Theorem 1.7 ((Approximation of model functions)).

Let

where , and let be a model function.

Set . Then there exists an entire function such that

(as ). If the domain is symmetric with respect to the real axis and , then can be chosen such that .

Dynamical approximation

In order to use Theorem 1.7 to prove Theorem 1.4, we observe that the approximation automatically preserves dynamical features. The key fact is that, given our quality of approximation, the functions and are quasiconformally equivalent near in the sense of [R2]:

Theorem 1.8 ((Quasiconformal equivalence)).

Let and be as in Theorem 1.7, and let be sufficiently large.

Then there exists a quasiconformal homeomorphism such that

for all with .

This map is asymptotically conformal at ; more precisely,

as .

By [R2], this implies that the functions are quasiconformally conjugate on the set of points whose orbits stay suitably large under iteration. In our setting we can even be sure (adapting ideas from [R2]) to obtain a global conjugacy on the Julia sets of the two functions, provided that the tract is sufficiently well inside the domain .

Theorem 1.9 ((Quasiconformal conjugacy)).

There is a universal constant with the following property.

Let be as in Theorem 1.7, with the additional property that .

Then, again setting , the function in Theorem 1.7 can be chosen such that and are quasiconformally conjugate near their Julia sets.

More precisely, there is a quasiconformal homeomorphism such that

whenever , and restricts to a homeomorphism between the Julia set (i.e., the set of points that remain in under iteration of ) and the Julia set of . Furthermore, the complex dilatation of equals zero almost everywhere on .

In order to prove Theorem 1.4, it will thus be sufficient to construct a model function for which the map contains hyperbolic subsets of dimension arbitrarily close to and which satisfies the hypotheses of Theorem 1.9. (Recall that quasiconformal mappings preserve sets of Hausdorff dimension .)

Further applications

Let us state two further new theorems that can be obtained from our approximation results via known constructions. (We refer to the articles in question for background on the questions answered by these examples.) The first is a strengthening of [RS, Theorems 8.2 and 8.3].

Theorem 1.10 ((Counterexamples of low growth to the strong Eremenko conjecture)).

There exists a disjoint-type transcendental entire function such that

  1. as ,

  2. has lower order , and

  3. the Julia set has no unbounded path-connected components.

{remark}

[Remark 1] We recall that any function must have lower order at least and that no function of finite (upper) order can satisfy (c) by [RS]. {remark}[Remark 2] The condition on the growth of implies that has Hausdorff dimension equal to two [BKS].

Our second application strengthens a counterexample from [RRS].

Theorem 1.11 ((Slowly escaping Devaney hairs)).

There exists a disjoint-type transcendental entire function such that

  1. and for some ;

  2. every connected component of is an arc connecting some finite endpoint to infinity, and is such a component;

  3. every belongs to the escaping set ;

  4. the real axis does not intersect the fast escaping set of points that escape to infinity “as fast as possible” in the sense of Bergweiler and Hinkkanen [BH].

We also note that the same construction as in the proof of Theorem 1.4, with different parameters, suggests a counterexample to the area conjecture of Epstein and Eremenko. However, while this construction yields a counterexample in the class of model functions, our approximation result does not allow us to construct such a counterexample in the class . This application and its background is discussed in [ER].

Some remarks about the proof

As already mentioned, the proof of Theorem 1.7 uses Cauchy integrals. More precisely, let be defined by

and consider the function

We will show that the integral converges absolutely for and that

as . It then follows that

is the desired entire function, where is the component of that is contained in .

We should comment that the constant appearing in the definition of is not best possible: Our proof shows that it can be replaced by any constant that is larger than , and with some more careful estimates, it could be reduced further. However, our proof does not yield the analog of Theorem 1.7 for a domain of the form

where is arbitrarily small.

We also note that, for the application in Theorem 1.4, it is important that the domain is of the form as above (compare Remark 1 after the proof of Theorem 7.6): e.g. it would not be sufficient to be able to approximate functions , where

Subsequent results

Motivated by our results, Chris Bishop [Bi2] has recently given a complete answer to Question 1.5, if one considers quasiconformal equivalence instead of uniform approximation: If , then there is a function , with a single tract, that is quasiconformally equivalent to near . Furthermore, there is a function such that a suitable restriction of is quasiconformally equivalent to . (In general, cannot be constructed to have only a single tract.) These results even hold for arbitrary unions of tracts that accumulate only at infinity. In particular, Bishop’s methods allow the construction of a counterexample to the area conjecture mentioned above, in the class , and indeed a counterexample [Bi1] to the stronger order conjecture of Adam Epstein, which asked whether the order of a transcendental entire function is invariant under quasiconformal equivalence.

Structure of the article

The first part of the paper deals with approximation and the proof of Theorem 1.7. In Section 2, we prove a technical result about the approximation of holomorphic functions using Cauchy integrals. (This covers a number of known constructions.) In Section 3, we collect some basic facts about hyperbolic geometry in plane domains; these are used in Section 4 to prove Theorem 1.7 in a slightly more general framework, using the results from Section 2. The short Section 5 is dedicated to verifying that our hypotheses in Theorem 1.7 indeed satisfy the assumptions used in Section 4.

The second part of the paper consists of Section 6, which establishes the results on quasiconformal equivalence and conjugacy.

Finally, Section 7 constructs the model function required for the proof of Theorem 1.4, while Section 8 briefly discusses Theorems 1.10 and 1.11.

We remark that the three parts of the paper can be read quite independently of each other (with the exception that the hyperbolic metric estimates of Section 3 will be used throughout).

Acknowledgments

I owe great thanks to Alexandre Eremenko, who introduced me to the method of approximation via Cauchy integrals by pointing me to the paper [GE], and who has shared many profound insights on this and related problems. I would also like to thank Adam Epstein, who led me to think about the area conjecture and to discover the basic structure of the example in Theorem 1.4, and Peter Hazard, stimulating conversations with whom resulted in the realization that this example could be adapted to yield functions with full hyperbolic dimension. Finally, I would like to thank Chris Bishop, Helena Mihaljević-Brandt, Phil Rippon, Gwyneth Stallard and Mariusz Urbański for interesting discussions about this work.

Basic notation

As usual, we denote by the complex plane. We also denote the right half plane by

and the (Euclidean) disk of radius around a point by

Euclidean distance is denoted ; e.g.  is the Euclidean distance between a set and the point .

As mentioned above, we set for . We also define

for all .

If is a transcendental entire function, we denote by its set of critical and asymptotic values. (Here is an asymptotic value if there is a curve with and for .) The closure of (in ) is denoted . An alternative definition of , which is the one we will be using, is as the smallest closed set that has the property that

is a covering map.

2 Approximation using Cauchy integrals

In this section, we prove a general technical result about the approximation of holomorphic functions by Cauchy integrals. In Section 4, this will be used to deduce our main approximation theorem (Theorem 1.7).

Theorem 2.1 ((Convergence of Cauchy integrals)).

Let be a simply-connected domain and let be holomorphic. Let be an injective and piecewise smooth curve such that as , and let be the component of that is contained in . We assume that runs around in clockwise direction.

Suppose furthermore that there are constants and such that the following hold for all (recall that ):

  1. ,

  2. ,

  3. , and

  4. if , then and .

Then

(2.1)

defines a holomorphic function for , and

extends to an entire function .

Furthermore, there is a constant such that

where depends only on ; more precisely, .

Proof.

We have

hence

(2.2)

This implies that the integral in (2.1) is absolutely convergent and defines a holomorphic function on . If , then we can modify the curve slightly to avoid the point , and thus see that the restriction of to has an analytic extension to a neighborhood of ; the same is true for the restriction . Using the residue theorem, we see that the two extensions differ exactly by the function in a neighborhood of , which shows that the function defined in the statement of the theorem does indeed extend to an entire function. (Compare also [RS, Claim 2 in Section 7].)

Thus it remains to prove that . The main problem is to estimate when is close to some point . In this case, we will modify to a curve that avoids the disk of radius around .

More precisely, let . If for all , then we set . Otherwise choose with such that is minimal. Let and be the smallest, respectively largest, values of for which , and set

where is an arc of chosen such that is homotopic to in .

We then have

Hence

(2.3)

To estimate the first integral, we bound from below in terms of . We have

(2.4)

Thus , and hence, by (a),

(2.5)

So, by choice of and (d), we can bound the first integral from (2.3):

(2.6)

Now we turn to estimating the second integral in (2.3). If , then we have . Using the definition of and monotonicity of the function , we see that

(2.7)

This estimate, together with (2.2), would be sufficient to prove that the integral in question, and hence , is bounded. In order to obtain the stronger fact that , we subdivide the remaining part of the curve once more. (We note that this stronger bound is not required for the applications that we have in mind.)

Define . For , we then have

We use this to estimate the integral over the curve

The idea is that is (at least) comparable to for all points on this curve. Indeed, suppose that . Then , and hence, by (2.7),

(2.8)

If , then , hence again . Thus, using (2.2):

(2.9)

It remains to deal with the part of the curve given by

We use (2.7), as well as (b) and (c) to obtain

(2.10)

Combining the estimates (2.6), (2.9) and (2.10), the proof is complete. ∎

The following proposition shows that any function approximating a universal covering must itself have a logarithmic singularity over infinity.

Proposition 2.2.

Let be a model function, and set . Suppose that is a holomorphic function with for some and all . Define

Then is a simply-connected domain and is a universal covering map.

Proof.

Let us set . Then is simply-connected, for all and . It follows from the minimum principle that is simply-connected. We can define a branch of . By continuity, we have as tends to a point in the boundary of (in ). We claim that as . Indeed, by assumption we have, for all ,

Furthermore, the argument of and differs by less than , and hence is contained in the union

Since is connected, it follows that

for some . Hence

for a suitable constant , which proves our claim that as .

So is a proper map, and hence has some well-defined degree . extends to a degree map from the boundary of (in the Riemann sphere) to . Since only has one preimage, it follows that . Thus is a conformal isomorphism, and is a universal covering map, as claimed. ∎

Before proving our main approximation result in Section 4, let us note that Theorem 2.1 includes the examples from [PS] and [S1].

Corollary 2.3.

Let and set

Also set

(where is defined on , and on ). Let be the boundary of , described in clockwise direction. Then

extends to an entire function . Furthermore, and

Proof.

It is easy to see that the parametrizations of and by arc-length satisfy the assumptions of Theorem 2.1, say with , and being the domain of definition of . Hence is indeed defined and extends to an entire function with the stated asymptotics. The fact that belongs to the class follows from Proposition 2.2. ∎

3 The hyperbolic metric of simply-connected domains

We frequently use the hyperbolic metric in a domain that omits more than two points. (For an introduction to the hyperbolic metric, see e.g. [BM].) We denote distance with respect to this metric by , and the density of the metric by . That is,

where the infimum is taken over all curves with and .

We shall routinely use a number of standard facts about the hyperbolic metric.

Proposition 3.1 ((Properties of the hyperbolic metric)).

  1. The hyperbolic metric in the right half plane is given by . In particular, for and for every .

  2. In the strip , we have for all .

  3. If , then for all .

  4. If are hyperbolic and is a conformal isomorphism, then is a hyperbolic isometry; i.e. .

  5. If is simply connected, then for all .

Let us make two more simple observations about the hyperbolic metric in simply-connected domains.

Lemma 3.2 ((Hyperbolic distance and Euclidean distance)).

Let be a simply-connected domain, and let . Then

Proof.

Set . Let be a curve connecting and , parametrized by Euclidean arc-length. So . Then we have . Thus, by Proposition 3.1 (e),

Lemma 3.3 ((Bounded hyperbolic diameter of Euclidean disks)).

Let be a simply-connected domain, let and let . Define .

If with , then .

Proof.

Set and let be the straight line segment connecting and , parametrized by arc-length. Then

and thus, again using Proposition 3.1 (e),

Finally, we will on occasion use the following version of the Ahlfors distortion theorem [A, Corollary to Theorem 4.8].

Theorem 3.4 ((Ahlfors distortion theorem)).

Let be a simply connected domain, and let with . Let be the maximal vertical line segments passing through resp. .

Set , and let be a conformal isomorphism such that and both separate from in (i.e., they connect the upper and lower boundaries of the strip ), and such that separates from (i.e., is to the left of in ).

For , let denote the shortest length of a vertical line segment at real part that separates from in .

If , then

We also note the following fact, which is closely related to the distortion theorem:

Lemma 3.5 ((Geodesics in quadrilaterals)).

Let be a simply-connected domain that is symmetric with respect to the real axis. Let and be two cross-cuts of that are symmetric with respect to the real axis, with , and suppose that the quadrilateral bounded by and in has modulus at least . (I.e., the extremal length of the family of curves connecting and in is at least .)

Then contains a geodesic of that is symmetric with respect to the real axis.

Proof.

Let denote the strip and let be a conformal isomorphism that takes to the real axis. Set ; then is a quadrilateral in , symmetric with respect to the real axis, of modulus at least . We must show that contains a vertical segment connecting the two boundary components of .

The exponential map takes to an annulus of modulus at least , slit along an interval of the positive real axis, which separates from . By Teichmüller’s modulus theorem [A, Theorem 4-7], the closure of this annulus contains a round circle centered at the origin, which completes the proof. ∎

4 Approximation of model functions

We now turn to proving Theorem 1.7. As already mentioned, this result is “best” possible with our method, in the sense that the domain is chosen as close to a right half plane as possible while still guaranteeing convergence of the Cauchy integral. However, sometimes it is convenient to use other image domains, e.g. because it might be possible to write down an explicit mapping function for these. We will therefore work in a somewhat more general setting. In particular, we recover the results of [RS] as a special case.

Standing Assumption 4.1 ((Assumption on and )).

is a simply connected domain containing the right half plane . Furthermore,

is a piecewise smooth injective curve for which there exist positive constants , , , and with such that, for all :

  1. .

  2. . (If belongs to the discrete set where is not differentiable, this means that both the left and right derivatives are bounded by .)

  3. . (Recall that denotes the hyperbolic distance in .)

  4. If with , then .

We refer to the pair of and as the initial configuration. The bounds below will depend on this initial choice.

{remark}

[Remark 1] The final two conditions may seem somewhat technical. Roughly, they mean that the curve stays within a comparable distance from both and the line ; compare Section 5. {remark}[Remark 2] It is not difficult to see that the choice

used in the statement of Theorem 1.7 and the curve

satisfy our standing assumption. For completeness, we provide the argument in Section 5. {remark}[Remark 3] In applications, the domain and the curve will be fixed, so dependence on the initial configuration will not usually be important. However, we note that our bounds will depend only on the constants to and , but not otherwise on and .

Standing Assumption 4.2 ((Model function)).

Furthermore,

is a model function in the sense of Definition 1.6 (where is the domain from Standing Assumption 4.1). We additionally assume, by way of normalization, that that , , and .

Let be a component of and let be the conformal isomorphism . We also set . Note that we have .

Finally, we set and . Let be the component of that is contained in .

We will now show that (under these assumptions), we can apply Theorem 2.1 to and a reparametrization of .

Lemma 4.3 ((Growth and distance to boundary)).

There are constants and , depending only on the initial configuration, such that

for all .

Proof.

We set . Using the fact that hyperbolic distances in are smaller than those in the half plane (recall Proposition 3.1), we see that