Models for the Eremenko–Lyubich class

The study of the dynamics of polynomials is now a major field of research, with many important and elegant results. The study of entire functions that are not polynomials – in other words transcendental entire functions – is somewhat less advanced, in part due to certain technical differences compared to polynomial or rational dynamics.In this paper we survey the dynamics of functions in the Eremenko-Lyubich class,B\mathcal {B}. Among transcendental entire functions, those in this class have properties that make their dynamics most “similar” to the dynamics of a polynomial, and so particularly accessible to detailed study. Many authors have worked in this field, and the dynamics of classB\mathcal {B}functions is now particularly well-understood and well-developed. There are many striking and unexpected results. Several powerful tools and techniques have been developed to help progress this work. There is also an increasing expectation that learning new results in transcendental dynamics will lead to a better understanding of the polynomial and rational cases.We consider the fundamentals of this field, review some of the most important results, techniques and ideas, and give stepping-stones to deeper inquiry.

Let f be a transcendental entire function in the Eremenko–Lyubich class B. We give a lower bound for the Hausdorff dimension of the Julia set of f that depends on the growth of f. This estimate is best possible and is obtained by proving a more general result concerning the size of the escaping set of a function with a logarithmic tract.

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.

Suppose that $f$ is a transcendental entire function. In 2011, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected and an example of a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the Eremenko–Lyubich class for which the escaping set is not a spider’s web. Finally, we give a novel topological criterion for certain sets to be a spider’s web.

Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and, since the order of functions in the class $\mathcal B$ is at least $1/2$, we achieve the smallest possible order. Finally, we can modify the construction to obtain functions of finite order in the class $\mathcal B$ with any number of grand orbits of wandering domains, including infinitely many.

W. Bergweiler proved in 1990 that the equation f∘g= Q has infinitely many solutions in the complex plane, provided f and g are transcendental entire functions and Q is a non-constant polynomial. We prove a related result, using standard notations from the Nevanlinna theory: Let f be a non-linear entire function of finite order of growth, let g be a transcendental entire function of finite lower order and let Q be a non-constant entire function such that T(r,Q)=S(r,g). If II denotes the canonical product formed with the zeros of f∘g Q, then T(r,φ) ≠ S(r,g). As an immediate consequencef∘g−Q has infinitely many zeros.

The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results on this topic have been obtained. In this paper, we study a transcendental entire function f (z) that shares a non-zero polynomial a (z) with f′(z), together with its linear differential polynomials of the form: L[f] = a2(z)f″(z) + a3 (z)f′′′(z) + … + am (z)f(m) (z) (am (z) $\not\equiv$ 0), where the coefficients ak (z) (k = 2, 3, ..., m) are rational functions.

We prove that any subset of $\overline {\mathbb {Q}}^m$ (closed under complex conjugation and which contains the origin) is the exceptional set of uncountably many transcendental entire functions over $\mathbb {C}^m$ with rational coefficients. This result solves a several variables version of a question posed by Mahler for transcendental entire functions [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer-Verlag, Berlin, 1976)].

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set J(f) are equal. It is also known that there is a family of hyperbolic transcendental meromorphic functions with infinitely many poles for which this result fails to be true. In this paper, new methods are used to show that there is a family of hyperbolic transcendental entire functions fK, K ∈ N, such that the box and packing dimensions of JfK are equal to two, even though as K → ∞ the Hausdorff dimension of JfK tends to one, the lowest possible value for the Hausdorff dimension of the Julia set of a transcendental entire function. 2000 Mathematics Subject Classification 30D05, 37F10, 37F15, 37F35, 37F50.

We introduce a new technique that allows us to make progress on two long standing conjectures in transcendental dynamics: Baker's conjecture that a transcendental entire function of order less than 1/2 has no unbounded Fatou components, and Eremenko's conjecture that all the components of the escaping set of an entire function are unbounded. We show that both conjectures hold for many transcendental entire functions whose zeros all lie on the negative real axis, in particular those of order less than 1/2. Our proofs use a classical distortion theorem based on contraction of the hyperbolic metric, together with new results which show that the images of certain curves must wind many times round the origin.

The link on extraneous non-repelling cycles of Schröder's methods of the first and second kind

we use symbolic dynamics to describe the set of allowable itineraries of orbits of the complex exponential family λexp(z). First, we show that the set of allowable itineraries is the same for every member of this family. We then show that the set of itineraries is the same for every map of finite exponential order. In addition, we study other transcendental entire functions, and show that they also have the same set of itineraries. Finally, we give an example of a class of functions with a different set of itineraries.

Let f and F be transcendental entire functions. We are concerned with a growth estimate of F ∘ f when F satisfies the condition $$\log M(r,F)=K(\log r)^p(1+O(1)),$$ where K is a positive constant and p > 1. It is shown that $$\log\log M(r,F(f))=p\log\log M(r,f)(1+O(1)).$$ We give an application to a certain functional equation.

Bergweiler and Kotus gave sharp upper bounds for the Hausdorff dimension of the escaping set of a meromorphic function in the Eremenko–Lyubich class, in terms of the order of the function and the maximal multiplicity of the poles. We show that these bounds are also sharp in the Speiser class. We apply this method also to construct meromorphic functions in the Speiser class with preassigned dimensions of the Julia set and the escaping set.

A classic theorem of Polya shows that 2 z is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of Gel’fond concerns entire functions that are integer valued on the setX a={a n:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed.