Baker’s conjecture and Eremenko’s conjecture for functions with negative zeros

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.

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.

Let f be a transcendental entire function, and let {U,V subset mathbb{C}} be disjoint simply-connected domains. Must one of {f^{-1}(U)} and {f^{-1}(V)} be disconnected? In 1970, Baker implicitly gave a positive answer to this question, in order to prove that a transcendental entire function cannot have two disjoint completely invariant domains. (A domain {Usubset mathbb{C}} is completely invariant under f if {f^{-1}(U)=U}.) It was recently observed by Julien Duval that Baker's argument, which has also been used in later generalisations and extensions of Baker's result, contains a flaw. We show that the answer to the above question is negative; so this flaw cannot be repaired. Indeed, for the function {f(z)= e^z+z}, there is a collection of infinitely many pairwise disjoint simply-connected domains, each with connected preimage. We also answer a long-standing question of Eremenko by giving an example of a transcendental meromorphic function, with infinitely many poles, which has the same property. Furthermore, we show that there exists a function f with the above properties such that additionally the set of singular values S(f) is bounded; in other words, f belongs to the Eremenko–Lyubich class. On the other hand, if S(f) is finite (or if certain additional hypotheses are imposed), many of the original results do hold. For the convenience of the research community, we also include a description of the error in Baker's proof, and a summary of other papers that are affected.

Let $f_i, i=1,2,\ldots,m $ be transcendental meromorphic functions of order less than $\frac{1}{2}$ with at most finitely many poles and at least one of them has positive lower order. Let $g = f_m \circ f_{m-1}\circ \cdots \circ f_1.$ Then either $g$ has no unbounded Fatou components or at least one unbounded Fatou component $g$ is multiply connected.

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.

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.

Entire functions with no unbounded Fatou components

Let f f be a transcendental entire function and let I ( f ) I(f) be the set of points whose iterates under f f tend to infinity. We show that I ( f ) I(f) has at least one unbounded component. In the case that f f has a Baker wandering domain, we show that I ( f ) I(f) is a connected unbounded set.

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.

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.

The fast escaping set, A(f), of a transcendental entire function f has begun to play a key role in transcendental dynamics. In many cases A(f) has the structure of a spider's web, which contains a sequence of fundamental loops. We investigate the structure of these fundamental loops for functions with a multiply connected Fatou component, and show that there exist transcendental entire functions for which some fundamental loops are analytic curves and approximately circles, while others are geometrically highly distorted. We do this by introducing a real-valued function which measures the rate of escape of points in A(f), and show that this function has a number of interesting properties.

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.

Let $\{ {S_k}\}$ be any sequence of sets in the complex plane, each of which has no finite limit point. The authors prove, answering affirmatively a question posed by P. Erdös, that there exists a sequence $\{ {n_k}\}$ of positive integers and a transcendental entire function $f(z)$ such that ${f^{({n_k})}}(z) = 0$ if $z \in {S_k}$.

Let f be a transcendental entire function. The fast escaping set, A(f), plays a key role in transcendental dynamics. The quite fast escaping set, Q(f), defined by an apparently weaker condition is equal to A(f) under certain conditions. Here we introduce Q2(f) defined by what appears to be an even weaker condition. Using a new regularity condition we show that functions of finite order and positive lower order satisfy Q2(f) = A(f). We also show that the finite composition of such functions satisfies Q2(f) = A(f). Finally, we construct a function for which Q2(f) ≠ Q(f) = A(f).

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