Walter Bergweiler

Iteration of meromorphic functions

This paper attempts to describe some of the results obtained in the iteration theory of transcendental meromorphic functions, not excluding the case of entire functions. The reader is not expected to be familiar with the iteration theory of rational functions. On the other hand, some aspects where the transcendental case is analogous to the rational case are treated rather briefly here. For example, we introduce the different types of components of the Fatou set that occur in the iteration of rational functions but omit a detailed description of these types. Instead, we concentrate on the types of components that are special to transcendental functions (Baker domains and wandering domains).

On semiconjugation of entire functions

Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g ∘ f = h ∘ g . We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g −1 ( J ( h ))= J ( f ). Here J (·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h , there are only countably many entire functions g such that g ∘ f = h ∘ g .

Invariant domains and singularities

Let U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U . Let P ( f ) be the closure of the set of the forward orbits of all critical and asymptotic values of f . We show that there exists a sequence p n ∈ P ( f ) such that dist( p n , U ) = o (| p n |), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist ( P ( f ), U ) > 0. In this example, U is bounded by a Jordan curve.

Bloch’s Principle

A heuristic principle attributed to André Bloch says that a family of holomorphic functions is likely to be normal if there are no non-constant entire functions with this property. We discuss this principle and survey recent results that have been obtained in connection with it. We pay special attention to properties related to exceptional values of derivatives and existence of fixed points and periodic points, but we also discuss some other instances of the principle.

Periodic points of entire functions: proof of a conjecture of Baker

Let f be an entire transcendental function and denote the nth iterate off by f n. Our main result is that if n ≥ 2, then there are infinitely many fixpoints of f n which are not fixpoints of fk for any k satisfying 1 ≤ k ≤ n. This had been conjectured by I. N. Baker in 1967. Actually, we prove that there are even infinitely many repelling fixpoints with this property. We also give a new proof of a conjecture of E Gross from 1966 which says that if h and g are entire transcendental functions, then the composite function hog has infinitely many fixpoints. We show that h∘g. has even infinitely many repelling fixpoints.

On the Julia set of analytic self-maps of the punctured plane

Let f be a non-constant and non-linear entire function, g an analytic self-map of C\{0}, and suppose that exp ◦f = g ◦ exp. It is shown that z is in the Julia set of f if and only if e is in the Julia set of g. 1991 Mathematics Subject Classification: 30D05, 58F23

A NEW PROOF OF THE AHLFORS FIVE ISLANDS THEOREM

We deduce the Ahlfors five islands theorem from a corresponding result of Nevanlinna concerning perfectly branched values, a rescaling lemma for non-normal families and an existence theorem for quasiconformal mappings. We also give a proof of Nevanlinna’s result based on the rescaling lemma and a version of Schwarz’s lemma.

On the zeros of certain homogeneous differential polynomials

E. Mues [10] proved in 1978 that if a ∈ C\{1} and if f is a transcendental entire function which is not of the form f(z) = exp(αz + β) where α, β ∈ C, then f(z)f ′′(z) − af ′(z)2 has at least one zero. The case a = 0 is due to W. K. Hayman [5, Theorem 5]. As shown by examples like f(z) = cos z the conclusion need not hold if a = 1, see [10] for further examples. If we allow f to be meromorphic, then we have further exceptional values for a. In fact, an easy computation shows that if a = (n + 1)/n for some n ∈ N and if f(z) = F (z)−n for an entire function F with the the property that F ′′ has no zeros, then f(z)f ′′(z)−af ′(z)2 = −nF (z)−2n−1F ′′(z) has no zeros. It seems reasonable to conjecture if f is a transcendental meromorphic function not of the form f(z) = exp(αz + β) and if a 6= 1 and a 6= (n + 1)/n, then f(z)f ′′(z) − af ′(z)2 has at least one zero. This has recently been proved by J. K. Langley [8] in the case that a = 0 and had been obtained earlier by Mues [9] in this case for functions of finite lower order.

On the nevanlinna characteristic of a composite function

Let f be a meromorphic function and g an entire function. T(r f) the Nevanlinna charachteristic of f and M (r g) the maximum modulus of g. We show that and discuss how far this inequality is best possible.

Zeros of Solutions of a Functional Equation

AbstractWe consider the zeros of transcendental entire solutions f of the functional equation