Norbert Steinmetz

TU Dortmund University | TUD · Faculty of Mathematics

The 4IM+1CM-Problem is determining all pairs (f, g) of meromorphic functions in the complex plane that are not Möbius transformations of each other and share five pairs of values, one of them CM (counting multiplicities). In the present paper it is shown that each such pair parameterises some algebraic curve K(x, y) = 0 of genus zero and bounded de...

In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in (Carleson and Gamelin in Complex dynamics, Springer, Berlin, 1993; Steinmetz in Math Ann 307:531–541, 1997), will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critic...

In this paper, the classification of rational functions whose Julia sets are Jordan arcs or curves, which started in [2, 6], will be completed. The method of proof is based on two quasi-conformal surgery procedures, which enables shifting the critical points in simply connected (super-)attracting and parabolic basins into a single critical point of...

Complex Numbers and Functions

The purpose of this paper is to determine the main properties of Laplace contour integrals $$\begin{aligned} \Lambda (z)=\frac{1}{2\pi i}\int _\mathfrak {C}\phi (t)e^{-zt}\,dt \end{aligned}$$ Λ ( z ) = 1 2 π i ∫ C ϕ ( t ) e - z t d t that solve linear differential equations $$\begin{aligned} L[w](z):=w^{(n)}+\sum _{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0. \e...

The aim of this paper is to classify the cubic polynomials $$\begin{aligned} H(z,x,y)=\sum _{j+k\le 3}a_{jk}(z)x^jy^k \end{aligned}$$ H ( z , x , y ) = ∑ j + k ≤ 3 a jk ( z ) x j y k over the field of algebraic functions such that the corresponding Hamiltonian system $$x'=H_y,$$ x ′ = H y , $$y'=-H_x$$ y ′ = - H x has at least one transcendental al...

The purpose of this paper is to determine the main properties of Laplace contour integrals Λ(z) = 1 2πi C φL(t)e −zt dt, that solve linear differential equations L[w](z) := w (n) + n−1 j=0 (aj + bjz)w (j) = 0. This concerns, in particular, the order of growth, asymptotic expansions , the Phragmén-Lindelöf indicator, the distribution of zeros, the e...

Using a construction due to Grunsky we will describe the proper mappings of non-degenerate multiply connected domains onto the unit disc.

This paper is engaged with Painlevé’s differential equation P34:2ww″ = w″2 + 2w2(2w − z) − α, also known as Ince’s equation XXXIV and closely related to Painlevé’s second differential equation \({{\rm{P}}_\Pi}:\varpi\prime\prime= \alpha + z\varpi + 2{\varpi^3}\). We will show that the transcendental solutions belong to the Yosida class \({\mathfrak...

The aim of this paper is to classify the cubic polynomials over the field of algebraic functions such that the corresponding Hamiltonian system has at least one transcendental algebroid solution. Ignoring trivial subcases, the investigations essentially lead to several nontrivial Hamiltonians which are closely related to Painlevé's equations PI, PI...

This paper is engaged with Painlevé's differential equation P34 : 2ww = w 2 + 2w 2 (2w − z) − α, also known as Ince's equation XXXIV and closely related to Painlevé's second differential equation PII : = α+z +2 3. We will show that the transcendental solutions belong to the Yosida class Y 1, 1 2 and have no deficient rational targets. We will also...

It is shown that two non-constant meromorphic functions that share two pairs of values counting multiplicities and share three other pairs of values ignoring multiplicities are Mobius transformations of each other. This result is sharp. Examples of meromorphic functions that share four or five pairs of values are discussed.

In this chapter we will discuss several topics in Complex Analysis which usually are not or only incomprehensively considered in lectures and textbooks, but are of particular interest in the field of Analytic and Algebraic Differential Equations. Our standard reference is Ahlfors’ forever young Complex Analysis.

The present chapter is devoted to applications of Nevanlinna Theory to general questions in the theory of entire and meromorphic functions. This concerns algebraic differential and functional equations, uniqueness of meromorphic functions, and the value distribution of differential polynomials. We will always consider meromorphic functions in the p...

One of the most difficult problems in the theory of Algebraic Differential Equations is to decide whether or not the solutions are meromorphic in the plane. In case this question has been answered satisfactorily, which by experience requires particular strategies adapted to the equations under consideration, there remain several major problems to b...

In this chapter Nevanlinna Theory, Cartan’s Theory of Entire Curves, and the Selberg–Valiron Theory of Algebroid Functions will be outlined, particularly with regard to the applications in the subsequent chapters and including recent developments in the context of the Second Main Theorem. Nevanlinna Theory provides the most effective tools in the m...

In this chapter we will extend the investigations of the previous chapter to second-order algebraic differential equations and two-dimensional Hamiltonian systems whose solutions are meromorphic functions. Having established the Painlevé property for distinguished equations and systems, we will draw a comprehensive picture of the solutions. This in...

In this chapter the theory of Normal Families will be deepened and enlarged, and applied to various problems in the fields of entire and meromorphic functions, distribution of zeros of differential polynomials, ordinary differential equations and functional equations. The Yosida classes, which play an outstanding part in the theory of algebraic dif...

It is shown that two non-constant meromorphic functions that share two pairs of values counting multiplicities and share three other pairs of values ignoring multiplicities are Möbius transformations of each other. This result is sharp. Examples of meromorphic functions that share four or five pairs of values are discussed.

We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painlevé's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).

This book offers a modern introduction to Nevanlinna theory and its intricate relation to the theory of normal families, algebraic functions, asymptotic series, and algebraic differential equations. Following a comprehensive treatment of Nevanlinna’s theory of value distribution, the author presents advances made since Hayman’s work on the value d...

It is shown that two non-constant meromorphic functions that share two pairs of values counting multiplicities and share three other pairs of values ignoring multiplicities are Mobius transformations of each other. This result is sharp. Examples of meromorphic functions that share four or �ve pairs of values are discussed.

In this paper we discuss the dynamical structure of the rational family $(f_t)$ given by $$f_t(z)=tz^{m}\Big(\frac{1-z}{1+z}\Big)^{n}\quad(m\ge 2,~t\ne 0).$$ Each map $f_t$ has two super-attracting immediate basins and two free critical points. We prove that for $0<|t|\le 1$ and $|t|\ge 1$, either of these basins is completely invariant and at leas...

We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus zero.

We utilise a recent approach via the so-called re-scaling method to derive a unified and comprehensive theory of the solutions to Painleve's differential equations (I), (II) and (IV), with emphasis on the most elaborate equation (IV).

We utilise recent results about the transcendental solutions to Riccati differential equations to provide a comprehensive description of the nature of the transcendental solutions to algebraic first order differential equations of genus zero.

This is a report on de Branges' proof of the Bieberbach conjecture based on two one-hour-lectures given at the Oberwolfach Conference on General Iequalities May 5-9 1986.

Based on the so-called re-scaling method, we will give a detailed description of the solutions to the Hamiltonian system (\ref{Hsystem}) below, which was discovered only recently by Kecker, and is strongly related to Painleve's fourth differential equation. In particular, the problem to determine those fourth Painleve transcendents with positive Ne...

We determine all pairs (f,g) of meromorphic functions that share four pairs of values (aν,bν), 1≤ν≤4, and a fifth pair (a5,b5) under some natural constraint.

We determine all pairs $(f,g)$ of meromorphic functions that share four pairs of values $(a_\nu,b_\nu)$, $1\le\nu\le 4$, and a fifth pair $(a_5,b_5)$ under some mild additional condition.

We utilise a new approach via the so-called re-scaling method to derive a thorough theory for polynomial Riccati differential equations in the complex domain.

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $${f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}$$ with free critical orbit $$\pm\sqrt{2}\mathop \rightarrow \limits^{(2)}0 \mathop \rightarrow \limits^{(4)}t \mathop\rightarrow \limits^{(1)}\cdots.$$ In...

We give a new proof of the fact that the solutions of Painlevé's differential equations I, II and IV are meromorphic functions in the complex plane. The method of proof is based on differential inequality techniques.

In a recent paper, Aimo Hinkkanen and Ilpo Laine proved that the transcendental solutions to Painleve's second differential equation w"=a+zw+w^3 have either order of growth 3 or else 3/2. We complete this result by proving that there exist no sub-normal solutions (order of growth 3/2) other than the so-called Airy solutions.

We discuss families of meromorphic functions $f_h$ obtained from single functions $f$ by the re-scaling process $f_h(z)=h^{-\alpha}f(h+h^{-\beta}z)$ generalising Yosida's process $f_h(z)=f(h+z)$. The main objective is to obtain information on the value distribution of the generating functions $f$ themselves. Among the most prominent generalised Yos...

The aim of this paper is to describe the origin, first solutions, further progress, the state of art, and a new ansatz in the treatment of a problem dating back to the 1920's, which still has not found a satisfactory solution and deserves to be better known.

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps $\ds f_t(z)=-\frac{t}{4}\frac{(z^{2}-2)^{2}}{z^{2}-1}$ with free critical orbit $\pm\sqrt{2}\xrightarrow{(2)}0\xrightarrow{(4)}t\xrightarrow{(1)}...$. In particular it is shown that for any escape parameter $t$ the boun...

It is proved that any family of analytic functions with spherical derivative uniformly bounded away from zero ist normal.

We discuss the dynamics as well as the structure of the parameter plane of certain families of rational maps with few critical orbits. Our paradigm is the family Rt(z) = 𝔱 (1 + (4/27)z3/(1 − z)), with dynamics governed by the behaviour of the postcritical orbit (R𝔱n(𝔱))n∈ℕ. In particular, it is shown that if 𝔱 escapes (that is, R𝔱n(𝔱) tends to infi...

The workshop Normal Families and Complex Dynamics , organised by Phil Rippon (Milton Keynes), Norbert Steinmetz (Dortmund) and Lawrence Zalcman (Ramat-Gan) was held February 18th–February 24th, 2007. Normal families and Nevanlinna theory. Almost half a century after it first appeared, Hayman's seminal work [ Picard values of meromorphic functions a...

In this note we prove that the so-called Sierpi\’nski holes in the parameter plane 0 < ¦λ¦ < ∞ of the McMullen family Fλ(z) = z m + λ/z ℓ, m ≥ 2 and ℓ ≥ 1 fixed, are simply connected, and determine the total number of these domains.

In this note we discuss the parameter plane and the dynamics of the rational family R(z)=z^m+lambda/z^{l} , with m>= 2 , l>= 1 , and 0

In this note we discuss the parameter plane and the dynamics of the rational family R(z) = zm + ‚=z', with m ‚ 2; ' ‚ 1, and 0 < j‚j < 1:

We prove several lower estimates for the Nevanlinna characteristic functions and the orders of growth of the Painlev e transcendents I, II and IV. In particular it is shown that (a) lim supr!1 T (r; w1)=r5=2 > 0 for any rst transcendent, (b) %(w2) 3 2 for most classes of second transcendents, (c) %(w4) 2 for several classes of fourth transcendents,...

We prove that any transcendental solution of Painlevé’s second equation w″ = α + zw + 2w 3, which has the form w = R(z,u), with R rational in both variables and non-linear with respect to u, is obtained by repeated application of the Bäcklund transformation to some solution of the Riccati equation U′ = ±(z/2 + U 2). In particular, \(\alpha = n+1/2,...

We consider the solutions of the First Painlevé Differential Equationω″=z+6w 2, commonly known as First Painlevé Transcendents. Our main results are the sharp order estimate λ(w)≤5/2, actually an equality, and sharp estimates for the spherical derivatives ofw andf(z)=z −1w(z 2), respectively:w#(z)=O(|z|3/4) andf#(z)=O(|z|3/2). We also determine in...

We give a new existence proof for a singular metric on a marked planar domain via First-order algebraic differential equations. This singular metric applies in complex dynamics to sub-hyperbolic rational functions.

Let , be a linear differential operator, whose coefficients are (constants or) 2i-periodic entire functions of order one, mean type. We will prove that any exceptional solution of L[L] = 0, i.e., any solution satisfying , has the form where q ≥ 1 is an integer, the c j's are complex constants and S and the P j's are polynomials. We give also a new...

We prove that, with two exceptions, the set of polynomials with Julia set J has the form fõ pn : n 2 N;õ 2 Üg; where p is one of these polynomials and Ü is the symmetry group of J : The exceptions occur when J is a circle or a straight line segment.

An elementary proof of the Riemann-Hurwitz Formula for plane domains is given, avoiding the concept of Euler-characteristic.

We prove a sharp order estimate for entire functions of completely regular growth, whose zeros are distributed near Þnitely many rays arg z = !j in terms of the angles !j. This result then leads imme- diately to a proof of a conjecture of Hellerstein and Rossi concerning the distribution of zeros of the solutions of linear dierential equations with...

A short proof is given of sulhvan's Classification Theorem of periodic stable domains in the theory of internation of rational maps.

On considere des equations differentielles lineaires homogenes a coefficients polynomiaux. On etudie la distribution des zeros des solutions, la densite des zeros des solutions individuelles et la densite des zeros d'un ensemble fondamental de solutions

It is shown that the second order differential equation where L is rational, has a non-rational meromorphic solution w=win the plane only if w(z)is also a solution of some Riccati equation with rational coefficients or of an equation = n=2, 3.4 or 6. In this case and respectively.

This is a report on de Branges’ proof of the Bieberbach conjecture based on two one-hour-lectures given at the Oberwolfach Conference on General Inequalities (May 5–9, 1986). The main objective was to present to the audience an idea of the function-theoretic background, especially of Löwner’s beautiful theory.

Über diese Arbeit wurde auf eiriern “Tag der Funktionentheorie“ am 15. und 16.6.1984 an der RWTH Aachen berichtet. Let Ω denote the set of continuous functions Q:[0,1)→R such that (a) (1−x)Q(x) is nonincreasing and (b) the solution of the initial value problem y″ +Q(x)y=0, y(0)=1, y′(0)=0 is Positive in [0, 1). We prove the following theorem: Let f...

It is shown that the order and lower order of an entire function with zeros restricted to k distinct rays differ at most by A, if either k ≤ 2 or if the zeros or the rays are regularly distributed.

Zusammenfassung. In dieser Arbeit werden die Pólyaschen Exponentialpolynome und ihre Quotienten unter dem Gesichtspunkt der Nevanlinnaschen Wertverteilungslehre untersucht. Falls eine sehr allgemeine Exponenten- und Koeffizientenbedingung erfüllt ist, dann hat man einfache geometrische Formeln für die charakteristischen Größen. Eine wesentliche Rol...

It is shown that the general solution of a homogeneous second-order algebraic differential equation has the form w(z)=(g1(z)/g2(z)) exp (g3(z)), where g1, g2 and g3 are entire functions of finite order.