Norbert Steinmetz

On Painlevé's equations I, II and IV

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

On the dynamics of the McMullen family ()=^{}+/^{ℓ}

In this note we discuss the parameter plane and the dynamics of the rational family R(z) = z + λ/z, with m ≥ 2, ` ≥ 1, and 0 < |λ| < ∞.

Value distribution of the Painlevé Transcendents

We consider the solutions of the First Painlevé Differential Equationω″=z+6w2, 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(z2), respectively:w#(z)=O(|z|3/4) andf#(z)=O(|z|3/2). We also determine in some detail the local asymptotic distribution of poles, zeros anda-points. The methods also apply to Painlevé’s Equations II and IV.

Sierpiński Curve Julia Sets of Rational Maps

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

The formula of riemann-hurwitz and iteration of rational functions

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

An old new class of meromorphic functions

Based on the so-called rescaling method, we give a detailed description of the solutions of the Hamiltonian system (1) below, which was discovered only recently by Kecker and is strongly related to Painlevé’s fourth differential equation. In particular, the problem of determining those fourth Painlevé transcendents with positive Nevanlinna deficiency δ(0,w) is completely resolved.

First order algebraic differential equations of genus zero

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.

On the Dynamics of the Rational Family f_{t}(z)=- t/4{(z^{2}- 2)^{2}/(z^{2}- 1)}

In this paper we discuss the dynamics as well as the structure of the parameter space of the one-parameter family of rational maps

On the dynamics of rational maps with two free critical points

{f_{t}(z)}= - {t\over 4} {(z^{2}- 2)^{2}\over {z^{2}- 1}}