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Algorithmische Methoden der konformen Abbildungen auf fraktale Gebiete

Kraetzer, Philipp

This doctoral thesis consists of two parts. The main subject of the first three chapters is the integral means spectrum of a conformal function f mapping the unit disc onto any bounded domain of the complex plain. The integral means spectrum of f is defined by          betaf(p) = limsup log (int |f'(z)|p |dz|) / -log(1-r) with a limes for r going to 1 and the integral for |z|=r. It measures the average growth of the derivative of f aproaching the boundary. The course of the universal integral means spectrum B(p) defined as the supremum of betaf over all bounded univalent functions is not entirely known. The author conjectured in [Complex Variables 31 (1996)] that B(p) = p2/4 for |p| < 2. The first three chapters introduce conformal maps constructed by using complex dynamics, the theory of lacunary series as well as geometric fractals as given by the snowflake. These functions are considered to have fast growing integral means. Numerical results are presented. The objective is to obtain lower bounds for the conjectured course of the universal integral means spectrum. Best results are achieved with functions having an image domain bounded by the Julia set of a quadratic polynomial. The fourth chapter deals with an other issue. We consider the trajectories of quadratic differentials and first give some well known results of their local behaviour. After that we develop an algorithm to compute precisely the trajectories for rational quadratic differentials. The regularisation of a quadratic differential near poles and zeroes of the function plays a key role in the algorithm. We consider some special cases related to function theoretic applications concerning extremal domains and present the graphic results of computer calculations.