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zur Erlangung des Doktorgrades

By Der Mathematische-naturwissenschaftlichen Fakultäten, Der Georg-august-universität Göttingen, Arnd Lauber, Koreferent Prof and Dr. Manfred Denker

Abstract

Tag der mündlichen Prüfung: This thesis deals with Baker domains and approximation of Julia sets, so it belongs to the area of ”Holomorphic Dynamical Systems”. Dynamical systems belong to the all-day life of scientists and engineers and are closely related to the term of iteration. Processes of iteration occur if the state of a system is changed by external influences at discrete points of time. Examples are the weather, turbulent flows in liquids or the development of populations. Moreover, iteration can be a tool to solve other mathematical problems or to approximate their solutions. Among various numerical methods we only mention Newton’s method to approximate roots of differentiable functions. All these dynamical systems have in common that they may develop in different directions. The boundary between different initial states of different developments is just the Julia set of the corresponding function. Julia sets were systematically analyzed for the first time around 1920 by the French matematicians P. Fatou and G. Julia, who concentrated on rational functions and observed that Julia sets are either very simple objects or extremly complicated. The development of powerful computers and new mathematical methods gave a boost to the research in this area in the 80’s, and since then also a theory of iteration of entire transcendental functions has been founded. The aim of this work is to describe what can happen to the Julia sets if an entire transcendental function satisfying a certain condition (having so-called Baker domains or wandering domains) is approximated by a sequence of polynomials or is perturbed holomorphically in a class of entire transcendental functions. 4 Organization of the paper This paper comes in four chapters: the Introduction, Preliminaries and Notation, the Results and the Proofs. All the results will be stated in chapter 3, and, for example, a result in section 3.1.1 will have its proof located in section 4.1.1

Year: 2009
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