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Theses

Construction of rational maps with prescribed dynamics

Abstract : In this thesis, we are interested in the existence criterions and the effective construction of rational maps with prescribed dynamics. We start by studying the same problem for some post-critically finite ramified coverings and we give a construction method from dynamical trees. Then we present a Thurston's theorem which provides a combinatorial characterization to go from the topological point of view to the analytical one. In particular, we generalize to non-post-critically finite maps a Levy's result which simplifies the Thurston's criterion in the polynomial case. We illustrate this generalization by a sufficient condition for existence of polynomials with a fixed Siegel disk of bounded type. Next we detail the construction by quasiconformal surgery of an example of non-post-critically finite rational map whose dynamics is described by a tree. More generally, we show that a result of Cui Guizhen and Tan Lei allows to construct a family of rational maps with disconnected Julia sets from some weighted Hubbard trees.
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Sébastien Godillon. Construction of rational maps with prescribed dynamics. Dynamical Systems [math.DS]. Université de Cergy Pontoise, 2010. English. ⟨tel-00796874⟩

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