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Connectivity of Julia sets of transcendental meromorphic functions
Taixés i Ventosa, Jordi
Fagella, Núria, dir. (Universitat de Barcelona. Grup de Sistemes Dinàmics)
Jarque i Ribera, Xavier, dir. (Universitat de Barcelona)

Publicació: [Barcelona] : Universitat de Barcelona 2011
Descripció: 1 recurs electrònic (113 p.)
Resum: Newton's method associated to a complex holomorphic function f is defined by the dynamical system Nf(z) = z – f(z) / f'(z). As a root-finding algorithm, a natural question is to understand the dynamics of Nf about its fixed points, as they correspond to the roots of the function f. In other words, we would like to understand the basins of attraction of Nf, i. e. , the sets of points that converge to a root of f under the iteration of Nf. Basins of attraction are actually just one type of stable component or component of the Fatou set, defined as the set of points for which the family of iterates is defined and normal locally. The Julia set or set of chaos is its complement (taken on the Riemann sphere). The study of the topology of these two sets is key in Holomorphic Dynamics. In 1990, Mitsuhiro Shishikura proved that, for any non-constant polynomial P, the Julia set of NP is connected. In fact, he obtained this result as a consequence of a much more general theorem for rational functions: If the Julia set of a rational function R is disconnected, then R has at least two weakly repelling fixed points. With the final goal of proving the transcendental version of this theorem, in this Thesis we see that: If a transcendental meromorphic function f has either a multiply-connected attractive basin, or a multiply-connected parabolic basin, or a multiply-connected Fatou component with simply-connected image, then f has at least one weakly repelling fixed point. Our proof for this result is mainly based in two techniques: quasiconformal surgery and the study of the existence of virtually repelling fixed points. We conclude the Thesis with an idea of the strategy for the proof of the case of Herman rings, as well as some ideas for the case of Baker domains, which is left as a subject for a future project.
Nota: Tesi doctoral - Universitat de Barcelona. Departament de Matemàtica Aplicada i Anàlisi, 2011
Drets: Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades. Creative Commons
Llengua: Anglès
Document: Tesis i dissertacions electròniques ; recerca ; doctoralThesis ; preprint
Matèria: Connectivitat ; Meromorfa ; Trascendent ; Conjunt de Julia ; Dinàmica complexa ; Sistemes dinàmics ; Holomorf ; Connectivity ; Meromorphic ; Transcendental ; Julia set ; Complex dynamics ; Dynamical systems ; Holomorphic

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Documents de recerca > Documents dels grups de recerca de la UAB > Centres i grups de recerca (producció científica) > Ciències > GSD (Grup de sistemes dinàmics)
Documents de recerca > Tesis doctorals

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