About the connectivity of Fatou components for some families of rational maps
[eng] Rational iteration is the study of the asymptotic behaviour of the sequences given by the iterates of a rational map on the Riemann sphere. According to Montel's theory on normal families, the phase space (also called the dynamical plane) is divided in two completely in variant sets know...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/202073 |
| Acceso en línea: | https://hdl.handle.net/2445/202073 http://hdl.handle.net/10803/688996 |
| Access Level: | acceso abierto |
| Palabra clave: | Anàlisi numèrica Mètodes iteratius (Matemàtica) Teoria de conjunts Pertorbacions singulars (Matemàtica) Algorismes Numerical analysis Iterative methods (Mathematics) Set theory Singular perturbations (Mathematics) Algorithms |
| Sumario: | [eng] Rational iteration is the study of the asymptotic behaviour of the sequences given by the iterates of a rational map on the Riemann sphere. According to Montel's theory on normal families, the phase space (also called the dynamical plane) is divided in two completely in variant sets known as the Fatou set (an open set where the dynamics is tame) and the Julia set (a closed set where the dynamics is chaotic). The main topic of this thesis is the study of the connectivity of the Fatou components for certain families of rational maps. On the one hand, we consider a family of singular perturbation and extend previous results on singular perturbations of Blaschke products. The main result is to show that the dynamical planes for the corresponding maps present Fatou components of arbitrarily large connectivity and determine precisely these connectivities. On the other hand, we consider a different problem related to root-finding algorithms. More precisely, we study the Chebyshev-Halley methods applied to a symmetric family of polynomials of arbitrary degree. The main goal is to show the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. Moreover, we also prove that the corresponding dynamical plane contains a connected component of the Julia set, which is a quasiconforrnal deformation of the Julia set of the map obtained by applying Newton's method. |
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