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...
| Autor: | |
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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 |
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About the connectivity of Fatou components for some families of rational mapsParaschiv, Dan AlexandruAnàlisi numèricaMètodes iteratius (Matemàtica)Teoria de conjuntsPertorbacions singulars (Matemàtica)AlgorismesNumerical analysisIterative methods (Mathematics)Set theorySingular perturbations (Mathematics)Algorithms[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.[cat] La iteració racional és l'estudi del comportament asimptòtic de les seqüencies donades pels iterats d'una funció racional sobre l'esfera de Riemann. Segons la teoria de Montel sobre les famílies normals, l'espai de fases (també anomenat pla dinàmic) es divideix en dos conjunts totalment invariants coneguts com a conjunt de Fatou (unió de components oberts on la dinàmica és essencialment senzilla) i el conjunt de Julia (un conjunt tancat on la dinàmica és caòtica). El tema principal d'aquesta tesi és l'estudi de la connectivitat de les components de Fatou pera determinar les famílies de funcions racionals. D'una banda, l'autor considera una familia de pertorbacions singulars i amplia els resultats anteriors sobre pertorbacions singulars dels productes de Blaschke. El resultat principal és mostrar que els plans dinàmics d'aquestes funcions presenten components de Fatou de connectivitat arbitràriament grans i determinen precisament aquestes connectivitats. D'altra banda, l’autor considera un problema diferent relacionat amb els algorismes de recerca d'arrel. Més precisament, estudia els mètodes Chebyshev-Halley aplicats a una família simètrica de polinomis de grau arbitrari. L'objectiu principal és mostrar l'existència de paràmetres de manera que les conques d'atracció immediates corresponents a les arrels de la unitat tinguin connectivitat infinita. A més, també demostra que el pla dinàmic corresponent conté una component connexa del conjunt de Julia, que és una deformació quasiconforme del conjunt de Julia de la funció obtinguda aplicant el mètode de Newton.Universitat de BarcelonaCanela Sánchez, JordiJarque i Ribera, XavierUniversitat de Barcelona. Departament de Matemàtiques i Informàtica2023info:eu-repo/semantics/doctoralThesisinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/202073http://hdl.handle.net/10803/688996Tesis Doctorals - Departament - Matemàtiques i Informàticareponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaIngléscc by (c) Paraschiv, Dan Alexandru, 2023http://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/2020732026-05-27T06:46:51Z |
| dc.title.none.fl_str_mv |
About the connectivity of Fatou components for some families of rational maps |
| title |
About the connectivity of Fatou components for some families of rational maps |
| spellingShingle |
About the connectivity of Fatou components for some families of rational maps Paraschiv, Dan Alexandru Anàlisi numèrica Mètodes iteratius (Matemàtica) Teoria de conjunts Pertorbacions singulars (Matemàtica) Algorismes Numerical analysis Iterative methods (Mathematics) Set theory Singular perturbations (Mathematics) Algorithms |
| title_short |
About the connectivity of Fatou components for some families of rational maps |
| title_full |
About the connectivity of Fatou components for some families of rational maps |
| title_fullStr |
About the connectivity of Fatou components for some families of rational maps |
| title_full_unstemmed |
About the connectivity of Fatou components for some families of rational maps |
| title_sort |
About the connectivity of Fatou components for some families of rational maps |
| dc.creator.none.fl_str_mv |
Paraschiv, Dan Alexandru |
| author |
Paraschiv, Dan Alexandru |
| author_facet |
Paraschiv, Dan Alexandru |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Canela Sánchez, Jordi Jarque i Ribera, Xavier Universitat de Barcelona. Departament de Matemàtiques i Informàtica |
| dc.subject.none.fl_str_mv |
Anàlisi numèrica Mètodes iteratius (Matemàtica) Teoria de conjunts Pertorbacions singulars (Matemàtica) Algorismes Numerical analysis Iterative methods (Mathematics) Set theory Singular perturbations (Mathematics) Algorithms |
| topic |
Anàlisi numèrica Mètodes iteratius (Matemàtica) Teoria de conjunts Pertorbacions singulars (Matemàtica) Algorismes Numerical analysis Iterative methods (Mathematics) Set theory Singular perturbations (Mathematics) Algorithms |
| description |
[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. |
| publishDate |
2023 |
| dc.date.none.fl_str_mv |
2023 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/doctoralThesis info:eu-repo/semantics/publishedVersion |
| format |
doctoralThesis |
| status_str |
publishedVersion |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2445/202073 http://hdl.handle.net/10803/688996 |
| url |
https://hdl.handle.net/2445/202073 http://hdl.handle.net/10803/688996 |
| dc.language.none.fl_str_mv |
Inglés |
| language_invalid_str_mv |
Inglés |
| dc.rights.none.fl_str_mv |
cc by (c) Paraschiv, Dan Alexandru, 2023 http://creativecommons.org/licenses/by/3.0/es/ info:eu-repo/semantics/openAccess |
| rights_invalid_str_mv |
cc by (c) Paraschiv, Dan Alexandru, 2023 http://creativecommons.org/licenses/by/3.0/es/ |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Universitat de Barcelona |
| publisher.none.fl_str_mv |
Universitat de Barcelona |
| dc.source.none.fl_str_mv |
Tesis Doctorals - Departament - Matemàtiques i Informàtica reponame:Dipòsit Digital de la UB instname:Universidad de Barcelona |
| instname_str |
Universidad de Barcelona |
| reponame_str |
Dipòsit Digital de la UB |
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Dipòsit Digital de la UB |
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1869423793224548352 |
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15.300724 |