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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2023.tde-22022024-164112
Document
Author
Full name
Artur Almeida Moura de Oliveira
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2023
Supervisor
Committee
Lomonaco, Luciana Luna Anna (President)
Faria, Edson de
Hazard, Peter Edward
Title in Portuguese
Um estudo sobre as Coordenadas de Fatou durante a bifurcação parabólica
Keywords in Portuguese
Bifurcação parabólica
Coordenadas de Fatou
Dinâmica complexa
Implosão parabólica
Abstract in Portuguese
A presente dissertação possui o objetivo de trazer detalhes sobre a implosão parabólica utilizando as Coordenadas de Fatou, estudando os conceitos e principais teoremas da bifurcação parabólica dentro da área da dinâmica complexa em C. Para um estudo completo, introduzimos conceitos básicos do tema, além de importantes resultados dos mapas quase-conformes, tais como a cirurgia quase-conforme, que foi de grande interesse para compreender a bifurcação parabólica, objetivo principal do trabalho. A área de dinâmica complexa foi primeiramente estudada por Gaston Julia, Pierre Fatou, Lucjan Böttcher, Gabriel Koenigs, Ernst Schröder, entre outros e, ultimamente Shishi- kura, Douady, Hubbard e Sullivan fizeram importantes contribuições para o avanço da Dinâmica Complexa Moderna. O trabalho usa como referência a obra de Shishikura, intitulada Bifurcação de pontos Fixos Parabólicos. Cada capítulo do trabalho está organizado por temas. O primeiro capítulo traz detalhes sobre os conceitos básicos da dinâmica complexa que são importantes para o entendimento da bifurcação de pontos fixos parabólicos. O segundo traz detalhes sobre os mapas quaseconformes. O terceiro sobre a dinâmica parabólica. O quarto traz o objetivo principal do tra- balho, a bifurcação parabólica. Por fim, nos último capítulos, os apêndices.
Title in English
A study about the Fatou Coordinates during the parabolic fixed points bifurcation
Keywords in English
Complex dynamics
Fatou coordinates
Parabolic bifurcation
Parabolic implosion
Abstract in English
The present dissertation has the objective of bringing details about the parabolic implosion using the Fatou coordinates, studying the concepts and main theorems of the parabolic bifurcation within the area of complex dynamics in the complex plane. For a complete study, we introduce basic concepts of the area, in addition to important results of the quasiconformal mappings, such as the quasiconformal surgery, which was of great interest to understand the parabolic bifurcation, the main objective of the work. The area of complex dynamics was first studied by Gaston Julia, Pierre Fatou, Lucjan Böttcher, Gabriel Koenigs, Ernst Schröder, among others, and lately Shishi- kura, Douady, Hubbard and Sullivan made important contributions to the advancement of Modern Complex Dynamics. The work uses as a reference the paper by Shishikura, entitled Bifurcation of Parabolic Fixed Points. Each chapter of this paper is organized by themes. The first chapter details the basic concepts of Complex Dynamics which are important to understanding the bifurcation of parabolic fixed points. The second brings details about the quasiconformal mappings. The third on Parabolic Dynamics. The fourth brings the main objective of the work: the parabolic bifurcation. And, in the last chapters, the appendices.
 
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Publishing Date
2024-02-27
 
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