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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2020.tde-25092020-010951
Document
Author
Full name
Pedro Iván Suarez Navarro
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2020
Supervisor
Committee
Vargas, Edson (President)
Bonnot, Sylvain Philippe Pierre
Brandão, Daniel Smania
Lomonaco, Luciana Luna Anna
Quiñones, Pablo Andrés Guarino
Title in Portuguese
Dinâmica de um produto de Blaschke
Keywords in Portuguese
Conjunto de Mandelbrot
Conjuntos de Julia
Dinâmica holomorfa
Partições de Yoccoz
Produtos de Blaschke
Abstract in Portuguese
Nesta tese nós estudamos as propriedades da dinâmica de um produtos Blaschke no espaço de fase e suas variações com o parâmetro. No plano dinâmico, nós provamos que os conjuntos de Julia são conexos, e construindo uma partição de Yoccoz no plano complexo provamos que os conjuntos Julia são localmente conexos. No plano de parâmetros, nós definimos e estudamos o conjunto de não escape, o Blasckebrot, e parametrizamos as componentes hiperbólicas no interior, as componentes de captura e provamos que o Blasckebrot é conexo. Provaremos a existência de parâmetros na família de produtos Blaschke, cujo conjunto de Julia é localmente homeomorfo a um conjunto de Julia da família de polinômios cúbicos uni-críticos.
Title in English
Dynamics of a Blaschke product
Keywords in English
Blaschke product
Holomorphic dynamics
Julia Set
Mandelbrot set
Puzzle of Yoccoz
Abstract in English
In this thesis we study the properties of the dynamics of a Blaschke product in the phase space and its variations with the parameter. In the dynamic plane, we prove that the Julia sets are connected, and by building a Yoccoz partition in the complex plane we prove that the Julia sets are locally connected. In the parameter plane, we define and study the non-escape set, the Blasckebrot, and parameterize the hyperbolic components, the capture components and prove that the Blasckebrot is connected. We will prove the existence of parameters in the Blaschke product family, whose Julia set is locally homeomorphic to a Julia set of the uni-critical cubic polynomial family
 
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Publishing Date
2021-01-20
 
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