Trabajo Fin de Grado
Iteraciones de funciones racionales sobre la esfera de Riemann
Autor/es | Martínez de la Fuente, Julia |
Director | García Vázquez, Juan Carlos |
Departamento | Universidad de Sevilla. Departamento de Análisis Matemático |
Fecha de publicación | 2022-06-19 |
Fecha de depósito | 2023-02-22 |
Titulación | Universidad de Sevilla. Grado en Matemáticas |
Resumen | The main aim of these notes is to provide some knowledge about Julia set of rational
functions. For it, we study iterations of rational functions.
In Chapter 1, we consider polynomials functions. In this way, we will ... The main aim of these notes is to provide some knowledge about Julia set of rational functions. For it, we study iterations of rational functions. In Chapter 1, we consider polynomials functions. In this way, we will study how the structure of their Julia set is. We will also study how Mandelbrot and Julia sets of a quadratic function are connected. In Chapter 2 we will provide a formal definition of rational functions and provide some important properties about them. We will study this properties in the complex sphere and prove various results involving concepts such as the Lipschitz condition, conjugacy and valency. These notions will be present throughout the remaining of these notes. The formal definition of Julia and Fatou sets of a rational function will appear in Chapter 3. Moreover, we will prove important properties and theorems such as Theorem 3.3.5 and its consequences. Finally, in Chapter 4, we will prove new consequences of Theorem 3.3.5 and talk about Property of Minimality of Julia set: it establishes that the Julia set is the smallest closed, completely invariant set with, at least, three points. At the end of this chapter, we will study an example about a rational function for which its Julia set is the complex sphere. |
Cita | Martínez de la Fuente, J. (2022). Iteraciones de funciones racionales sobre la esfera de Riemann. (Trabajo Fin de Grado Inédito). Universidad de Sevilla, Sevilla. |
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