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Title: The Denjoy-Wolff theorem, extensions and applications
Author: Jové Campabadal, Anna
Director/Tutor: Fagella Rabionet, Núria
Keywords: Equacions funcionals
Treballs de fi de grau
Funcions analítiques
Dominis de Siegel
Mètodes iteratius (Matemàtica)
Sistemes dinàmics complexos
Funcions meromorfes
Functional equations
Bachelor's theses
Analytic functions
Siegel domains
Iterative methods (Mathematics)
Complex dynamical systems
Meromorphic functions
Issue Date: 22-Jun-2020
Abstract: [en] The aim of this project is to prove the Denjoy-Wolff Theorem, which deals with iteration of holomorphic self-maps of the unit disk D. It claims that either the map is conjugate to a rotation about the origin or all the points converge to a unique point in D under iteration. We will also prove that there always exists a fundamental set, an invariant subset reached by all the compact sets in a finite number of iterations and where the map is one-to-one. Fundamental sets can be classified in four different types, up to conformal conjugation. Finally, we will use this results to classify the periodic Fatou components of entire maps. For each of them, we can find a fundamental set. In the case of attracting or parabolic components or Siegel disks, the dynamics in the fundamental set is determined up to conformal conjugation. However, in the case of Baker domains three different types can occur and we will present some examples of them.
Note: Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2020, Director: Núria Fagella Rabionet
Appears in Collections:Treballs Finals de Grau (TFG) - Matemàtiques

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