Density of hyperbolicity in families of complex rational maps

Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2025. Director: Kostiantyn Drach

Detalles Bibliográficos
Autor: Timoner Vaquer, Francesc
Tipo de recurso: tesis de maestría
Fecha de publicación:2025
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/228236
Acceso en línea:https://hdl.handle.net/2445/228236
Access Level:acceso abierto
Palabra clave:Sistemes dinàmics complexos
Funcions de variables complexes
Aplicacions quasiconformes
Treballs de fi de màster
Anàlisi de Fourier
Sistemes dinàmics diferenciables
Francesc Timoner Vaquer
Complex dynamical systems
Functions of complex variables
Quasiconformal mappings
Master's thesis
Fourier analysis
Differentiable dynamical systems
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spelling Density of hyperbolicity in families of complex rational mapsTimoner Vaquer, FrancescSistemes dinàmics complexosFuncions de variables complexesAplicacions quasiconformesTreballs de fi de màsterAnàlisi de FourierSistemes dinàmics diferenciablesFrancesc Timoner VaquerComplex dynamical systemsFunctions of complex variablesQuasiconformal mappingsMaster's thesisFourier analysisDifferentiable dynamical systemsTreballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2025. Director: Kostiantyn DrachIn this work, we address the fundamental open problem of whether hyperbolic rational maps, the ones for which every critical point lies in the basin of an attracting cycle, are dense in the space of rational maps of the same degree. By this, we mean if any such map can be uniformly approximated on compact sets by hyperbolic ones. Conjecturally, the answer is ’yes’, and this is known as the Density of Hyperbolicity Conjecture. After reviewing key tools from complex dynamics such as puzzle pieces constructions, quasi-conformal conjugacies, Böttcher coordinates and holomorphic motions, we introduce complex box mappings as a natural extension of polynomiallike maps and discuss their rigidity under combinatorial equivalence. Focusing on non-renormalisable polynomials without neutral periodic points, we reproduce, clarify and check the Kozlovski–van Strien result that such polynomials admit approximating hyperbolic maps by constructing dynamically natural box mappings and applying topological and rigidity results. In conclusion, we outline how this framework, with a careful setting, promises to extend beyond the polynomial case to prove density of hyperbolicity in broader families such as Newton and McMullen maps, thereby sketching a clear path for future advances in complex dynamics.Drach, Kostiantyn2025info:eu-repo/semantics/masterThesisapplication/pdfhttps://hdl.handle.net/2445/228236Màster Oficial - Matemàtica Avançadareponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaIngléscc by-nc-nd (c) Francesc Timoner Vaquer, 2025http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/2282362026-05-27T06:46:51Z
dc.title.none.fl_str_mv Density of hyperbolicity in families of complex rational maps
title Density of hyperbolicity in families of complex rational maps
spellingShingle Density of hyperbolicity in families of complex rational maps
Timoner Vaquer, Francesc
Sistemes dinàmics complexos
Funcions de variables complexes
Aplicacions quasiconformes
Treballs de fi de màster
Anàlisi de Fourier
Sistemes dinàmics diferenciables
Francesc Timoner Vaquer
Complex dynamical systems
Functions of complex variables
Quasiconformal mappings
Master's thesis
Fourier analysis
Differentiable dynamical systems
title_short Density of hyperbolicity in families of complex rational maps
title_full Density of hyperbolicity in families of complex rational maps
title_fullStr Density of hyperbolicity in families of complex rational maps
title_full_unstemmed Density of hyperbolicity in families of complex rational maps
title_sort Density of hyperbolicity in families of complex rational maps
dc.creator.none.fl_str_mv Timoner Vaquer, Francesc
author Timoner Vaquer, Francesc
author_facet Timoner Vaquer, Francesc
author_role author
dc.contributor.none.fl_str_mv Drach, Kostiantyn
dc.subject.none.fl_str_mv Sistemes dinàmics complexos
Funcions de variables complexes
Aplicacions quasiconformes
Treballs de fi de màster
Anàlisi de Fourier
Sistemes dinàmics diferenciables
Francesc Timoner Vaquer
Complex dynamical systems
Functions of complex variables
Quasiconformal mappings
Master's thesis
Fourier analysis
Differentiable dynamical systems
topic Sistemes dinàmics complexos
Funcions de variables complexes
Aplicacions quasiconformes
Treballs de fi de màster
Anàlisi de Fourier
Sistemes dinàmics diferenciables
Francesc Timoner Vaquer
Complex dynamical systems
Functions of complex variables
Quasiconformal mappings
Master's thesis
Fourier analysis
Differentiable dynamical systems
description Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2025. Director: Kostiantyn Drach
publishDate 2025
dc.date.none.fl_str_mv 2025
dc.type.none.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/228236
url https://hdl.handle.net/2445/228236
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.rights.none.fl_str_mv cc by-nc-nd (c) Francesc Timoner Vaquer, 2025
http://creativecommons.org/licenses/by-nc-nd/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc by-nc-nd (c) Francesc Timoner Vaquer, 2025
http://creativecommons.org/licenses/by-nc-nd/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv Màster Oficial - Matemàtica Avançada
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869406298145030145
score 15,812429

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