Systems of iterative functional equations : theory and applications

Bibliographic Details
Main Author: Serpa, Maria Cristina Gonçalves Silveira de, 1974-
Publication Date: 2015
Language: eng
Source: Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
Download full: http://hdl.handle.net/10451/22975
Summary: Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015
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spelling Systems of iterative functional equations : theory and applicationsTeses de doutoramento - 2015Domínio/Área Científica::Ciências Naturais::MatemáticasTese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015We formulate a general theoretical framework for systems of iterative functional equations between general spaces. We find general necessary conditions for the existence of solutions such as compatibility conditions (essential hypotheses to ensure problems are well-defined). For topological spaces we characterize continuity of solutions; for metric spaces we find sufficient conditions for existence and uniqueness. For a number of systems we construct explicit formulae for the solution, including affine and other general non-linear cases. We provide an extended list of examples. We construct, as a particular case, an explicit formula for the fractal interpolation functions with variable parameters. Conjugacy equations arise from the problem of identifying dynamical systems from the topological point of view. When conjugacies exist they cannot, in general, be expected to be smooth. We show that even in the simplest cases, e.g. piecewise affine maps, solutions of functional equations arising from conjugacy problems may have exotic properties. We provide a general construction for finding solutions, including an explicit formula showing how, in certain cases, a solution can be constructively determined. We establish combinatorial properties of the dynamics of piecewise increasing, continuous, expanding maps of the interval such as description/enumeration of periodic and pre-periodic points and length of pre-periodic itineraries. We include a relation between the dynamics of a family of circle maps and the properties of combinatorial objects such as necklaces and words. We provide some examples. We show the relevance of this for the representation of rational numbers. There are many possible proofs of Fermat's little theorem. We exemplify those using necklaces and dynamical systems. Both methods lead to generalizations. A natural result from these proofs is a bijection between aperiodic necklaces and circle maps. The representation of numbers plays an important role in much of this work. Starting from the classical base p representation we present other type of representation of numbers: signed base p representation, Q-representation and finite base p representation of rationals. There is an extended p representation that generalizes some of the listed representations. We consider the concept of bold play in gambling, where the game has a unique win pay-off. The probability that a gambler reaches his goal using the bold play strategy is the solution of a functional equation. We compare with the timid play strategy and extend to the game with multiple pay-offs.Centro de Matemática e Aplicações Fundamentais da ULisboa; Fundação da Faculdade de Ciências da ULisboaBuescu, Jorge, 1964-Repositório da Universidade de LisboaSerpa, Maria Cristina Gonçalves Silveira de, 1974-2016-03-09T19:06:11Z201520152015-01-01T00:00:00Zdoctoral thesisinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/10451/22975TID:101443382enginfo:eu-repo/semantics/openAccessreponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiainstacron:RCAAP2025-03-17T13:27:18Zoai:repositorio.ulisboa.pt:10451/22975Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireinfo@rcaap.ptopendoar:https://opendoar.ac.uk/repository/71602025-05-29T02:44:58.018803Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologiafalse
dc.title.none.fl_str_mv Systems of iterative functional equations : theory and applications
title Systems of iterative functional equations : theory and applications
spellingShingle Systems of iterative functional equations : theory and applications
Serpa, Maria Cristina Gonçalves Silveira de, 1974-
Teses de doutoramento - 2015
Domínio/Área Científica::Ciências Naturais::Matemáticas
title_short Systems of iterative functional equations : theory and applications
title_full Systems of iterative functional equations : theory and applications
title_fullStr Systems of iterative functional equations : theory and applications
title_full_unstemmed Systems of iterative functional equations : theory and applications
title_sort Systems of iterative functional equations : theory and applications
author Serpa, Maria Cristina Gonçalves Silveira de, 1974-
author_facet Serpa, Maria Cristina Gonçalves Silveira de, 1974-
author_role author
dc.contributor.none.fl_str_mv Buescu, Jorge, 1964-
Repositório da Universidade de Lisboa
dc.contributor.author.fl_str_mv Serpa, Maria Cristina Gonçalves Silveira de, 1974-
dc.subject.por.fl_str_mv Teses de doutoramento - 2015
Domínio/Área Científica::Ciências Naturais::Matemáticas
topic Teses de doutoramento - 2015
Domínio/Área Científica::Ciências Naturais::Matemáticas
description Tese de doutoramento, Matemática (Análise Matemática), Universidade de Lisboa, Faculdade de Ciências, 2015
publishDate 2015
dc.date.none.fl_str_mv 2015
2015
2015-01-01T00:00:00Z
2016-03-09T19:06:11Z
dc.type.driver.fl_str_mv doctoral thesis
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10451/22975
TID:101443382
url http://hdl.handle.net/10451/22975
identifier_str_mv TID:101443382
dc.language.iso.fl_str_mv eng
language eng
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
instname:FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
instacron:RCAAP
instname_str FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
instacron_str RCAAP
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reponame_str Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
collection Repositórios Científicos de Acesso Aberto de Portugal (RCAAP)
repository.name.fl_str_mv Repositórios Científicos de Acesso Aberto de Portugal (RCAAP) - FCCN, serviços digitais da FCT – Fundação para a Ciência e a Tecnologia
repository.mail.fl_str_mv info@rcaap.pt
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