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Systems of Iterative Functional Equations: Theory and Applications UNIVERSIDADE DE LISBOA FACULDADE DE CIÊNCIAS Systems of Iterative Functional Equations: Theory and Applications Doutoramento em Matemática Especialidade: Análise Matemática Maria Cristina Gonçalves Silveira de Serpa Tese orientada por: Professor Doutor Jorge Sebastião de Lemos Carvalhão Buescu Documento especialmente elaborado para a obtenção do grau de doutor 2015 Systems of Iterative Functional Equations: |- |- Theory and Applications -Faculdade de Ci^enciasda Universidade de Lisboa ||{ | |- |- |- |- |- Departamento de Matem´atica Maria Cristina Gon¸calves Silveira de Serpa | ||- ||- |-2015 | ||- |- -Documento especialmente elaborado para a obten¸c~aodo grau de doutor | ||- ||{Documento Definitivo| ||- |{ Doutoramento em Matem´atica- Especialidade em An´aliseMatem´atica|- | Tese orientada por: Professor Doutor Jorge Sebasti~aode Lemos Carvalh~aoBuescu 1 Abstract We formulate a general theoretical framework for systems of iterative functio- nal 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 exis- tence 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 sys- tems 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 increa- sing, 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 genera- lizations. 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 repre- sentation 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. Keywords: functional equation, conjugacy equation, fractal interpolation, representation of numbers, bold play 2 Resumo Na primeira parte deste trabalho ´eformulado um enquadramento te´orico para determinados sistemas de equa¸c~oesfuncionais iterativas entre espa¸cosgerais. Formulam-se condi¸c~oesnecess´ariaspara a exist^enciade solu¸c~ao,tais como as condi¸c~oesde compatibilidade (hip´oteses essenciais para assegurar que os prob- lemas deste tipo est~aobem definidos). Estas condi¸c~oes s~aodefinidas a partir da informa¸c~aodo sistema nos pontos de contacto (obtidos por condi¸c~oesiniciais ou por resolu¸c~aoparcial do sistema). Para espa¸costopol´ogicoscaracteriza-se a continuidade de solu¸c~oes(aqui a informa¸c~aorelevante do sistema est´anos pontos em contacto e nos pontos limite em contacto); para espa¸cosm´etricos estabelecem-se condi¸c~oessuficientes de exist^enciae unicidade de solu¸c~ao(as hip´otesesnecess´ariase suficientes t^emem conta que a informa¸c~aocontida nas v´ariasequa¸c~oes cobre integralmente e de forma coerente o dom´ıniodo sistema e ainda que existe contractividade das equa¸c~oes).Para v´ariostipos de sistemas constroem-se f´ormulas expl´ıcitas, incluindo os casos afins, bem como outros casos n~aolineares gerais. Exibe-se uma extensa lista de exemplos: sistemas contractivos, em especial afins, complexo-conjugados afins, n~aocont´ınuos, de dimens~aosuperior a 2, com fun¸c~oesargumento n~aopadr~aoe os gerais n~ao lineares. Os exemplos cl´assicosincluem, por exemplo, os de de Rham, de Gir- gensohn e de Zdun. Os mais trabalhados s~aoos do tipo afim. Apresenta-se em primeiro lugar o caso dos sistemas de equa¸c~oesde conjuga¸c~ao(n~aodependem da vari´avel independente x). Mais geralmente, estudam-se tamb´emos sistemas que dependem explicitamente da vari´avel independente x. Constr´oi-se,como um caso particular, uma f´ormula expl´ıcita para as fun¸c~oes de interpola¸c~aofractal com par^ametrosvari´aveis, que ´euma generaliza¸c~aodo caso original estabelecido por Barnsley (caso afim). Aqui inclui-se tanto o caso em que os dados a inter- polar est~aouniformemente distribu´ıdoscomo o caso contr´ario.No que se refere aos sistemas de equa¸c~oesde conjuga¸c~ao,estebelece-se a rela¸c~aoexistente entre determinadas equa¸c~oesde conjuga¸c~aoe os vulgarmente denominados sistemas de fun¸c~oesiteradas (IFS), que permitem estabelecer caracter´ısticasfractais das solu¸c~oesdestas equa¸c~oes.Introduz-se o conceito de sistemas de equa¸c~oesincom- pletos. Para estes sistemas a informa¸c~aosobre as solu¸c~oesn~aopermite determi- nar uma ´unicasolu¸c~ao,mas por vezes ainda assim ´eposs´ıvel obter explicitamente a imagem da(s) solu¸c~ao(~oes)para um subconjunto do contradom´ınio. As equa¸c~oesde conjuga¸c~aoresultam do problema de identifica¸c~aode sistemas din^amicosdo ponto de vista topol´ogico.Quando existem conjuga¸c~oes,n~ao´ede esperar que estas sejam, em geral, suaves. Em primeiro lugar faz-se uma s´ıntese dos m´etodos existentes para resolu¸c~aodeste tipo de equa¸c~oes.Um dos m´etodos ´eum ponto de partida para o desenvolvimento do que se segue. Mostra-se que mesmo em casos muito simples, por exemplo em aplica¸c~oesafins por tro¸cos, existem solu¸c~oesde equa¸c~oesfuncionais que t^em propriedades ex´oticas:podem ser singulares, ou fractais. Apresenta-se uma constru¸c~aogeral que permite obter solu¸c~oesdeste tipo de equa¸c~oes.Esta constru¸c~ao´eresultante de sistemas cujos dom´ıniosest~aodefinidos por parti¸c~oes. E´ atrav´esdestas parti¸c~oese dos resulta- dos para sistemas de equa¸c~oesfuncionais que se obt^emsolu¸c~oespara as fun¸c~oes 3 de conjuga¸c~ao. Incluem-se f´ormulas expl´ıcitas,indicando que a solu¸c~aopode ser determinada construtivamente. Atrav´esdestes m´etodos tamb´emse indica o caminho a seguir, no caso de se pretender encontrar solu¸c~oeshomeom´orficas de equa¸c~oesde conjuga¸c~aotopol´ogica.Isto porque, para a teoria dos sistemas din^amicos,os sistemas que t^emsolu¸c~oeshomeom´orficast^emrelev^anciapara analisar aplica¸c~oestopologicamente conjugadas. Em geral existem m´ultiplas solu¸c~oes;no caso de homeomorfismos ´eposs´ıvel existirem duas solu¸c~oes:uma crescente e uma decrescente. Em determinadas situa¸c~oes´eposs´ıvel obter um conjunto, designado de conjunto aglutinador de uma equa¸c~aode conjuga¸c~ao, que engloba todas as poss´ıveis solu¸c~oesdestas equa¸c~oes.O cl´assico Cantor dust ´eum exemplo deste tipo de conjunto. Depois de uma exposi¸c~aote´oricafundamentada e generalizada sobre os sis- temas de equa¸c~oesfuncionais iterativos em estudo s~aodadas algumas aplica¸c~oes de natureza din^amica,combinatorial, de teoria dos n´umerose matem´aticarecre- ativa. As aplica¸c~oes das equa¸c~oesfuncionais `asfun¸c~oesfractais s~aoum t´opicocom bastantes desenvolvimentos cient´ıficos em diversas ´areascomo sejam a Medic- ina, a F´ısica,a Engenharia, a Hidrologia, a Sismologia, a Biologia e a Econo- mia. A disponibiliza¸c~aode uma f´ormula expl´ıcita para este tipo de fun¸c~oes permite o c´alculopontual preciso, que se apresenta como uma vantagem em compara¸c~aocom os m´etodos num´ericosusualmente utilizados para estimar a solu¸c~ao. Ilustra-se a utilidade de fun¸c~oesfractais com par^ametrosvari´aveis atrav´esde gr´aficosexemplificativos. Estabelecem-se propriedades combinatoriais da din^amicade determinadas aplica¸c~oescont´ınuas e expansivas por tro¸cos,como a descri¸c~aoe enumera¸c~ao dos pontos e ´orbitas peri´odicase pr´e-peri´odicas e o comprimento dos respectivos itiner´arios.Inclui-se a rela¸c~aoentre a din^amicadas aplica¸c~oesdo c´ırculoe algu- mas propriedades de objectos combinat´oricos,como sejam os colares e palavras. Estes conceitos s~aoilustrados com exemplos, em especial de forma recreativa. Mostra-se a relev^anciadeste assunto com a representa¸c~aodos n´umerosracionais. Existem muitas formas de demonstrar o pequeno teorema de Fermat, de en- tre as quais est~aoas que utilizam colares e sistemas din^amicos.Exemplificam-se algumas destas demonstra¸c~oes.Estes m´etodos levam a generaliza¸c~oes.Apresenta- se uma perspectiva hist´oricada forma como foram surgindo as diversas for- mula¸c~oese demonstra¸c~oes. Um resultado natural destas demonstra¸c~oes´ea exist^enciade uma bijec¸c~aoentre o conjunto dos colares aperi´odicose o conjunto das ´orbitasperi´odicasde aplica¸c~oesdo c´ırculo. A representa¸c~aode n´umerostem um papel importante na maior parte do trabalho aqui desenvolvido.
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