Thesis (MSc) -- University of Stellenbosch, 2000. / ENGLISH ABSTRACT: A discrete dynamical system is a pair (X, cf;) comprising a non-empty set X and a map cf; : X ---+ X. A study is made of the effect of repeated application of cf; on X, whereby points and subsets of X are classified according to their behaviour under iteration. These subsets include the JULIA and FATOU sets of the map and the sets of periodic and preperiodic points, and many interesting questions arise in the study of their properties. Such questions have been extensively studied in the case of complex dynamics, but much recent work has focussed on non-archimedean dynamical systems, when X is projective space over some field equipped with a non-archimedean metric. This work has uncovered many parallels to complex dynamics alongside more striking differences. In this thesis, various aspects of the theory of non-archimedean dynamics are presented, with particular reference to JULIA and FATOU sets and the relationship between good reduction of a map and the empty JULIA set. We also discuss questions of the finiteness of the sets of periodic points in special contexts. / AFRIKAANSE OPSOMMING: 'n Paar (X, <jJ) bestaande uit 'n nie-leë versameling X tesame met 'n afbeelding <jJ: X -+ X vorm 'n diskrete dinamiese sisteem. In die bestudering van so 'n sisteem lê die klem op die uitwerking op elemente van X van herhaalde toepassing van <jJ op die versameling. Elemente en subversamelings van X word geklasifiseer volgens dinamiese kriteria en op hierdie wyse ontstaan die JULIA en FATOU versamelings van die afbeelding en die versamelings van periodiese en preperiodiese punte. Interessante vrae oor die eienskappe van hierdie versamelings kom na vore. In die geval van komplekse dinamika is sulke vrae reeds deeglik bestudeer, maar onlangse werk is op nie-archimediese dinamiese sisteme gedoen, waar X 'n projektiewe ruimte is oor 'n liggaam wat met 'n nie-archimediese norm toegerus is. Hierdie werk het baie ooreenkomste maar ook treffende verskille met die komplekse dinamika uitgewys. In hierdie tesis word daar ondersoek oor verskeie aspekte van die teorie van nie-archimediese dinamika ingestel, in besonder met betrekking tot die JULIA en FATOU versamelings en die verband tussen goeie reduksie van 'n afbeelding en die leë JULIA versameling. Vrae oor die eindigheid van versamelings van periodiese punte in spesiale kontekste word ook aangebied.

Estudaremos sistemas dinâmicos complexos da esfera de Riemann, e empregaremos técnicas do Formalismo Termodinâmico incluindo a fórmula de Bowen para provar que a dimensão de Hausdorff \'dim IND. H\' J( \'f IND. lâmbda\' ) do conjunto de Julia J( \'f IND. lâmbda\' ) de uma família holomorfa de funções racionais hiperbólicas f \'lambda\' define uma função real analítica do parâmetro \'lambda\' . Este resultado foi provado por Ruelle [44] em 1981. Daremos uma prova alternativa usando movimentos holomorfos. Trata-se de uma técnica inovadora, originalmente desenvolvida por Mañé, Sad e Sullivan no trabalho [31] sobre estabilidade estrutural de sistemas dinâmicos complexos / We shall study complex dynamical systems in the Riemann sphere and prove that the Hausdorff dimension \'dim IND. H\' J( \'f IND. Lãmbda\' ) of the Julia set J( \'f IND. lâmbda\' ) of an holomorphic family of hyperbolic rational maps \'f IND. lâmbda\' defines a real analytic map of the parameter \'lâmbda\': This result was proved in 1981 by D. Ruelle (see [44]). We give an alternative proof using holomorphic motions (see [31]), which was originally developed to study the structural stability problem of complex dynamical systems. Throughout this work, we shall use several tools of Thermodynamic Formalism, including Bowens formula

Der Ziel dieser Arbeit ist der Hausdorff Konvergenz der Juliamengen zu beweisen, als wir eine Familie von ganzen transzendenten Funktionen, die ein einziges Bakergebiet enthalten, approximieren. Als erstes geben wir eine vollständige dynamische Beschreibung der approximierenden transzendenten Funktionen und zeigen die Existenz von invarianten Gebiete unter der Iterierte. Insbesondere besitzen die approximierenden Funktionen ein Attraktionsgebiet, das gegen das Bakergebiet als Kernel im Sinn von Carathéodory konvergiert. Letztlich beweisen wir Hausdorff Konvergenz auf zwei Wege. Einerseits zeigen wir unter bestimmten Bedingungen der Fatoumenge der Grenzfunktion die Hausdorff Konvergenz der Juliamengen. Anderseits zeigen wir unter verschiedenen Bedingungen der Fatoumenge der Grenzfunktion die Hausdorff Konvergenz der ausgefüllten Juliamengen, die bezüglich der Bakergebiet oder der Attraktionsgebiet definiert sind.

Orientador: Ali Messaoudi / Banca: Eduardo Garibaldi / Banca: Maria Gorete Carreira Andrade / Resumo: Neste trabalho, estudaremos as propriedades dinâmicas de endomorfismos do plano complexo C. Provaremos e o teorema de Montel e mostraremos algumas propriedades topológicas do conjunto de Julia J(f), onde f : C "seta" C é uma aplicação racional de grau > ou = 2 / Abstract: In this work, we will study the dynamical properties of endomorfisms of complex plane C. We will also prove Montel's theorem and show some topological properties of Julia set J(f), where f : C 'seta' C is a rational map of degree > ou = 2. / Mestre

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-08-10Bitstream added on 2014-06-13T18:07:02Z : No. of bitstreams: 1 marchioli_ab_me_sjrp.pdf: 317494 bytes, checksum: 518683b62d488d3433a0bee79ecd4f53 (MD5) / Neste trabalho, estudaremos as propriedades dinâmicas de endomorfismos do plano complexo C. Provaremos e o teorema de Montel e mostraremos algumas propriedades topológicas do conjunto de Julia J(f), onde f : C seta C é uma aplicação racional de grau > ou = 2 / In this work, we will study the dynamical properties of endomorfisms of complex plane C. We will also prove Montel's theorem and show some topological properties of Julia set J(f), where f : C 'seta' C is a rational map of degree > ou = 2.

Estudaremos sistemas dinâmicos complexos da esfera de Riemann, e empregaremos técnicas do Formalismo Termodinâmico incluindo a fórmula de Bowen para provar que a dimensão de Hausdorff \'dim IND. H\' J( \'f IND. lâmbda\' ) do conjunto de Julia J( \'f IND. lâmbda\' ) de uma família holomorfa de funções racionais hiperbólicas f \'lambda\' define uma função real analítica do parâmetro \'lambda\' . Este resultado foi provado por Ruelle [44] em 1981. Daremos uma prova alternativa usando movimentos holomorfos. Trata-se de uma técnica inovadora, originalmente desenvolvida por Mañé, Sad e Sullivan no trabalho [31] sobre estabilidade estrutural de sistemas dinâmicos complexos / We shall study complex dynamical systems in the Riemann sphere and prove that the Hausdorff dimension \'dim IND. H\' J( \'f IND. Lãmbda\' ) of the Julia set J( \'f IND. lâmbda\' ) of an holomorphic family of hyperbolic rational maps \'f IND. lâmbda\' defines a real analytic map of the parameter \'lâmbda\': This result was proved in 1981 by D. Ruelle (see [44]). We give an alternative proof using holomorphic motions (see [31]), which was originally developed to study the structural stability problem of complex dynamical systems. Throughout this work, we shall use several tools of Thermodynamic Formalism, including Bowens formula

We generalize the notions of structural stability and hyperbolicity for the family of (multivalued) complex maps Hc(z) = zr + c; where r > 1 is rational and zr = exp r log z: We discovered that Hc is structurally stable at every hyperbolic parameter satisfying the escaping condition. Surprisingly, there may be infinitely many attracting periodic points for Hc. The set of such points gives rise to the dual Julia set, which is a Cantor set coming from a Conformal Iterated Funcion System. Both the Julia set and its dual are projections of holomorphic motions of dynamical systems (single valued maps) defined on compact subsets of Banach spaces, denoted by Xc and Wc, respectively. For c close to zero: (1) we show that Jc is a union of quasiconformal arcs around the unit circle; (2) the set Xc is an holomorphic motion of the solenoid X0; (3) using the formalism of Gibbs states we exhibit an upper bound for the Hausdorff dimension of Jc; which implies that Jc has zero Lebesgue measure. / Generalizamos as noções de estabilidade estrutural e hiperbolicidade para a família de correspondências holomorfas Hc(z) = zr + c; onde r > 1 é racional e zr = exp r log z: Descobrimos que Hc é estruturalmente estável em todos os parâmetros hiperbólicos satisfazendo a condição de fuga. Tipicamente Hc possui infinitos pontos periódicos atratores, fato totalmente inesperado, uma vez que este número é sempre finito para aplicações racionais. O conjunto de tais pontos dá origem ao chamado conjunto de Julia dual, que é um conjunto de Cantor proveniente de um Conformal Iterated Function System. Tanto o conjunto de Julia e quanto seu dual são projeções de movimentos holomorfos de sistemas definidos em subconjuntos compactos denotados por Xc e Wc; respectivamente de um espaço de Banach. Para todo c próximo de zero: (1) mostramos que Jc é reunião de arcos quase-conformes próximos do círculo unitário; (2) o conjunto Xc é um movimento holomorfo do solenóide X0; (3) utilizando o formalismo dos estados de Gibbs, exibimos um limitante superior para a dimensão de Hausdorff de Jc. Consequentemente, Jc possui medida de Lebesgue nula.

Šiame darbe buvo analizuojama Niutono fraktalų Žulija aibės. Dažniausiai Žulija ir užpildytų Žulija aibių vaizdai gaunami, panaudojant "pabėgimo laiko" algoritmą. Norėdami šį algoritmą naudoti kompleksinio daugianario šaknų vizualizacijai, turime nurodyti iteracijų skaiči��, algoritmo tikslumą, žingsnį bei kompleksiniu Niutono metodu rasti daugianario šaknis. Taikant Niutono metodą, buvo susidurta su pradinių taškų parinkimo problema. Tyrimo metu patvirtinta, kad pakanka Niutono iteracinę funkciją taikyti taškams z, kurių modulis 2. Darbe buvo pasiūlytas šaknų lokalizacijos srities nustatymo būdas. Naudojant PL-algoritmą, pasirinktu žingsniu pereiname visus taškus, kurie patenka į šią sritį. Taip gauname Niutono-Rafsono fraktalus ir lygiagrečiai analizuojame Žulija aibes bei užpildytas Žulija aibes. / Julia sets and filled Julia sets of Newton‘s fractals are analyzed in this work. The Escape Time Algorithm provides us with a means for "seeing" the filled Julia sets of Newton‘s fractals, but roots, (zeros) of the polynomial under investigation should be known. The Newton‘s method for finding roots of an algebraic equation is well known. Here in the paper the complex Newton method for finding roots of a complex polynomial is presented. The main difficulties, associated with implementation of this method in practice, are discussed, namely: construction of the set of initial points (first approximations of the roots), finding the basin of attraction for a particular root and so forth. Some experimental results are presented.

Dans cette thèse, on s'intéresse aux systèmes dynamiques holomorphes dépendants de paramètres. Notre objectif est de contribuer à une théorie de la stabilité et des bifurcations en plusieurs variables complexes, généralisant celle des applications rationnelles fondées sur les travaux de Mané, Sad, Sullivan et Lyubich. Pour une famille d'applications d'allure polynomiale, on prouve l'équivalence de plusieurs notions de stabilité, entre autres une version asymptotique du mouvement holomorphe des cycles répulsifs et d'un sous-ensemble de l'ensemble de Julia de mesure pleine. Cela peut etre considéré comme une généralisation mesurable à plusieurs variables du célèbre lambda-lemme et nous permet de dégager un concept cohérent de stabilité dans ce cadre. Après avoir compris les bifurcations holomorphes, on s'intéresse à la continuité Hausdorff des ensembles de Julia. Nous relions cette propriété à l'existence de disques de Siegel dans l'ensemble de Julia, et donnons un exemple de ce phénomène. Finalement, on étudie la continuité du point de vue de l'implosion parabolique. Nous établissons un théorème de Lavaurs deux-dimensionel, ce qui nous permet d'étudier des phénomènes de discontinuité pour des perturbations d'applications tangentes à l'identité. / In this thesis we study holomorphic dynamical systems depending on parameters. Our main goal is to contribute to the establishment of a theory of stability and bifurcation in several complex variables, generalizing the one for rational maps based on the seminal works of Mané, Sad, Sullivan and Lyubich. For a family of polynomial like maps, we prove the equivalence of several notions of stability, among the others an asymptotic version of the holomorphic motion of the repelling cycles and of a full-measure subset of the Julia set. This can be seen as a measurable several variables generalization of the celebrated lambda-lemma and allows us to give a coherent definition of stability in this setting. Once holomorphic bifurcations are understood, we turn our attention to the Hausdorff continuity of Julia sets. We relate this property to the existence of Siegel discs in the Julia set, and give an example of such phenomenon. Finally, we approach the continuity from the point of view of parabolic implosion and we prove a two-dimensional Lavaurs Theorem, which allows us to study discontinuities for perturbations of maps tangent to the identity.

Submitted by Marlene Santos (marlene.bc.ufg@gmail.com) on 2016-10-03T21:11:40Z No. of bitstreams: 2 Dissertação - Márcio Vaiz dos Reis - 2016.pdf: 2097960 bytes, checksum: 296b1790b8c8fe50c0e91d2d5ee204c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2016-10-04T10:46:49Z (GMT) No. of bitstreams: 2 Dissertação - Márcio Vaiz dos Reis - 2016.pdf: 2097960 bytes, checksum: 296b1790b8c8fe50c0e91d2d5ee204c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-10-04T10:46:49Z (GMT). No. of bitstreams: 2 Dissertação - Márcio Vaiz dos Reis - 2016.pdf: 2097960 bytes, checksum: 296b1790b8c8fe50c0e91d2d5ee204c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-08-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The purpose of this dissertation is to present an introductory approach to the complex dynamics and fractal geometry, especially the Mandelbrot set. With the goal to simplify and stimulate the introduction of complex number in high school, the approach adopted was: the definition of the Mandelbrot set and its characteristics; the relationship between the Mandelbrot set and Julia set; how to find by using the Mandelbrot set. One of the tools used to help the teaching was Geogeobra, a dynamic software that allows the student to build the Mandelbrot set. Through this study, it is expected to motivate the learning of complex numbers by using fractal obtained by the study of function ( ) . Obtaining, as a result, a differentiated and motivating way of learning for a better understanding and intellectual development of the students. / Esse trabalho apresenta uma abordagem introdutória para a dinâmica complexa e a geometria fractal, em especial o conjunto de Mandelbrot. Com objetivo de facilitar e motivar a interação dos alunos com o ensino dos números complexos, a abordagem adotada foi: a definição do conjunto de Mandelbrot e suas características; a relação entre o conjunto de Mandelbrot e o conjunto de Julia; a relação do conjunto de Mandelbrot e o número . Uma das ferramentas utilizadas para auxiliar o professor foi o Geogeobra, um software dinâmico que permite o aluno a construção do conjunto de Mandelbrot. Por meio deste trabalho, espera-se motivar o ensino dos números complexos através do fractal obtido pelo estudo da função ( ) . Obtendo assim, como resultado, uma forma diferenciada e motivadora do aprendizado do aluno, garantindo um melhor entendimento e desenvolvimento intelectual.