Complex dynamics is the study of iteration of functions which map the complex plane onto itself. In general, their dynamics are quite complicated and hard to explain but for some simple classes of functions many interesting results can be proved. For example, one often studies the class of rational functions (i.e. quotients of polynomials) or, even more specifically, polynomials. Each such function f partitions the extended complex plane C into two regions, one where iteration of the function is chaotic and one where it is not. The nonchaotic region, called the Fatou Set, is the set of all points z such that, under iteration by f, the point z and all its neighbors do approximately the same thing. The remainder of the complex plane is called the Julia set and consists of those points which do not behave like all closely neighboring points. The Julia set of a polynomial typically has a complicated, self similar structure. Many questions can be asked about this structure. The one that we seek to investigate is the notion of the dimension of the Julia set. While the dimension of a line segment, disc, or cube is familiar, there are sets for which no integer dimension seems reasonable. The notion of Hausdorff dimension gives a reasonable way of assigning appropriate non-integer dimensions to such sets. Our goal is to investigate the behavior of the Hausdorff dimension of the Julia sets of a certain simple class of polynomials, namely fd,c(z) = zd + c. In particular, we seek to determine for what values of c and d the Hausdorff dimension of the Julia set varies continuously with c. Roughly speaking, given a fixed integer d > 1 and some complex c, do nearby values of c have Julia sets with Hausdorff dimension relatively close to each other? We find that for most values of c, the Hausdorff dimension of the Julia set does indeed vary continuously with c. However, we shall also construct an infinite set of discontinuities for each d. Our results are summarized in Theorem 10, Chapter 2. In Chapter 1 we state and briefly explain the terminology and definitions we use for the remainder of the paper. In Chapter 2 we will state the main theorems we prove later and deduce from them the desired continuity properties. In Chapters 3 we prove the major results of this paper.

There are many iterative techniques to find a root or zero of a given function. For any iterative technique, it is often of interest to know which initial seeds lead to which roots. When the iterative technique used is Newton’s Method, this is known as Cayley’s Problem. In this thesis, I investigate two extensions of Cayley’s Problem. In particular, I study generalizations of Newton’s Method, in both C and R2, and the associated fractal structures that arise from using more sophisticated numerical approximation techniques.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2009. / Title from PDF title page (viewed Sept. 2, 2009). Additional advisors: Alexander Blokh, Lex G. Oversteegen, Purushotham Bangalore, Vo Thanh Liem, Kyle Siegrist. Degree earned with the cooperation of additional faculty from the University of Alabama and the University of Alabama in Huntsville. Includes bibliographical references.

A Julia set for a complex function 𝑓 is the set of all points in the complex plane where the iterates of 𝑓 do not form a normal family. A picture of the Julia set for a function can be generated with a computer by coloring pixels (which we consider to be small squares) based on the behavior of the point at the center of each pixel. We consider the accuracy of computer generated pictures of Julia sets. Such a picture is said to be accurate if each colored pixel actually contains some point in the Julia set. We extend previous work to show that the pictures generated by an algorithm for the family λe² are accurate, for appropriate choices of parameters in the algorithm. We observe that the Julia set for meromorphic functions with polynomial Schwarzian derivative is the closure of those points which go to infinity under iteration, and use this as a basis for an algorithm to generate pictures for such functions. A pixel in our algorithm will be colored if the center point becomes larger than some specified bound upon iteration. We show that using our algorithm, the pictures of Julia sets generated for the family λtan(z) for positive real λ are also accurate. We conclude with a cautionary example of a Julia set whose picture will be inaccurate for some apparently reasonable choices of parameters, demonstrating that some care must be exercised in using such algorithms. In general, more information about the nature of the function may be needed. / Ph. D.

Made available in DSpace on 2015-03-26T13:45:36Z (GMT). No. of bitstreams: 1 texto completo.pdf: 680613 bytes, checksum: d49992ace83b65d0a439badc8cc946f3 (MD5) Previous issue date: 2014-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we study some geometric properties of Julia sets and filled-in Julia sets of polynomials. In addition, we seek a form of measure the Julia set, for this we use the Hausdorff measure and determine a lower bound to the Hausdorff dimension of the Julia set. / Neste trabalho estudamos algumas propriedades geométricas do Conjunto de Julia e do e Conjunto de Julia Cheio. Além disso, procuramos uma forma de mensurar o conjunto de Julia, para isso utilizamos a medida de Hausdorff e determinamos uma cota inferior para a dimensão de Hausdorff do conjunto de Julia.

Cette thèse est constituée de cinq parties distinctes. La première partie est consacrée au problème de rigidité quasi-symétrique associé à un nouveau modèle de tapis de Sierpinski, qui ne sont pas quasi-symétriquement équivalent aux tapis de Sierpinski usuels. La seconde partie est une discussion portant sur la géométrie quasi-symétrique des ensembles de tapis de Julia, incluant en outre le quasi-cercle uniforme, ainsi que certaines propriétés de séparation uniforme. Lors de la troisième partie, nous déterminerons une condition permettant de savoir quand deux rayons externes d'un polynôme tendent vers un même point. Comme application, nous montrerons également la monotonie de l'entropie associée à une famille de polynômes quadratiques. La quatrième partie est inspirée du travail récent de Cui Guizhen et Tan Lei. En utilisant des outils classiques (module d'anneau et chirurgie quasi-conforme), nous étudierons la convergence de certains rayons en campagne locus espace des paramètres. Enfin, la dernière partie pore sur la famille des transformations de renormalisations générées. Plus précisément, cette partie abordera la connexité de ces ensembles de Julia, et le lieu de confinement dans l'espace des paramètres, ainsi que la formule asymptotique de la dimension d'Hausdorff des ensembles de Julia. / The thesis consists of five parts. The first part is concerned with the quasisymmetric rigidity of a new Sierpinski carpet, which are not quasis-ymmetrically equivalent to the standard Sierpinski carpets. The second part discusses the quasisymmetrically geometry of the carpet Julia sets, including the uniformly quasicircle and uniformly separated properties. The third part is to determine when two external rays of a polynomial land at the same point. As an application, we also show the monotonicity of core-entropy on a family of quadratic polynomials. In the fourth part, following Cui and Tan's work, we use the classic tools modulus of annulus and quasi-conformal surgery to study the landing of some parameter rays in shift locus parameter space. The last part discusses a family of generated renormal-ization transformations. Specifically, it is on the connec-tivity of its Julia sets and the non-escaping locus in its parameter space, the asymptotic formula of the Hausdorff dimention of the Julia sets.

Orientador: Ali Messaoudi / Banca: Vanderlei Minori Horita / Banca: Daniel Smania Brandão / Banca: Christian Mauduit / Banca: Glauco Valle da Silva Coelho / Resumo: Em 2000, Killeen e Taylor definiram a máquina de somar estocástica em base 2. Eles mostraram que o espectro do op erador de transi cão (agindo em l∞( N)), associado a essa máquina, e igual ao conjunto de Julia cheio de uma função quadrática. Nesse trabalho, estudamos outras propriedades espectrais e topológicass da máquina de Killeen e Taylor, e também das suas extensões à l∞(Z) e a outras bases não constantes. Esse estudo envolve conjuntos de Julia de funções quadráticas e também conjuntos de Julia cheios de endomor smos de C2 . Finalmente estudamos algumas propriedades aritméticas e topológicas de uma classe de fractais de Rauzy. Em particular estudamos o azulejamento periódico do plano complexo C induzido por eles. / Abstract: In 2000, Killeen and Taylor de ned the sto hastic adding machine in base 2. They proved that the sp ectrum of the transition op erator (acting in l∞(N )) asso ciated to this machine is equal to the lled Julia set of a quadratic polynomial map. In this work, we study other sp ectral and top ological prop erties of Killeen and Taylor machine, and also of its extensions to l∞( Z) and to other non constant bases. This study envolves Julia sets of quadratic maps and also lled Julia sets of endomorphisms of C2 . Finally we study some arithmetical and topological prop erties of a class of Rauzy fractals. In particular we study the p erio dictiling of complex plane C induced by this class. / Doutor

On s'intéresse dans ce travail à la description des enrichissements des disques de Siegel d'une fraction rationnelle f. Dans un premier temps nous étudions les enrichissements qui sont définis sur un ouvert de la grande orbite d'un disque de Siegel donné. Ce sont nécessairement des applications qui commutent à f là où les compositions ont un sens. Ce sont donc des applications linéaires en coordonnées linéarisantes. Le résultat principal de ce travail est que l'on peut obtenir toutes les applications linéaires en coordonnées linéarisantes définies sur un sous-disque du disque de Siegel de f. Pour démontrer ce résultat nous utilisons la compacité des applications linéarisantes normalisées, le théorème des fonctions implicites dans l'espace des fractions rationnelles de degré fixé et une étude du comportement du rayon d'univalence des applications linéarisantes. Nous identifions également les approches donnant lieu à des enrichissements définis ou à valeurs dans le disque de Siegel tout entier (enrichissements maximaux). Au passage nous généralisons aux limites avec ordre de contact fini par rapport au cercle unité un théorème de JC.Yoccoz sur le comportement du rayon d'univalence pour la famille quadratique lorsque le paramètre converge vers un nombre complexe de module un et d'argument un nombre de Brjuno.Ensuite, nous nous intéressons au cas où f a plusieurs cycles de disques de Siegel. Nous utilisons le théorème de transversalité d'A.Epstein pour décrire les enrichissements de f dans ce cas là. La linéarisabilité de f et la convergence des applications linéarisantes permet de transférer le problème de la description des enrichissements de Siegel de f à un problème de limite géométrique de sous-semigroupes de l'ensemble des nombres complexes non-nuls engendrés par un élément. Nous donnons dans ce travail un modèle topologique de l'adhérence de cet ensemble de sous-semigroupes. Nous déduisons de ces résultats une interprétation en terme de convergence géométrique de dynamiques de polynômes quadratiques et une description des points d'accumulation, pour la topologie de Hausdorff sur les compacts non-vides, des ensembles de Julia lorsque le paramètre tend vers un paramètre de Siegel. / In this work we are interested in giving the description of Siegel discs enrichments of a rational map f. We first study the case of enrichments that are defined on an open subset of the grand orbit of a given Siegel disc. These maps commute with f where it makes sense. Thus they are linear in linearizing coordinates. The main result of this work is that we can obtain all linear maps in linearizing coordinates that are defined in a subdisc of the Siegel disc. For this we use the compactness of the set of normalized linearizing maps, the implicit functions theorem in the space of rational maps with fixed degree and a study on the behaviour on the univalent radius of the linearizing maps. We identify approaches giving enrichments that are defined or take values on the whole Siegel disc (maximal enrichments). We generalize to finite order of contact approaches with respect to the unit circle a theorem of JC.Yoccoz on the behaviour of the univalent radius for the quadratic family when the parameter converges to a complex number of modulus one with argument a Brjuno number.We then focus on the case where f has more than one Siegel disc. We make use of A.Epstein's transversality theorem to describe Siegel enrichments of f in this case. The linearisability of f and the convergence of the linearizing maps reduces the problem of Siegel enrichments description to a geometric limit problem on one generated closed sub-semigroups ofthe set of non zero complex numbers. We give in this work a topological model fot the closure of this set of sub-semigroups.We deduce from these results an interpretation in terms of geometric convergence of quadratic polynomial dynamics and we describe the accumulation points (for the Hausdorff topology on non empty compact subsets) of Julia sets when the parameter converges to a Siegel parameter.

Indiana University-Purdue University Indianapolis (IUPUI) / This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will \twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.