Fractals is a relatively new mathematical topic which received thorough treatment only starting with 1960's. Fractals can be observed everywhere in nature and in day-to-day life. To give a few examples, common fractals are the spiral cactus, the romanesco broccoli, human brain and the outline of the Swedish map. Fractal dimension is a dimension which need not take integer values. In fractal geometry, a fractal dimension is a ratio providing an index of the complexity of fractal pattern with regard to how the local geometry changes with the scale at which it is measured. In recent years, fractal analysis is used increasingly in many areas of engineering and technology. Among others, fractal analysis is used in signal and image compression, computer and video design, neuroscience and fractal based cancer modelling and diagnosing.   This study consists of two main parts. The first part of the study aims to understand the appearance of an irregular Cantor set generated by the chaotic dynamical system generated by the logistic function on the unit interval [0,1]. In order to understand this irregular Cantor set, we studied the topological properties of the Cantor Middle-thirds set and the generalised Cantor sets, all of which have zero length. The necessity to compare these sets with regard to their size led us to the second part of this paper, namely the dimension studies of fractals. More complex fractals were presented in the second part, three definitions of dimension were introduced. The fractal dimension of the irregular Cantor set generated by the logistic mapping was estimated and we found that the Hausdorff dimension has the widest scope and greatest flexibility in the fractal studies. / Fraktaler är ett relativt nytt ämne inom matematik som fick sitt stora genomslag först efter 60-talet.  En fraktal är ett självliknande mönster med struktur i alla skalor. Några vardagliga exempel på fraktaler är spiralkaktus, romanescobroccoli, mänskliga hjärnan, blodkärlen och Sveriges fastlandskust. Bråktalsdimension är en typ av dimension där dimensionsindexet tillåts att anta alla icke-negativa reella tal. Inom fraktalgeometri kan dimensionsindexet betraktas som ett komplexitetsindex av mönstret med avseende på hur den lokala geometrin förändras beroende på vilken skala mönstret betraktas i. Under det senaste decenniet har fraktalanalysen använts alltmer flitigt inom tekniska och vetenskapliga tillämpningar. Bland annat har fraktalanalysen använts i signal- och bildkompression, dator- och videoformgivning, neurovetenskap och fraktalbaserad cancerdiagnos.   Denna studie består av två huvuddelar. Den första delen fokuserar på att förstår hur en fraktal kan uppstå i ett kaotiskt dynamiskt system. För att vara mer specifik studerades den logistiska funktionen och hur denna ickelinjära avbildning genererar en oregelbunden Cantormängd på intervalet [0,1]. Vidare, för att förstå den oregelbundna Cantormängden studerades Cantormängden (eng. the Cantor Middle-Thirds set) och de generaliserade Cantormängderna, vilka alla har noll längd. För att kunna jämföra de olika Cantormängderna med avseende på storlek, leds denna studie vidare till dimensionsanalys av fraktaler som är huvudämnet i den andra delen av denna studie. Olika topologiska fraktaler presenterades, tre olika definitioner av dimension introducerades, bland annat lådräkningsdimensionen och Hausdorffdimensionen. Slutligen approximerades dimensionen av den oregelbundna Cantormängden med hjälp av Hausdorffdimensionen. Denna studie demonstrerar att Hausdorffdimensionen har större omfattning och mer flexibilitet för fraktalstudier.

Cette thèse est consacrée à l’étude du phénomène de Newhouse et des bifurcations en dynamique holomorphe à plusieurs variables. Elle comporte trois Théorèmes principaux. Le premier de ces trois résultats est un Gap Lemma complexe. En dynamique réelle, le Gap Lemma de Newhouse donne un critère sur le produit des épaisseurs de deux ensembles de Cantor dynamiques pour prouver que leur intersection est non vide. On en donne une généralisation partielle au cas des ensembles de Cantor dynamiques dans C. Plus précisément, on introduit une notion d’épaisseur pour un ensemble de Cantor dynamique planaire et on fournit un critère sur le produit de deux épaisseurs afin d’obtenir une intersection entre deux ensembles de Cantor dynamiques. On montre également que l’épaisseur est une quantité qui varie continûment, ce qui permet d’obtenir des intersections persistantes d’ensembles de Cantor dynamiques. Le second Théorème de cette thèse démontre l’existence du phénomène de Newhouse dans l’espace des automorphismes polynomiaux de degré d pour n’importe quel degré d ≥ 2 dans C^{3}. Au contraire de la situation dans C^{2}, le degré est ici connu et optimal. Le point clef de la preuve est l’introduction dans le domaine complexe d’un outil issu de la dynamique réelle : le blender de Bonatti et Diaz. On formalise le concept de blender complexe et on donne un automorphisme polynomial de C^{3} de degré 2 possédant un blender. Puis, on l’utilise afin de construire successivement des tangences persistantes et des sous-ensembles résiduels d’automorphismes ayant une infinité de puits. Enfin, le dernier résultat porte sur les bifurcations d’endomorphismes holomorphes de P^{2}(C) très particuliers, appelés exemples de Lattès, semi-conjugués à une application affine sur un tore. Dujardin a conjecturé que ces derniers étaient accumulés par des ouverts de bifurcations. On montre que tout exemple de Lattès de degré suffisamment élevé est accumulé par de telles bifurcations robustes. Ceci implique en particulier que tout exemple de Lattès possède un itéré dans l’adhérence de l’intérieur du lieu de bifurcation. La démonstration est basée sur l’obtention d’intersections persistantes entre l’ensemble postcritique et un ensemble hyperbolique répulsif contenu dans l’ensemble de Julia. La preuve est divisée en deux parties : on donne tout d’abord un toy-model qui permet d’obtenir des intersections persistantes entre l’ensemble limite d’un certain type d’IFS, appelé IFS correcteur, et une courbe. Ensuite, dans un second temps, on perturbe l’exemple de Lattès pour créer simultanément un IFS correcteur dans l’ensemble de Julia et une courbe bien orientée dans l’ensemble postcritique / In this PhD thesis, we study Newhouse’s phenomenon and bifurcations in the context of dynamics in several complex variables. We prove three main Theorems. The first one is a complex Gap Lemma. In real dynamics, Newhouse’s Gap Lemma gives a criterion on the product of the thicknesses of two dynamical Cantor sets K and L to show that K ∩ L is not empty. We show a partial generalization of this result for dynamical Cantor sets in C. A relevant notion of thickness in this case is defined and we give some criterion on the product of two thicknesses to show that two dynamical Cantor sets in C must intersect. We also show that the thickness varies continuously, which generates persistent intersections of dynamical Cantor sets. In the second Theorem, we show that there exists a polynomial automorphism f of C^{3} of degree 2 such that for every automorphism g sufficiently close to f, g admits a tangency between the stable and unstable laminations of some hyperbolic set. As a consequence, for each d ≥ 2, there exists an open set of polynomial automorphisms of degree at most d in which the automorphisms having infinitely many sinks are dense. In contrary to the case of C^{2}, the degree is known. To prove these results, we give a complex analogous to the notion of blender introduced by Bonatti and Diaz. In particular, we use a blender to produce robust tangencies. In the third and last result, we study the phenomenon of robust bifurcations in the space of holomorphic maps of P^{2}(C). We prove that any Lattès example of sufficiently high degree belongs to the closure of the interior of the bifurcation locus. This gives a partial answer to a conjecture of Dujardin. In particular, every Lattès map has an iterate with this property. To show this, we design a method creating robust intersections between the limit set of a particular type of iterated functions system in C^{2} with a well-oriented complex curve. Then we show that any Lattès map of sufficiently high degree can be perturbed so that the perturbed map exhibits this geometry

[pt] Um diagrama de Bratteli B é um objeto combinatório representado por um grafo dividido em infinitos níveis, cada um com número finito de vértices e de arestas entre vértices de níveis consecutivos. Além disso, todo vértice possui ligação com vértices dos níveis precedente e sucessor. Estudamos, do ponto de vista topológico, o espaço dos caminhos infinitos formados pelas arestas de um diagrama de Bratteli, denotado por XB. Estabelecemos uma relação de equivalência neste espaço, denominada AF. Quando é possível definir uma ordem parcial em XB o diagrama é dito ordenado; neste caso, definimos um homeomorfismo em XB denominado de função de Bratteli-Vershik. Consideramos sistemas dinâmicos minimais definidos em conjuntos de Cantor e associamos a estes diagramas de Bratteli ordenados. Um exemplo paradigmático de um conjunto de Cantor é o espaço das sequências infinitas formadas por 00s e 10s, munido de uma métrica apropriada. Neste espaço são definidas as funções odômetro. Definimos a relação de equivalência orbital, na qual duas sequências são equivalentes se estão na mesma órbita do odômetro, e a relação de equivalência de caudas, onde duas sequências são equivalentes se a partir de alguma entrada elas são iguais. Estudamos como estas duas relações estão relacionadas. Provamos que o odômetro diádico é um homeomorfismo minimal definido em um conjunto de Cantor e, portanto, pode ser associado a um diagrama de Bratteli ordenado. Uma relação de equivalência é dita étale quando admite uma topologia gerada por uma ação local. Dois exemplos são as relações AF e orbital. Dada uma relação de equivalência étale R em um espaço X, definimos um invariante algébrico D(X,R). Construímos o grupo de dimensão de um diagrama de Bratteli. Provamos, então, que dado um diagrama de Bratteli B, seu grupo de dimensão é isomorfo a D(XB,RB), onde RB é relação AF de B. Finalmente, estudamos sob quais condições um grupo abeliano ordenado é o grupo de dimensão de um diagrama de Bratteli. Esta dissertação é baseada no livro de Ian F. Putnam Cantor minimal systems, publicado em University Lecture Series, 70. American Mathematical Society, Providence, RI, 2018. [6]. / [en] A Bratteli diagram B is a combinatorial object represented by a graph divided into infinite levels, each level with a finite number of vertices and edges between vertices of consecutive levels. Moreover, every vertex is connected to vertices of the preceding and successor levels. We study, from a topological point of view, the space of infinite paths formed by the edges of a Bratteli diagram, denoted by XB. We establish an equivalence relation on this space, called the AF relation. When it is possible to define a partial order in XB the Bratteli diagram is called ordered; in this case, we define a homeomorphism on XB called the Bratteli-Vershik function. We consider minimal dynamic systems defined on Cantor sets and associate to these systems ordered Bratteli diagrams. A paradigmatic example of a Cantor set is the space of the infinite sequences formed by 00s and 10s, equipped with an appropriate metric. In this space, are defined the odometer functions. We define the orbital equivalence relation, in which two elements of the Cantor set are equivalent if they are in the same orbit of the odometer, and the tail equivalence relation, where two sequences are equivalents if they differ in only finitely many entries. We study how these equivalence relations are related. We prove that the dyadic odometer is a minimal homeomorphism and, therefore, it can be associated to a ordered Bratteli diagram. An equivalence relation is called étale if it admits a topology generated by a local action. Two examples are the AF equivalence relation and the orbital equivalence relation above. Given an étale equivalence relation R on a space X, we define an algebraic invariant D(X,R). We construct the dimension group of a Bratteli diagram. Then, we prove that given a Bratteli diagram B, its dimension group is isomorphic to D(XB,RB), where RB is the AF equivalence relation of B. Finally, we study under which conditions an ordered abelian group is the dimension group for some Bratteli diagram. This master thesis is based on the book by Ian F. Putnam Cantor minimal systems, published in University Lecture Series, 70. American Mathematical Society, Providence, RI, 2018. [6].

<p>Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics.</p><p>In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared.</p><p>A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.</p>

Fractal analysis is an important tool when we need to study geometrical objects less regular than ordinary ones, e.g. a set with a non-integer dimension value. It has developed intensively over the last 30 years which gives a hint to its young age as a branch within mathematics. In this thesis we take a look at some basic measure theory needed to introduce certain definitions of fractal dimensions, which can be used to measure a set's fractal degree. Comparisons of these definitions are done and we investigate when they coincide. With these tools different fractals are studied and compared. A key idea in this thesis has been to sum up different names and definitions referring to similar concepts.

This Master's thesis deals with history of Fractal geometry and describes the fractal science development. In the begining there are essential Fractal science terms explained. Then description of fractal types and typical or most known examples of them are mentioned. Fractal knowledge application besides computer graphics area is discussed. Thesis informs about fractal geometry practical usage. Few present software packages or more programs which can be used for making fractal pictures are described in this work. Some of theirs capabilities are described. Thesis' practical part consists of slides, demonstrational program and poster. Electronical slides represents brief scheme usable for fractal geometry realm lectures. Program generates selected fractal types. Thesis results are projected on poster.

In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.