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Mucheroni, Laís Fernandes [UNESP]. "Dimensão de Hausdorff e algumas aplicações." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/151653.
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Intuitivamente, um ponto tem dimensão 0, uma reta tem dimensão 1, um plano tem dimensão 2 e um cubo tem dimensão 3. Porém, na geometria fractal encontramos objetos matemáticos que possuem dimensão fracionária. Esses objetos são denominados fractais cujo nome vem do verbo "frangere", em latim, que significa quebrar, fragmentar. Neste trabalho faremos um estudo sobre o conceito de dimensão, definindo dimensão topológica e dimensão de Hausdorff. O objetivo deste trabalho é, além de apresentar as definições de dimensão, também apresentar algumas aplicações da dimensão de Hausdorff na geometria fractal.
We know, intuitively, that the dimension of a dot is 0, the dimension of a line is 1, the dimension of a square is 2 and the dimension of a cube is 3. However, in the fractal geometry we have objects with a fractional dimension. This objects are called fractals whose name comes from the verb frangere, in Latin, that means breaking, fragmenting. In this work we will study about the concept of dimension, defining topological dimension and Hausdorff dimension. The purpose of this work, besides presenting the definitions of dimension, is to show an application of the Hausdorff dimension on the fractal geometry.
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Reeve, Russell Lynn. "Estimating the Hausdorff dimension." Diss., Virginia Tech, 1990. http://hdl.handle.net/10919/37744.
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Spear, Donald W. "Hausdorff, Packing and Capacity Dimensions." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc330990/.
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In this thesis, Hausdorff, packing and capacity dimensions are studied by evaluating sets in the Euclidean space R^. Also the lower entropy dimension is calculated for some Cantor sets. By incorporating technics of Munroe and of Saint Raymond and Tricot, outer measures are created. A Vitali covering theorem for packings is proved. Methods (by Taylor and Tricot, Kahane and Salem, and Schweiger) for determining the Hausdorff and capacity dimensions of sets using probability measures are discussed and extended. The packing pre-measure and measure are shown to be scaled after an affine transformation.
A Cantor set constructed by L.D. Pitt is shown to be dimensionless using methods developed in this thesis. A Cantor set is constructed for which all four dimensions are different. Graph directed constructions (compositions of similitudes follow a path in a directed graph) used by Mauldin and Willjams are presented. Mauldin and Williams calculate the Hausdorff dimension, or, of the object of a graph directed construction and show that if the graph is strongly connected, then the a—Hausdorff measure is positive and finite. Similar results will be shown for the packing dimension and the packing measure. When the graph is strongly connected, there is a constant so that the constant times the Hausdorff measure is greater than or equal to the packing measure when a subset of the realization is evaluated. Self—affine Sierpinski carpets, which have been analyzed by McMullen with respect to their Hausdorff dimension and capacity dimension, are analyzed with respect to their packing dimension. Conditions under which the Hausdorff measure of the construction object is positive and finite are given.
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Serantola, Leonardo Pereira. "Dimensão generalizada de Hausdorff /." São José do Rio Preto, 2019. http://hdl.handle.net/11449/191015.
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Orientador: Márcio Ricardo Alves GouveiaResumo: O presente trabalho trata de conceitos relacionados com a medida generalizada de Hausdorff, onde o principal objetivo consiste na obtenção de conjuntos cuja dimensão seja um número positivo não inteiro. Ele começa com uma definição sobre as propriedades que uma função de conjunto deve satisfazer para ser considerada uma medida de Carathéodory, suas implicações e consequências. Após a explicação destes conceitos iniciais, dá-se alguns exemplos de funções de conjunto contínuas e monótonas com a apresentação da função de escala logarítmica, que é peça chave para o desenvolvimento de conjuntos de medidas positivas não inteiras, além da introdução da medida de Hausdorff com seus desdobramentos. Algumas hipóteses sobre funções côncavas são apresentadas juntamente com fórmulas deduzidas com bases nestas hipóteses e na concavidade da função. Utiliza-se a função de escala logarítima para a determinação da dimensão de vários conjuntos, inclusive o conjunto de Cantor. Posteriormente, há uma adaptação dos conceitos trabalhados para o tratamento de dimensões relacionadas à números diádicos irracionais. Por fim, os conceitos tratados sobre a reta real são estendidos para produtos cartesianos, com especial enfoque para conjuntos planares.
Abstract: The present work deals with concepts related to the generalized Hausdorff measure, where the main objective is to obtain sets whose dimension is a positive non integer number. It begins with a definition of the properties that a set function must satisfy to be considered a Carathéodory measure, their implications and consequences. Following the explanation of these initial concepts, some examples of continuous and monotonous set functions are given with the presentation of the logarithmic scale function, which is key to the development of non-integer positive measure sets, in addition to the introduction of the Hausdorff measure with its developments. Some assumptions about concave functions are presented together with formulas derived from these assumptions and the concavity of the function. The logarithmic scale function is used to determine the dimension of various sets, including the Cantor set. Later, there is an adaptation of the concepts worked for the treatment of dimensions related to irrational dyadic numbers. Finally, the concepts treated on the real line are extended to Cartesian products, with special focus on planar sets.
Mestre
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Dufloux, Laurent. "Dimension de Hausdorff des ensembles limites." Thesis, Sorbonne Paris Cité, 2015. http://www.theses.fr/2015USPCD022/document.
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Soit G le groupe SO°(1, n) (n ≥ 3) ou PU(1, n) (n ≥ 2) et fixons une décomposition d'Iwasawa G = KAN. Soit ɼ un sous-groupe discret de G, que nous supposons Zariski-dense et de mesure de Bowen-Margulis-Sullivan finie. Lorsque G = SO°(1, n), nous étudions la géométrie de la mesure de Bowen-Margulis-Sullivan le long des sous-groupes fermés connexes de N, en lien avec la dichotomie de Mohammadi-Oh. Nous établissons des résultats déterministes sur la dimension des projections de la mesure de Patterson- Sullivan. Lorsque G = PU(1, n), nous relions la géométrie de la mesure de Bowen- Margulis-Sullivan le long du centre du groupe de Heisenberg au problème du calcul de la dimension de Hausdorff de l'ensemble limite relativement à la distance sphérique au bord. Nous calculons cette dimension pour certains groupes de SchottkyLet G be the group SO° (1,n) (n ≥ 3) or PU(1, n) (n ≥ 2) and fix some Iwasawa decomposition G = KAN. Let ɼ be a discrete subgroup of G.We assume that ɼ is Zariski-dense with finite Bowen-Margulis-Sullivan measure. When G = SO°(1,n), we investigate the geometry of the Bowen-Margulis-Sullivan measure elong connected closed subgroups of N. This is related to the Mohammadi-Oh dichotomy. We then prove deterministic results on the dimension of projections of Patterson-Sullivan measure. When G = PU(1,n), we relate the geometry of Bowen-Margulis-Sullivan measure along the center of Heisenberg group to the problem of computing the Hausdorff dimension of the limit set with respect to the spherical metric on the boudary. We construct some Schottky subgroups for wich we are able to compute this dimension
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Zürcher, Thomas Zürcher Thomas. "Hausdorff dimension and regularity of Sobolev functions /." [S.l.] : [s.n.], 2009. http://www.ub.unibe.ch/content/bibliotheken_sammlungen/sondersammlungen/dissen_bestellformular/index_ger.html.
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Martin, Charles 1966. "Hausdorff dimension of harmonic measures in Rd." Thesis, McGill University, 1993. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=69764.
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In 1986, J. Bourgain showed that, for a given dimension d $ ge$ 2, there exists $ rho sb{d}$ $<$ d such that any harmonic measure in $ Re sp{d}$ is supported by a set of Hausdorff dimension at most $ rho sb{d}$.This thesis presents a detailed and comprehensive exposition of Bourgain's theorem. Formal definitions of harmonic measures and Hausdorff dimensions are provided and all the required preliminary results about these concepts are rigorously proved. Generally speaking, the proof of the theorem itself is similar to the one originally presented by Bourgain. However, a more structured approach and an increased level of details make the argument easier to follow and understand.
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Siebert, Kitzeln B. "A modern presentation of "dimension and outer measure"." Columbus, Ohio : Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1211395297.
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Satiracoo, Pairote. "Hausdorff dimension of attractors of infinitely renormalizable maps." Thesis, University of Warwick, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.400147.
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Brandão, Daniela Teresa Quaresma Santos. "Dimensões fractais e dimensão de correlação." Master's thesis, Universidade de Évora, 2008. http://hdl.handle.net/10174/17740.
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O objetivo deste trabalho é o estudo da dimensão fractal, nomeadamente a dimensão de Hausdorff, dimensão de capacidade e dimensão de correlação, relacionando-as e efetuando o cálculo em alguns exemplos. Sempre que se considera indispensável, são apresentadas noções introdutórias para uma melhor compreensão dos conceitos analisados. O Capítulo 2 é dedicado ao estudo da dimensão de Hausdorff, introduzindo, previamente, uma noção de medida, de Hausdorff. No Capítulo 3 analisamos a dimensão de capacidade, suas propriedades e inconvenientes, relacionando, no final, esta dimensão com a dimensão de Hausdorff. O Capítulo 4 estuda técnicas para calcular dimensões. São estudados subconjuntos de medida. Finita, sistemas de funções iteradas, conjuntos auto-semelhantes e auto-afins e dimensões de gráficos. O Capítulo 5 é dedicado à dimensão de correlação. Estuda o expoente de correlação Introduzido por Grassberger e Procaccia. São analisadas funções de dimensão 1 e no plano. Terminamos com o estudo de séries temporais de variável única. ABSTRACT: The aim of this work is the study of the fractal dimension, namely the Hausdorff dimension, the box-counting dimension and the correlation dimension, relating and computing them in some examples. Everytime it is necessary we introduce the basic concepts to a better understanding of the concepts analysed in this work. Chapter 2 is dedicated to the study of the Hausdorff dimension, introducing first the notion of Hausdorff measure. Chapter 3 is concerned with the box-counting dimension, its properties and problems. Then we relate this dimension With Hausdorff dimension studied in Chapter 2. Chapter 4 is dedicated to the techniques for calculating dimensions. We study subsets of finite measure, iterated function schemes, self-similar and self-affine sets and dimensions of graphs. Finally, in Chapter 5 we present the correlation dimension. We study the correlation exponent, introduced by Grassberger and Procaccia. We finish this Chapter with a study of single-variable time series.
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Rydell, Simon. "Worm simulation of Hausdorff dimension of critical loop fluctuations." Thesis, KTH, Fysik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-239619.
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Tiozzo, Giulio. "Entropy, Dimension and Combinatorial Moduli for One-Dimensional Dynamical Systems." Thesis, Harvard University, 2013. http://dissertations.umi.com/gsas.harvard:10891.
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The goal of this thesis is to provide a unified framework in which to analyze the dynamics of two seemingly unrelated families of one-dimensional dynamical systems, namely the family of quadratic polynomials and continued fractions. We develop a combinatorial calculus to describe the bifurcation set of both families and prove they are isomorphic. As a corollary, we establish a series of results describing the behavior of entropy as a function of the parameter. One of the most important applications is the relation between the topological entropy of quadratic polynomials and the Hausdorff dimension of sets of external rays landing on principal veins of the Mandelbrot set.Mathematics
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Beffara, Vincent. "Mouvement brownien plan, SLE, invariance conforme et dimensions fractales." Paris 11, 2003. http://www.theses.fr/2003PA112039.
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Cette thèse est consacrée à l'étude de quelques propriétés géométriques du mouvement brownien plan et du processus SLE (ou processus de Loewner stochastique). On prouve qu'il existe presque sûrement sur la trajectoire brownienne plane des points "pivots", i. E. Des points de coupure autour desquels l'une des moitiés de la trajectoire peut pivoter d'un angle strictement positif sans jamais rencontrer l'autre moitié; l'ensemble des point pivots d'angle donné (suffisamment petit) est alors de dimension de Hausdorff strictement positive. Concernant le SLE, le principal résultat obtenu dans cette thèse est le calcul de la dimension de Hausdorff de la courbe qui l'engendre (qui est égale à un plus la huitième partie du paramètre), et ceci pour tout paramètre positif différent de quatre et inférieur à huit. On s'intéresse également au problème de la généralisation du processus SLE à des surfaces non simplement connexes; on montre que cela est faisable pour deux valeurs particulières du paramètre (six et huit tiers), mais que l'on perd la propriété d'universalité du SLE usuelThis thesis is dedicated to the study of various geometric properties of planar Brownian motion and the SLE process (also known as stochastic Loewner evolution). We prove that, on a typical planar Brownian path, there almost surely exist "pivoting" points, i. E. Cut-points around which one half of the curve can rotate by a positive angle without ever intersecting the other half of the path; the set of all pivoting points of a given positive (small enough) angle is then of positive Hausdorff dimension. About SLE, the main result we obtain in this thesis is the computation of the Hausdorff dimension of the curve generating it (the dimension is equal to one plus one eighth of the parameter), for any positive parameter smaller than eight and different from four. We also study the problem of the generalization of the SLE process to non-simply connected surfaces; we show that the construction is doable for two particular values of the parameter (six and eight thirds), but the universality property of usual SLE 1S then lost
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Lopez, Marco Antonio. "Hausdorff Dimension of Shrinking-Target Sets Under Non-Autonomous Systems." Thesis, University of North Texas, 2018. https://digital.library.unt.edu/ark:/67531/metadc1248505/.
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For a dynamical system on a metric space a shrinking-target set consists of those points whose orbit hit a given ball of shrinking radius infinitely often. Historically such sets originate in Diophantine approximation, in which case they describe the set of well-approximable numbers. One aspect of such sets that is often studied is their Hausdorff dimension. We will show that an analogue of Bowen's dimension formula holds for such sets when they are generated by conformal non-autonomous iterated function systems satisfying some natural assumptions.
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Yang, Lei. "HAUSDORFF DIMENSION OF DIVERGENT GEODESICS ON PRODUCT OF HYPERBOLIC SPACES." The Ohio State University, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=osu1401466357.
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Silva, Alex Pereira da. "Um estudo da teoria das dimensões aplicado a sistemas dinâmicos." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-07082015-113917/.
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Este trabalho se propõe a estudar o comportamento assintótico dos sistemas dinâmicos autônomos respaldado na Teoria das Dimensões. Mais precisamente, vamos compreender de que maneira nos é útil limitar a dimensão fractal do atrator global de um semigrupo a fim de estudar a dinâmica em dimensão finita, sem que se perca informações sobre a dinâmica ao fazê-lo. Para tanto, o Teorema de Mañé tem um papel decisivo junto às propriedades da dimensão de Hausdorff e a da dimensão fractal; nos permitindo encontrar uma projeção cuja restrição ao atrator é injetora sobre um espaço de dimensão finita. Constatamos ainda que esta abordagem por projeções se aplica largamente a semigrupos originados de equações diferenciais em espaços de Banach de dimensão infinita.In this work, we study the asymptotic behavior of autonomous dynamical systems supported on the Dimension Theory. More precisely, we understand how fractal dimension finiteness of the global attractor of a semigroup can be used to study the dynamics in finite dimension, without losing information on the dynamics in doing so. For this purpose, the Mañés Theorem plays a decisive role considering the Hausdorff dimension properties and the fractal dimension; thanks to which we managed to find a projection whose restriction to the attractor is an injective application over a finite dimensional space. Besides, we also acknowledge that this projections approach is largely applied to semigroups arrising from differential equations in infinite dimensional Banach spaces.
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Sumi, Hiroki. "Dynamics of rational semigroups and Hausdorff dimension of the Julia sets." Kyoto University, 1999. http://hdl.handle.net/2433/157102.
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本文データは平成22年度国立国会図書館の学位論文(博士)のデジタル化実施により作成された画像ファイルを基にpdf変換したものであるKyoto University (京都大学)
0048
新制・課程博士
博士(人間・環境学)
甲第7909号
人博第52号
10||134(吉田南総合図書館)
新制||人||14(附属図書館)
UT51-99-G503
京都大学大学院人間・環境学研究科人間・環境学専攻
(主査)教授 宇敷 重廣, 教授 淺野 潔, 助教授 木上 淳
学位規則第4条第1項該当
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Lima, Carlos Alberto Siqueira. "Dinâmica complexa e formalismo termodinâmico." Universidade de São Paulo, 2011. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-06062011-152648/.
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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 complexosWe 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
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Brucks, Karen M. (Karen Marie) 1957. "Dynamics of One-Dimensional Maps: Symbols, Uniqueness, and Dimension." Thesis, North Texas State University, 1988. https://digital.library.unt.edu/ark:/67531/metadc332102/.
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This dissertation is a study of the dynamics of one-dimensional unimodal maps and is mainly concerned with those maps which are trapezoidal. The trapezoidal function,
f_e, is defined for eΣ(0,1/2) by f_e(x)=x/e for xΣ[0,e], f_e(x)=1 for xΣ(e,1-e), and f_e(x)=(1-x)/e for xΣ[1-e,1]. We study the symbolic dynamics of the kneading sequences and relate them to the analytic dynamics of these maps. Chapter one is an overview of the present theory of Metropolis, Stein, and Stein (MSS). In Chapter two a formula is given that counts the number of MSS sequences of length n. Next, the number of distinct primitive colorings of n beads with two colors, as counted by Gilbert and Riordan, is shown to equal the number of MSS sequences of length n. An algorithm is given that produces a bisection between these two quantities for each n. Lastly, the number of negative orbits of size n for the function f(z)=z^2-2, as counted by P.J. Myrberg, is shown to equal the number of MSS sequences of length n. For an MSS sequence P, let H_ϖ(P) be the unique common extension of the harmonics of P. In Chapter three it is proved that there is exactly one J(P)Σ[0,1] such that the itinerary of λ(P) under the map is λ(P)f_e is H_ϖ(P).
In Chapter four it is shown that only period doubling or period halving bifurcations can occur for the family λf_e, λΣ[0,1]. Results concerning how the size of a stable orbit changes as bifurcations of the family λf_e occur are given.
Let λΣ[0,1] be such that 1/2 is a periodic point of λf_e. In this case 1/2 is superstable. Chapter five investigates the boundary of the basin of attraction of this stable orbit. An algorithm is given that yields a graph directed construction such that the object constructed is the basin boundary. From this we analyze the Hausdorff dimension and measure in that dimension of the boundary. The dimension is related to the simple β-numbers, as defined by Parry.
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Inui, Kanji. "Study of the fractals generated by contractive mappings and their dimensions." Kyoto University, 2020. http://hdl.handle.net/2433/253370.
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Kyoto University (京都大学)0048
新制・課程博士
博士(人間・環境学)
甲第22534号
人博第937号
新制||人||223(附属図書館)
2019||人博||937(吉田南総合図書館)
京都大学大学院人間・環境学研究科共生人間学専攻
(主査)教授 角 大輝, 教授 上木 直昌, 准教授 木坂 正史
学位規則第4条第1項該当
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Schubert, Hendrik [Verfasser]. "Über die Hausdorff-Dimension der Juliamenge von Funktionen endlicher Ordnung / Hendrik Schubert." Kiel : Universitätsbibliothek Kiel, 2008. http://d-nb.info/1019659858/34.
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Sönmez, Ercan [Verfasser]. "Hausdorff dimension results for operator-self-similar stable random fields / Ercan Sönmez." Düsseldorf : Universitäts- und Landesbibliothek der Heinrich-Heine-Universität Düsseldorf, 2017. http://d-nb.info/1128293935/34.
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Cohen, Dolav. "An exploration of fractal dimension." Kansas State University, 2013. http://hdl.handle.net/2097/16194.
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Master of ScienceDepartment of Mathematics
Hrant Hakobyan
When studying geometrical objects less regular than ordinary ones, fractal analysis becomes a valuable tool. Over the last 30 years, this small branch of mathematics has developed extensively. Fractals can be de fined as those sets which have non-integral Hausdor ff dimension. In this thesis, we take a look at some basic measure theory needed to introduce certain de finitions of fractal dimensions, which can be used to measure a set's fractal degree. We introduce Minkowski dimension and Hausdor ff dimension as well as explore some examples where they coincide. Then we look at the dimension of a measure and some very useful applications. We conclude with a well known result of Bedford and McMullen about the Hausdor ff dimension of self-a ffine sets.
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Levesley, Jason. "Inhomogeneous and non-linear metric diophantine approximation." Thesis, University of York, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298490.
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Nilsson, Anders. "Dimensions and projections." Licentiate thesis, Umeå University, Mathematics and Mathematical Statistics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-939.
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This thesis concerns dimensions and projections of sets that could be described as fractals. The background is applied problems regarding analysis of human tissue. One way to characterize such complicated structures is to estimate the dimension. The existence of different types of dimensions makes it important to know about their properties and relations to each other. Furthermore, since medical images often are constructed by x-ray, it is natural to study projections.
This thesis consists of an introduction and a summary, followed by three papers.
Paper I, Anders Nilsson, Dimensions and Projections: An Overview and Relevant Examples, 2006. Manuscript.
Paper II, Anders Nilsson and Peter Wingren, Homogeneity and Non-coincidence of Hausdorff- and Box Dimensions for Subsets of ℝn, 2006. Submitted.
Paper III, Anders Nilsson and Fredrik Georgsson, Projective Properties of Fractal Sets, 2006. To be published in Chaos, Solitons and Fractals.
The first paper is an overview of dimensions and projections, together with illustrative examples constructed by the author. Some of the most frequently used types of dimensions are defined, i.e. Hausdorff dimension, lower and upper box dimension, and packing dimension. Some of their properties are shown, and how they are related to each other. Furthermore, theoretical results concerning projections are presented, as well as a computer experiment involving projections and estimations of box dimension.
The second paper concerns sets for which different types of dimensions give different values. Given three arbitrary and different numbers in (0,n), a compact set in ℝn is constructed with these numbers as its Hausdorff dimension, lower box dimension and upper box dimension. Most important in this construction, is that the resulted set is homogeneous in the sense that these dimension properties also hold for every non-empty and relatively open subset.
The third paper is about sets in space and their projections onto planes. Connections between the dimensions of the orthogonal projections and the dimension of the original set are discussed, as well as the connection between orthogonal projection and the type of projection corresponding to realistic x-ray. It is shown that the estimated box dimension of the orthogonal projected set and the realistic projected set can, for all practical purposes, be considered equal.
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Akter, Hasina. "Real Analyticity of Hausdorff Dimension of Disconnected Julia Sets of Cubic Parabolic Polynomials." Thesis, University of North Texas, 2012. https://digital.library.unt.edu/ark:/67531/metadc271768/.
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Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .
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Catalan, Thiago Aparecido. "Resultados genéricos sobre entropia e dimensão de Hausdorff para difeomorfismos conservativos sobre superfícies." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-10062008-142348/.
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Apresentamos duas propriedades genéricas para difeomorfismos conservativos da classe \'C POT.1\' sobre uma superfície compacta de dimensão dois. Obtemos uma limitação inferior para entropia topológica de difeomorfismos genéricos, e mostramos que tais difeomorfismos sempre possuem conjuntos invariantes fechados com órbitas densas e dimensão de Hausdorff doisWe present two generic properties of \'C POT.1\" area preserving diffeomorphisms of a two dimensional compact oriented surface. We obtain a lower bound for the topological entropy of a generic diffeomorphisms, and we show that such a diffeomorphism always has closed invariant sets with dense orbits and Hausdorff dimension two
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Haas, Stephen. "The Hausdorff Dimension of the Julia Set of Polynomials of the Form zd + c." Scholarship @ Claremont, 2003. https://scholarship.claremont.edu/hmc_theses/148.
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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.
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Ben, Nasr Fathi. "Étude de mesures aléatoires et calculs de dimensions de Hausdorff." Paris 11, 1986. http://www.theses.fr/1986PA112141.
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Ce travail comprend deux parties. La première traite du calcul de la dimension de Hausdorff de certains ensembles du plan dont les recouvrements naturels se font au moyen de rectangles qui s'aplatissent à mesure que leur diamètre tend vers zéro. Or on sait que les mesures et dimension de Hausdorff se définissent au moyen de recouvrements par des boules. Il se pose donc le problème de passer d'un recouvrement économique par des rectangles à un recouvrement économique par des boules. Les ensembles que nous étudions sont définis par des propriétés des développements dans des bases éventuellement différentes des coordonnées de leurs points. Dans certains cas nous savons déterminer la dimension de Hausdorff de ces ensembles, dans d'autres nous en obtenons seulement un encadrement. Cette étude tire son origine de résultats d'Eggleston qu'elle généralise. Nous étudions aussi des ensembles aléatoires obtenus en effectuant des partages aléatoires successifs à la manière de Cantor, et déterminons leur dimension de Hausdorff, généralisant et améliorant ainsi des résultats de J. Peyrière. Dans la seconde partie nous définissons et étudions une variante d'un modèle de turbulence dû à B. Mandelbrot et étudié par J. P. Kahane et J. Peyrière : une mesure aléatoire est définie par un produit infini de fonctions aléatoires. Nous donnons une condition nécessaire et suffisante de non-dégénérescence de ce processus. Nous déterminons aussi à quelle condition certains moments sont finis et donnons la dimension minimum des boréliens qui portent une partie de cette mesureThis thesis is divided into two parts. The first one deals with the determination of the Hausdorff dimension of some planar sets of which the natural coverings are made of rectangles which become thinner and thinner as their diameter tends to zero. But we know that measures and Hausdorff dimension are defined by the mean of cave rings by balls. So the problem to pass from economical coverings by rectangles to economical coverings by balls is posed. The sets we are studying are defined by properties of the expansions in two different bases of the coordinates of their points. In certain cases we determine the Hausdorff dimension of these sets, which in ether cases we only obtain lower and upper bounds for it. This study sterns results by Eggleston which we generalize. We also determine the dimension of sets obtained by Cantor like constructions, generalizing and improving results by Peyrière. In the second part we define and study a modification of a model of turbulence due to B. Mandelbrot and studied by J. P. Kahane and J. Peyrière: a random measure is defined by an infinite product of random functions. We give a necessary and sufficient condition of non-degeneracy of this process. We also determine under what conditions within moments are finite and get the minimum dimension of sets which carry a part of this measure
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Santos, Filipe André Paulino. "Análise da difusão através de métodos probabilísticos e dimensão de Hausdorff." Master's thesis, Instituto Superior de Economia e Gestão, 2012. http://hdl.handle.net/10400.5/10848.
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Mestrado em Matemática FinanceiraA presente tese consiste numa exposição teórica da relação entre os processos de difusão e os métodos probabilísticos que podem ser usados para os descrever. Centramos-nos em processos estocásticos de aplicabilidade em Finanças como o passeio aleatório para o caso discreto e o movimento browniano visto como o seu limite no caso contínuo. É feita a modelação do fenómeno do calor culminando na resolução ou verificação da equação do calor, intimamente ligada com a equação de Black-Scholes como patente na fórmula de Feynman-Kac. Na segunda metade é introduzida a geometria fractal, tendo como principal conceito a dimensão de Hausdorff. Esta dimensão é de extrema importância para o estudo das trajectórias do movimento browniano e de todos os outros os processos utilizados em Finanças que exibem o mesmo comportamento fractal. Além de todo um conjunto de ferramentas e técnicas para a análise de fractais, é feito o cálculo rigoroso da dimensão de Hausdorff do gráfico das trajectórias do movimento browniano. São ainda obtidos resultados sobre a diferenciabilidade por intervalos dessas mesmas trajectórias.
The present thesis is a theoretical exposition of the relation between the diffusion processes and the probabilistic methods that can be used to describe them. We focus in stochastic processes with applicability in Finance, like the random walk for the discrete case and the Brownian motion seen as his continuous time limit. Then we do the modelling of the heat phenomenon culminating in the resolution or verification of the heat equation which is deeply connected with the Black-Scholes equation as in the Feynman-Kac formula. On the second half we introduce the fractal geometry, having as main concept the Hausdorff dimension. This is of great importance for the study of Brownian motion trajectories and all other processes used in Finance presenting the same fractal behaviour. Besides the introduction of a whole set of tools and techniques for fractal analysis, we do the rigorous Hausdorff dimension computation for the Brownian motion trajectories. Moreover, we obtain results about the differentiability by intervals for these same trajectories.
31
Roueff, François. "Dimension de Hausdorff du graphe d'une fonction continue : une étude analytique et statistique." Paris, ENST, 2000. http://www.theses.fr/2000ENST0032.
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Parmi les autres dimensions fractales, la dimension de Hausdorff est souvent considérée comme la plus canonique. Malheureusement, elle est aussi celle dont le calcul pose le plus de problèmes, notamment dans le cas particulier qui nous a interessé dans cette thèse : le graphe d'une fonction continue f, définie sur r d. Deux types de résultats sont connus dans ce cadre. Des résultats déterministes : on sait majorer la dimension de Hausdorff du graphe de f par des indices de régularite plus accessibles. Des résultats probabilistes : on sait calculer la dimension de Hausdorff presque sure du graphe de f pour f appartenant à des classes précises de fonctions aléatoires. Un exeme bien connu de ces classes de fonctions aléatoires sont les trajectoires continues des processus de Adler. Dns un premier temps, nous généralisons ces résultats en utilisant une décomposition de f en série d'ondelettes. Les résultats obtenus établissent en particulier comment la régularite d'une fonction continue f en terme d'espaces de Besov estreliée à ladimension de Hausdorff de son graphe en général. Nous posons par ailleurs le problème des généralisations possibles des processus de Adler au cas non-gaussien et/ou non-stationnaire et mettons notre approche en parallèle avec des modéles déjà proposes dans le cadre de l'analyse multifractal. Dans un deuxieme temps, nous nous intéressons a l'estimation du paramètre de Hurst d'un processus de Adler dans un modèle semi-paramétrique à partir d'un échantillon composé de n observations du processus, équi-reparties dans 0, 1. Nous déterminons des classes de tels processus sur lesquelles la vitesse de convergence en moyenne quadratique de l'estimateur utilisé est optimale au sens minimax.
32
Roueff, François. "Dimension de Hausdorff du graphe d'une fonction continue : une étude analytique et statistique /." Paris : École nationale supérieure des télécommunications, 2001. http://catalogue.bnf.fr/ark:/12148/cb37629396n.
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Attia, Najmeddine. "Comportement asymptotique de marches aléatoires de branchement dans Rd et dimension de Hausdorff." Paris 13, 2012. http://scbd-sto.univ-paris13.fr/intranet/edgalilee_th_2012_attia.pdf.
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Nous calculons presque sûrement, simultanément, les dimensions de Hausdorff des ensembles de branches infinies de la frontière d’un arbre de Galton-Watson super-critique (muni d’une métrique aléatoire) le long desquelles les moyennes empiriques d’une marche aléatoire de branchement vectorielle admettent un ensemble donné de points limites. Cela va au-delà de l’analyse multifractale, question pour laquelle nous complétons les travaux antérieurs en considérant les ensembles associés à des niveaux situés dans la frontière du domaine d’étude. Nous utilisons une méthode originale dans ce contexte, consistant à construire des mesures de Mandelbrot inhomogènes appropriées. Cette méthode est inspirée de l’approche utilisée pour résoudre des questions similaires dans le contexte de la dynamique hyperboliques pour les moyennes de Birkhoff de potentiels continus. Elle exploite des idées provenant du chaos multiplicatif et de la théorie de la percolation pour estimer la dimension inférieure de Hausdorff des mesures de Mandelbrot inhomogènes. Cette méthode permet de renforcer l’analyse multifractale en raffinant les ensembles de niveaux de telle sorte qu’ils contiennent des branches infinies le long desquels on observe une version quantifiée de la loi des grands nombres d’Erdös Renyi ; de plus elle permet d’obtenir une loi de type 0-∞ pour les mesures de Hausdorff de ces ensembles. Nos résultats donnent naturellement des informations géométriques et de grandes déviations sur l’hétérogénéité du processus de naissance le long des différentes branches infinies de l’arbre de Galton-WatsonWe compute almost surely (simultaneaously) the Hausdorff dimensions of the sets of infinite branches of the boundary of a super-critical Galton-Watson tree (endowed with a random metric)along which the averages of a vector valued branching random walk have a given set of limit points. This goes beyond multifractal analysis, for which we complete the previous works on the subject by considering the sets associated with levels in the boundary of the domain of study. Our method is inspired by some approach used to solve similar questions in the different context of hyperbolic dynamics for the Birkhoff averages of continuous potentials. It also exploits ideas from multiplicative chaos and percolation theories, which are used to estimate the lower Hausdorff dimension of a family of inhomogeneous Mandelbrot measures. This method also makes it possible to strengthen the multifractal analysis of the branching random walk averages by refining the level sets so that they contain branches over which a quantified version of the Erdös Renyi law of large numbers holds, and yields a 0-∞ law for the Hausdorff measures of these sets. Our results naturally give geometric and large deviations information on the heterogeneity of the birth process along different infinite branches of the Galton-Watson tree
34
Gauthier, Thomas. "Dimension de Hausdorff de lieux de bifurcations maximales en dynamique des fractions rationnelles." Phd thesis, Université Paul Sabatier - Toulouse III, 2011. http://tel.archives-ouvertes.fr/tel-00646407.
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Dans l'espace $\mathcal{M}_d$ des modules des fractions rationnelles de degré $d$, le lieu de bifurcation est le support d'un $(1,1)$-courant positif fermé $T_{\textup{bif}}$ appelé \emph{courant de bifurcation}. Ce courant induit une mesure $\mu_{\textup{bif}}=(T_{\textup{bif}})^{2d-2}$ dont le support est le siége de bifurcations maximales. Notre principal résultat est que le support de $\mu_{\textup{bif}}$ est de dimension de Hausdorff totale $2(2d-2)$. Il s'ensuit que l'ensemble des fractions rationnelles de degré $d$ possédant $2d-2$ cycles neutres distincts est dense dans un ensemble de dimension de Hausdorff totale. Remarquons que jusqu'alors, seule l'existence de telles fractions rationnelles (Shishikura) était connue. Mentionnons que pour notre démonstration, nous établissons au préalable que les fractions rationnelles $(2d-2)$-Misiurewicz appartiennent au support de $\mu_{\textup{bif}}$. \par Le dernier chapitre, indépendant du reste de la thése, traite de l'espace $\mathcal{M}_2$. Nous montrons que, dans ce cas, le courant $T_{\textup{bif}}$ se prolonge naturellement á $\p^2$ en un $(1,1)$-courant positif fermé dont nous calculons les nombres de Lelong. Nous montrons aussi que le support de la mesure $\mu_{\textup{bif}}$ est non-borné dans $\mathcal{M}_2$.
35
Gauthier, Thomas. "Dimension de Hausdorff de lieux de bifurcations maximales en dynamique des fractions rationnelles." Phd thesis, Toulouse 3, 2011. http://thesesups.ups-tlse.fr/1477/.
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Dans l'espace Md des modules des fractions rationnelles de degré d,le lieu de bifurcation est le support d'un (1, 1)-courant positif fermé Tbif appelé courant de bifurcation. Ce courant induit une mesure µbif := (Tbif)2d-2 dont le support est le siège de bifurcations maximales. Notre principal résultat est que le support de µbif est de dimension de Hausdor. Totale 2(2d - 2). Il s'ensuit que l'ensemble des fractions rationnelles de degré d possédant 2d - 2cycles neutres distincts est dense dans un ensemble de dimension de Hausdor. Totale. Remarquons que jusqu'alors, seule l'existence de telles fractions rationnelles (Shishikura) était connue. Mentionnons que pour notre démonstration, nous établissons au préalable que les fractions rationnelles (2d - 2)-Misiurewicz appartiennent au support de µbif. Le dernier chapitre, indépendant du reste de la thèse, traite de l'espace M2. Nous montrons que, dans ce cas, le courant Tbif se prolonge naturellement à P2 en un (1, 1)-courant positif fermé dont nous calculons les nombres de Lelong. Nous montrons aussi que le support de la mesure µbif est non-borné dans M2In the moduli space Md of degree d rational maps, the bifurcation locus is the support of a closed (1, 1) positive current Tbif called bifurcation current. This current gives rise to a measure µbif := (Tbif)2d-2 whose support is the seat of strong bifurcations. Our main result says that supp(µbif)has maximal Hausdor. Dimension 2(2d-2). It follows that the set of degree d rational maps having 2d-2distinct neutral cycles is dense in a set of full Hausdor. Dimension. Note that previously, only the existence of such rational maps (Shishikura) was known. Let us mention that for our proof, we. Rst establish that the (2d - 2)-Misiurewicz rational maps belong to the support of µbif. The last chapter, which is independent of the rest of the thesis, deals with the space M2. We prove that, in this case, the current Tbif naturally extends to a (1, 1)-closed positive current on P2 which we calculate the Lelong numbers. We also show that the support of µbif is unbounded in M2
36
Franz, Astrid. "Abschätzungen der Hausdorff-Dimension invarianter Mengen dynamischer Systeme auf Mannigfaltigkeiten unter besonderer Berücksichtigung nicht invertierbarer Abbildungen." [S.l. : s.n.], 1999. http://www.bsz-bw.de/cgi-bin/xvms.cgi?SWB10324702.
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Kiefer, Richard. "Multiple points on the Brownian frontier." Berlin mbv, 2009. http://d-nb.info/993935737/04.
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Snigireva, Nina. "Inhomogeneous self-similar sets and measures." Thesis, St Andrews, 2008. http://hdl.handle.net/10023/X682.
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Luu, Tien Duc. "Régularité des cônes et d’ensembles minimaux de dimension 3 dans R4." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112301/document.
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On étudie dans cette thèse la régularité des cônes et d'ensembles de dimension 3 dans l'espace Euclidien de dimension 4.Dans la première partie, on étudie d'abord la régularité Bi-Hölderienne des cônes minimaux de dimension 3 dans l'espace Euclidien de dimension 4. Ceci nous permet ensuite de montrer qu'il existe un difféomorphisme locale entre un cône minimal de dimension 3 dans l'espace Euclidien de dimension 4 et un cône minimal de dimension 3, de type P, Y ou T, loin d'origine. La méthode est la même que pour les ensembles minimaux de dimension 2. On construit des compétiteurs et on se ramène aux situations connues des ensembles minimaux de dimension 2 dans l'espace Euclidien de dimension 3.Dans la deuxième partie, on utilise le résultat de la première partie pour donner quelques résultats de régularité Bi-Hölderienne pour les ensembles minimaux de dimension 3 dans l'espace Euclidien de dimension 4. On s'intéresse aussi aux ensembles minimaux de Mumford-Shah et on obtient un résultat de l'existence d'un point de type TIn this thesis we study the problems of regularity of three-dimensional minimal cones and sets in l'espace Euclidien de dimension 4In the first part we study the Hölder regularity for minimal cones of dimension 3 in l'espace Euclidien de dimension 4. Then we use this for showing that there exists a local diffeomorphic mapping between a minimal cone of dimension 3 and a minimal cone of dimension 3 of type P, Y or T, away from the origin. The techniques used here are the same as the ones for the regularity of two-dimensional minimal sets. We construct some competitors to reduce to the known situation of two-dimensional minimal sets in l'espace Euclidien de dimension 3.In the second part, we use the first part to give somme results of the Hölder regularity for three-dimensional minimal sets in l'espace Euclidien de dimension 4. We interested also in Mumford-Shah minimal sets and we get a result of the existence of a T-point
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Akman, Murat. "On the Dimension of a Certain Measure Arising from a Quasilinear Elliptic Partial Differential Equation." UKnowledge, 2014. http://uknowledge.uky.edu/math_etds/12.
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We study the Hausdorff dimension of a certain Borel measure associated to a positive weak solution of a certain quasilinear elliptic partial differential equation in a simply connected domain in the plane. We also assume that the solution vanishes on the boundary of the domain. Then it is shown that the Hausdorff dimension of this measure is less than one, equal to one, greater than one depending on the homogeneity of the certain function. This work generalizes the work of Makarov when the partial differential equation is the usual Laplace's equation and the work of Lewis and his coauthors when it is the p-Laplace's equation.
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Fraser, Jonathan M. "Dimension theory and fractal constructions based on self-affine carpets." Thesis, University of St Andrews, 2013. http://hdl.handle.net/10023/3869.
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The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.
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Franz, Astrid. "Abschätzungen der Hausdorff-Dimension invarianter Mengen dynamischer Systeme auf Mannigfaltigkeiten unter besonderer Berücksichtigung nicht invertierbarer Abbildungen." Doctoral thesis, Universitätsbibliothek Chemnitz, 1999. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199900135.
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Die exakte Bestimmung der Dimension invarianter Mengen
dynamischer Systeme ist nur in Ausnahmesituationen möglich.
In der vorliegenden Arbeit wird untersucht, wie unter
Ausnutzung von speziellen Eigenschaften des dynamischen
Systems die Hausdorff-Dimension zugehöriger invarianter
Mengen nach oben und unten abgeschätzt werden kann.
Es wird gezeigt, wie der Grad der Nichtinjektivität der
Abbildung, die das dynamische System erzeugt, in die
Beschreibung des Deformationsverhaltens von k-Volumina
einbezogen werden kann, so daß eine Abschwächung der
Kontraktionsbedingung für Hausdorff-Maße erreicht werden
kann. Dazu werden äußere Hausdorff-Integrale über beliebige
nichtnegative Funktionen betrachtet, die im Falle der
Integration über charakteristische Funktionen den
gewichteten Hausdorff-Maßen entsprechen. Schrankensätze, die
sich als verallgemeinerte Transformationssätze für Integrale
ergeben, charakterisieren das Verhalten der äußeren
Integrale bei Transformationen. Diese Schrankensätze eignen
sich, um Kontraktionsbedingungen für die äußeren
Hausdorff-Maße und damit Oberschranken für die
Hausdorff-Dimension zu formulieren.
Ein weiterer Teil der Arbeit ist der Abschwächung des
Konzepts der hyperbolischen Mengen gewidmet. Es werden
Mengen mit einer äquivarianten Zerlegung des
Tangentialbündels betrachtet, d. h. mit einer Zerlegung, die
unter der Tangentialabbildung invariant bleibt. Solch eine
Zerlegung ermöglicht die Betrachtung der auf die jeweiligen
Teilbündel eingeschränkten Tangentialabbildung, entweder in
der ursprünglichen Zeitrichtung oder in umgekehrter
Zeitrichtung. Im Gegensatz zu hyperbolischen Mengen werden
hier aber keine Voraussetzungen bezüglich der Streckungs-
und Stauchungseigenschaften der Tangentialabbildung in den
Teilräumen gestellt. Unter diesen abgeschwächten Bedingungen
können für invariante Mengen von Diffeomorphismen und
Flüssen ähnliche obere Dimensionsschranken wie für
hyperbolische Mengen erreicht werden, die sowohl in der
Sprache der Singulärwerte als auch der globalen
Lyapunov-Exponenten der Tangentialabbildung und der
topologischen Entropie der Abbildung formuliert werden
können. Es wird außerdem gezeigt, daß sich die für Systeme
mit einer äquivarianten Zerlegung des Tangentialbündels
angewandte Beweistechnik auch auf eine spezielle Klasse
nicht injektiver Abbildungen, die sogenannten
k-1-Endomorphismen, anwenden läßt.
Untere Dimensionsschranken für invariante Mengen dynamischer
Systeme lassen sich in der Regel nur durch das Ausnutzen von
Zusatzstrukturen des Systems ableiten. Die Klasse der
k-1-Endomorphismen weist solche speziellen Strukturen auf.
Die Eigenschaften der invarianten Mengen solcher
Endomorphismen ermöglichen die Konstruktion von Minoranten
für die Hausdorff-Maße ohne Verwendung potentialtheoretischer
Hilfsmittel, aus denen sich eine untere Schranke für die
Hausdorff-Dimension ableiten läßt.
Eine breite Palette von Beispielsystemen demonstriert die
Leistungsfähigkeit der hergeleiteten Abschätzungen der
Hausdorff-Dimension. Insbesondere zählen hierzu
Hufeisenabbildungen, iterierte Funktionensysteme und
Julia-Mengen quadratischer Polynome in der komplexen Ebene.
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Attia, Najmeddine. "Comportement asymptotique de marches aléatoires de branchement dans $\mathbb{R}^d$ et dimension de Hausdorff." Phd thesis, Université Paris-Nord - Paris XIII, 2012. http://tel.archives-ouvertes.fr/tel-00841496.
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Nous calculons presque sûrement, simultanément, les dimensions de Hausdorff des ensembles de branches infinies de la frontière d'un arbre de Galton-Watson super-critique (muni d'une métrique aléatoire) le long desquelles les moyennes empiriques d'une marche aléatoire de branchement vectorielle admettent un ensemble donné de points limites. Cela va au-delà de l'analyse multifractale, question pour laquelle nous complétons les travaux antérieurs en considérant les ensembles associés à des niveaux situés dans la frontière du domaine d'étude. Nous utilisons une méthode originale dans ce contexte, consistant à construire des mesures de Mandelbrot inhomogènes appropriées. Cette méthode est inspirée de l'approche utilisée pour résoudre des questions similaires dans le contexte de la dynamique hyperboliques pour les moyennes de Birkhoff de potentiels continus. Elle exploite des idées provenant du chaos multiplicatif et de la théorie de la percolation pour estimer la dimension inférieure de Hausdorff des mesures de Mandelbrot inhomogènes. Cette méthode permet de renforcer l'analyse multifractale en raffinant les ensembles de niveaux de telle sorte qu'ils contiennent des branches infinies le long desquels on observe une version quantifiée de la loi des grands nombres d'Erdös Renyi ; de plus elle permet d'obtenir une loi de type $0-\infty$ pour les mesures de Hausdorff de ces ensembles. Nos résultats donnent naturellement des informations géométriques et de grandes déviations sur l'hétérogénéité du processus de naissance le long des différentes branches infinies de l'arbre de Galton-Watson.
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Megeney, Alison Claire Verne. "The Besicovitch-Hausdorff dimension of the residual set of packings of convex bodies in R'n." Thesis, University College London (University of London), 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.392996.
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Hassan, S. A. "Bounds for the Hausdorff dimension of exceptional sets arising in the theory of Diophantine approximation." Thesis, University of York, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234934.
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Dickinson, Henrietta. "The Hausdorff dimension of some exceptional sets arising in the theory of metric Diophantine approximation." Thesis, University of York, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.317873.
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Farkas, Ábel. "Dimension and measure theory of self-similar structures with no separation condition." Thesis, University of St Andrews, 2015. http://hdl.handle.net/10023/7854.
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We introduce methods to cope with self-similar sets when we do not assume any separation condition. For a self-similar set K ⊆ ℝᵈ we establish a similarity dimension-like formula for Hausdorff dimension regardless of any separation condition. By the application of this result we deduce that the Hausdorff measure and Hausdorff content of K are equal, which implies that K is Ahlfors regular if and only if Hᵗ (K) > 0 where t = dim[sub]H K. We further show that if t = dim[sub]H K < 1 then Hᵗ (K) > 0 is also equivalent to the weak separation property. Regarding Hausdorff dimension, we give a dimension approximation method that provides a tool to generalise results on non-overlapping self-similar sets to overlapping self-similar sets. We investigate how the Hausdorff dimension and measure of a self-similar set K ⊆ ℝᵈ behave under linear mappings. This depends on the nature of the group T generated by the orthogonal parts of the defining maps of K. We show that if T is finite then every linear image of K is a graph directed attractor and there exists at least one projection of K such that the dimension drops under projection. In general, with no restrictions on T we establish that Hᵗ (L ∘ O(K)) = Hᵗ (L(K)) for every element O of the closure of T , where L is a linear map and t = dim[sub]H K. We also prove that for disjoint subsets A and B of K we have that Hᵗ (L(A) ∩ L(B)) = 0. Hochman and Shmerkin showed that if T is dense in SO(d; ℝ) and the strong separation condition is satisfied then dim[sub]H (g(K)) = min {dim[sub]H K; l} for every continuously differentiable map g of rank l. We deduce the same result without any separation condition and we generalize a result of Eroğlu by obtaining that Hᵗ (g(K)) = 0. We show that for the attractor (K1, … ,Kq) of a graph directed iterated function system, for each 1 ≤ j ≤ q and ε > 0 there exists a self-similar set K ⊆ Kj that satisfies the strong separation condition and dim[sub]H Kj - ε < dim[sub]H K. We show that we can further assume convenient conditions on the orthogonal parts and similarity ratios of the defining similarities of K. Using this property we obtain results on a range of topics including on dimensions of projections, intersections, distance sets and sums and products of sets. We study the situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result here shows that this equality holds for any subset of a set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self-similar or graph directed self-similar set, regardless of separation conditions. The main tool in the proof is an exhaustion lemma for Hausdorff measure based on the Vitali's Covering Theorem. We also give several examples showing that one cannot hope for the equality to hold in general if one moves in a number of the natural directions away from `self-similar'. Finally we consider an analogous version of the problem for packing measure. In this case we need the strong separation condition and can only prove that the packing measure and δ-approximate packing pre-measure coincide for sufficiently small δ > 0.
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Higgens, Thomas. "Classicalness and the Hausdorff dimension of limit sets of divergent sequences of genus two Schottky groups." Thesis, University of Southampton, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.432718.
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Mielke, Jöran [Verfasser], Ludwig [Akademischer Betreuer] Staiger, and Vasco [Akademischer Betreuer] Brattka. "Verfeinerung der Hausdorff-Dimension und Komplexität von ω-Sprachen / Jöran Mielke. Betreuer: Ludwig Staiger ; Vasco Brattka." Halle, Saale : Universitäts- und Landesbibliothek Sachsen-Anhalt, 2010. http://d-nb.info/1024976246/34.
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Liberato, Serginei José do Carmo. "Algumas Propriedades Geométricas do Conjunto de Julia." Universidade Federal de Viçosa, 2014. http://locus.ufv.br/handle/123456789/4928.
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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.