Cette thèse s’intéresse principalement à l’existence et à la régularité desensembles minimaux. On commence par montrer, dans le chapitre 3, que le problème de Plateau étudié par Reifenberg admet au moins une solution. C’est-à-dire que, si l’onse donne un ensemble compact B⊂R^n et un sous-groupe L du groupe d’homologie de Čech H_(d-1) (B;G) de dimension (d-1) sur un groupe abelien G, on montre qu’il existe un ensemble compact E⊃B tel que L est contenu dans le noyau de l’homomorphisme H_(d-1) (B;G)→H_(d-1) (E;G) induit par l’application d’inclusion B→E, et pour lequel la mesure de Hausdorff H^d (E∖B) est minimale (sous ces contraintes). Ensuite, on montre au chapitre 4, que pour tout ensemble presque minimal glissant E de dimension 2, dans un domaine régulier Σ ressemblant localement à un demi espace, associé à la frontière glissante ∂Σ, et tel que E⊃∂Σ, il se trouve qu’à la frontière E est localement équivalent, par un homéomorphisme biHöldérien qui préserve la frontière, à un cône minimal glissant contenu dans un demi plan Ω, avec frontière glissante ∂Ω. De plus les seuls cônes minimaux possibles dans ce cas sont ∂Ω seul, ou son union avec un cône de type P_+ ou Y_+. / This thesis focuses on the existence and regularity of minimal sets. First we show, in Chapter 3, that there exists (at least) a minimizerfor Reifenberg Plateau problems. That is, Given a compact set B⊂R^n, and a subgroup L of the Čech homology group H_(d-1) (B;G) of dimension (d-1)over an abelian group G, we will show that there exists a compact set E⊃B such that L is contained in the kernel of the homomorphism H_(d-1) (B;G)→H_(d-1) (E;G) induced by the natural inclusion map B→E, and such that the Hausdorff measure H^d (E∖B) is minimal under these constraints. Next we will show, in Chapter 4, that if E is a sliding almost minimal set of dimension 2, in a smooth domain Σ that looks locally like a half space, and with sliding boundary , and if in addition E⊃∂Σ, then, near every point of the boundary ∂Σ, E is locally biHölder equivalent to a sliding minimal cone (in a half space Ω, and with sliding boundary ∂Ω). In addition the only possible sliding minimal cones in this case are ∂Ω or the union of ∂Ω with a cone of type P_+ or Y_+.

Sei V ein holomorphes Vektorbündel über einer q-konvexen Mannigfaltigkeit X. Die Andreotti-Grauert-Theorie sagt, dass die r-te Kohomologiegruppe holomorpher Schnitte mit Werten in V endlich-dimensional ist und dass die Kohomologie verschwindet, falls X q-vollständig ist. Ist E ein holomorphes Banachbündel über X, dann ist bekannt, dass die erste Kohomologiegruppe verschwindet, falls X Steinsch ist. Kapitel I gibt einen ausführlichen Überblick über die Arbeit. In Kapitel II wird gezeigt, dass es holomorphe Hilbertbündel über 1-konvexen Mannigfaltigkeiten gibt, für die die erste Kohomologie nicht Hausdorffsch ist. In Kapitel III wird folgender Endlichkeitssatz gezeigt: Ist E ein holomorph triviales Banachbündel oder ein holomorphes Banachbündel von kompaktem Typ mit kompakter Approximationseigenschaft über einer q-konvexen Mannigfaltigkeit X, und ist V ein holomorphes Vektorbündel über X, für das die q-te Kohomologie verschwindet, dann gilt: Die q-te Kohomologie für das Tensorprodukt von V und E ist endlich-dimensional. Ist X q-vollständig, dann verschwindet die r-te Kohomologie, falls r größer oder gleich q ist. Für r größer q kann dies auch für beliebige holomorphe Banachbündel E gezeigt werden. Im Anhang wird skizziert, wie der Ansatz der L2-Methode im Fall r gleich q für Hilbertbündel zu einem Verschwindungssatz führen könnte. / Let V be a holomorphic vector bundle over a q-convex manifold X. The Andreotti-Grauert theory says that the r-th cohomology group of holomorphic section with values in V is finite dimensional and that the cohomology is vanishing if X is q-complete. If E is a holomorphic Banach bundle over X, it is known that the first cohomology group vanishes if X is Stein. Chapter I gives a detailed overview of the work. In chapter II it is shown that there are holomorphic Hilbert bundles over 1-convex manifolds such that the first cohomology of the bundle is not Hausdorff. In chapter III the following finiteness theorem is shown: If E is a holomorphically trivial Banach bundle or a holomorphic Banach bundle of compact type with the compact approximation property over a q-convex manifold X, and if V is a holomorphic vector bundle over X such that the q-th cohomology vanishes, then the following holds true: The q-th cohomology for the tensor product of V and E is finite dimensional. If X is q-complete, then the r-th cohomology vanishes if r is greater or equal q. If r is greater than q, this is shown also for arbitrary holomorphic Banach bundles E. In the appendix it is sketched how for r equal q the L2 method could yield a vanishing theorem for Hilbert bundles.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste presente trabalho, faremos o estudo de importantes propriedades geomÃtricas e analÃticas de soluÃÃes de equaÃÃes diferenciais parciais elÃpticas totalmente nÃo-lineares do tipo: singulares e degeneradas. O estudo de processos de combustÃo que se degeneram ao longo do conjunto de anulamento da densidade de um gÃs, um caso particular de problemas do tipo "quenching", apresentam em sua modelagem equaÃÃes singulares que estÃo descritas neste trabalho. Nesta primeira parte iremos obter propriedades de uma soluÃÃo minimal, que vÃo desde o controle completo Ãtimo, atà a obtenÃÃo de estimativas de Hausdorff da fronteira livre singular. Por fim, iremos obter a regularidade Ãtima de soluÃÃes de equaÃÃes em que suas propriedades de difusÃo(elipticidade) se deterioram na ordem de uma potÃncia do seu gradiente ao longo do conjunto em que tal taxa de variaÃÃo se anula. / In this work we study important geometric and analytic properties to solutions of fully nonlinear elliptic partial differential equations, both singular and degenerate types. The study of combustion processes that degenerate along the null-set of the density of a gas, a particular case of quenching problems, present in their modeling, equations described in this work. In this first part we obtain properties of a minimal solution, since the complete optimal control until the Hausdorff estimates of the singular free boundary. Ultimately, we obtain the optimal regularity to equation solutions where their diffusion property (elipticity) deterorate in a power of their gradient along the set where such rate of variation nullifies.

Orientador: Márcio Ricardo Alves Gouveia / Resumo: 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

Orientador: Prof. Dr. Daniel Miranda Machado / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2015.

Submitted by Bruna Rodrigues (bruna92rodrigues@yahoo.com.br) on 2016-10-10T13:51:29Z No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:33:51Z (GMT) No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-10-21T13:34:02Z (GMT) No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) / Made available in DSpace on 2016-10-21T13:34:12Z (GMT). No. of bitstreams: 1 DissWM.pdf: 576272 bytes, checksum: 2aed9c1a9025d2818155f1b5a16ffc54 (MD5) Previous issue date: 2016-02-17 / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / We present a study of rank-one perturbations of self-adjoint and unitary operators, in particular related to the so-called Kotani’s trick. In Chapter 2 we recall relevant definitions and elementary facts. In Chapter 3 we present recent results by Marx, which involve self-adjoint operators and Hausdorff measures. A generalization of such results to the set of unitary operators is described in Chapter 4. / Apresentaremos um estudo realizados sobre pertubações de posto 1 de operadores auto-adjuntos e unitários, particularmente relacionados ao chamado truque de Kotani. O Capítulo 2 consiste na exposição de definições elementares e fatos básicos que serão utilizados nos outros capítulos. O Capítulo 3 consiste na exposição de um estudo feito de um artigo recente de Marx sobre o tema, envolvendo operadores autoadjuntos e medidas de Hausdorff. Uma generalização dos resultados obtidos no artigo já mencionado para o contexto de operadores unitários é discutida no Capítulo 4.

Dans cette thèse, on étudie diverses propriétés dimensionnelles de la régularité de processus de difusions à sauts, solution d’une classe d’équations différentielles stochastiques à sauts. En particulier, on décrit la fluctuation de la régularité höldérienne de ces processus et celle de la dimension locale pour la mesure d’occupation qui leur est associée en calculant leur spectre multifractal. La dimension de Hausdorff de l’image et du graphe de ces processus ont aussi étudiées.Dans le dernier chapitre, on applique une nouvelle notion de dimension de grande échelle pour décrire l’asymptote à l’infini du temps de séjour d’un mouvement brownien en dimension 1 sous des frontières glissantes / In this dissertation, we study various dimension properties of the regularity of jump di usion processes, solution of a class of stochastic di erential equations with jumps. In particular, we de- scribe the uctuation of the Hölder regularity of these processes and that of the local dimensions of the associated occupation measure by computing their multifractal spepctra. e Hausdor dimension of the range and the graph of these processes are also calculated.In the last chapter, we use a new notion of “large scale” dimension in order to describe the asymptotics of the sojourn set of a Brownian motion under moving boundaries

Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C. At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some beautiful applications regarding the Julia sets (defined in Chapter 8) of a certain family of polynomials, or a single one. We will be able to provide an explicit formula for computing the capacity of a Julia set, which in some sense, gives us a finer measurement of the set. In turn, this provides us with a sharp estimate for the diameter of the Julia set. Further if we pick any point w from the Julia set, then the inverse images q−n(w) span the whole Julia set. In fact, the point-mass measures with support at the discrete set consisting of roots of the polynomial, (qn-w) will eventually converge to the equilibrium measure of the Julia set, in the weak*-sense. This provides us with a very effective insight into the analytic structure of the set. Hausdorff dimension is one of the most effective notions of fractal dimension in use. With the help of potential theory and some ergodic theory, we can show that for a certain holomorphic family of polynomials varying over a simply connected domain D, one can gain nice control over how the Hausdorff dimensions of the respective Julia sets change with the parameter λ in D.

We study the geometric and topological properties of strange non-chaotic attractors created in non-smooth saddle-node bifurcations of quasiperiodically forced interval maps. By interpreting the attractors as limit objects of the iterates of a continuous curve and controlling the geometry of the latter, we determine their Hausdorff and box-counting dimension and show that these take distinct values. Moreover, the same approach allows us to describe the topological structure of the attractors and to prove their minimality.

In the first part of this thesis we transfer a result of Guillopé et al. concerning the number of zeros of the Selberg zeta function for convex cocompact Schottky groups to the setting of certain types of graph directed Markov systems (GDMS). For these systems the zeta function will be a type of Ruelle zeta function. We show that for a finitely generated primitive conformal GDMS S, which satisfies the strong separation condition (SSC) and the nestedness condition (NC), we have for each c>0 that the following holds, for each w \in\$C$ with Re(w)>-c, |\Im(w)|>1 and for all k \in\$N$ sufficiently large: log | zeta(w) | <<e {delta(S).log(Im|w|)} and card{w \in\ Q(k) | zeta(w)=0} << k {delta(S)}. Here, Q(k)\subset\%C$ denotes a certain box of height k, and delta(S) refers to the Hausdorff dimension of the limit set of S. In the second part of this thesis we show that in any dimension m \in\$N$ there are GDMSs for which the Hausdorff dimension of the uniformly radial limit set is equal to a given arbitrary number d \in\(0,m) and the Hausdorff dimension of the Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m). Furthermore, we derive various relations between the exponents of convergence and the Hausdorff dimensions of certain different types of limit sets for iterated function systems (IFS), GDMSs, pseudo GDMSs and normal subsystems of finitely generated GDMSs. Finally, we apply our results to Kleinian groups and generalise a result of Patterson by showing that in any dimension m \in\$N$ there are Kleinian groups for which the Hausdorff dimension of their uniformly radial limit set is less than a given arbitrary number d \in\ (0,m) and the Hausdorff dimension of their Jørgensen limit set is equal to a given arbitrary number j \in\ [0,m).