This work details an O(n^2) algorithm for computing the spectra of discrete Schroedinger operators with periodic potentials. Spectra of these objects enhance our understanding of fundamental aperiodic physical systems and contain rich theoretical structure of interest to the mathematical community. Previous work on the Harper model led to an O(n^2) algorithm relying on properties not satisfied by other aperiodic operators. Physicists working with the Fibonacci Hamiltonian, a popular quasicrystal model, have instead used a problematic dynamical map approach or a sluggish O(n^3) procedure for their calculations. The algorithm presented in this work, a blend of well-established eigenvalue/vector algorithms, provides researchers with a more robust computational tool of general utility. Application to the Fibonacci Hamiltonian in the sparsely studied intermediate coupling regime reveals structure in canonical coverings of the spectrum that will prove useful in motivating conjectures regarding band combinatorics and fractal dimensions.

Cette thèse porte sur la construction de nouveaux exemples de surfaces minimales dérivées de la famille de surfaces de Costa-Hoffman-Meeks. Il s'agit d'une famille de surfaces minimales complètes plongées avec trois bouts et genre k > 0. Soit M_k la surface de Costa_Hoffman_Meeks de genre k. Dans le chapitre 1, j'ai démontré que M_k est non dégénérée pour k > 37. J'ai donc étendu les résultats de S. Nayatani qui assuraient que la surface M_k est non dégénérée seulement pour k=1,...,37. Ce résultat permet de montrer dans les chapitres 2 et 3 l'existence de nouveaux exemples de surfaces minimales de genre g arbitraire à l'aide d'une procédure de collage d'autres surfaces déjà connues (parmi lesquelles y figure la surface M_k). Sans ceci, ces résultats ne seraient valables que pour k < 38. En particulier dans le chapitre 2, j'ai démontré l'existence, dans H^2 x R, (H^2 étant le plan hyperbolique) d'une famille de surfaces minimales plongées inspirées de M_k, pour tout k > 0. Ce résultat peut être censé un cas particulier d'un théorème générale de désingularisation de l'intersection de deux surfaces minimales annoncé par N. Kapouleas et jamais publié. Le chapitre 3 est consacré à la construction de trois familles de surfaces minimales simplement périodiques plongées dans R^3 dont le quotient a genre arbitraire. Les résultats présentés dans ce chapitre (obtenus en collaborations avec L. Hauswirth et M. Rodríguez) généralisent plusieurs anciennes constructions / This thesis is devoted to the construction of new examples of minimal surfaces derived from the family of surfaces if Costa-Hoffman-Meeks. Surfaces in this family are complete embedded with 3 ends and genus k > 0. Let M_k denote the surface of Costa-Hoffman-Meeks of genus k. In chapter 1 I showed M_k is non degenerate for k > 37. So I extended the results of S. Nayatani which insured M_k is non degenerate only for k=1,...,37. That allows to prove in chapters 2 and 3 the existence of new examples of minimal surfaces by a gluing procedure involving already known surfaces (among which figures M_k). Without it theses results would hold only for k < 38. In particular in chapter 2 I showed the existence in H^2 x R (where H^2 denotes the hyperbolic plane) of a family of surfaces inspired to M_k, for all k > 0, which are complete and embedded. This result can be considered as a particular case of a general theorem of desingularization of the intersection of two minimal surfaces announced by N. Kapouleas and never published. Chapter 3 is devoted to the construction of 3 families of singly periodic minimal surfaces, embedded in R^3, whose quotient has an arbitrary value of the genus. The results showed in this chapter (obtained in collaboration with L. Hauswirth and M. Rodríguez) generalize many previous constructions

In this thesis we prove the genericity of the set of metrics on a manifold with boundary M^{n+1}, such that all free boundary constant mean curvature (CMC) embeddings \\varphi: \\Sigma^n \\to M^{n+1}, being \\Sigma a manifold with boundary, are non-degenerate (Bumpy Metrics), (Theorem 2.4.1). We also give sufficient conditions to obtain a free boundary CMC deformation of a CMC inmersion (Theorems 3.2.1 and 3.2.2), and a stability criterion for this type of immersions (Theorem 3.3.3 and Corollary 3.3.5). In addition, given a one-parametric family, {\\varphi _t : \\Sigma \\to M} , of free boundary CMC immersions, we give criteria for the existence of smooth bifurcated branches of free boundary CMC immersions for the family {\\varphi_t}, via the implicit function theorem when the kernel of the Jacobi operator J is non-trivial, (Theorems 4.2.3 and 4.3.2), and we study stability and instability problems for hypersurfaces in this bifurcated branches (Theorems 5.3.1 and 5.3.3). / Nesta tese, provamos a genericidade do conjunto de métricas em uma variedade com fronteira M^{n+1}, de modo que todos os mergulhos de curvatura média constante (CMC) e fronteira livre \\varphi : \\Sigma^n \\to M^{n+1}, sendo \\Sigma uma variedade com fronteira, sejam não-degenerados (Métricas Bumpy), (Teorema 2.4.1). Nós também fornecemos condições suficientes para obter uma deformação CMC e fronteira livre de uma imersão CMC (Teoremas 3.2.1 and 3.2.2), e um critério de estabilidade para este tipo de imersões (Teorema 3.3.3 and Corolario 3.3.5). Além disso, dada uma família 1-paramétrica, {\\varphi _t : \\Sigma \\to M} , de imersões de CMC e fronteira livre, damos os critérios para a existência de ramos de bifurcação suaves de imersões CMC e fronteira livre para a familia {\\varphi_t}, por meio de o teorema da função implícita quando o kernel do operador Jacobi J é não-trivial, (Teoremas 4.2.3 and 4.3.2), e estudamos o problema da estabilidade e instabilidade para hipersuperfícies em naqueles ramos de bifurcação (Teoremas 5.3.1 and 5.3.3).

In this thesis we prove the genericity of the set of metrics on a manifold with boundary M^{n+1}, such that all free boundary constant mean curvature (CMC) embeddings \\varphi: \\Sigma^n \\to M^{n+1}, being \\Sigma a manifold with boundary, are non-degenerate (Bumpy Metrics), (Theorem 2.4.1). We also give sufficient conditions to obtain a free boundary CMC deformation of a CMC inmersion (Theorems 3.2.1 and 3.2.2), and a stability criterion for this type of immersions (Theorem 3.3.3 and Corollary 3.3.5). In addition, given a one-parametric family, {\\varphi _t : \\Sigma \\to M} , of free boundary CMC immersions, we give criteria for the existence of smooth bifurcated branches of free boundary CMC immersions for the family {\\varphi_t}, via the implicit function theorem when the kernel of the Jacobi operator J is non-trivial, (Theorems 4.2.3 and 4.3.2), and we study stability and instability problems for hypersurfaces in this bifurcated branches (Theorems 5.3.1 and 5.3.3). / Nesta tese, provamos a genericidade do conjunto de métricas em uma variedade com fronteira M^{n+1}, de modo que todos os mergulhos de curvatura média constante (CMC) e fronteira livre \\varphi : \\Sigma^n \\to M^{n+1}, sendo \\Sigma uma variedade com fronteira, sejam não-degenerados (Métricas Bumpy), (Teorema 2.4.1). Nós também fornecemos condições suficientes para obter uma deformação CMC e fronteira livre de uma imersão CMC (Teoremas 3.2.1 and 3.2.2), e um critério de estabilidade para este tipo de imersões (Teorema 3.3.3 and Corolario 3.3.5). Além disso, dada uma família 1-paramétrica, {\\varphi _t : \\Sigma \\to M} , de imersões de CMC e fronteira livre, damos os critérios para a existência de ramos de bifurcação suaves de imersões CMC e fronteira livre para a familia {\\varphi_t}, por meio de o teorema da função implícita quando o kernel do operador Jacobi J é não-trivial, (Teoremas 4.2.3 and 4.3.2), e estudamos o problema da estabilidade e instabilidade para hipersuperfícies em naqueles ramos de bifurcação (Teoremas 5.3.1 and 5.3.3).