L'objectiu bàsic d'aquesta tesi és l'estudi de la dinàmica a l'entorn de tors de dimensió baixa de sistemes hamiltonians analítics. Per aquest estudi l'eina fonamental és l'ús de formes normals al voltant d'aquests objectes.La formulació dels resultats d'aquesta memòria s'ha fet de manera adient per a la seva aplicació a models de mecànica celeste del món real. Per aquest motiu els resultats no es redueixen només al cas autònom, sinó que algun d'ells pren en consideració la possiblitat que les diferents perturbacions pugin dependre del temps de forma periòdica o quasiperiòdica. Aquests resultats s'apliquen per descriure la dinàmica d'alguns problemes d'interes per la Astronàutica. Per tant, els resultats obtinguts inclouen també aplicacions numèriques.Els resultats assolits en cadascun del capítols de la memòria es poden sintetitzar de la forma següent:Capítol 1.- Estudi de la dinàmica entorn d'un tor parcialment el.líptic d'un sistema Hamiltonià autònom. Es donen cotes inferiors pel temps de difusió entorn d'un tor totalment el.líptic, així com estimacions, en el cas general, de la densitat de tors invariants (de qualsevol dimensió) al voltant del tor inicial. Les estimacions en la velocitat de difusió i en la proximitat a 1 d'aquesta densitat, són exponencialment petites respecte la distància al tor inicial.Capítol 2.- Computació numèrica de formes normals al voltant d'òrbites periòdiques. Es desenvolupa un mètode per a calcular formes normals al voltant d'òrbites periòdiques el.líptiques de sistemes hamiltonians. Aquesta metodologia és aplicada numèricament a una òrbita periòdica del Problema Restringit de tres Cossos espaial. Els resultats d'aquest capítol es poden veure com una implementació numèrica del Capítol 1.Capítol 3.- Persistència de tors de dimensió baixa sota perturbacions quasiperiòdiques. Es mostra que un tor de dimensió baixa d'un sistema hamiltonià sotmès a una perturbació quasiperiòdica és pot continuar respecte el paràmetre perturbatiu, tot afegint a les freqüències bàsiques inicials les de la perturbació, excepte per un conjunt de mesura petita pel paràmetre. Al igual que en el Capítol 1 també s'estima la densitat de tors en el problema perturbat. En ambdós casos, les cotes obtingudes per la mesura dels tors pels qual no és possible provar existència són de tipus exponencialment petit.Apèndix. Es presenta un resultat obtingut de forma conjunta amb Rafael Ramírez-Ros sobre la reducció a coeficients constants de sistemes d'equacions lineals autònoms perturbats quasiperiòdicament. Es mostra que tal reducció és possible excepte un reste exponencialment petit en el tamany de la perturbació.

This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. Results for one-dimensional invariant tori in two and three-dimensional state space and for two-dimensional invariant tori in three and four-dimensional maps are presented. / Master of Science

The dynamics of Hamiltonian systems (e.g. planetary motion, electron dynamics in nano-structures, molecular dynamics) can be investigated by symplectic maps. While a lot of work has been done for 2D maps, much less is known for higher dimensions. For a generic 4D map regular 2D-tori are organized around a skeleton of families of elliptic 1D-tori, which can be visualized by 3D phase-space slices. An analysis of the different bifurcations of the families of 1D-tori in phase space and in frequency space by computing the involved hyperbolic and elliptic 1D-tori is presented. Applying known results of normal form analysis, both the local and the global structure can be understood: Close to a bifurcation of a 1D-torus, the phase-space structures are surprisingly similar to bifurcations of periodic orbits in 2D maps. Far away the phase-space structures can be explained by remnants of broken resonant 2D-tori. / Die Dynamik Hamilton'scher Syteme (z.B. Planetenbewegung, Elektronenbewegung in Nanostrukturen, Moleküldynamik) kann mit Hilfe symplektischer Abbildungen untersucht werden. Bezüglich 2D Abbildungen wurde bereits umfassende Forschungsarbeit geleistet, doch für Systeme höherer Dimension ist noch vieles unverstanden. In einer generischen 4D Abbildung sind reguläre 2D-Tori um ein Skelett aus Familien von elliptischen 1D-Tori organisiert, was in 3D Phasenraumschnitten visualisiert werden kann. Durch die Berechnung der beteiligten hyperbolischen und elliptischen 1D-Tori werden die verschiedenen Bifurkationen der Familien von 1D-Tori im Phasenraum und im Frequenzraum analysiert. Die Anwendung bekannter Ergebnisse aus Normalformanalysen ermöglicht das Verständnis sowohl des lokalen, als auch des globalen Regimes. Nahe an der Bifurkation eines 1D-Torus sind die Phasenraumstrukturen denen von Bifurkationen periodischer Orbits in 2D Abbildungen überraschend ähnlich. Weit entfernt können die Phasenraumstrukturen als Überreste eines zerplatzten resonanten 2D-Torus erklärt werden.

The dynamics of Hamiltonian systems (e.g. planetary motion, electron dynamics in nano-structures, molecular dynamics) can be investigated by symplectic maps. While a lot of work has been done for 2D maps, much less is known for higher dimensions. For a generic 4D map regular 2D-tori are organized around a skeleton of families of elliptic 1D-tori, which can be visualized by 3D phase-space slices. An analysis of the different bifurcations of the families of 1D-tori in phase space and in frequency space by computing the involved hyperbolic and elliptic 1D-tori is presented. Applying known results of normal form analysis, both the local and the global structure can be understood: Close to a bifurcation of a 1D-torus, the phase-space structures are surprisingly similar to bifurcations of periodic orbits in 2D maps. Far away the phase-space structures can be explained by remnants of broken resonant 2D-tori. / Die Dynamik Hamilton'scher Syteme (z.B. Planetenbewegung, Elektronenbewegung in Nanostrukturen, Moleküldynamik) kann mit Hilfe symplektischer Abbildungen untersucht werden. Bezüglich 2D Abbildungen wurde bereits umfassende Forschungsarbeit geleistet, doch für Systeme höherer Dimension ist noch vieles unverstanden. In einer generischen 4D Abbildung sind reguläre 2D-Tori um ein Skelett aus Familien von elliptischen 1D-Tori organisiert, was in 3D Phasenraumschnitten visualisiert werden kann. Durch die Berechnung der beteiligten hyperbolischen und elliptischen 1D-Tori werden die verschiedenen Bifurkationen der Familien von 1D-Tori im Phasenraum und im Frequenzraum analysiert. Die Anwendung bekannter Ergebnisse aus Normalformanalysen ermöglicht das Verständnis sowohl des lokalen, als auch des globalen Regimes. Nahe an der Bifurkation eines 1D-Torus sind die Phasenraumstrukturen denen von Bifurkationen periodischer Orbits in 2D Abbildungen überraschend ähnlich. Weit entfernt können die Phasenraumstrukturen als Überreste eines zerplatzten resonanten 2D-Torus erklärt werden.

This thesis is concerned with numerical techniques for resolving and continuing closed, compact invariant manifolds in parameter-dependent dynamical systems with specific emphasis on invariant tori under flows. In the first part, we review several numerical methods of continuing invariant tori and concentrate on one choice called the ``orthogonality condition'. We show that the orthogonality condition is equivalent to another condition on the smooth level and show that they both descend from the same geometrical relationship. Then we show that for hyperbolic, periodic orbits in the plane, the linearization of the orthogonality condition yields a scalar system whose characteristic multiplier is the same as the non-unity multiplier of the orbit. In the second part, we demonstrate that one class of discretizations of the orthogonality condition for periodic orbits represents a natural extension of collocation. Using this viewpoint, we give sufficient conditions for convergence of a periodic orbit. The stability argument does not extend to higher-dimensional tori, however, and we prove that the method is unconditionally unstable for some common types of two-tori embedded in R^3 with even numbers of points in both angular directions. In the third part, we develop several numerical examples and demonstrate that the convergence properties of the method and discretization can be quite complicated. In the fourth and final part, we extend the method to the general case of p-tori in R^n in a different way from previous implementations and solve the continuation problem for a three-torus embedded in R^8.

All classes of Active Galactic Nuclei (AGN) are fundamentally powered by accretion of gas onto a supermassive black hole. The process converts the potential energy of the infalling matter to X-ray and ultraviolet (UV) radiation, releasing up to several 1012 solar luminosities. Observations show that the accreting "central engines" in AGN are surrounded by dusty matter. The dust occupies a "torus" around the AGN which is comprised of discrete clumps. If the AGN radiation is propagating through the torus on its way to an observer, it will be heavily re-processed by the dust, i.e. converted from UV to infrared (IR) wavelengths. Much of the information about the input radiation is lost in this conversion process while an imprint of the dusty torus is left in the released IR photons. Our group was the first to formulate a consistent treatment of radiative transfer in a clumpy medium an important improvement over simpler models with smooth dust distributions previously used by researchers. Our code CLUMPY computes spectral energy distributions (SED) for any set of model parameters values. Fitting these models to observed AGN SEDs allows us to determine important quantities, such as the torus size, the spatial distribution of clumps, the torus covering factor, or the intrinsic AGN luminosity. Detailed modeling also permits us to study the complex behavior of certain spectral features. IR radiative transfer introduces degeneracies to the solution space: different parameter values can yield similar SEDs. The geometry of the torus further exacerbates the problem. Knowing the amount of parameter degeneracy present in our models is important for quantifying the confidence in data fits. When matching the models to observed SEDs we must employ modern statistical methods. In my research I use Bayesian statistics to determine the likely ranges of parameter values. I have developed all tools required for fitting observed SEDs with our large model database: the latest implementation of CLUMPY, the fit algorithms, the Markov Chain Monte Carlo sampler, and the Bayesian estimator. In collaboration with observing groups we have applied our methods to a multitude of real-life AGN.

Un problema clásico en geometría lorentziana es la descripción de las inmersiones isométricas entre los espacios lorentzianos de curvatura constante. En este trabajo nos centramos en la clasificación de las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter tridimensional. Damos una fórmula de representación de estas inmersiones en términos de pares de curvas (con posibles singularidades) en el plano hiperbólico. Esto nos permite resolver los problemas propuestos por Dajczer y Nomizu en 1981. De entre todas las inmersiones isométricas del plano lorentziano en el espacio anti-de Sitter, algunas de ellas corresponden a toros lorentzianos (los ejemplos más sencillos son los toros de Hopf). Como aplicación de nuestra anterior descripción, probamos que todos estos toros pueden obtenerse a partir de dos curvas cerradas en el espacio hiperbólico. Finalmente, demostramos que los toros de Hopf son los únicos toros llanos lorentzianos inmersos en una amplia familia de sumersiones de Killing lorentzianas tridimensionales. / A classical problem in Lorentzian geometry is the description of the isometric immersions between Lorentzian spaces of constant curvature. We investigate the problem of classifying the isometric immersion from the Lorentz plane into the three-dimensional anti-de Sitter space, providing a representation formula of these isometric immersions in terms of pairs of curves (possibly with singularities) in the hyperbolic plane. We then give an answer to the open problems proposed by Dajczer and Nomizu in 1981. Among all isometric immersions of the Lorentz plane into the anti-de Sitter space, some of them are actually Lorentzian tori (the basic examples are the Hopf tori). As an application of our previous description, we prove that any such torus can be recovered from two closed curves in the hyperbolic plane. Finally, we prove that Lorentzian Hopf tori are the only immersed Lorentzian flat tori in a wide family of Lorentzian three-dimensional Killing submersions.

Dans cette thèse, nous résolvons la conjecture de Fuglede sur le corps des nombres p-adiques, et étudions certaines propriétés aléatoires des suites liées à la conjecture de Sarnak, ainsi que leur propriétés oscillantes. Dans la première partie, nous prouvons d'abord la conjecture de Fuglede pour des ensembles ouverts compacts dans Q_p. Celle-ci indique qu'un ensemble ouvert compact dans Q_p est un ensemble spectral si et seulement s'il pave Q_p par translation. Il est également prouvé qu'un ensemble ouvert compact est un ensemble spectral (ou une tuile) si et seulement s'il est p-homogène. Nous caractérisons les ensembles spectraux dans Z / p^n Z ( p>1 premier, n>0 entier) par la propriété de pavage et aussi par leur homogénéité. Finalement, nous montrons la conjecture de Fuglede dans Q_p sans la restriction d'être ouvert compact en montrant que tout ensemble spectral ou toute tuile doivent être ouvert et compact à un ensemble de mesure nulle près. Dans la seconde partie, nous donnons d'abord plusieurs définitions équivalentes d'une suite oscillante en termes de disjonction de différents systèmes dynamiques sur des tores. Ensuite, nous définissons la propriété de Chowla et la propriété de Sarnak pour des suites numériques prenant des valeurs 0 ou des nombres complexes de module 1. Nous prouvons que la propriété de Chowla implique la propriété de Sarnak. Il est également prouvé que pour Lebesgue presque tout b> 1, la suite (e^{2 pi b^n})_{n in N} partage la propriété de Chowla et est par conséquent orthogonale à tout système dynamique topologique d'entropie nulle. Nous discutons également si les échantillons d'une suite aléatoire donnée ont presque sûrement la propriété de Chowla. Nous construisons certaines suites aléatoires dépendantes ayant presque sûrement la propriété de Chowla / In this thesis, we solve Fuglede's conjecture on the field of p-adic numbers, and study some randomness and the oscillating properties of sequences related to Sarnak's conjecture. In the first part, we first prove Fuglede's conjecture for compact open sets in the field Q_p which states that a compact open set in Q_p is a spectral set if and only if it tiles Q_p by translation. It is also proved that a compact open set is a spectral set (or a tile) if and only if it is p-homogeneous. We characterize spectral sets in Z/p^n Z (p>1 prime, n>0 integer) by tiling property and also by homogeneity. Finally, we prove Fuglede's conjecture in Q_p without the assumption of compact open sets and also show that the spectral sets (or tiles) are the sets which differ by null sets from compact open sets. In the second part, we first give several equivalent definitions of oscillating sequences in terms of their disjointness from different dynamical systems on tori. Then we define Chowla property and Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that Chowla property implies Sarnak property. It is also proved that for Lebesgue almost every b>1, the sequence (e^{2 pi b^n})_{n in N} shares Chowla property and consequently is orthogonal to all topological dynamical systems of zero entropy. We also discuss whether the samples of a given random sequence have almost surely Chowla property. Some dependent random sequences having almost surely Chowla property are constructed

Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP⁺05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.

Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than conventional analogues. The systems we study, called torus-based cryptosystems, were analyzed by Karl Rubin and Alice Silverberg in 2003 [RS03]. They interpreted the XTR [LV00] and LUC [SL93] cryptosystems in terms of quotients of algebraic tori and birational parameterizations, and they also presented CEILIDH, a new torus-based cryptosystem. This thesis introduces the geometry of algebraic tori, uses it to explain the XTR, LUC, and CEILIDH cryptosystems, and presents torus-based extensions of van Dijk, Woodruff, et al. [vDW04, vDGP⁺05] that require even less bandwidth. In addition, a new algorithm of Granger and Vercauteren [GV05] that attacks the security of torus-based cryptosystems is presented. Finally, we list some open research problems.