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The development and use of cocycles for analysis of non-autonomous behaviour is a technique that has been known for several years. Initially developed as an extension to semi-group theory for studying rion-autonornous behaviour, it was extensively used in analysing random dynamical systems [2, 9, 10, 12]. Many of the results regarding asymptotic behaviour developed for random dynamical systems, including the concept of cocycle attractors were successfully transferred and reinterpreted for deterministic non-autonomous systems primarily by P. Kloeden and B. Schmalfuss [20, 21, 28, 29]. The theory concerning cocycle attractors was later developed in various contexts specific to particular classes of dynamical systems [6, 7, 13], although a comprehensive understanding of cocycle attractors (redefined as pullback attractors within this thesis) and their role in the stability of non-autonomous dynamical systems was still at this stage incomplete. It was this purpose that motivated Chapters 1-3 to define and formalise the concept of stability within non-autonomous dynamical systems. The approach taken incorporates the elements of classical asymptotic theory, and refines the notion of pullback attraction with further development towards a study of pull-back stability arid pullback asymptotic stability. In a comprehensive manner, it clearly establishes both pullback and forward (classical) stability theory as fundamentally unique and essential components of non-autonomous stability. Many of the introductory theorems and examples highlight the key properties arid differences between pullback and forward stability. The theory also cohesively retains all the properties of classical asymptotic stability theory in an autonomous environment. These chapters are intended as a fundamental framework from which further research in the various fields of non-autonomous dynamical systems may be extended. A preliminary version of a Lyapunov-like theory that characterises pullback attraction is created as a tool for examining non-autonomous behaviour in Chapter 5. The nature of its usefulness however is at this stage restricted to the converse theorem of asymptotic stability. Chapter 7 introduces the theory of Loci Dynamics. A transformation is made to an alternative dynamical system where forward asymptotic (classical asymptotic) behaviour characterises pullback attraction to a particular point in the original dynamical system. This has the advantage in that certain conventional techniques for a forward analysis may be applied. The remainder of the thesis, Chapters 4, 6 and Section 7.3, investigates the effects of perturbations and discretisations on non-autonomous dynamical systems known to possess structures that exhibit some form of stability or attraction. Chapter 4 investigates autonomous systems with semi-group attractors, that have been non-autonomously perturbed, whilst Chapter 6 observes the effects of discretisation on non-autonomous dynamical systems that exhibit properties of forward asymptotic stability. Chapter 7 explores the same problem of discretisation, but for pullback asymptotically stable systems. The theory of Loci Dynamics is used to analyse the nature of the discretisation, but establishment of results directly analogous to those discovered in Chapter 6 is shown to be unachievable. Instead a case by case analysis is provided for specific classes of dynamical systems, for which the results generate a numerical approximation of the pullback attraction in the original continuous dynamical system. The nature of the results regarding discretisation provide a non-autonomous extension to the work initiated by A. Stuart and J. Humphries [34, 35] for the numerical approximation of semi-group attractors within autonomous systems. . Of particular importance is the effect on the system's asymptotic behaviour over non-finite intervals of discretisation.

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Thesis advisor: Benjamin V. Howard / Given a field <italic>F</italic>, an algebraic closure <italic>K</italic> and an <italic>F</italic>-vector space <italic>V</italic>, we can tensor the space <italic>V</italic> with the algebraic closure <italic>K</italic>. Two quadratic spaces of the same dimension become isomorphic when tensored with an algebraic closure. The failure of this isomorphism over <italic>F</italic> is measured by the Hasse invariant. This paper explains how the determinants and Hasse Invariants of quadratic forms are related to certain cohomology classes constructed from specific short exact sequences. In particular, the Hasse Invariant is defined as an element of the Brauer group. / Thesis (MA) — Boston College, 2010. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

Le résultat principal de cette thèse est une solution locale du problème des coquecigrues. Par problème des coquecigrues, nous parlons du problème d'intégration des algèbres de Leibniz. Cette question consiste à trouver une généralisation du troisième théorème de Lie pour les algèbres de Leibniz. Ce théorème établit que pour toute algèbre de Lie g, il existe un groupe de Lie G dont l'espace tangent en 1 est muni d'une structure d'algèbre de Lie isomorphe à g. La sructure d'algèbre de Leibniz généralise celle d'algèbre de Lie, nous cherchons donc une structure algébrique généralisant celle de groupe et répondant à la même question. Nous résolvons ce prob- lème en intégrant localement toute algèbre de Leibniz en un rack de Lie augmenté local. Un rack de Lie étant une variété munie d'un produit satisfaisant plusieurs axiomes qui généralisent des propriétés de la conjugaison dans un groupe. En particulier, ce produit est autodistributif. Notre approche de ce problème est basée sur une preuve donnée par E.Cartan dans le cas des groupes et algèbres de Lie, et consiste à associer à toute algèbres de Leibniz une extension abélienne d'une algèbre de Lie par un module antisymétrique. Cette extension est caractérisée par une classe dans le second groupe de cohomologie de Leibniz, et nous associons à tous représentant de cette classe un cocyle de rack de Lie local qui nous permet de construire un rack de Lie augmenté local répondant au problème. Pour construire ce cocycle, nous généralisons une méthode d'intégration d'un cocycle d'algèbre de Lie en cocycle de groupe de Lie due à W.T.Van Est.

The Multiplicative Ergodic Theorem is a powerful tool for studying certain types of dynamical systems, involving real matrix cocycles. It gives a block diagonalization of these cocycles, according to the Lyapunov exponents. We ask if it is always possible to refine the diagonalization to a block upper-triangularization, and if not over the real numbers, then over the complex numbers. After building up to the posing of the question, we prove that there are counterexamples to this statement, and give concrete examples of matrix cocycles which cannot be block upper-triangularized. / Graduate / 0405 / jahoran@uvic.ca

The concept of hyperreflexivity has previously been defined for subspaces of $B(X,Y)$, where $X$ and $Y$ are Banach spaces. We extend this concept to the subspaces of $B^n(X,Y)$, the space of bounded $n$-linear maps from $X\times\cdots\times X=X^{(n)}$ into $Y$, for any $n\in \mathbb{N}$. If $A$ is a Banach algebra and $X$ a Banach $A$-bimodule, we obtain sufficient conditions under which $\Zc^n(A,X)$, the space of all bounded $n$-cocycles from $A$ into $X$, is hyperreflexive. To do so, we define two notions related to a Banach algebra: The strong property $(\B)$ and bounded local units (b.l.u). We show that there are sufficiently many Banach algebras which have both properties. We will prove that all C$^*$-algebras and group algebras have the strong property $(\B).$ We also prove that finite CSL algebras and finite nest algebras have this property. We further show that for an arbitrary Banach algebra $A$ and each $n\geq 2$, $M_n(A)$ has the strong property $(\B)$ whenever it is equipped with a Banach algebra norm. In particular, this implies that all Banach algebras are embedded into a Banach algebra with the strong property $(\B)$. With regard to bounded local units, we show that all $C^*$-algebras and many group algebras have b.l.u. We investigate the hereditary properties of both notions to construct more example of Banach algebras with these properties. We apply our approach and show that the bounded $n$-cocycle spaces related to Banach algebras with the strong property $(\B)$ and b.l.u. are hyperreflexive provided that the space of the corresponding $n+1$-coboundaries are closed. This includes nuclear C$^*$-algebras, many group algebras, matrix spaces of certain Banach algebras and finite CSL and nest algebras. We finish the thesis with introducing {\it the hyperreflexivity constant}. We make our results more precise with finding an upper bound for the hyperreflexivity constant of the bounded $n$-cocycle spaces.

Cette thèse porte sur l'étude des cocycles quasi-périodiques à valeurs dans des groupes de Lie compacts semi-simples. Nous nous restreindrons au cas des cocycles à une fréquence. Nous démontrons que, pour un ensemble de mesure de Lebesgue pleine de fréquences, l'ensemble des cocycles $C^{\infty }$ qui sont $C^{\infty }$-réductibles sont $C^{\infty }$-denses. Le premier pas sera l'obtention de deux invariants de la dynamique, qu'on appellera énergie et degré, qui distinguent en particulier les cocycles réductibles des cocycles non-réductibles. On entamera ensuite la démonstration du théorème principal. Nous démontrons dans un second temps qu'un algorithme dit de renormalisation permet de ramener l'étude de tout cocycle à celle des perturbations de modèles simples indexés par le degré. Nous analysons ensuite ces perturbations par des méthodes inspirés de la théorie K.A.M.. En particulier, nous démontrons qu'un cocycle $C^{\infty }$ mesurablement réductible à une constante diophantienne est alors $C^{\infty }$-réductible.

L'objectif de cette thèse est de traiter de deux aspects différents de la théorie de l'exposant de Lyapunov de cocycles de Schrödinger définis par une dynamique ergodique. Dans la première partie, on s'intéresse aux estimées de grandes déviations de type Bourgain & Goldstein pour des cocycles quasi-périodiques, puis pour ceux définis par le doublement de l'angle. Après avoir montré que seule une estimée par dessus sur une bande complexe est nécessaire pour avoir la minoration, on redémontre cette inégalité pour une dynamique quasi-périodique en utilisant des techniques de mouvement brownien en lien avec des fonctions sous-harmoniques. Ensuite on adapte la méthode au cas du doublement de l'angle pour lequel on prouve des estimées de grandes déviations sur les branches inverses de cette dynamique. Dans la deuxième partie sont étudiés des cocycles de Schrödinger dont la dynamique est une somme de dynamiques quasi-périodique et aléatoire. On démontre que, dans le régime perturbatif, les développements asymptotiques de l'exposant de Lyapunov attaché à ces cocycles sont similaires à ceux déjà démontrés dans le cas aléatoire par Figotin & Pastur ou Sadel & Schulz-Baldes. L'analyse se fait en fonction du caractère diophantien ou résonant de l'énergie par rapport à la fréquence diophantienne de la partie quasi-périodique du potentiel. / In this thesis we are interested in the Lyapunov exponent of ergodic Schrödinger cocycles. These cocycles occur in the analysis of solutions to the Schrödinger equation where the potential is defined with ergodic dynamics. We study two distinct aspects related to the the Lyapunov exponent for different kinds of dynamics. First we focus on a large deviation theorem for quasi-periodic cocycles and then for potentials defined by the doubling map. We prove that estimates of Bourgain & Goldstein type are granted if an upper estimate involved in the theorem is true on a strip of the complex plane. Then we establish a new technique to prove this upper bound in the quasi-periodic setting, based on subharmonic arguments suggested by Avila, Jitomirskaya & Sadel. We adapt afterwards the method to the doubling map and prove a large deviation theorem for the inverse branches of this dynamics. In the second part, we establish an asymptotic development similar to the results of Figotin & Pastur and Sadel & Schulz-Baldes for the Lyapunov exponent of Schrödinger cocycles at small coupling when the potential is a mixture of quasi-periodic and random. The analysis distinguishes the cases when the energy is either diophantine or resonant with respect to the frequency of the quasi-periodic part of the potential.

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