In this thesis examples of spectral triples, which represent fractal sets, are examined and new insights into their noncommutative geometries are obtained. Firstly, starting with Connes' spectral triple for a non-empty compact totally disconnected subset E of {R} with no isolated points, we develop a noncommutative coarse multifractal formalism. Specifically, we show how multifractal properties of a measure supported on E can be expressed in terms of a spectral triple and the Dixmier trace of certain operators. If E satisfies a given porosity condition, then we prove that the coarse multifractal box-counting dimension can be recovered. We show that for a self-similar measure μ, given by an iterated function system S defined on a compact subset of {R} satisfying the strong separation condition, our noncommutative coarse multifractal formalism gives rise to a noncommutative integral which recovers the self-similar multifractal measure ν associated to μ, and we establish a relationship between the noncommutative volume of such a noncommutative integral and the measure theoretical entropy of ν with respect to S. Secondly, motivated by the results of Antonescu-Ivan and Christensen, we construct a family of (1, +)-summable spectral triples for a one-sided topologically exact subshift of finite type (∑{{A}} {{N}}, σ). These spectral triples are constructed using equilibrium measures obtained from the Perron-Frobenius-Ruelle operator, whose potential function is non-arithemetic and Hölder continuous. We show that the Connes' pseudo-metric, given by any one of these spectral triples, is a metric and that the metric topology agrees with the weak*-topology on the state space {S}(C(∑{{A}} {{N}}); {C}). For each equilibrium measure ν[subscript(φ)] we show that the noncommuative volume of the associated spectral triple is equal to the reciprocal of the measure theoretical entropy of ν[subscript(φ)] with respect to the left shift σ (where it is assumed, without loss of generality, that the pressure of the potential function is equal to zero). We also show that the measure ν[subscript(φ)] can be fully recovered from the noncommutative integration theory.

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, <em>i. e. </em> in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, <em>i. e. </em> develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.

In this thesis, the problem of well-posedness of nonlinear viscoelasticity under the assumptions allowing for phase transformations in solids is considered. In one space dimension we prove existence and uniqueness of the solutions for the quasistatic version of the model using approximating sequences corresponding to the case when initial data takes finitely many values. This special case also provides upper and lower bounds for the solutions which are interesting in their own rights. We also show equivalence of the existence theory we develop with that of gradient flows when the stored-energy function is assumed to be -convex. Asymptotic behaviour of the solutions as time goes to infinity is then investigated and stabilization results are obtained by means of a new argument. Finally, we look at the problem from the viewpoint of curves of maximal slope and follow a time-discretization approach. We introduce a three-dimensional method based on composition of time-increments, as a result of which we are able to deal with the physical requirement of frame-indifference. In order to test this method and distinguish the difficulties for possible generalizations, we look at the problem in a convex setting. At the end we are able to obtain convergence of the minimization scheme as time step goes to zero.

Cette thèse est consacrée à l'étude de certaines questions de convergence et de récurrence multiples en théorie ergodique. Nous distinguons les systèmes munis d'une transformation et ceux munis de plusieurs transformations qui commutent. Dans les premiers, le mécanisme de facteurs caractéristiques et les nilsystèmes jouent un rôle important dans l'étude de convergence et de récurrence multiples. À l'aide de ces outils, nous étendons les résultats sur la convergence de moyennes ergodiquesmultiples pondérées de Host et Kra pour le cas linéaire au cas polynômial. En conséquence, nous montrons que pour toute fonction $f$ mesurable bornée sur un système ergodique, la suite $(f(T^n x))$ est universellement bonne pour presque tout $x$. Quand il y a plusieurs transformations qui commutent, à l'aide de la machinerie des systèmes magiques introduite récemment par Host et développée dans cette thèse, nous étendons les résultats sur la convergence de moyennes ergodiques multiples sur les cubes de Host et Kra avec une transformation à plusieurs transformations qui commutent. Nous obtenons aussi un résultat de récurrence multiple quantitatif pour deux transformations qui commutent, similaire en faveur du cas d'une transformation établi par Bergelson, Host et Kra / This thesis is devoted to the study of some questions of multiple convergence and recurrence in ergodic theory. We distinguish between systems endowed with a single transformation and systems endowed with several commuting transformations. In the former, characteristic factors and nilsystemsplay an important role in the study of multiple convergence and recurrence. Using these tools, we extend results on convergence of weighted multiple ergodic averages of Host and Kra for the linear case to the polynomial case. As a consequence, we show that for any bounded measurable function $f$ on an ergodic system, the sequence $f(T^n x)$ is universally good for almost every $x$. In systems endowed with several commuting transformations, we use the machinery of magic systems introduced recently by Host and further properties of magic systems developed in this thesis,to extend results of Host and Kra on convergence of multiple ergodic averages along cubes with a single transformation to commuting transformations. We obtain a quantitative multiple recurrence result for two commuting transformations, similar in flavour to that of a single transformationestablished by Bergelson, Host and Kra, but with a different conclusion

L'objectif de cette thèse est d'étudier les comportements typiques d'une famille de transformations du cylindre discret Z×T (où T=R/Z est le cercle de longueur un). Appliquez une rotation à chaque cercle du cylindre puis coupez tous les cercles en deux et déplacez une moitié de chaque cercle d'un niveau vers le bas et une moitié d'un niveau vers le haut. Nous utilisons pour cela des résultats existants en théorie des échanges d'intervalles et en théorie des surfaces de translation compactes. Tout d'abord, nous avons prouvé que pour presque toute suite bi-infinie de rotations, le système obtenu est conservatif (c'est à dire il n'y a pas d'ensemble errant de mesure strictement positive). Ensuite, nous avons prouvé que pour un ensemble Gδ-dense de paramètres, le système est en même temps conservatif, minimal et ergodique. Ce système a un rapport heuristique avec une famille de billards planaires, ainsi qu'une traduction dans des flots sur des surfaces de translation de genre infini. Cela est expliqué dans la thèse. / The main objective of this thesis is the study of the typical dynamical behaviour of a family of transformations on the discrete cylinder Z×T (where T=R/Z is the length one circle). Apply a rotation to every single circle of the cylinder then cut every circle in two and move half of each circle one level down and the other half one level up. To achieve this goal, we use existing results about interval exchange transformations and about compact translation surfaces. First, we proved that for almost every bi-infinite sequence of rotations, the obtained system is conservative (i.e. there is not wandering set of positive measure). Next, we proved that for a Gδ-dense set of parameters, the described system is ergodic, minimal and conservative. This system is heuristically related to a family of planar billiards, it has also a translation into flows on infinite genus translation surfaces.

Neste trabalho, provamos um caso particular do Teorema de Ratner de classificação de medidas, que nos diz que se X =&Gamma;\\G é um espaço homogêneo, onde G é um grupo de Lie e &Gamma; é um lattice de G, então dado um subgrupo unipotente U de G, conseguimos classificar as medidas ergódicas com relação a ação por translação do grupo U em X. Além do Teorema de Ratner de classificação de medidas, falamos sobre o Teorema de Ratner de equidistribuição e o Teorema de Ratner do fecho da órbita, que nos dizem como são as órbitas pela ação por translação do grupo U e como é sua dinâmica em X, do ponto de vista da Teoria Ergódica. Embora estes últimos resultados não sejam provados nesta dissertação, exibimos uma importante aplicação do Teorema de Ratner do fecho da órbita em teoria dos números, provando a Conjectura de Oppeinheim, também conhecida como Teorema de Margullis. / In this work, we prove a particular case of the Ratners measure classification theorem, which tell us that if X = &Gamma;\\G is an homogeneous space, where G is a Lie group and &Gamma; is a lattice of G, then given any unipotent group U of G, we can classify the measures that are ergodic with respect to the translation group action of U in X In addition to the Ratners measure classification theorem, we talk about the Ratners equidistribuition theorem and the Ratners orbit closure theorem, which tell us how the orbit due the action by translation by the group U are and how the dynamics in X is, in an Ergodic Theory point of view. While we didnt prove the last two Ratners theorems, we exhibit an important application of the Ratners orbit closure theorem in number theory, proving the Oppeinheim Conjecture, also know as Margullis Theorem.

In this thesis we study actions of countable groups on measure spaces underthe assumption that the dynamics are non-singular, with particular reference topointwise ergodic theorems and their relationship to the critical dimensions ofthe action.

Systems whose time evolutions are entirely deterministic can nevertheless be studied probabilistically, <em>i. e. </em> in terms of the evolution of probability distributions rather than individual trajectories. This approach is central to the dynamics of ensembles (statistical mechanics) and systems with uncertainty in the initial conditions. It is also the basis of ergodic theory--the study of probabilistic invariants of dynamical systems--which provides one framework for understanding chaotic systems whose time evolutions are erratic and for practical purposes unpredictable. Delay differential equations (DDEs) are a particular class of deterministic systems, distinguished by an explicit dependence of the dynamics on past states. DDEs arise in diverse applications including mathematics, biology and economics. A probabilistic approach to DDEs is lacking. The main problems we consider in developing such an approach are (1) to characterize the evolution of probability distributions for DDEs, <em>i. e. </em> develop an analog of the Perron-Frobenius operator; (2) to characterize invariant probability distributions for DDEs; and (3) to develop a framework for the application of ergodic theory to delay equations, with a view to a probabilistic understanding of DDEs whose time evolutions are chaotic. We develop a variety of approaches to each of these problems, employing both analytical and numerical methods. In transient chaos, a system evolves erratically during a transient period that is followed by asymptotically regular behavior. Transient chaos in delay equations has not been reported or investigated before. We find numerical evidence of transient chaos (fractal basins of attraction and long chaotic transients) in some DDEs, including the Mackey-Glass equation. Transient chaos in DDEs can be analyzed numerically using a modification of the "stagger-and-step" algorithm applied to a discretized version of the DDE.

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

An ergodic hidden Markov model (EHMM) can be useful in extracting underlying structure embedded in connected speech without the need for a time-aligned transcribed corpus. In this research, we present a query-by-example (QbE) spoken term detection system based on an ergodic hidden Markov model of speech. An EHMM-based representation of speech is not invariant to speaker-dependent variations due to the unsupervised nature of the training. Consequently, a single phoneme may be mapped to a number of EHMM states. The effects of speaker-dependent and context-induced variation in speech on its EHMM-based representation have been studied and used to devise schemes to minimize these variations. Speaker-invariance can be introduced into the system by identifying states with similar perceptual characteristics. In this research, two unsupervised clustering schemes have been proposed to identify perceptually similar states in an EHMM. A search framework, consisting of a graphical keyword modeling scheme and a modified Viterbi algorithm, has also been implemented. An EHMM-based QbE system has been compared to the state-of-the-art and has been demonstrated to have higher precisions than those based on static clustering schemes.