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.

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.

The aim of this thesis is to seek a low-dimensional description of baroclinic instability in general, and of the Martian atmosphere in particular, where both forcing and spatial resonance are relevant to the dynamics of the system being analysed. The Proper Orthogonal Decomposition (POD) is used to determine a basis for the modal decomposition of climatic simulations of Mars, obtained by using two General Circulation Models (GCMs): (a) a simple GCM, which is an idealised model in which the meteorological primitive equations are solved on a sphere with simplified physical parameters and (b) the Martian GCM, a more realistic model in which a comprehensive range of the relevant Martian physical parameters and topography are represented. Results of these analyses are presented for a range of Martian seasons and climatic conditions. The effects of using different forms of energy norm in performing the analysis is considered, with the objective of providing analyses which represents the physically most significant components of the circulation, with optimal efficiency. Reduced low-dimensional models that replicated the full simple GCM streamfunction simulations are formulated by projecting the spherical quasi-geostrophic equations onto the PODs of the large-scale calculations. The resulting models are analysed by using a combination of solution continuation and numerical integration methods. A thorough analysis of the models reveals that a 6-D POD model is capable of reproducing the amplitude, frequency and behaviour of the leading oscillatory structures of the simple GCM, to within a 1% error. Such an excellent reproduction of the original system is shown to be due to (1) an accurate vertical formulation scheme, (2) the use of the correct norm, (3) a sufficiently high level of truncation and (4) the fact that the original system is a steady wave flow. The behaviour of the various regimes observed in the low-order models are comparable with observations from studies of large-scale waves and instabilities in planetary atmospheres, including a range of hydrodynamical experiments on baroclinic wave interactions of a stratified fluid in cylindrical containers.

We report progress made on an analytic investigation of low-frequency cardiorespiratory variability in humans. The work is based on an existing physiological model of chemically-mediated blood-gas control via the central and peripheral chemoreceptors, that of Grodins, Buell & Bart (1967). Scaling and simplification of the Grodins model yields a rich variety of dynamical subsets; the thesis focusses on the dynamics obtained under the normoxic assumption (i.e., when oxygen is decoupled from the system). In general, the method of asymptotic reduction yields submodels that validate or invalidate numerous (and more heuristic) extant efforts in the literature. Some of the physiologically-relevant behaviour obtained here has therefore been reported before, but a large number of features are reported for the first time. A particular novelty is the explicit demonstration of cardiorespiratory coupling via chemosensory control. The physiology and literature reviewed in Chapters 1 and 2 set the stage for the investigation. Chapter 3 scales and simplifies the Grodins model; Chapters 4, 5, 6 consider carbon dioxide dynamics at the central chemoreceptor. Chapter 7 begins analysis of the dynamics mediated by the peripheral receptor. Essentially all of the dynamical behaviour is due to the effect of time delays occurring within the conservation relations (which are ordinary differential equations). The pathophysiology highlighted by the analysis is considerable, and includes central nervous system disorders, heart failure, metabolic diseases, lung disorders, vascular pathologies, physiological changes during sleep, and ascent to high altitude. Chapter 8 concludes the thesis with a summary of achievements and directions for further work.

In this thesis we consider mathematical problems related to different aspects of hard sphere systems. In the first part we study planar billiards, which arise in the context of hard sphere systems when only one or two spheres are present. In particular we investigate the possibility of elliptic periodic orbits in the general construction of hyperbolic billiards. We show that if non-absolutely focusing components are present there can be elliptic periodic orbits with arbitrarily long free paths. Furthermore, we show that smooth stadium like billiards have elliptic periodic orbits for a large range of separation distances. In the second part we consider hard sphere systems with a large number of particles, which we model by the Boltzmann equation. We develop a new approach to derive hydrodynamic limits, which is based on classical methods of geometric singular perturbation theory of ordinary differential equations. This method provides new geometric and dynamical interpretations of hydrodynamic limits, in particular, for the of the dissipative Boltzmann equation.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [51]-52).