We investigate the genericity of measure-preserving actions of the free group Fn, on possibly countably infinitely many generators, acting on a standard probability space. Specifically, we endow the space of all measure-preserving actions of Fn acting on a standard probability space with the weak topology and explore what properties may be verified on a comeager set in this topology. In this setting we show an analog of the classical Rokhlin Lemma. From this result we conclude that every action of Fn may be approximated by actions which factor through a finite group. Using this finite approximation we show the actions of Fn, which are rigid and hence fail to be mixing, are generic. Combined with a recent result of Kerr and Li, we obtain that a generic action of Fn is weak mixing but not mixing. We also show a generic action of Fn has sigma-entropy at most zero. With some additional work, we show the finite approximation result may be used to that show for any action of Fn, the crossed product embeds into the tracial ultraproduct of the hyperfinite II1 factor. We conclude by showing the finite approximation result may be transferred to a subspace of the space of all topological actions of Fn on the Cantor set. Within this class, we show the set of actions with sigma-entropy at most zero is generic.

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Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 68-69). Also available for download via the World Wide Web; free to University of Oregon users.

The main aim of this thesis is to design efficient and novel numerical algorithms for a class of deterministic and stochastic dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become, therefore, inefficient when the much longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. This dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three different papers. The framework of the heterogeneous multiscale method (HMM) is considered as a general strategy both for the design and the analysis of multiscale methods. In Chapter 2, we consider a new class of multiscale methods that use a technique related to the construction of a Poincaré map. The main idea is to construct effective paths in the state space whose projection onto the slow subspace shows the correct dynamics. More precisely, we trace the evolution of the invariant manifold M(t), identified by the level sets of slow variables, by introducing a slowly evolving effective path which crosses M(t). The path is locally constructed through interpolation of neighboring points generated from our developed map. This map is qualitatively similar to a Poincaré map, but its construction is based on the procedure which solves two split equations successively backward and forward in time only over a short period. This algorithm does not require an explicit form of any slow variables. In Chapter 3, we present efficient techniques for numerical averaging over the invariant torus defined by ergodic dynamical systems which may not be mixing. These techniques are necessary, for example, in the numerical approximation of the effective slow behavior of highly oscillatory ordinary differential equations in weak near-resonance. In this case, the torus is embedded in a higher dimensional space and is given implicitly as the intersection of level sets of some slow variables, e.g. action variables. In particular, a parametrization of the torus may not be available. Our method constructs an appropriate coordinate system on lifted copies of the torus and uses an iterated convolution with respect to one-dimensional averaging kernels. Non-uniform invariant measures are approximated using a discretization of the Frobenius-Perron operator. These two numerical averaging strategies play a central role in designing multiscale algorithms for dynamical systems, whose fast dynamics is restricted not to a circle, but to the tori. The efficiency of these methods is illustrated by numerical examples. In Chapter 4, we generalize the classical two-scale averaging theory to multiple time scale problems. When more than two time scales are considered, the effective behavior may be described by the new type of slow variables which do not have formally bounded derivatives. Therefore, it is necessary to develop a theory to understand them. Such theory should be applied in the design of multiscale algorithms. In this context, we develop an iterated averaging theory for highly oscillatory dynamical systems involving three separated time scales. The relevant multiscale algorithm is constructed as a family of multilevel solvers which resolve the different time scales and efficiently computes the effective behavior of the slowest time scale. / text

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The theoretical basis for the Lyapunov exponents of continuous- and discrete-time dynamical systems is developed, with the inclusion of the statement and proof of the Multiplicative Ergodic Theorem of Oseledec. The numerical challenges and algorithms to approximate Lyapunov exponents and vectors are described, with multiple illustrative examples. A novel generalized impulsive collision rule is derived for particle systems interacting pairwise. This collision rule is constructed to address the question of whether or not the quantitative measures of chaos (e.g. Lyapunov exponents and Kolmogorov-Sinai entropy) can be reduced in these systems. Major results from previous studies of hard-disk systems, which interact via elastic collisions, are summarized and used as a framework for the study of the generalized collision rule. Numerical comparisons between the elastic and new generalized rules reveal many qualitatively different features between the two rules. Chaos reduction in the new rule through appropriate parameter choice is demonstrated, but not without affecting the structural properties of the Lyapunov spectra (e.g. symmetry and conjugate-pairing) and of the tangent space decomposition (e.g. hyperbolicity and domination of the Oseledec splitting). A novel measure of the degree of domination of the Oseledec splitting is developed for assessing the impact of fluctuations in the local Lyapunov exponents on the observation of coherent structures in perturbation vectors corresponding to slowly growing (or contracting) modes. The qualitatively different features observed between the dynamics of generalized and elastic collisions are discussed in the context of numerical simulations. Source code and complete descriptions for the simulation models used are provided.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

This thesis introduces the notion of using stability analysis in terms of two measures for impulsive switched systems. Impulsive switched systems are defined in the context of hybrid system theory and the motivation for the study of these systems is presented. The motivation for studying stability in two measures is also given, along with the definitions of stability, uniform stability, and uniform asymptotic stability in one and two measures. The results presented are a sets of sufficient stability criteria for linear and nonlinear systems. For autonomous linear systems, there are criteria for stability and asymptotic stability using a particular family of choices for the two measures. There is an additional stronger set of criteria for asymptotic stability using one measure, for comparison. There is also a proposed method for finding the asymptotic stability of a non-autonomous system in one measure. The method for extending these criteria to linearized systems is also presented, along with stability criteria for such systems. The criteria for nonlinear systems cover stability, uniform stability, and uniform asymptotic stability, considering state-based and time-based switching rules in different ways. The sufficient stability criteria that were found were used to solve four instructive examples. These examples show how the criteria are applied, how they compare, and what the shortcomings are in certain situations. It was found that the method of using two measures produced stricter stability requirements than a similar method for one measure. It was still found to be a useful result that could be applied to the stability analysis of an actual impulsive switched system.

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The past few years have witnessed a growth in the study of the long-time behaviour of physical, biological and economic systems using measure-theoretic and probabilistic methods. In this dissertation we present a study of the evolution of dynamical systems that display various types of irregular behaviour for large times. Large systems, containing many elements, like e.g. bacteria populations or ensembles of gas particles, are very difficult to analyse and contain elements of uncertainty. Also, in general, it is not necessary to know the evolution of each bacteria or each gas particle. Therefore we replace the "pointwise" description of the evolution of the system with that of the evolution of suitable averages of the population like e.g. the gas or the bacteria spatial density. In particular cases, when the quantity in the evolution that we analyse has the probabilistic interpretation, say, the probability of finding the particle in certain state at certain time, we will be talking about the evolution of (probability) densities. We begin with the establishment of results for discrete time systems and this is later followed with analogous results for continuous time systems. We observe that in many cases the system has two important properties: at each step it is determined by a non-negative function (for example the spatial density or the probability density) and the overall quantity of the elements remains preserved. Because of these properties the most suitable framework to investigate such systems is the theory of Markov operators. We shall discuss three levels of "chaotic" behaviour that are known as ergodicity, mixing and exactness. They can be described as follows: ergodicity means that the only invariant sets are trivial, mixing means that for any set A the sequence of sets S-n(A) becomes, asymptotically, independent of any other set B, and exactness implies that if we start with any set of positive measure, then, after a long time the points of this set will spread and completely fill the state space. In this dissertation we describe an application of two operators related to the generating Markov operator to study and characterize the abovementioned properties of the evolution system. However, a system may also display regular behaviour. We refer to this as the asymptotic stability of the Markov operator generating this system and we provide some criteria characterizing this property. Finally, we demonstrate the use of the above theory by applying it to a system that is modeled by the linear Boltzmann equation. / Thesis (M.Sc.)-University of Natal, Durban, 2001.