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We apply anisotropic Banach space methods together with operator renewal theory to obtain polynomial rates of decay of correlations for a class of generalized baker's transformations. The polynomial rates were proved for a smaller class of observables in a 2013 paper of Bose and Murray by fundamentally different methods. Our approach provides a direct analysis of the Frobenius-Perron operator associated to a generalized baker's transformation in contrast to the paper of Bose and Murray where decay rates are obtained for a factor map and lifted to the full map. / Graduate

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [10]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II 1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [59]. We also obtain a product version of this result: any maximal abelian ∗-subalgebra of any II 1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana's cocycle superrigidity theorem [36], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article [11] which has already been submitted for publication.

Ce travail met en jeu plusieurs systèmes dynamiques sur des tores en dimension finie, pour lesquels on sait établir des théorèmes limites, qui permettent de préciser leur comportement stochastique. On généralise d'abord le théorème limite local usuel sur un sous-shift de type fini, en ajoutant un terme de perturbation, en reprenant la preuve classique, par des techniques d'opérateurs. On en déduit un théorème limite local pour les sommes de « Riesz-Raïkov unitaires étendues », et des observables höldériennes. Pour cela, on reprend une méthode employée par Bernard Petit, en utilisant des codages symboliques, et le théorème limite local avec perturbation. Puis, on présente plusieurs situations de composées d'automorphismes hyperboliques du tore en dimension deux pour lesquelles on sait établir un théorème limite central quelque soit le choix de la composée. En particulier, on aborde le cas des matrices à coefficients entiers positifs.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009. / Committee Chair: Bunimovich, Leonid; Committee Member: Bonetto, Federico; Committee Member: Chow, Shui-Nee; Committee Member: Cvitanovic, Predrag; Committee Member: Weiss, Howard. Part of the SMARTech Electronic Thesis and Dissertation Collection.

We study compensation functions for an infinite-to-one factor code $\pi : X \to Y$ where $X$ is a shift of finite type. The $p$-Dini condition is given as a way of measuring the smoothness of a continuous function, with $1$-Dini corresponding to functions with summable variation. Two types of compensation functions are defined in terms of this condition. Given a fully-supported invariant measure $\nu$ on $Y$, we show that the relative equilibrium states of a $1$-Dini function $f$ over $\nu$ are themselves fully supported, and have positive relative entropy. We then show that there exists a compensation function which is $p$-Dini for all $p > 1$ which has relative equilibrium states supported on a finite-to-one subfactor. / Graduate / 0405 / antoniol@uvic.ca

The theory of nonlinear mass-spring chains has a history stretching back to the now famous numerical simulations of Fermi, Pasta and Ulam. The unexpected results of that experiment have led to many new fields of study. Despite this, the mathematics of the lattice equations have proved sufficiently rich to attract continued attention to the present day. This work is concerned with the motions of an infinite one dimensional lattice with nearest-neighbour interactions governed by a generic potential. The Hamiltonian of such a system may be written $H = \sum_{i=-\infty}^{\infty} \, \Bigl(\frac{1}{2}p_i^2 + V(q_{i+1}-q_i)\Bigr)$, in terms of the momenta $p_i$ and the displacements $q_i$ of the lattice sites. All sites are assumed to be of equal mass. Certain generic conditions are placed on the potential $V$. Of particular interest are the solitary wave solutions which are known to exist upon such lattices. The KdV equation has long been known to emerge in a formal manner from the lattice equations as a continuum limit. More recently, the lattice's localized nonlinear modes have been rigorously approximated by the KdV's well-studied soliton solution, in the lattice's long wavelength regime. To date, however, little is known about how, and to what extent, lattice solitary waves differ from KdV solitons. It is proved in this work that a solution (which we prove to be unique) to a particular linear ordinary differential equation provides a correction to the KdV approximation. This gives, in an explicit way, the lowest order effect of lattice discreteness upon lattice solitary waves. It is also shown how such discreteness effects are propagated along the lattice both in isolation (single soliton case), and in the presence of another soliton correction (the bisoliton case). In the latter case their interaction is studied and the impact of lattice discreteness upon lattice solitary wave interactions is observed. This is possible by virtue of the discovery of an evolution equation for discreteness effects on the lattice. This equation is proved to have appropriate unique solutions and is found to be strikingly similar to corresponding equations known in both the theories of shallow water waves and ion-acoustic waves.

The goal of this thesis is to investigate model imperfection in the context of forecasting. We focus on an electronic circuit built in a laboratory and then enclosed to reduce environmental effects. The non-dimensionalised model equations, obtained by applying Kirchhoff’s current and voltage laws, are the Moore-Spiegel Equations [47], but they exhibit a large disparity with the circuit. At parameter values used in the circuit, they yield a periodic trajectory whilst the circuit exhibits chaotic behaviour. Therefore, alternative models for the circuit are sought. The models we consider are local and global prediction models constructed from data. We acknowledge that all our models have errors and then seek to quantify how these errors are distributed across the circuit attractor. To this end, q-pling times of initial uncertainties are computed for the various models. A q-pling time is the time for an initial uncertainty to increase by a factor of q [67], where q is a real number. Whereas it is expected that different models should have different q-pling time distributions, it is found that the diversity in our models can be increased by constructing them in different coordinate spaces. To forecast the future dynamics of the circuit using any of the models, we make probabilistic forecasts [8]. The question of how to choose the spread of the initial ensemble is addressed by the use of skill scores [8, 9]. Finally, the diversity in our models is exploited by combining probabilistic forecasts from them so as to minimise some skill score. It is found that the skill of combined not-so-good models can be increased by combining them as discussed in this thesis.

Physical processes are often modelled using nonlinear dynamical systems. If such models are relevant then they should be capable of demonstrating behaviour observed in the physical process. In this thesis a new measure of model optimality is introduced: the distribution of ɩ-shadowing times defines the durations over which there exists a model trajectory consistent with the observations. By recognising the uncertainty present in every observation, including the initial condition, ɩ-shadowing distinguishes model sensitivity from model error; a perfect model will always be accepted as optimal. The traditional root mean square measure may confuse sensitivity and error, and rank an imperfect model over a perfect one. In a perfect model scenario a good variational assimilation technique will yield an ɩ-shadowing trajectory but this is not the case given an imperfect model; the inability of the model to ɩ-shadow provides information on model error, facilitating the definition of an alternative assimilation technique and enabling model improvement. While the ɩ-shadowing time of a model defines a limit of predictability, it does not validate the model as a predictor. Ensemble forecasting provides the preferred approach for evaluating the uncertainty in predictions, yet questions remain as to how best to construct ensembles. The formation of ensembles is contrasted in perfect and imperfect model scenarios in systems ranging from the analytically tractable to the Earth's atmosphere, thereby addressing the question of whether the apparent simplicity often observed in very high-dimensional weather models fails `even in or only in' low-dimensional chaotic systems. Simple tests of the consistency between constrained ensembles and their methods of formulation are proposed and illustrated. Specifically, the commonly held belief that initial uncertainties in the state of the atmosphere of realistic amplitude behave linearly for two days is tested in operational numerical weather prediction models and found wanting: nonlinear effects are often important on time scales of 24 hours. Through the kind consideration of the European Centre for Medium-range Weather Forecasting, the modifications suggested by this are tested in an operational model.