Many spectral questions about one-dimensional Schrödinger operators with quasi-periodic potentials can be reduced to dynamical questions about certain quasi-periodic SL(2, R)-valued cocycles. This connection has recently been employed to prove a number of long-standing conjectures. The aim of this mini-workshop was to bring together people from both spectral theory and dynamical systems in order to further develop and exploit the dynamical approach to quasi-periodic Schrödinger operators.

This article stresses the importance of extending the Nyquist servomechanism analysis to include the transient response and points out one method of obtaining the transient response from the conventional frequency response data which is usually the end point of the Nyquist analysis.The basis for the present "rules of thumb" (phase and gain margins) used when only frequency data are considered is reviewed briefly by comparing the characteristics of a simple feedback servomechanism with those of a linear second-order system. It is shown that, although these rules are pointed toward obtaining a good transient response by considering the system frequency characteristics, they are often inadequate. Once this is realized, the value of actually obtaining the transient response is apparent.The method outlined for obtaining the transient response to a given input consists of representing the input as a Fourier series, operating on the series with the system transfer function, and summing the resulting series. Response to a step input is given as an example, and the necessary procedure for carrying out the analysis is given and discussed.

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Generally speaking, the term nonautonomous dynamics refers to the systematic use of dynamical tools to study the solutions of differential or difference equations with time-varying coefficients. The nature of the time variance may range from periodicity at one extreme, through Bohr almost periodicity, Birkhoff recurrence, Poisson recurrence etc. to stochasticity at the other extreme. The "dynamical tools" include almost everywhere Lyapunov exponents, exponential splittings, rotation numbers, and the theory of cocycles, but are by no means limited to these. Of course in practise one uses whatever "works" in the context of a given problem, so one usually finds dynamical methods used in conjunction with those of numerical analysis, spectral theory, the calculus of variations, and many other fields. The reader will find illustrations of this fact in all the papers of the present volume. Each of the papers presented here contains a detailed introduction, which describes its contents and provides sufficient background material to orient the reader with respect to the related literature and, more generally, to the scientific context in which it was conceived. Nevertheless, we present here a brief description of each contribution and some comments concerning the relations between them. It is often convenient to study a nonautonomous differential/difference system by introducing a so-called driving flow; one obtains a family of systems whose solutions define a cocycle over that flow. This situation can be achieved via a Bebutov-type construction if the coefficients of the system are bounded; that is to say, one closes the set of translates of the coefficient function(s) in some topology. The result of this construction is sometimes called the Bebutov hull of the system. It should also be said that a good number of problems are posed with a driving flow already present; e.g. this is often the case in what is known as stochastic dynamics. In any case, the cocycle gives rise to a skew-product flow, which lives on the product of the phase space of the driving flow with the phase space of the differential/difference system. Several of the papers in this volume treat problems formulated in the driving flow/cocycle framework, as we now indicate. Campos, Obaya and Tarallo study some consequences of their Fredholm alternative theory for linear nonhomogeneous ODEs with Birkhoff recurrent coefficients, and generalize a well-known result of Cieutat and Haraux. Cong and Son consider bounded linear random ODEs, that is, the coefficients are bounded and are driven by an ergodic flow on a probability space. In this context they prove the openness and density of the integral separation property in the space of coefficients, in the topology of the L ∞ norm. Damanik, Fillman, Lukic and Yessen use dynamical methods to study the spectral theory of CMV matrices; these are intimately related to the (Szego) cocycles which arise in the study of orthogonal polynomials on the unit circle. As a preliminary to this they give a unified treatment of th...