A stack of N 1 transparent plates with randomly varying thicknesses (e.g. viewgraphs) reflects light perfectly, as a result of the accumulation of reflections from interfaces at the air gaps separating the plates. Two theories of this effect are discordant. The naive ray theory assumes that the random phases associated with the thickness variations make all the reflections incoherent, and predicts that the transmitted intensity decays as 1/N. This theory is wrong because some distinct multiply reflected waves have identical path lengths and so superpose coherently. The true decay is exponential: exact averaging of the logarithm of the transmitted intensity over the random phases, assuming these are uniformly distributed modulo 2π, gives the transmitted intensity as τ 2N = exp{−2N log(1/τ)}, where τ is the intensity transmittance of a single interface. Transparent mirrors are naked-eye examples of the localization of light, for which the localization length (inverse decay exponent) can be calculated exactly. Experiments confirm the exponential decay. Zusammenfassung. Ein Stapel von N 1 transparenten Platten mit beliebig variierender Dicke (z.B.Overhead-Folien) reflektiert Licht perfekt. Diese perfekte Reflexion ist das Ergebnis der Akkumulation von Reflexionen an den Grenzflächen der Luftspalte zwischen den Platten. Die zwei Theorien dieses Effektes stimmen nichtüberein. Die naive Strahlen-Theorie nimmt an, daß die beliebigen Phasen, die mit den Variationen der Plattendicke verknüpft sind, zu absolut inkoherenten Reflexionen führen und sagt einen 1/N Verlauf für die durchgelassene Intensität voraus. Diese Theorie ist falsch, da einige der mehrfach reflektierten Wellen identische Weglängen haben und daher koherentüberlagern. Der wahre Verlauf ist exponential: exakte Mittelung des Logarithmus der durchgelassenen Intensitätüber die beliebigen Phasen, unter der Annahme sie seien gleichmässig verteilt modulo 2π , führt zur durchgelassenen Intensität τ 2N = exp{−2N log(1/τ)}, wo τ die durchgelassene Intensität einer einzelnen Grenzfläche ist. Durchsichtige Spiegel sind Beispiele der Lokalisation des Lichtes für das unbewaffenete Auge, für die die Lokalisationslänge (inverser Exponentialkoeffizient) exakt berechnet werden kann. Experimente bestätigen den exponentialen Verlauf.

In this paper we consider the discrete one-dimensional Schro¨dinger operator with quasiperiodic potential v n ¼ lvðx þ noÞ: We assume that the frequency o satisfies a strong Diophantine condition and that the function v belongs to a Gevrey class, and it satisfies a transversality condition. Under these assumptions we prove-in the perturbative regime-that for large disorder l and for most frequencies o the operator satisfies Anderson localization. Moreover, we show that the associated Lyapunov exponent is positive for all energies, and that the Lyapunov exponent and the integrated density of states are continuous functions with a certain modulus of continuity. We also prove a partial nonperturbative result assuming that the function v belongs to some particular Gevrey classes. r

The "Atlantis Studies in Dynamical Systems" publishes monographs in the area of dynamical systems, written by leading experts in the field and useful for both students and researchers. Books with a theoretical nature will be published alongside books emphasizing applications. ISBN 978-94-6239-123-9 ISBN 978-94-6239-124-6 (eBook) DOI 10.2991/978-94-6239-124-6 Library of Congress Control Number: 2016933219 © Atlantis Press and the author(s) 2016 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paperIn memory of João Santos Guerreiro and Ricardo Mañé, professors whose friendship and intelligence I miss Pedro DuarteTo Florin Popovici and Șerban Strătilă who taught me to seek and to appreciate good mathematical exposition Silvius Klein PrefaceThe aim of this monograph is to present a general method of proving continuity of the Lyapunov exponents (LE) of linear cocycles.The method consists of an inductive procedure that establishes continuity of relevant quantities for finite, larger and larger number of iterates of the system. This leads to continuity of the limit quantities, the LE. The inductive procedure is based upon a deterministic result on the composition of a long chain of linear maps called the Avalanche Principle (AP). A geometric approach is used to derive a general version of this principle.The main assumption required by this method is the availability of appropriate large deviation type (LDT) estimates for quantities related to the iterates of the base and fiber dynamics associated with the linear cocycle. Crucial for our approach is the uniformity in the data of these estimates.We derive such LDT estimates for various models of random cocycles (over Bernoulli and Markov systems) and quasi-periodic cocycles (defined by one or multivariable torus translations). The random model, treated under an irreducibility assumption, uses an existing functional analytic approach which we adapt so that it provides the required uniformity of the estimates. The quasi-periodic model uses harmonic analysis and it involves the study of (pluri) subharmonic functions.This method has its origins in a paper of M. Goldstein and W. Schlag which proves continuity of the Lyapunov exponent for the one-parameter family of quasi-periodic Schrödinger cocycles, assuming a uniform lower bound on the exponent. This is where the first version of the Avalanche Principle appeared, along with the use and proof of the relevant LDT estimate.The present work expands upon their approach in both depth and breadth. Moreover, it reduces the general problem of proving continuity of the LE to one of a different nature-proving LDT estimates. This may be treated independently and by means specific to the underlying base dynamic of the the cocycle.Our geometric approach to the AP also gives ri...

We consider an m-dimensional analytic cocycle). Assuming that the d × d upper left corner block of A is typically large enough, we prove that the d largest Lyapunov exponents associated with this cocycle are bounded away from zero. The result is uniform relative to certain measurements on the matrix blocks forming the cocycle. As an application of this result we obtain nonperturbative (in the spirit of Sorets-Spencer theorem) positive lower bounds of the nonnegative Lyapunov exponents for various models of band lattice Schrödinger operators.

An analytic quasi-periodic cocycle is a linear cocycle over a fixed ergodic torus translation of one or several variables, where the fiber action depends analytically on the base point. Consider the space of all such cocycles of any given dimension and endow it with the uniform norm. Assume that the translation vector satisfies a generic Diophantine condition. We prove large deviation type estimates for the iterates of such cocycles, which, moreover, are stable under small perturbations of the cocycle. As a consequence of these uniform estimates, we establish continuity properties of the Lyapunov exponents regarded as functions on this space of cocycles. This result builds upon our previous work on this topic and its proof uses an abstract continuity theorem of the Lyapunov exponents which we derived in a recent monograph. The new feature of this paper is extending the availability of such results to cocycles that are identically singular (i.e. non-invertible anywhere), in the several variables torus translation setting. This feature is exactly what allows us, through a simple limiting argument, to obtain criteria for the positivity and simplicity of the Lyapunov exponents of such cocycles. Specializing to the family of cocycles corresponding to a block Jacobi operator, we derive consequences on the continuity, positivity and simplicity of its Lyapunov exponents, and on the continuity of its integrated density of states.