Lyapunov Exponents of Linear Cocycles

Abstract

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...

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“…We will thus focus on the more interesting situation where E ∈ [−2, 2], which will be handled as an application of Theorem 5.10. Here, we also mention that while Theorem 5.1 in [24] does predict the qualitative continuity of the Lyapunov exponent as λ → 0 + , i.e. lim λ→0 + L λ (E) = L λ=0 (E) , for each E ∈ [−2, 2] , (5.50) conclusions about the modulus of continuity based on [24] are not possible since L λ=0 (E) = 0 for every E ∈ [−2, 2].…”

“…Here, we also mention that while Theorem 5.1 in [24] does predict the qualitative continuity of the Lyapunov exponent as λ → 0 + , i.e. lim λ→0 + L λ (E) = L λ=0 (E) , for each E ∈ [−2, 2] , (5.50) conclusions about the modulus of continuity based on [24] are not possible since L λ=0 (E) = 0 for every E ∈ [−2, 2]. In order to apply Theorem 5.10 to the weak-disorder limit for the model described in (5.1), hypothesis [H3] needs to be verified.…”

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