Lyapunov Exponents of Linear Cocycles Continuity via Large Deviatio
The aim of this monograph is to present a general method of proving continuity of Lyapunov exponents of linear cocycles. The method uses an inductive procedure based on a general, geometric version of the Avalanche Principle. The main assumption required
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Pedro Duarte Silvius Klein
Lyapunov Exponents of Linear Cocycles Continuity via Large Deviations · Volume 3
Atlantis Studies in Dynamical Systems Volume 3
Series editors Henk Broer, Groningen, The Netherlands Boris Hasselblatt, Medford, USA
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.
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Pedro Duarte Silvius Klein •
Lyapunov Exponents of Linear Cocycles Continuity via Large Deviations
Pedro Duarte Faculdade de Ciências Universidade de Lisboa Lisbon Portugal
Silvius Klein Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) Trondheim Norway
Atlantis Studies in Dynamical Systems ISBN 978-94-6239-123-9 ISBN 978-94-6239-124-6 DOI 10.2991/978-94-6239-124-6
(eBook)
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 paper
In memory of João Santos Guerreiro and Ricardo Mañé, professors whose friendship and intelligence I miss Pedro Duarte To Florin Popovici and Șerban Strătilă who taught me to seek and to appreciate good mathematical exposition Silvius Klein
Preface
The 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 a
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