Atlantis Studies in Dynamical Systems Series Editors: H. Broer · B. Hasselblatt 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 StudiesinDynamical Systems”publishes monographs inthearea 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. More information about this series at http://www.atlantis-press.com Pedro Duarte Silvius Klein (cid:129) Lyapunov Exponents of Linear Cocycles Continuity via Large Deviations PedroDuarte Silvius Klein Faculdade deCiências Department ofMathematical Sciences Universidade deLisboa NorwegianUniversity of Scienceand Lisbon Technology (NTNU) Portugal Trondheim Norway Atlantis Studies inDynamical Systems ISBN978-94-6239-123-9 ISBN978-94-6239-124-6 (eBook) DOI 10.2991/978-94-6239-124-6 LibraryofCongressControlNumber:2016933219 ©AtlantisPressandtheauthor(s)2016 Thisbook,oranypartsthereof,maynotbereproducedforcommercialpurposesinanyformorbyany means, electronic or mechanical, including photocopying, recording or any information storage and retrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher. Printedonacid-freepaper 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 relevantquantitiesforfinite,largerandlargernumberofiteratesofthesystem.This leadstocontinuityofthelimitquantities,theLE.Theinductiveprocedureisbased uponadeterministicresultonthecompositionofalongchainoflinearmapscalled 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 largedeviationtype(LDT)estimatesforquantitiesrelatedtotheiteratesofthebase 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 multivariabletorustranslations).Therandommodel,treatedunderanirreducibility assumption,usesanexistingfunctionalanalyticapproachwhichweadaptsothatit 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.ThisiswherethefirstversionoftheAvalanchePrincipleappeared,along with the use and proof of the relevant LDT estimate. The present work expands upon their approach in both depth and breadth. Moreover,itreducesthegeneralproblemofprovingcontinuityoftheLEtooneof adifferent nature—provingLDTestimates.Thismaybetreatedindependentlyand by means specific to the underlying base dynamic of the the cocycle. Our geometric approach to the AP also gives rise to a mechanism for studying the most expanding singular direction of the composition of a long chain of linear maps. This allows us to obtain a new proof of the Multiplicative Ergodic vii viii Preface Theorem of Oseledets. Moreover, assuming the availability of the same LDT estimates, this extension of the AP leads to continuity properties of the Oseledets filtration and decomposition. Most of the results presented in this research monograph are new. We assume thereadertohaveacertaindegreeoffamiliaritywithbasicdynamicalsystemsand ergodic theory notions. The relevant concepts and definitions needed for the for- mulation of the main results are introduced in Chap. 1. While each subsequent chapter is to some extent self-contained and it may be read independently of the rest, all the arguments in this work are based upon the results in Chaps. 2 and 3. Besides the formulation and the proof of the AP, Chap. 2 contains Lipschitz esti- mates on certain Grassmann geometrical quantities that are crucial in Chap. 4, where we study the Oseledets filtration and decomposition and their continuity properties.InChap.3weestablishtheabstractcontinuitytheorem(ACT)oftheLE and some other related technical results. In Chaps. 5 and 6, under appropriate assumptions, we derive the relevant LDT estimates for random and respectively quasi-periodiccocycles.ThegeneralresultsinChaps.3and4arethenapplicableto these models, and they imply continuity properties of the LE and of the Oseledets filtration and decomposition for the corresponding spaces of cocycles. Our work concludes in Chap. 7 with a list of related open problems, some of which may be treated using the methods described in this monograph. Thefirstauthor wassupportedbyNationalFunding fromFCT—Fundaçãopara a Ciência e a Tecnologia, under the project: UID/MAT/04561/2013. The second author was supported by the Norwegian Research Council project no. 213638, “Discrete Models in Mathematical Analysis”. Both authors are grateful totheFaculty ofSciences of theUniversity of Lisbon (FCUL) and to the Norwegian University of Science and Technology (NTNU) for the support received and for facilitating their collaboration on this monograph. We would like to thank José Pedro Gaivão and Wilhelm Schlag for reading through parts of the manuscript. And last but not least, many thanks to Teresa, Zé, Jaime, Daniel and Jaqueline for their understanding. Lisbon Pedro Duarte Trondheim Silvius Klein January 2016 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 The Main Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 The Continuity Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Large Deviations Type Estimates. . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Summary of Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.6 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2 Estimates on Grassmann Manifolds . . . . . . . . . . . . . . . . . . . . . . . 23 2.1 Grassmann Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.2 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.3 Grassmann Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.4 Flag Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Singular Value Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 Singular Value Decomposition . . . . . . . . . . . . . . . . . . . 32 2.2.2 Gaps and Most Expanding Directions . . . . . . . . . . . . . . 35 2.2.3 Angles and Expansion. . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Lipschitz Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 Projective Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.2 Operations on Flag manifolds. . . . . . . . . . . . . . . . . . . . 50 2.3.3 Dependence on the Linear Map. . . . . . . . . . . . . . . . . . . 57 2.4 Avalanche Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.4.1 Contractive Shadowing . . . . . . . . . . . . . . . . . . . . . . . . 64 2.4.2 Statement and Proof of the AP . . . . . . . . . . . . . . . . . . . 68 2.4.3 Consequences of the AP. . . . . . . . . . . . . . . . . . . . . . . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 ix x Contents 3 Abstract Continuity of Lyapunov Exponents. . . . . . . . . . . . . . . . . 81 3.1 Definitions, the Abstract Setup and Statement . . . . . . . . . . . . . . 81 3.1.1 Cocycles and Observables . . . . . . . . . . . . . . . . . . . . . . 82 3.1.2 Large Deviations Type Estimates. . . . . . . . . . . . . . . . . . 84 3.1.3 Abstract Continuity Theorem of the Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2 Upper Semicontinuity of the Top Lyapunov Exponent . . . . . . . . 86 3.3 Finite Scale Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.4 The Inductive Step Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.5 General Continuity Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.6 Modulus of Continuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4 The Oseledets Filtration and Decomposition . . . . . . . . . . . . . . . . . 113 4.1 Introduction and Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.2 The Ergodic Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.1 Review of Grassmann Geometry Concepts and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.2 The Ergodic Theorems of Birkhoff and Kingman . . . . . . 118 4.2.3 The Multiplicative Ergodic Theorem . . . . . . . . . . . . . . . 120 4.3 Abstract Continuity Theorem of the Oseledets Filtration. . . . . . . 141 4.3.1 Continuity of the Most Expanding Direction. . . . . . . . . . 142 4.3.2 Spaces of Measurable Filtrations and Decompositions . . . 149 4.3.3 Continuity of the Oseledets Filtration. . . . . . . . . . . . . . . 153 4.3.4 Continuity of the Oseledets Decomposition. . . . . . . . . . . 154 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5 Large Deviations for Random Cocycles. . . . . . . . . . . . . . . . . . . . . 161 5.1 Introduction and Statements . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.2 The Spectral Method. . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2 An Abstract Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.2.1 The Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.2.2 An Abstract Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.3 The Proof of LDT Estimates. . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.1 Base LDT Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.2 Fiber LDT Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.4 Deriving Continuity of the Lyapunov Exponents. . . . . . . . . . . . 201 5.4.1 Proof of the Continuity . . . . . . . . . . . . . . . . . . . . . . . . 201 5.4.2 Some Generalizations. . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.4.3 Method Limitations. . . . . . . . . . . . . . . . . . . . . . . . . . . 204 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205