Lyapunov Exponents of Linear Cocycles: Continuity via Large Deviations

271 Pages·2016·2.975 MB·English

by  Pedro Duarte, Silvius Klein (auth.)

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
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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