Lectures on Lyapunov exponents

Marcelo VianaIMPA – Rio de Janeiro

v

To Tania, Miguel and Anita,for their understanding.

Contents

Preface pagexi

1 Introduction 11.1 Existence of Lyapunov exponents 11.2 Pinching and twisting 21.3 Continuity of Lyapunov exponents 31.4 Notes 31.5 Exercises 4

2 Linear cocycles 62.1 Examples 7

2.1.1 Products of random matrices 72.1.2 Derivative cocycles 82.1.3 Schrodinger cocycles 9

2.2 Hyperbolic cocycles 102.2.1 Definition and properties 102.2.2 Stability and continuity 142.2.3 Obstructions to hyperbolicity 16

2.3 Notes 182.4 Exercises 19

3 Extremal Lyapunov exponents 203.1 Subadditive ergodic theorem 20

3.1.1 Preparing the proof 213.1.2 Fundamental lemma 233.1.3 Estimatingϕ− 243.1.4 Boundingϕ+ from above 26

3.2 Theorem of Furstenberg and Kesten 283.3 Herman’s formula 293.4 Theorem of Oseledets in dimension 2 30

vii

viii Contents

3.4.1 One-sided theorem 303.4.2 Two-sided theorem 34

3.5 Notes 363.6 Exercises 36

4 Multiplicative ergodic theorem 384.1 Statements 384.2 Proof of the one-sided theorem 40

4.2.1 Constructing the Oseledets flag 404.2.2 Measurability 414.2.3 Time averages of skew products 444.2.4 Applications to linear cocycles 474.2.5 Dimension reduction 484.2.6 Completion of the proof 52

4.3 Proof of the two-sided theorem 534.3.1 Upgrading to a decomposition 534.3.2 Subexponential decay of angles 554.3.3 Consequences of subexponential decay 56

4.4 Two useful constructions 594.4.1 Inducing and Lyapunov exponents 594.4.2 Invariant cones 61

4.5 Notes 634.6 Exercises 64

5 Stationary measures 675.1 Random transformations 675.2 Stationary measures 705.3 Ergodic stationary measures 755.4 Invertible random transformations 77

5.4.1 Lift of an invariant measure 795.4.2 s-states andu-states 81

5.5 Disintegrations ofs-states andu-states 855.5.1 Conditional probabilities 855.5.2 Martingale construction 865.5.3 Remarks on 2-dimensional linear cocycles 89

5.6 Notes 915.7 Exercises 91

6 Exponents and invariant measures 966.1 Representation of Lyapunov exponents 976.2 Furstenberg’s formula 102

6.2.1 Irreducible cocycles 102

Contents ix

6.2.2 Continuity of exponents for irreducible cocycles 1036.3 Theorem of Furstenberg 105

6.3.1 Non-atomic measures 1066.3.2 Convergence to a Dirac mass 1086.3.3 Proof of Theorem 6.11 111

6.4 Notes 1126.5 Exercises 113

7 Invariance principle 1157.1 Statement and proof 1167.2 Entropy is smaller than exponents 117

7.2.1 The volume case 1187.2.2 Proof of Proposition 7.4. 119

7.3 Furstenberg’s criterion 1247.4 Lyapunov exponents of typical cocycles 125

7.4.1 Eigenvalues and eigenspaces 1267.4.2 Proof of Theorem 7.12 128

7.5 Notes 1307.6 Exercises 131

8 Simplicity 1338.1 Pinching and twisting 1338.2 Proof of the simplicity criterion 1348.3 Invariant section 137

8.3.1 Grassmannian structures 1378.3.2 Linear arrangements and the twisting property 1398.3.3 Control of eccentricity 1408.3.4 Convergence of conditional probabilities 143

8.4 Notes 1478.5 Exercises 147

9 Generic cocycles 1509.1 Semi-continuity 1519.2 Theorem of Mane–Bochi 153

9.2.1 Interchanging the Oseledets subspaces 1559.2.2 Coboundary sets 1579.2.3 Proof of Theorem 9.5 1609.2.4 Derivative cocycles and higher dimensions 161

9.3 Holder examples of discontinuity 1649.4 Notes 1689.5 Exercises 169

x Contents

10 Continuity 17110.1 Invariant subspaces 17210.2 Expanding points in projective space 17410.3 Proof of the continuity theorem 17610.4 Couplings and energy 17810.5 Conclusion of the proof 181

10.5.1 Proof of Proposition 10.9 18310.6 Final comments 18610.7 Notes 18910.8 Exercises 189References 191Index 198

Preface

1. The study of characteristic exponents originated from the fundamental workof Aleksandr Mikhailovich Lyapunov [85] on the stability ofsolutions of dif-ferential equations. Consider a linear equation

v(t) = B(t) ·v(t) (1)

whereB(·) is a bounded function fromR to the space ofd×d matrices. By thegeneral theory of differential equations, there exists a so-called fundamentalmatrix At , t ∈R such thatv(t) = At ·v0 is the unique solution of (1) with initialconditionv(0) = v0. If the characteristic exponents

λ (v) = limsupt→∞

1t

log‖At ·v‖ (2)

are negative, for allv 6= 0, then the trivial solutionv(t) ≡ 0 is asymptoticallystable, and even exponentially asymptotically stable. Thestability theorem ofLyapunov asserts that, under an additional regularity condition, stability re-mains valid for nonlinear perturbations

w(t) = B(t) ·w(t)+F(t,w) with ‖F(t,w)‖ ≤ const‖w‖1+ε .

That is, the trivial solutionw(t)≡ 0 is still exponentially asymptotically stable.The regularity condition of Lyapunov means, essentially, that the limit in

(2) does exist, even if one replaces vectorsv by l -vectorsv1 ∧ ·· · ∧ vl ; thatis, elements of thek-exterior power ofRd, for any 0≤ l ≤ d. This is usuallydifficult to check in specific situations. But the multiplicative ergodic theoremof Oseledets asserts that Lyapunov regularity holds with full probability, ingreat generality. In particular, it holds on almost every flow trajectory, relativeto any probability measure invariant under the flow.

2. The work of Furstenberg, Kesten, Oseledets, Kingman, Ledrappier, Guiv-arc’h, Raugi, Gol’dsheid, Margulis and other mathematicians, mostly in the

xi

xii Preface

1960s–80s, built the study of Lyapunov characteristic exponents into a veryactive research field in its own right, and one with an unusually vast array of in-teractions with other areas of Mathematics and Physics, such as stochastic pro-cesses (random matrices and, more generally, random walks on groups), spec-tral theory (Schrodinger-type operators) and smooth dynamics (non-uniformhyperbolicity), to mention just a few.

My own involvement with the subject goes back to the late 20thcenturyand was initially motivated by my work with Christian Bonatti and Jose F.Alves on the ergodic theory of partially hyperbolic diffeomorphisms and, soonafterwards, with Jairo Bochi on the dependence of Lyapunov exponents on theunderlying dynamical system. The way these two projects unfolded very muchinspired the choice of topics in the present book.

3. A diffeomorphismf : M → M is calledpartially hyperbolicif there existsa D f -invariant decomposition

TM = Es⊕Ec⊕Eu

of the tangent bundle such thatEs is uniformly contracted andEu is uniformlyexpanded by the derivativeD f , whereas the behavior ofD f along thecenterbundle Ec lies somewhere in between. It soon became apparent that to improveour understanding of such systems one should try to get a better hold of thebehavior ofD f | Ec and, in particular, of its Lyapunov exponents. In doingthis, we turned to the classical linear theory for inspiration.

That program proved to be very fruitful, as much in the linearcontext (e.g.the proof of the Zorich–Kontsevich conjecture, by Artur Avila and myself) asin the setting of partially hyperbolic dynamics we had in mind originally (e.gthe rigidity results by Artur Avila, Amie Wilkinson and myself), and remainsvery active to date, with important contributions from several mathematicians.

4. Before that, in the early 1980s, Ricardo Mane came to the surprising conclu-sion that generic (a residual subset of) volume-preservingC1 diffeomorphismson any surface have zero Lyapunov exponents, or else they areglobally hyper-bolic (Anosov); in fact, the second alternative is possibleonly if the surface isthe torusT2. This discovery went against the intuition drawn from the classicaltheory of Furstenberg.

Although Mane did not write a complete proof of his findings, his approachwas successfully completed by Bochi almost two decades later. Moreover, theconclusions were extended to arbitrary dimension, both in the volume-preserv-ing and in the symplectic case, by Bochi and myself.

Preface xiii

5. In this monograph I have sought to cover the fundamental aspects of theclassical theory (mostly in Chapters 1 through 6), as well asto introduce someof the more recent developments (Chapters 7 through 10).

The text started from a graduate course that I taught at IMPA during the(southern hemisphere) summer term of 2010. The very first draft consisted oflecture notes taken by Carlos Bocker, Jose Regis Varao and Samuel Feitosa.The unpublished notes [9] and [28], by Artur Avila and Jairo Bochi were im-portant for setting up the first part of the course.

The material was reviewed and expanded later that year, in myseminar,with the help of graduate students and post-docs of IMPA’s Dynamics group.I taught the course again in early 2014, and I took that occasion to add someproofs, to reorganize the exercises and to include historicnotes in each of thechapters. Chapter 10 was completely rewritten and this preface was also muchexpanded.

6. The diagram below describes the logical connections between the ten chap-ters. The first two form an introductory cycle. In Chapter 1 weoffer a glimpseof what is going to come by stating three main results, whose proofs will ap-pear, respectively, in Chapters 3, 6 and 10. In Chapter 2 we introduce the no-tion of linear cocycle, upon which is built the rest of the text. We examine moreclosely the particular case of hyperbolic cocycles, especially in dimension 2,as this will be useful in Chapter 9.

Ch. 1Ch. 2

Ch. 3 Ch. 4 Ch. 5Ch. 6

Ch. 7 Ch. 8 Ch. 9 Ch. 10

In the next four chapters we present the main classical results, includingthe Furstenberg–Kesten theorem and the subadditive ergodic theorem of King-man (Chapter 3), the multiplicative ergodic theorem of Oseledets (Chapter 4),Ledrappier’s exponent representation theorem, Furstenberg’s formula for ex-ponents of irreducible cocycles and Furstenberg’s simplicity theorem in dimen-sion 2 (Chapter 6). The proof of the multiplicative ergodic theorem is based onthe subadditive ergodic theorem and also heralds the connection between Lya-

xiv Preface

punov exponents and invariant/stationary measures that lies at the heart of theresults in Chapter 6. In Chapter 5 we provide general tools todevelop thatconnection, in both the invertible and the non-invertible case.

7. The last four chapters are devoted to more advanced material. The maingoal there is to provide a friendly introduction to the existing research litera-ture. Thus, the emphasis is on transparency rather than generality or complete-ness. This means that, as a rule, we choose to state the results in the simplestpossible (yet relevant) setting, with suitable referencesgiven for stronger state-ments.

Chapter 7 introduces the invariance principle and exploitssome of its con-sequences, in the context of locally constant linear cocycles. This includesFurstenberg’s criterion forλ− = λ+, that extends Furstenberg’s simplicity the-orem to arbitrary dimension. The invariance principle has been used recentlyto analyze much more general dynamical systems, linear and nonlinear, whoseLyapunov exponents vanish. A finer extension of Furstenberg’s theorem ap-pears in Chapter 8, where we present a criterion for simplicity of the wholeLyapunov spectrum.

Then, in Chapter 9, we turn our attention to the contrasting Mane–Bochiphenomenon of systems whose Lyapunov spectra are generically not sim-ple. We prove an instance of the Mane–Bochi theorem, for continuous linearcocycles. Moreover, we explain how those methods can be adapted to con-struct examples of discontinuous dependence of Lyapunov exponents on thecocycle, even in the Holder-continuous category. Having raised the issue of(dis)continuity, in Chapter 10 we prove that for products ofrandom matricesin GL(2) the Lyapunov exponents do depend continuously on the cocycle data.

8. Each chapter ends with set of notes and a list of exercises. Some of the ex-ercises are actually used in the proofs. They should be viewed as an invitationfor the reader to take an active part in the arguments. Throughout, it is assumedthat the reader is familiar with the basic ideas of Measure Theory, DifferentialTopology and Ergodic Theory. All that is needed can be found,for instance,in my book with Krerley Oliveira,Fundamentos da Teoria Ergodica [114]; atranslation into English is under way.

I thank David Tranah, of Cambridge University Press, for hisinterest inthis book and for patiently waiting for the writing to be completed. I am alsograteful to Vaughn Climenhaga, and David himself, for a careful revision ofthe manuscript that very much helped improve the presentation.

Rio de Janeiro, March, 2014Marcelo Viana

1

Introduction

This chapter is a kind of overture. Simplified statements of three theoremsare presented that set the tone for the whole text. Much broader versions ofthese theorems will appear later and several other themes around them will beintroduced and developed as we move on. At this initial stagewe choose tofocus on the following special, yet significant, setting.

LetA1, . . . ,Am be invertible 2×2 real matrices and letp1, . . . , pm be positivenumbers withp1+ · · ·+ pm = 1. Consider

Ln = Ln−1 · · ·L1L0, n≥ 1,

where theL j are independent random variables with identical probability dis-tributions, such that

the probability of{L j = Ai} is equal topi

for all j ≥ 0 and i = 1, . . . ,m. In brief, our goal is to describe the (almostcertain) behavior ofLn asn→ ∞.

1.1 Existence of Lyapunov exponents

We begin with the following seminal result of Furstenberg and Kesten [56]:

Theorem 1.1 There exist real numbersλ+ andλ− such that

limn

1n

log‖Ln‖= λ+ and limn

1n

log‖(Ln)−1‖−1 = λ−

with full probability.

The numbersλ+ andλ− are calledextremal Lyapunov exponents. Clearly,

λ+ ≥ λ− (1.1)

1

2 Introduction

because‖B‖ ≥ ‖B−1‖−1 for any invertible matrixB. If B has determinant±1then we even have‖B‖ ≥ 1≥ ‖B−1‖−1. Hence,

λ+ ≥ 0≥ λ− (1.2)

when all matricesAi , 1≤ i ≤ m have determinant±1.

1.2 Pinching and twisting

Next, we discuss conditions for the inequalities (1.1) and (1.2) to be strict.Let B be themonoidgenerated by the matricesAi , i = 1, . . . ,m; that is, the

set of all productsAk1 · · ·Akn with 1≤ k j ≤ mandn≥ 0 (for n= 0 interpret theproduct to be the identity matrix). We say thatB is pinchingif for any constantκ > 1 there exists someB∈ B such that

‖B‖> κ‖B−1‖−1. (1.3)

This means that the images of the unit circle under the elements of B areellipses with arbitrarily large eccentricity. See Figure 1.1.

B

1

‖B‖‖B−1‖−1

Figure 1.1 Eccentricity and pinching

We say that the monoidB is twisting if given any vector linesF, G1, . . . ,Gn ⊂ R2 there existsB∈ B such that

B(F) /∈ {G1, . . . ,Gn}. (1.4)

The following result is a variation of a theorem of Furstenberg [54]:

Theorem 1.2 AssumeB is pinching and twisting. Thenλ− < λ+. In partic-ular, if |detAi | = 1 for all 1 ≤ i ≤ m then both extremal Lyapunov exponentsare different from zero.

1.3 Continuity of Lyapunov exponents 3

1.3 Continuity of Lyapunov exponents

The extremal Lyapunov exponentsλ+ andλ− may be viewed as functions ofthe data

A1, . . . ,Am, p1, . . . , pm.

Let the matricesA j vary in the linear group GL(2) of invertible 2×2 matricesand the probability vectors(p1, . . . , pm) vary in the open simplex

∆m = {(p1, . . . , pm) : p1 > 0, . . . , pm > 0 andp1+ · · ·+ pm= 1}.

The following result is part of a theorem of Bocker and Viana [35]:

Theorem 1.3 The extremal Lyapunov exponentsλ± depend continuously on(A1, . . . ,Am, p1, . . . , pm) ∈ GL(2)m×∆m at all points.

Example 1.4 Let m= 2, with

A1 =

(

σ 00 σ−1

)

and A2 = Rθ A1R−θ , Rθ =

(

cosθ −sinθsinθ cosθ

)

for someσ > 1 andθ ∈ R. By Theorem 1.3, the Lyapunov exponentsλ±

depend continuously on the parameterσ andθ . Moreover, using Theorem 1.2,we haveλ+ = 0 if and only if p1 = p2 = 1/2 andθ = π/2+nπ for somen∈Z.

1.4 Notes

Theorem 1.1 is a special case of the theorem of Furstenberg and Kesten [56],which is valid in any dimensiond ≥ 2. The full statement and the proof willappear in Chapter 3: we will deduce this theorem from an even more generalstatement, the subadditive ergodic theorem of Kingman [74]. Kingman’s theo-rem will also be used in Chapter 4 to prove the fundamental result of the theoryof Lyapunov exponents, the multiplicative ergodic theoremof Oseledets [92].

It is natural to ask whether the type of asymptotic behavior prescribed byTheorem 1.1 for the norm‖Ln‖ and conorm‖(Ln)−1‖−1 extends to the indi-vidual matrix coefficients Lni, j . Furstenberg, Kesten [56] proved that this is soif the coefficients of the matricesAi , 1≤ i ≤ mare all strictly positive. The ex-ample in Exercise 1.3 shows that this assumption cannot be removed. On theother hand, the theorem of Oseledets theorem does contain such a descriptionfor the matrixcolumn vectors.

Theorem 1.2 is also the tip of a series of fundamental results, which are tobe discussed in Chapters 6 through 8. The full statement and proof of Fursten-berg’s theorem for 2-dimensional cocycles (Furstenberg [54]) will be given in

4 Introduction

Chapter 6. The extension to any dimension will be stated and proved in Chap-ter 7: it will be deduced from the invariance principle (Ledrappier [81], Bon-atti, Gomez-Mont and Viana [37], Avila and Viana [16], Avila, Santamaria andViana [13]), a general tool that has several other applications, both for linearand nonlinear systems.

In dimension larger than 2, there is a more ambitious problem: rather thanasking whenλ− < λ+, one wants to know whenall the Lyapunov exponentsare distinct. That will be the subject of Chapter 8, which is based on Avila andViana [14, 15].

Furstenberg and Kifer [57] proved continuity of the Lyapunov exponents ofproducts of random matrices, restricted to the (almost) irreducible case. A vari-ation of their argument will be given in Section 6.2.2. The reducible case re-quires a delicate analysis of the random walk defined by the cocycle in projec-tive space. That was carried out by Bocker and Viana [35], in the 2-dimensionalcase, using certain discretizations of projective space. At the time of writing,Avila, Eskin and Viana [12] are extending the statement of the theorem to arbi-trary dimension, using a very different strategy. The proofof Theorem 1.3 thatwe present in Chapter 10 is based on this more recent approach.

The problem of the dependence of Lyapunov exponents on the data canbe formulated in the broader context of linear cocycles thatwe are going tointroduce in Chapter 2. We will see in Chapter 9 that, in contrast, continuityoften breaks down in that generality.

1.5 Exercises

The following elementary notions are used in some of the exercises that fol-low. We call a 2×2 matrixhyperbolicif it has two distinct real eigenvalues,parabolicif it has a unique real eigenvalue, with a one-dimensional eigenspace,andelliptic if it has two distinct complex eigenvalues. Multiples of theidentitybelong to neither of these three classes.

Exercise 1.1 Show that, in dimensiond = 2, if |detAi |= 1 for all 1≤ i ≤ mthenλ++λ− = 0.

Exercise 1.2 Calculate the extremal Lyapunov exponents form= 2 andp1,p2 > 0 with p1+ p2 = 1 and

(1) A1 =

(

σ 00 σ−1

)

andA2 =

(

σ−1 00 σ

)

, whereσ > 1;

1.5 Exercises 5

(2) A1 =

(

σ 00 σ−1

)

andA2 =

(

0 −11 0

)

, whereσ > 1.

Exercise 1.3 (Furstenberg and Kesten [56]) Takem= 2 with p1 = p2 = 1/2and

A1 =

(

2 00 1

)

and A2 =

(

0 11 0

)

.

Show that limn(1/n) log|Lni, j | does not exist for anyi, j, with full probability.

Exercise 1.4 Show that if some matrixAi , 1≤ i ≤ m is either hyperbolic orparabolic then the monoidB is pinching.

Exercise 1.5 Show that the monoidB may be pinching even if all the matri-cesAi , 1≤ i ≤ mare elliptic.

Exercise 1.6 Suppose that there exists 1≤ i ≤ m such thatAi is conjugate toan irrational rotation. Conclude thatB is twisting.

Exercise 1.7 Suppose that there exist 1≤ i, j ≤ m such thatAi andA j areeither hyperbolic or parabolic and that they have no common eigenspace. Con-clude thatB is twisting (and pinching).

Exercise 1.8 Let Ai , i = 1,2 be as in the second part of Exercise 1.2. Checkthat

λ+(A1,A2,1,0) 6= limp2→0

λ+(A1,A2,1− p2, p2).

Thus, the hypothesisp1 > 0, . . . , pm > 0 cannot be removed in Theorem 1.3.

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Index

∗

convolution, 174, 179Bt

adjoint linear operator, 107C(V,a)

cone of widtha aroundV, 61C0(P)

space of continuous functions, 44D f

derivative of a smooth map, 8Es, Eu

stable and unstable bundles, 10, 57Eδ (ξ ), eδ (η)

δ -energy of a measure, 178, 179Ev, Ex

slices of a set in a product space, 67, 91G(d), G(k,d)

Grassmannian, 16, 38L1(µ)

space of integrable functions, 28Rθ

rotation, 3V⊥

xorthogonal complement, 48

Ws(x), Wu(x)(global) stable and unstable sets, 187

Wsloc(x), Wu

loc(x)local stable and unstable sets, 187

∆m

open simplex, 3∆r

r-neighborhood of the diagonal, 180Λl E

exterior l-power, 57Λl

dEdecomposablel-vectors, 57

GL(d), GL(d,C)linear group, 6

PRd, PCd

projective space, real and complex, 17SL(d), SL(d,C)

special linear group, 6, 7Td

d-dimensional torus, 17B(Y)

Borel σ -algebra, 41G (d)

space of compactly supported measures onGL(d), 171

K (Y)space of compact subsets, 41

M (µ)space of measures projecting down toµ , 44

P(d)space of probability measures inPRd, 174

ecc(B)eccentricity of a linear map, 133

ℓ2

space of square integrable sequences, 9λ(x,v)

Lyapunov exponent function, 40λ+, λ−

extremal Lyapunov exponents, 1, 6PF

projective cocycle, 68projw u

orthogonal projection, 175α

generic element of GL(d)N, 172ϕ+, ϕ−

positive and negative parts of a function, 20p∗η , p1 ∗ p2

14

Index 15

convolution, 174, 179sn(x)

most contracted vector, 31un(x)

most expanded vector, 31

δ -energy of a measure, 178, 179ε-concentrated measure, 144C0 topology, 153C1 topology, 161l-vector, 57

decomposable, 57L∞ cocycle, 168Lp cocycle, 168p-expanding point, 175p-invariant point, 175s-state, 81, 83, 86, 94, 149su-state, 81u-state, 81, 83, 86, 94, 149

additive sequence, 21adjoint

cocycle, 108monoid, 148operator, 69, 107

Anderson localization, 18angle between subspaces, 39Anosov diffeomorphism, 17, 161aperiodic measure, 153, 155, 158

backward stationary measure, 78, 101Baire space, 152barycenter, 130Bernoulli shift, 7

characteristic exponents, xicoboundary set, 158, 159, 170cocycle

L∞, 168Lp, 168continuous, 154, 168derivative, 161fiber-bunched, 187Grassmannian, 137Holder continuous, 165, 187inverse, 146invertible, 34, 39irreducible, 102locally constant, 7projective, 68, 92, 137strongly irreducible, 102, 103

cohomological condition, 105complete

flag, 38

probability space, 38conditional

expectation, 86probabilities, 85

cone, 61, 65conformal barycenter, 130conorm, 20continuity theorem, 3, 171contraction property, 189convergence

pointwise, 74weak∗ , 74

convolution, 174, 179, 189diagonal, 183

coupling, 178cross-ratio, 61cylinder, 82, 87, 89, 170

decomposablel-vector, 57derivative cocycle, 8, 161determinant, 58diagonal

convolution, 183push-forward, 183

dimension of a vector bundle, 8disintegration, 85, 87dominated decomposition, 65, 162, 163duality, 69

eccentricity of a linear map, 133elliptic matrix, 4energy of a measure, 178, 179ergodic

decomposition, 95decomposition theorem, 77measure, 159, 167stationary measure, 76theorem of Birkhoff, 27

essentiallyinvariant function, 21, 36invariant set, 21

expectation, 86exponents representation theorem, 97exterior

power, 57, 151product, 57, 137

extremalLyapunov exponents, 1measure, 92

fiber-bunched cocycle, 187fibered

entropy, 117

16 Index

Jacobian, 117first return

map, 59, 158time, 59

flag, 38complete, 38of Oseledets, 39, 41

formulaof Furstenberg, 102of Herman, 29

forward stationary measure, 78, 101Furstenberg’s

criterion, 2, 105, 124formula, 102

geometrichyperplane, 137subspace, 137

graph transform, 53Grassmannian, 16, 38

cocycle, 137manifold, 16, 38, 57

Holdercontinuity, 164norm, 164

Hamiltonian operator, 18Hausdorff distance, 190Herman’s formula, 29Hilbert metric, 61holonomy

stable, 188unstable, 188

homogeneous measure, 143hyperbolic

cocycle, 17, 153linear cocycle, 10, 15matrix, 4

hyperbolicitynon-uniform, 19uniform, 10, 15

hyperplane, 137section, 134, 137

inducing, 59, 166interchanging subspaces, 155, 160, 166, 167invariance principle, 115, 116invariant

function, 21holonomies, 187measure, 70, 79, 97set, 21

inverse

cocycle, 146monoid, 139

invertible cocycle, 34, 39invertible extension, 99, 101irreducibility, 102

strong, 102, 134irreducible cocycle, 102, 103

strongly, 102

Lebesgue space, 158lift

of a stationary measure, 81, 88, 89of an invariant measure, 79–81

lineararrangement, 137, 139, 143

proper, 139cocycle, 6

hyperbolic, 10, 15invertible extension, 101on a vector bundle, 8

derivative cocycle, 8group, 6

special, 6, 7invariance principle, 116Schrodinger cocycle, 9section, 137

localstable set, 187unstable set, 187

local product structure, 91locally constant

cocycle, 7, 96skew product, 67

Lyapunovexponents, 39, 40, 53, 60

extremal, 1spectrum, 39, 133

martingale, 86, 88convergence theorem, 86

mass of a measure, 178matrix

elliptic, 4hyperbolic, 4parabolic, 4

meager subset, 152measurable

section, 99sub-bundle, 64, 98

measurebackward stationary, 78ergodic, 76extremal, 92

Index 17

forward stationary, 78invariant, 70non-atomic, 102, 106stationary, 70

monoid, 2, 134most contracted

subspace, 134, 140vector, 11, 19, 31

most expandedsubspace, 134, 140vector, 11, 19, 31

multiplicative ergodic theoremone-sided, 30, 38two-sided, 34, 39

multiplicity of a Lyapunov exponent, 39, 133

natural extension, 65, 99negative part of a function, 20non-atomic measure, 102, 106non-compactness condition, 105, 132, 134normalized restriction, 59, 113

open simplex, 3orbital sum, 21orthogonal complement, 48Oseledets

decomposition, 34, 39, 60, 97flag, 39, 41, 53, 97sub-bundles, 39subspaces, 39, 40

parabolic matrix, 4parallelizable manifold, 8pinching, 2, 113, 134, 139, 148point

p-expanding, 175p-invariant, 175

pointwiseconvergence, 74topology, 72

positive part of a function, 20potential, 9probabilistic repeller, 174probability space

complete, 38separable, 38

projection theorem, 41projective

action, 106cocycle, 68, 92, 137space, 17

proper linear arrangement, 139

quasi-periodic Schrodinger cocycle, 9

Radon–Nikodym derivative, 117random

matrices, 7, 68Schrodinger cocycle, 9transformation, 67, 97transformation (invertible), 77transformation (one-sided), 70, 76

residual subset, 152, 153, 168rotation, 3

s-holonomy, 188Schrodinger

cocycle, 9quasi-periodic, 9random, 9

operator, 9second return map, 154, 158, 167self-coupling, 179

optimal, 179symmetric, 179

semi-continuity, 151, 152, 160separable

metric space, 41probability space, 38

sequenceadditive, 21subadditive, 21super-additive, 45

sequential compactness, 83simple Lyapunov spectrum, 39, 133simplicity criterion, 134singular

measures, 117values, 133

slices of a set in a product space, 68small subset, 126special linear group, 6, 7stabilizer of a measure, 108, 131stable

bundle, 10, 57, 135holonomy, 188set, 187

stationaryfunction, 75measure, 70, 101, 115, 171set, 75

strong irreducibility, 102, 103, 105, 134subadditive

ergodic theorem, 20, 21sequence, 21

subexponential decay, 31, 34, 35, 39subharmonic function, 29, 30

18 Index

superadditive sequence, 45support of a measure, 105symmetric self-coupling, 179

optimal, 179symplectic

diffeomorphism, 163form, 163

theoreminvariance principle, 116martingale convergence, 86, 87of Baire, 169of Birkhoff, 27of continuity, 3, 171of ergodic decomposition, 77of Furstenberg, 2, 105of Furstenberg (formula), 102of Furstenberg-Kesten, 1, 20, 28of Herman (formula), 29of Kingman, 21of Kingman (subadditive), 20of Ledrappier, 97, 116of Mane–Bochi, 153of Mane-Bochi, 161of Oseledets (one-sided), 30, 38of Oseledets (two-sided), 34, 39of Prohorov, 83of Rokhlin, 85of simplicity, 134of Thieullen, 106projection, 41subadditive, 21

topological degree, 17topology

C0, 153C1, 161pointwise, 72uniform, 72weak∗ , 72

total variation norm, 72transition

operator, 68operator (adjoint), 69probabilities, 68

transitive action, 162twisting, 2, 113, 134, 139, 148

u-holonomy, 188uniform

convergence, 83hyperbolicity, 10, 15integrability, 103topology, 72

unstablebundle, 10, 57, 135holonomy, 188set, 187

vectorbundle, 8

dimension, 8most contracted, 31most expanded, 31

volumeof a decomposablel-vector, 58

weak∗

convergence, 74, 83topology, 72