Selected Title s i n Thi s Serie s

23 Lui s Barreir a an d Yako v B . Pesin , Lyapuno v exponent s an d smoot h ergodi c theory , 2002

22 Yve s Meyer , Oscillatin g pattern s i n imag e processin g an d nonlinea r evolutio n equations ,

2001

21 Bojk o Bakalo v an d Alexande r Kirillov , Jr. , Lecture s o n tenso r categorie s an d

modular functors , 200 1

20 Aliso n M . Etheridge , A n introductio n t o superprocesses , 200 0

19 R . A . Minlos , Introductio n t o mathematica l statistica l physics , 200 0

18 Hirak u Nakajima , Lecture s o n Hilber t scheme s o f points o n surfaces , 199 9

17 Marce l Berger , Riemannia n geometr y durin g th e secon d hal f o f the twentiet h century ,

2000

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adic group s (wit h

notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w examples o f Frobenius extensions , 199 9

13 Yako v M . Eliashber g an d Willia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 Stephe n Gelbart , Lecture s o n the Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 And y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf f an d Dietma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plane curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a round dro p become s a curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0 1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometry t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/023

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Lyapunov Exponent s an d Smooth Ergodi c Theor y

University

LECTURE Series

Volume 2 3

Lyapunov Exponent s an d Smooth Ergodi c Theor y

Luis Barreir a Yakov B . Pesi n

American Mathematica l Societ y Providence, Rhod e Islan d

Editorial Boar d

Jerry L . Bon a (Chair ) Nige l J . Hitchi n Jean-Luc Brylinsk i Nicola i Reshetikhi n

2000 Mathematics Subject Classification. Primar y 37D25 , 37C40 .

ABSTRACT. Thi s boo k provide s a systemati c introductio n t o smoot h ergodi c theory , includin g the genera l theor y o f Lyapuno v exponents , nonunifor m hyperboli c theory , stabl e manifol d the -ory emphasizin g absolut e continuit y o f invarian t foliations , an d th e ergodi c theor y o f dynamica l systems wit h nonzer o Lyapuno v exponents . Th e boo k ca n b e use d a s a primar y textboo k fo r a special topic s cours e o n nonunifor m hyperbolicit y o r a s supplementar y readin g fo r a basi c cours e on dynamica l systems .

Library o f Congres s Cataloging-in-Publicatio n D a t a

Barreira, Luis , 1968 -Lyapunov exponent s an d smoot h ergodi c theor y / Lui s Barreir a an d Yako v B . Pesin .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 23 ) Includes bibliographica l reference s an d index . ISBN 0-8218-2921- 1 (alk . paper ) 1. Ergodi c theory . 2 . Lyapuno v exponents . I . Pesin , Ya . B . II . Title . III . Universit y lec -

ture serie s (Providence , R.I. ) ; 23.

QA611.5.B37 200 1 515'.42—dc21 200104588 2

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

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except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t URL : http://www.ams.org /

10 9 8 7 6 5 4 3 2 1 0 7 0 6 0 5 0 4 0 3 0 2

To m y parent s an d m y sister , Luis Barreir a

To m y wif e Natasha , Yasha Pesi n

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Contents

Preface x i

Introduction 1

Chapter 1 . Lyapuno v Stabilit y Theor y o f Differentia l Equation s 5 1.1. Lyapuno v Exponent s fo r Differentia l Equation s 6 1.2. Abstrac t Theor y o f Lyapuno v Exponent s 9 1.3. Forwar d an d Backwar d Regularit y 1 6 1.4. Stabilit y Theor y o f Nonautonomous Differentia l Equation s 2 6 1.5. Lyapuno v Regularit y an d th e Oseledet s Decompositio n 3 1

Chapter 2 . Element s o f Nonuniform Hyperboli c Theor y 3 5 2.1. Dynamica l System s wit h Nonzer o Lyapuno v Exponent s 3 6 2.2. Nonunifor m Hyperbolicit y an d Regula r Set s 4 5 2.3. Holde r Continuit y o f Invarian t Distribution s 4 8 2.4. Proo f o f the Multiplicativ e Ergodi c Theore m 5 1

Chapter 3 . Example s o f Nonuniformly Hyperboli c System s 6 1 3.1. Anoso v Diffeomorphism s 6 1 3.2. Diffeomorphism s wit h Nonzer o Lyapuno v Exponent s o n

Surfaces 6 6 3.3. A Flow wit h Nonzer o Lyapuno v Exponent s 7 1 3.4. Geodesi c Flow s o n Compac t Manifold s o f Nonpositiv e

Curvature 7 4

Chapter 4 . Loca l Manifol d Theor y 8 1 4.1. Existenc e o f Loca l Stabl e Manifold s 8 1 4.2. Basi c Propertie s o f Stabl e an d Unstabl e Manifold s 9 4 4.3. Absolut e Continuit y Propert y 9 9 4.4. Computin g th e Jacobia n o f the Holonom y Ma p 10 9 4.5. Partia l Hyperbolicit y 11 1

Chapter 5 . Ergodi c Propertie s o f Smoot h Hyperboli c Measure s 11 5 5.1. Absolut e Continuit y an d Smoot h Invarian t Measure s 11 5 5.2. Ergodicit y o f Smo 5.1. Absolute Continuity and Smooth Invariant Measuresoth Hyperboli c Measure s 11 7 5.3. Loca l Ergodicit y 12 2

IX

x CONTENT S

5.4. Th e Entrop y Formul a 13 0 5.5. SRB-Measure s an d Genera l Hyperboli c Measure s 13 8 5.6. Geodesi c Flow s on Compac t Surface s o f Nonpositiv e

Curvature 14 0

Bibliography 14 5

Index 147

Preface

This boo k provide s a systemati c introductio n t o th e cor e o f smooth er -godic theory . Despit e a n impressiv e amoun t o f literatur e i n th e fiel d ther e is n o textboo k whic h contain s a sufficientl y complet e presentatio n o f th e theory. Thi s boo k attempt s t o fill in thi s gap . W e describe th e genera l (ab -stract) theor y o f Lyapuno v exponent s an d it s application s t o th e stabilit y theory o f differentia l equations , th e stabl e manifol d theory , absolut e conti -nuity o f stable manifolds , an d th e ergodic theory o f dynamical system s wit h nonzero Lyapuno v exponent s (includin g geodesi c flows).

The book i s a revised an d considerabl y expande d versio n o f our Lectures on Lyapunov Exponents and Smooth Ergodic Theory [4] . W e ad d mor e ex -amples o f dynamica l system s wit h nonzer o Lyapuno v exponents , includin g diffeomorphisms o n two-dimensiona l tor i an d o n spheres . Furthermore , w e substantially expan d th e expositio n o f the crucia l absolut e continuit y prop -erty. I n particular, we include an example of a foliation tha t i s not absolutel y continuous and establis h the formula fo r the Jacobian o f the holonomy map . We also add a complete proof o f the Multiplicativ e Ergodi c Theore m a s well as provide mor e detail s i n the proof s o f severa l basi c results . Finally , a fe w more figures ar e adde d t o illustrat e th e exposition .

We hop e tha t thes e improvement s mak e th e boo k mor e accessibl e t o graduate student s o r anyon e wh o wishes t o acquir e a working knowledg e of smooth ergodi c theor y an d t o lear n ho w t o us e it s tools . Indeed , th e boo k can b e use d a s a primar y textboo k fo r a specia l topic s cours e o n nonuni -form hyperboli c theor y o r a s supplementar y readin g fo r a basi c cours e o n dynamical systems .

This boo k i s self-contained. W e only assum e tha t th e reade r ha s a basi c knowledge of real analysis, measure theory, differentia l equations , and topol -ogy. W e presen t th e basi c concept s o f smoot h ergodi c theor y an d provid e complete proo f o f al l mai n results . W e also state som e result s whos e proof s require mor e advance d technique s whic h excee d th e scop e o f th e book . I n our opinio n thi s give s th e reade r a broade r vie w o f smoot h ergodi c theor y and ma y hel p stimulat e furthe r study . Thi s wil l als o provid e nonexpert s with a broade r perspectiv e o f the field.

While writin g thi s boo k w e consulte d wit h Anatol e Kato k o n severa l topics an d w e woul d lik e t o than k hi m fo r hi s man y valuabl e comments .

xi

xii PREFAC E

We also would lik e to than k Mish a Brin , Bori s Hasselblatt , Jor g Schmeling , and Howi e Weiss for man y usefu l remark s o n mathematica l structure , style , references, etc . I t i s als o ou r pleasur e t o than k Hi e Ugarcovici an d Alistai r Windsor, graduat e student s a t Pen n State , wh o rea d th e tex t thoroughl y and helpe d u s correct som e typos an d mistake s an d improv e the exposition . We especially than k Natash a Pesin , a n experience d editor , fo r he r editoria l assistance.

August 200 1

Luis Barreir a Yako v B . Pesi n Lisboa, Portuga l Stat e College , PA , US A

Bibliography

D. Anosov , Tangential fields of transversal foliations in Y-systems, Math . Note s 2 (1967), no . 5 , 818-823 . D. Anosov , Geodesic flows on closed Riemann manifolds with negative curvature, Proc. Steklo v Inst . Math . 9 0 (1969) , 1-235 . D. Anoso v an d Ya . Sinai , Certain smooth ergodic systems, Russia n Math . Survey s 2 2 (1967), no . 5 , 103-167 . L. Barreira and Ya . Pesin, Lectures on Lyapunov exponents and smooth ergodic theory, in Smooth ergodic theory and its applications, A . Katok , R . d e l a Llave , Ya . Pesi n and H . Weis s eds. , Proc . Symp . Pur e Math. , Amer . Math . Soc , 2001 . L. Barreira , Ya . Pesi n an d J . Schmeling , Dimension and product structure of hyper-bolic measures, Ann . o f Math . (2 ) 14 9 (1999) , 755-783 . L. Barreira an d J . Schmeling , Sets of "non-typical" points have full topological entropy and full Hausdorff dimension, Israe l J . Math . 11 6 (2000) , 29-70 . J. Bochi , Genericity of zero Lyapunov exponents, preprint , 2001 . M. Brin , Holder continuity of invariant distributions, i n Smooth ergodic theory and its applications, A . Katok , R . d e l a Llave , Ya . Pesi n an d H . Weis s eds. , Proc . Symp . Pure Math. , Amer . Math . Soc , 2001 . M. Bri n an d Ya . Pesin , Partially hyperbolic dynamical systems, Math . USSR-Izv . 8 (1974), 177-218 . K. Burns , C . Pugh , M . Shub , A . Wilkinson , Recent results about stable ergodicity, i n Smooth ergodic theory and its applications, A . Katok , R . d e l a Llave , Ya . Pesi n an d H. Weis s eds. , Proc . Symp . Pur e Math. , Amer . Math . Soc , 2001 . D. Bylov , R . Vinograd , D . Grobman an d V . Nemyckii , Theory of Lyapunov exponents and its application to problems of stability, Izdat . "Nauka" , Moscow , 1966 , in Russian . D. Dolgopyat , H . H u an d Ya . Pesin , An example of a smooth hyperbolic measure with countably many ergodic components, i n Smooth ergodic theory and its applications, A. Katok , R . d e l a Llave , Ya . Pesi n an d H . Weis s eds. , Proc . Symp . Pur e Math. , Amer. Math . Soc , 2001 . D. Dolgopya t an d Ya . Pesin , On the existence of Bernoulli diffeomorphisms with nonzero Lyapunov exponents on compact smooth manifolds, preprint , 2001 . P. Eberlein , Geodesic flows on negatively curved manifolds I, Ann . o f Math . (2 ) 9 5 (1972), 492-510 . P. Eberlein , When is a geodesic flow of Anosov type? I, J . Differentia l Geom . 8 (1973), 437-463 ; II, J . Differentia l Geom . 8 (1973) , 565-577 . P. Eberlein , Geodesic flows in manifolds of nonpositive curvature, i n Smooth ergodic theory and its applications, A . Katok , R . d e l a Llave , Ya . Pesi n an d H . Weis s eds. , Proc Symp . Pur e Math. , Amer . Math . Soc , 2001 . B. Hasselblatt , Regularity of the Anosov splitting and of horospheric foliations, Er -godic Theor y Dynam . System s 1 4 (1994) , 645-666 . G. Hedlund , The dynamics of geodesic flows, Bull . Amer. Math . So c 4 5 (1939) , no. 4, 241-260. E. Hopf, Statistik der geoddtischen Linien in Mannigfaltigkeiten negativer Krummung, Ber. Verh . Sachs . Akad . Wiss . Leipzi g 9 1 (1939) , 261-304 .

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H. Hu , Statistical properties of some almost hyperbolic systems, i n Smooth ergodic theory and its applications, A . Katok , R . d e l a Llave , Ya . Pesi n an d H . Weis s eds. , Proc. Symp . Pur e Math. , Amer . Math . Soc , 2001 . A. Katok , Bernoulli dijfeomorphism on surfaces, Ann . o f Math . (2 ) 11 0 (1979) , 529 -547. A. Katok an d K. Burns, Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems, Ergodi c Theor y Dynam . System s 14 (1994) , 757-785 . A. Kato k an d L . Mendoza , Dynamical systems with nonuniformly hyperbolic behav-ior, i n Introduction to the modern theory of dynamical systems b y A . Kato k an d B. Hasselblatt , Cambridg e Universit y Press , 1995 . F. Ledrappie r an d J.-M . Strelcyn , A proof of the estimate from below in Pesin's entropy formula, Ergodi c Theor y Dynam . System s 2 (1982) , 203-219 . F. Ledrappie r an d L.-S . Young, The metric entropy of diffeomorphisms. L Character-ization of measures satisfying Pesin's entropy formula, Ann . o f Math. (2 ) 12 2 (1985) , 509-539. A. Lyapunov , The general problem of the stability of motion, Taylo r & ; Francis, 1992 . I. Malkin , A theorem on stability via the first approximation, Dokladi , Akademi i Nau k USSR 7 6 (1951) , no . 6 , 783-784 . J. Milnor , Fubini foiled: Katok's paradoxical example in measure theory, Math . Intel -ligencer 1 9 (1997) , 30-32 . V. Oseledets , A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems, Trans . Mosco w Math . Soc . 1 9 (1968) , 197-221 . O. Perron , Die Ordnungszahlen linearer Differentialgleichungssyteme, Math . Zs . 3 1 (1930), 748-766 . Ya. Pesin , An example of a nonergodic flow with nonzero characteristic exponents, Func. Anal , an d it s Appl . 8 (1974) , no . 3 , 263-264 . Ya. Pesin , Families of invariant manifolds corresponding to nonzero characteristic exponents, Math . USSR-Izv . 4 0 (1976) , no . 6 , 1261-1305 . Ya. Pesin , Characteristic Ljapunov exponents, and smooth ergodic theory, Russia n Math. Survey s 3 2 (1977) , no . 4 , 55-114 . Ya. Pesin, Geodesic flows on closed Riemannian manifolds without focal points, Math . USSR-Izv. 1 1 (1977) , no . 6 , 1195-1228 . Ya. Pesin , Geodesic flows with hyperbolic behaviour of the trajectories and objects connected with them, Russia n Math . Survey s 3 6 (1981) , no . 4 , 1-59 . Ya. Pesin , Dimension theory in dynamical systems: contemporary views and applica-tions, Chicag o Lecture s i n Mathematic s Series , Universit y o f Chicag o Press , 1997 . C. Pugh , The C l+OL hypothesis in Pesin theory, Inst . Haute s Etude s Sci . Publ . Math . 59 (1984) , 143-161 . D. Ruelle , An inequality for the entropy of differentiable maps, Bol . Soc . Brasil . Mat . 9 (1978) , no . 1 , 83-87 . Ya. Sinai , Dynamical systems with countably-multiple Lebesgue spectrum II, Amer . Math. Soc . Trans . (2 ) 6 8 (1966) , 34-88 . S. Smale , Differentiable dynamical systems, Bull . Amer . Math . Soc . 7 3 (1967) , 747 -817.

Index

A, 7 7 A(W\W2), 10 1 r m ( t ) , 1 7 rv

m(t), 1 7 A, 4 0 Ai, 11 7 A?', 12 1 X+W, 6 X+(z,v) , 3 6 X+, 31 , 37 X~(v),31 X~(x,t;),37 x r . 3 1 , 3 7 7(X,X), 1 4 A-lemma, 9 6 X(/)L17 TT(X,X)> 1 4

Tr(y), 10 1 TTI(M), 14 0

7Tfc(y), 11 4

A, 4 0 X + H , 1 6 X ~ H , 3 1 x ( A 1 7

A+( t ) , 7 6 i4~(t), 7 6 absolute continuity , 99 , 11 5

theorem, 102 , 11 3 absolutely continuou s

measure, 10 1 transformation, 10 1

Anosov diffeomorphism, 46 , 61 , 62 flow, 6 5

asymptotic geodesies , 14 1 automorphism

Bernoulli - , 12 1 hyperbolic tora l - , 6 2

backward regular

Lyapunov exponent , 3 1 point, 3 7

regularity, 1 6 bases, dua l - , 1 4 basis

construction o f norma l - , 1 2 normal - , 1 1 ordered - , 1 1

Bernoulli automorphism, 12 1 flow, 12 2

Besicovich coverin g lemma , 10 3

C s(x), 12 8 Cu(x), 12 8 canonical metric , 7 5 Cauchy matrix , 2 6 characteristic exponent , see Lyapuno v

exponent chart, foliatio n coordinat e - , 9 9 coefficient

Perron - , 1 4 regularity - , 1 4

cohomologous functions , 6 4 compliance o f nitrations , 3 2 components, ergodi c - , 11 7 conditional entropy , 13 0 coordinate chart , foliatio n - , 9 9

((5, g)-foliation, 12 3 with smoot h leaves , 12 3

dv(x), 13 8 du(x), 13 8 d'f, 3 8 d*f, 3 8 d c i (Vi , 1*0,95 decomposition, Oseledet s - , 31 , 32, 3 9 diffeomorphism

147

148 INDEX

Anosov-, 46 , 61 , 62 Lyapunov exponen t specifie d b y - , 3 6 nonuniformly partiall y

hyperbolic - , 11 3 structurally stabl e - , 6 4 uniformly partiall y hyperboli c - , 1 1 1 with nonzer o Lyapuno v exponents , 6 6

differential equatio n Lyapunov exponen t fo r - , 6 Lyapunov stabilit y theor y o f - , 5 stability theor y o f nonautonomou s - ,

26 direct product , 11 2 d i s t ( A £ ) , 4 9 dist(v,A), 4 8 distribution, 4 9

Holder continuou s - , 4 9 dual

bases, 1 4 Lyapunov exponents , 1 4

dynamical syste m nonuniformly hyperboli c - , 4 8 with nonzero Lyapunov exponents , 36 ,

40

E?(x), E+(v), E~(v), Es(x), Eu(x), Ei{x),

39 76 76 40 40 38

entropy, 13 0

of measurabl e partition , 13 0 ergodic

components, 11 7 measure, 39 , 11 7 properties, 11 5 smooth - theory , 1 , 115 , 117 , 13 0

ergodicity, 11 7 local - , 12 2 of smoot h hyperboli c measures , 11 7

exact dimensional measure , 13 8 Lyapunov exponent , 2 6

exponent, see Lyapuno v exponen t

7,18 / , 1 8 filtration, 1 0

associated t o Lyapuno v exponent , 1 1 of loca l manifolds , 11 4

nitrations, complianc e o f - , 3 2

flat stri p theorem , 14 1 flow, 36

Anosov - , 6 5 Bernoulli - , 12 2 frame - ,11 2 geodesic - , 74 , 14 0 time-t ma p of- , 11 2 uniformly partiall y hyperboli c - , 11 1 with nonzer o Lyapuno v exponents , 7 1

foliation, 9 9 (*,</)-, 12 3 (5, q)— with smoot h leaves , 12 3 coordinate chart , 9 9 nonabsolutely continuous , 10 8 smooth - , 10 0 with smoot h leaves , 99 , 12 3

forward regular

Lyapunov exponent , 26 , 3 1 point, 3 7

regularity, 9 , 1 6 frame flow, 11 2 function, tempere d - , 4 2 functions, cohomologou s - , 6 4

G s 2 , 7 0 Gj2, 6 8 geodesic flow, 74 , 14 0 geodesies, asymptoti c - , 14 1 global

leaf, 99 , 12 3 stable manifold , 62 , 98 , 9 9 unstable manifold , 62 , 98 , 9 9 weakly

stable manifold , 9 9 unstable manifold , 9 9

graph transfor m property , 9 7

# M ( 0 , 13 0 tf/x(£IC), 13 0 K(T), 13 0

Mr,o, 13° Holder continuou s distribution , 4 9 Hamiltonian, 6 9

with respec t t o th e area , 6 9 holonomy map , 101 , 114 horocycle, 14 2 hyperbolic

measure, 4 0 nonuniformly - set , 45 , 47 uniformly - set , 4 6

hyperbolic theory , nonunifor m - , 3 5

ideal boundary , 14 1

INDEX 149

implicit functio n theorem , 8 9 inclination lemma , 9 6 infinitesimal eventuall y

strict Lyapuno v function , 12 8 uniform Lyapuno v function , 12 9

inner product , Lyapuno v - , 8 3 invariant distributions , Holde r continu -

ity o f - , 4 8

J(T)(x), 10 2 Js(7r)(y), 10 2 Jacobi equation , 7 5 Jacobian, 10 1

Kx(vuv2), 7 4 fc+, 31 *+(*), 3 7 fcr, 3 1

fcr(*),37

£(x) , 10 1 £k(x), 11 4 leaf

global - , 99 , 12 3 local - , 99 , 12 3

limit solution

negative - , 7 6 positive - , 7 6

linear filtration , see filtratio n local

ergodicity, 12 2 leaf, 99 , 12 3 smooth submanifold , 30 , 8 1 stable manifold , 9 9 unstable manifold , 94 , 99

lower bound o f th e entropy , 13 4 pointwise dimension , 13 8

Lyapunov exponent, 6 , 9

backward regula r - , 3 1 exact - , 2 6 filtration associate d t o - , 1 1 for differentia l equations , 6 forward regula r - , 26 , 3 1 multiplicity o f value o f - , 11 , 31, 37 normalization property , 9 specified b y difTeomorphism , 3 6 values o f - , 1 0

exponents abstract theor y o f - , 9 dual - , 1 4

dynamical syste m wit h nonzer o - , 40

regular pai r o f - , 1 5 function

infinitesimal eventuall y stric t - , 12 8 infinitesimal eventuall y unifor m - ,

129 inner product , 8 3 norm, 8 4 regular

Lyapunov exponent , 3 2 pair o f Lyapuno v exponents , 3 2 point, 3 7

regularity, 3 1 spectrum, 11 , 37

of measure , 4 0 stability

theorem, 9 theory, 5

manifold o f nonpositiv e curvature , 7 4 map, time- t - o f flow , 11 2 measurable, partition , 13 0 measure

ergodic - , 11 7 hyperbolic, 4 0 physical - , 13 9 smooth, 1 smooth - , 11 5 SRB—, 138 , 13 9

metric, canonica l - , 7 5 multiplicative ergodi c theorem , 35 , 39

proof of- , 5 1 multiplicity o f valu e o f Lyapuno v expo -

nent, 11 , 31, 37

vs(w), 11 5 i/w , 10 2 vs (w), 11 6 negative limi t solution , 7 6 nonabsolutely continuou s foliation , 10 8 nonautonomous differentia l equation ,

stability theor y o f - , 2 6 nonpositive curvature , 7 4

manifold o f - , 7 4 nonuniform

hyperbolic theory , 30 , 35 , 8 1 hyperbolicity, 2 , 3 , 45 , 6 1

nonuniformly hyperbolic

dynamical system , 48 , 6 1 set, 45 , 4 7 system, 48 , 6 1

150 INDEX

partially hyperboli c diffeomorphism, 11 3

nonzero Lyapuno v exponent s diffeomorphism wit h - , 6 6 dynamical syste m wit h - , 36 , 40 flow with - , 7 1

norm, Lyapuno v - , 8 4 normal basis , 1 1

construction of- , 1 2 normalization property , 9

order o f perturbation , 7 ordered basis , 1 1 Oseledets decomposition , 31 , 32, 39

pair o f Lyapuno v exponents , Lyapuno v regular - , 3 2

partially hyperboli c diffeomorphis m nonuniformly - , 11 3 uniformly - , 11 1

partially hyperboli c flow, uniformly - , 11 1

partition, measurabl e - , 13 0 Perron coefficient , 1 4 perturbation, 7

order o f - , 7 Pesin set , 4 7 physical measure , 13 9 point

at infinity , 14 1 backward regula r - , 3 7 forward regula r - , 3 7 Lyapunov regula r - , 3 7

positive limi t solution , 7 6 product

direct - ,11 2 skew-, 11 2

proof o f multiplicativ e ergodi c theorem , 51

Q, 4 7 Q(x), 11 8 Q£(x), 10 1 Q£, 4 7

ft, 4 5 ft£, 4 6 ri, 9 5 regular

backward -Lyapunov exponent , 3 1 point, 3 7

forward -Lyapunov exponent , 26 , 3 1

point, 3 7 Lyapunov -

Lyapunov exponent , 3 2 pair o f Lyapuno v exponent , 3 2 point, 3 7

pair o f Lyapuno v exponents , 1 5 set, 45 , 46

of leve l I, 4 6 regularity

backward - , 1 6 coefficient, 1 4 forward - , 9 , 1 6 Lyapunov - , 3 1

Riemannian volume , 10 2 roof function , 6 6

50(n,IR), 5 7 Spx(i/) ,40 Sp X

+ (x ) , 3 7 S p x - ( * ) , 3 7 s+, 3 1 s~, 3 1 set

nonuniformly hyperboli c - , 4 5 Pesin - , 4 7 regular - , 4 6 uniformly hyperboli c - , 4 6

size o f loca l stabl e manifold , 8 2 skew product , 11 2 smooth

ergodic theory , 1 , 115 , 117 , 13 0 foliation, 10 0 measure, 1 , 11 5

spectral decompositio n theorem , 12 1 spectrum, see Lyapuno v spectru m

Lyapunov - , 3 7 Lyapunov - o f measure , 4 0

SRB-measure, 138 , 13 9 stability

Lyapunov - theorem , 9 theory, 5 , 2 6

stable manifold

global - , 62 , 98 , 9 9 global weakl y - , 9 9 local - , 9 9 theorem, 81 , 113 theorem fo r flows, 9 8

subspace, 40 , 46 , 7 7 structurally stabl e diffeomorphism , 6 4 submanifold, loca l smoot h - , 30 , 8 1 subspace

stable - , 40 , 46 , 7 7

unstable - , 40 , 46 , 7 7 subspaces, transverse , 4 9

tempered function , 4 2 theorem

absolute continuit y - , 102 , 11 3 flat stri p - , 1 4 1 implicit functio n - , 8 9 Lyapunov stabilit y - , 9 multiplicative ergodi c - , 35 , 39 proof o f multiplicativ e ergodi c - , 5 1 spectral decompositio n - , 12 1 stable manifol d - , 81 , 113 stable manifol d - fo r flows , 9 8

theory Lyapunov stabilit y - , 5 nonuniform hyperboli c - , 30 , 35 , 8 1 smooth ergodi c - , 1 , 115 , 117 , 13 0 stability - , 5 , 2 6

time-t ma p o f flow, 11 2 topologically transitive , 12 6 transitive, topologicall y - , 12 6 transversal t o family , 10 1 transverse subspaces , 4 9

uniformly hyperbolic set , 4 6 partially hyperboli c

diffeomorphism, 11 1 flow, 11 1

transverse submanifold , 10 1 unstable

manifold global - , 62 , 98 , 99 global weakl y - , 9 9 local - , 94 , 9 9

subspace, 40 , 46 , 7 7 upper

bound o f th e entropy , 13 1 pointwise dimension , 13 8

Vi", 3 1 V+(x), 3 7 VT,31 VT(s),3 7 V+ , 3 1 V+, 3 7 V", 3 1 V - , 3 7 values o f Lyapuno v exponent , 1 0 variational

differential equation , 3 6 system o f equations , 5

volume, Riemannia n - , 10 2

INDEX 15 1

WL 10 2 W^(w,q), 10 2 Ws(x), 9 8 Wu{x), 9 8 Ws0(x), 9 9 Wu0(x), 9 9

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