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 this gap . W e describe the 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 of stable manifolds, an d th e ergodic theory o f dynamical system s wit h nonzero Lyapunov exponent s (includin g geodesi c flows). The book i s a revised an d considerably expanded versio n of 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 that i s not absolutel y continuous and establish the formula fo r the Jacobian o f the holonomy map . We also add a complete proof of the Multiplicativ e Ergodi c Theorem a s well as provide mor e detail s i n the proof s o f several 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 book i s self-contained. W e only assum e that th e reade r ha s a basi c knowledge of real analysis, measure theory, differential 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 all main results . W e also state som e results 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 broader 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

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