Introduction Smooth ergodi c theory studie s the ergodic properties o f smooth dynam - ical system s o n Riemannia n manifold s wit h respec t t o "natural " invarian t measures. Amon g thes e measure s mos t importan t ar e smoot h measures , i.e., measure s tha t ar e equivalen t t o th e Riemannia n volume . Ther e ar e various classe s of smooth dynamica l system s whos e stud y require s differen t techniques. I n thi s boo k w e concentrate o n system s whos e trajectorie s ar e hyperbolic in some sense. Roughl y speaking, this means that th e behavior of trajectories nea r a given orbi t resemble s th e behavio r o f trajectories nea r a saddle point . I n particular, t o every hyperboli c trajector y on e can associat e two complementar y subspace s suc h tha t th e syste m act s a s a contractio n along on e o f the m (calle d th e stabl e subspace ) an d a s a n expansio n alon g the othe r (calle d th e unstabl e subspace) . A hyperboli c trajector y i s unstabl e - almos t ever y nearb y trajector y moves awa y fro m i t wit h time . I f th e se t o f hyperboli c trajectorie s i s suf - ficiently larg e (fo r example , ha s positiv e o r ful l measure) , thi s instabilit y forces trajectorie s t o becom e separated . I f th e phas e spac e o f the syste m i s compact, th e trajectorie s mi x together becaus e there i s not enoug h roo m t o separate them. Thi s is one of the main reasons why systems with hyperboli c trajectories o n compac t phas e space s exhibi t chaoti c behavior . Indeed , hy - perbolic theory provides a mathematical foundation fo r the paradigm that i s widely known a s "deterministi c chaos " - th e appearanc e o f irregular chaoti c motions i n purel y deterministi c dynamica l systems . Thi s paradig m assert s that conclusion s abou t globa l propertie s o f a nonlinea r dynamica l syste m with sufficientl y stron g hyperboli c behavio r ca n b e deduce d fro m studyin g the linearize d system s alon g it s trajectories . The study of hyperbolic phenomena originated in seminal works of Artin, Morse, Hedlund , an d Hop f o n th e instabilit y an d ergodi c propertie s o f geo- desic flow s o n compac t surface s (se e th e surve y [18 ] fo r a detaile d descrip - tion o f result s obtaine d a t thi s tim e an d fo r references) . Later , hyperboli c behavior wa s observe d i n othe r situation s (fo r example , Smal e horseshoe s and hyperboli c tora l automorphism) . Th e systemati c stud y o f hyperboli c dynamical systems was initiated b y Smale (wh o mainly considered th e prob- lem o f structura l stabilit y o f hyperboli c systems se e [40] ) an d b y Anoso v and Sina i (wh o were mainly concerned wit h ergodic properties of hyperboli c l http://dx.doi.org/10.1090/ulect/023/01
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