INTRODUCTION 3
theory t o produc e a variet y o f often stunnin g counterexample s bot h wit h zer o en -
tropy (ran k on e mixing, minima l self-joinings , etc ; see e.g. [Rud , JR2] ) an d wit h
positive entropy , (man y nonisomorphi c if-automorphism s wit h th e sam e entropy ,
etc, see e.g. [OSh]) , and, on the other hand , b y the development o f the approxima -
tion b y conjugation s construction s [AK ] whic h provide d th e first genera l metho d
for constructin g smoot h dynamica l system s with interestin g an d controlle d ergodi c
properties o n a broa d clas s o f manifold s [AK] ; se e Sectio n 8 fo r a mor e detaile d
discussion.
0.3. Th e stor y an d purpos e o f these notes . Th e present wor k is based o n
our notes "Construction s in Ergodic Theory" whic h were mostly written in 1982-83.
The firs t tw o o f the projecte d fou r part s wer e finished a t th e tim e whil e th e thir d
and fourt h part s wer e left unfinishe d b y reasons explained below . Th e presen t tex t
includes expande d an d update d version s o f Part s I an d I I an d th e sectio n o f Par t
IV which deal s with combinatoria l measurabl e (a s opposed th e smooth ) setting.* )
We dea l primaril y wit h zero-entrop y constructions . Th e origin s o f thes e con -
structions ca n be found i n [KS1] and [AK1]; their prominen t featur e i s a presenc e
of a n exac t o r approximat e grou p structure , althoug h i n th e forme r cas e i t ma y
be hidden . Thi s distinguishe s ou r construction s fro m "cuttin g an d stacking " typ e
constructions whic h ten d t o hav e les s symmetry . Ou r mai n goa l i s t o provid e a
detailed an d well-illustrate d presentatio n o f methods whic h hav e been applie d t o a
variety o f problems an d ca n b e applie d t o man y more . Th e natur e o f those meth -
ods, especiall y th e approximatio n b y conjugatio n metho d develope d i n [AK1], i s
such tha t the y allo w almos t unlimite d variation s an d an y attemp t t o provid e a n
all-encompassing framewor k woul d exclud e som e interestin g applications . I n th e
hindsight i t wa s th e struggl e t o find a prope r framework , bot h elegan t an d suffi -
ciently comprehensive according to the understanding a t the time, for the treatmen t
of realization o f various ergodi c propertie s i n smoot h an d analyti c categorie s (Par t
III) an d fo r th e approximatio n b y conjugatio n metho d (Par t IV ) whic h impede d
completion o f the origina l projec t .
Part I I is dedicated t o cohomological constructions . Whil e the development s of
the las t decad e especiall y thos e dealin g wit h action s o f groups othe r tha n Z an d R
changed th e fac e an d t o a certain exten t eve n th e basi c perceptio n o f the are a th e
program outline d an d illustrate d i n Par t I I o f "Construction s i n ergodi c theory "
has prove d t o b e fundamentall y sound . A n update d versio n o f thi s par t appeare d
as [KR] ; with furthe r addition s o f discussio n o f som e recen t result s i t i s include d
into th e presen t wor k a s Par t II .
The program o f Part II I has been advance d i n two directions: (i ) construction s
of Bernoull i geodesi c flows an d othe r Hamiltonia n system s i n a larg e variet y o f
setting including all compact manifolds in dimension two and three [KB] and (ii ) the
development o f the theory of stable ergodicity since mid-1990's with the Dolgopyat -
Pesin construction o f a Bernoulli system with nonvanishing Lyapunov exponents on
any manifol d [DP ] a s on e o f the crownin g achievements . Whil e construction s an d
methods discusse d i n Part II I still form th e basi s of the subjec t i t seem s clear tha t
its up-to-dat e presentatio n shoul d tak e a different for m an d wit h th e prevalenc e of
*)Parts o f th e materia l include d i n thi s wor k originate d fro m wor k presente d an d discusse d
in Mosco w a t th e semina r ru n b y D.V . Anoso v an d th e autho r a t th e Steklo v Institut e (MIAN )
and late r a t th e Centra l Economics-Mathematic s Institut e (CEMI) .
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