2
INTRODUCTION
This constructiv e approac h play s a n eve n greate r rol e i n th e question s con -
cerning realization s o f variou s particula r conjugac y classe s o f measure-preservin g
systems o r som e systems wit h particula r propertie s withi n mor e specialize d frame -
works. Premie r example s o f those framework s ar e smoot h o r real-analyti c system s
on compac t manifold s preservin g a n absolutel y continuou s o r smoot h measure .
Other classe s includ e abelia n grou p extension s ove r som e simpl e systems , inter -
val exchange transformations, time-change s o f some simple flows, etc. Thi s remar k
provides a bridg e betwee n purel y measurabl e ergodi c theor y consideration s an d
other area s o f dynamics.* ) Thi s interpla y i s quit e eviden t i n th e secon d par t o f
these notes .
Prevalence o f example s an d counterexample s ove r "theorems " somewha t cu -
riously correlate s wit h th e fac t (recognize d b y man y experts ) tha t ther e ar e n o
adequate up-to-dat e textbook s i n ergodic theory, either a t th e introductor y o r at a
more comprehensiv e level , a variety o f book s excellen t i n thei r ow n righ t notwith -
standing [CRW , Pa , Na , Pe , F2 , H , W] . Least o f all , th e presen t note s aspir e
to fill any o f these gap s o r eve n t o b e a seed fo r suc h a text .
0.2. A littl e history . I n th e prehistor y o f ergodi c theor y th e 1930 wor k o f
L.G. Shnirelma n [LSh ] shoul d b e noted . Shnirelma n constructe d th e first non -
trivial exampl e o f complicate d dynamica l behavio r unrelate d t o an y topologica l
complexity: perturbation s o f rotation s o f th e dis c wit h dens e orbits . Althoug h
not are a preservin g th e Schnirelma n exampl e wa s on e o f th e chie f source s fo r th e
approximation b y conjugatio n constructions .
Soon afterward s J . vo n Neuman n i n th e foundin g tex t o f ergodi c theor y [N ]
provided th e first example s o f weakly mixin g transformation s an d flows (se e mor e
detailed comment s i n Sectio n 5. 5 o f Par t II) . In th e 1940's P.R . Halmo s an d V.A .
Rokhlin realize d tha t combinatoria l construction s togethe r wit h th e Bair e categor y
theorem ca n b e use d fo r provin g bot h existenc e an d genericit y o f measur e pre -
serving transformation s wit h certai n properties ; i n particular , existenc e o f weakl y
mixing but no t mixin g transformations wa s first established indirectl y by genericity
arguments, (althoug h vo n Neumann's origina l examples in fact hav e that property ,
see [Kcl]) . Se e [H ] fo r th e proof s an d discussio n o f thei r result s an d Sectio n 2
below fo r a n exampl e o f a mor e elaborat e implementatio n o f essentiall y th e sam e
scheme; mor e discussio n ca n b e foun d i n Section s 6 and 7 .
In th e mid-1960's combinatoria l construction s wer e brought explicitl y int o th e
context o f ergodi c theor y b y tw o independen t an d almos t simultaneou s develop -
ments:
(i) th e "Chaco n example " [CI ] whic h stil l provide s a n interestin g an d challengin g
instance o f a constructio n wit h goo d rescalin g propertie s bu t wit h n o apparen t
symmetry comin g fro m a group structure , an d
(ii) th e metho d o f periodic approximatio n introduce d b y a group of Moscow math -
ematicians (se e [KSl ] an d reference s thereof)**) , whic h serve d a s th e foundatio n
for a systematic stud y o f "Liouvillean " behavio r i n dynamic s characterize d b y ab -
normally fas t recurrenc e an d accompanie d b y various form s o f instability .
These developments were followed i n the late 1960's and early 1970's by, on the
one hand , a systemati c us e o f th e "cuttin g an d stacking " construction s i n ergodi c
*)For a n attemp t t o classif y larg e part s o f moder n dynamic s fro m a unifie d structura l poin t
of vie w se e [HaK ]
**)The rol e o f V.I . Oseledet s a t th e earl y stage s mus t b e emphasize d
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