positive entropy constructions would probably not look quite in place in the contex t
of the presen t work .
A number o f recent development s revive d the interes t i n the approximatio n b y
conjugations method . Th e centra l ne w observatio n i s possibilit y o f mixin g withi n
a modifie d contex t o f the metho d [Fa2] . Th e treatment propose d i n the Par t I V of
"Constructions i n Ergodi c Theory " ha s becom e outdated . W e decided t o kee p th e
section whic h deal s wit h combinatoria l aspect s o f th e metho d i n th e measurabl e
setting sinc e it i s closely related t o content s o f Part I . It appear s a s Sectio n 8 . Th e
recent pape r [FaK ] contain s bot h a n up-to-dat e surve y o f th e approximatio n b y
conjugations metho d an d proof s o f a number o f new representative results .
These note s ma y serv e variou s purposes . O n th e on e hand , mos t o f th e ma -
terial i s presente d i n details , wit h attentio n t o motivatio n an d ke y idea s bu t als o
with sufiicient details . Require d background consist s mostly of the material usuall y
covered i n the first-yea r graduat e course s i n mahtematics i n mos t U S universities ,
namely standar d rea l analysis, poin t se t topology , measur e theory , functiona l anal -
ysis, includin g basi c theor y o f Banac h an d Hilber t space s an d spectra l theor y fo r
bounded unitary operators in a Hilbert space , and basic theory of Fourier series. W e
tried t o make these parts fairly elementar y an d self-contained , ofte n a t th e expens e
of generality , emphasizin g method s rathe r tha n mos t genera l results . Familiarit y
with basi c ergodi c theory i s quite useful , wit h Sectio n 3 of [HaK ] containin g mos t
necessary fact s an d som e proofs . Thu s thes e note s ma y b e use d a s a basi s o f a
graduate cours e roughl y a t th e secon d yea r leve l as well a s for independen t study .
On th e othe r hand , w e presen t a surve y o f a certai n area . W e mentio n man y
results an d concept s withou t detaile d definition s an d proofs . Thos e part s ma y b e
omitted b y a studen t o n th e firs t reading . The y ar e provide d wit h reference s wit h
the exceptio n o f case s wher e detaile d treatment s wer e no t published . Thos e part s
may b e o f interest t o specialist s a s well as to mor e advance d students .
0.4. Acknowledgements . Th e original notes "Construction s in Ergodic The-
ory" wer e written i n collaboration o f E. A . Robinson , Jr. , the n a Ph.D . studen t a t
the University of Maryland. Hi s help was invaluable; without hi s participation thes e
notes woul d no t hav e appeared . A t th e firs t stag e o f revisio n i n th e mid-1990's A .
Mezhirov, then a Ph.D. student a t the Pennsylvania Stat e University, provided con -
siderable hel p i n improving an d updatin g th e text . Th e tex t wa s carefully checke d
by A. Windsor whos e comment s an d criticis m helpe d t o correc t a number o f errors
and improve d presentations . I . Ugarcovic i provide d valuabl e hel p wit h th e fina l
typesetting o f the text .
Over th e cours e o f twenty year s a fairl y larg e numbe r o f mathematician s wh o
read variou s version s o f "Construction s o f Ergodi c Theory " mad e numerou s ob -
servations, suggestion s an d corrections . Whil e i t woul d b e difficul t t o mentio n
everyone b y nam e th e autho r expresse s dee p gratitud e t o al l of them .
The autho r wa s supporte d b y th e NS F Grant s MC S 79-0304 6 an d MC S 82 -
04024 durin g th e writin g o f origina l note s an d b y th e NS F Gran t DMS-00-71339
during th e completio n o f this book .
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