0. Introductio n
This boo k i s based o n the note s "Construction s i n Ergodic Theory " writte n i n
collaboration wit h E . A . Robinson , Jr .
0.1. Theorem s an d construction s an d ergodi c theory . Ergodi c theory ,
which i s sometime s als o calle d measurabl e dynamics , i s primaril y concerne d wit h
the propertie s o f measure-preservin g (and , t o a lesse r extent , non-singular ) trans -
formations an d flow s o f a standar d measur e space , usuall y calle d Lebesgu e space ,
invariant u p t o a measure-preservin g (correspondingly , measurabl e non-singular )
conjugacy o r u p t o natura l weake r equivalenc e relation s suc h a s monoton e (Kaku -
tani) equivalence . Severa l centra l bodie s o f result s i n ergodi c theor y dea l wit h
description o f particularl y importan t equivalenc e classe s (e.g . standard , loosel y
Bernoulli; se e [Kl] , [ORW]) , o r th e classificatio n o f specia l classe s o f system s
(such a s pure-poin t spectru m [CFS] , [HaK ] o r Bernoull i [Ol , T]) . Fo r a genera l
overview o f the subjec t se e [HaK] , Sectio n 3.
There ar e other classe s which are not a priori define d i n isomorphism invarian t
terms wher e measurabl e structur e exhibit s certai n rigidit y an d a s a resul t classifi -
cation u p t o isomorphis m turn s ou t t o b e accessible . Horocycl e flows o n surface s
of constan t negativ e curvatur e [Rat ] an d othe r homogeneou s unipoten t system s
are characteristic example s o f this phenomenon whic h is even mor e pronounced fo r
actions o f higher-rank abelia n group s (se e [KKS , KalSp , KitSch]) , an d become s
prevalent fo r action s o f group s whic h ar e themselve s "rigid " suc h a s highe r ran k
semisimple Li e groups o r lattice s i n such group s (se e e.g. [Zl , Fu]) .
However, ther e i s a widel y share d perceptio n tha t i n th e classica l cas e o f
measure-preserving automorphism s an d flows, i.e . action s o f Z an d M , most natu -
rally defined classe s of measure-preserving system s cannot b e classified (if-systems ,
superficially loo k so close to Bernoulli, provide a n outstanding exampl e [02 , OSh ,
OS2]), an d tha t mos t propertie s o r combination s o f properties whic h ar e no t pro -
hibited by some basic and mostly fairly elementar y general conditions can in fact b e
realized. Thus , classica l ergodi c theor y contain s relativel y fe w genera l "theorems "
and plent y o f interesting "examples " an d "counter-examples".* )
This general observation explains why various constructions play such an impor-
tant role in ergodic theory. A very large part o f those constructions is combinatorial-
approximational i n nature. Th e genera l ide a i s to produc e a n approximat e versio n
of a desire d propert y fo r a n appropriat e finite objec t an d the n concatenat e suc h
objects a t variou s scale s to produc e a desirable syste m a s a limi t o f some sort .
*)Two outstandin g ope n problem s whic h ma y ye t tur n int o rea l "theorems " ar e th e multipl e
mixing proble m (fo r dee p result s connectin g spectra l an d othe r propertie s wit h multipl e mixin g
see [Ho , Kal] ) an d th e simpl e Lebesgu e spectru m proble m wher e i t stil l look s likel y tha t absenc e
of a n exampl e i s due mor e t o a lac k o f ingenuit y tha n t o dee p structura l reasons .
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http://dx.doi.org/10.1090/ulect/030/01
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