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STUDENT MATHEMATICAL LIBRARY Volume 42
Invitation t o Ergodic Theory C. E. Silva
ojSSSAgb
#AMS AMERICAN MATHEMATICA L SOCIET Y
Providence, Rhode Islan d
http://dx.doi.org/10.1090/stml/042
Editorial Boar d Gerald B . Follan d Bra d G . Osgoo d Robin Forma n (Chair ) Michae l S ta rb i r d
2000 Mathematics Subject Classification. P r i m a r y 37A05 , 37A25 , 37A30 , 37A40, 37B10 , 28A05 , 28A12 , 28A20 , 28D05 , 54E52 .
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Silva, Cesa r Ernest o [date ] Invitation t o ergodi e theor y / C.E . Silva .
p. cm . — (Studen t mathematica l library , ISS N 1520-912 1 ; v. 42 ) Includes bibliographica l reference s an d index . ISBN 978-0-8218-4420- 5 (alk . paper ) 1. Ergodi e theory . I . Title .
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Contents
Preface
Chapter
Chapter
§2.1.
§2.2.
§2.3.
§2.4.
§2.5.
§2.6.
§2.7.
§2.8.
Chapter
§3.1.
§3.2.
§3.3.
1. Introductio n
2. Lebesgu e Measur e
Lebesgue Oute r Measur e
The Canto r Se t an d Nul l Set s
Lebesgue Measurabl e Set s
Countable Additivit y
Sigma-Algebras an d Measur e Space s
The Bore l Sigma-Algebr a
Approximation wit h Semi-ring s
Measures fro m Oute r Measure s
3. Recurrenc e an d Ergodicit y
An Example : Th e Baker' s Transformatio n
Rotation Transformation s
The Doublin g Map : A Bernoull i Noninvertibl e
vii
1
5
5
10
17
23
26
34
38
47
59
60
67
Transformation 7 5
§3.4. Measure-Preservin g Transformation s 8 3
§3.5. Recurrenc e 8 6
in
IV Contents
§3.6. Almos t Everywher e an d Invarian t Set s 9 1
§3.7. Ergodi c Transformation s 9 5
§3.8. Th e Dyadi c Odomete r 10 2
§3.9. Infinit e Measure-Preservin g Transformation s 10 9
§3.10. Factor s an d Isomorphis m 11 5
§3.11. Th e Induce d Transformatio n 12 0
§3.12. Symboli c Space s 12 3
§3.13. Symboli c System s 12 7
Chapter 4 . Th e Lebesgu e Integra l 13 1
§4.1. Th e Rieman n Integra l 13 1
§4.2. Measurabl e Function s 13 4
§4.3. Th e Lebesgu e Integra l o f Simple Function s 14 1
§4.4. Th e Lebesgu e Integra l o f Nonnegative Function s 14 5
§4.5. Application : Th e Gaus s Transformatio n 15 0
§4.6. Lebesgu e Integrabl e Function s 15 5
§4.7. Th e Lebesgu e Spaces : L\ L 2 an d L°° 15 9
§4.8. Eigenvalue s 16 6
§4.9. Produc t Measur e 17 0
Chapter 5 . Th e Ergodi c Theore m 17 5
§5.1. Th e Birkhof f Ergodi c Theore m 17 6
§5.2. Norma l Number s 18 8
§5.3. Wey l Equidistribution 19 1
§5.4. Th e Mea n Ergodi c Theore m 19 2
Chapter 6 . Mixin g Notion s 20 1
§6.1. Introductio n 20 1
§6.2. Wea k Mixin g 20 5
§6.3. Approximatio n 20 9
§6.4. Characterization s o f Weak Mixin g 21 4
§6.5. Chacon' s Transformatio n 21 8
§6.6. Mixin g 22 6
Contents v
§6.7. Rigidit y an d Mil d Mixin g 22 7
§6.8. Whe n Approximatio n Fail s 23 1
Appendix A . Se t Notatio n an d th e Completenes s o f R 23 5
Appendix B . Topolog y o f R an d Metri c Space s 24 1
Bibliographical Note s 25 1
Bibliography 25 5
Index 25 9
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Preface
This boo k provide s a n introductio n t o th e growin g field o f ergodi c theory, als o know n a s measurabl e dynamics . I t cover s topic s suc h a s recurrence, ergodicity , th e ergodi c theore m an d mixing . I t i s aime d at student s wh o have completed a basic course i n undergraduate rea l analysis coverin g topic s suc h a s basi c compactnes s propertie s an d open an d close d set s i n the rea l line . Measur e theor y i s not assume d and i s develope d a s needed . Reader s les s familia r wit h thes e topic s will find a discussion o f the relevant materia l fro m rea l analysi s in th e appendices.
I have used early versions of this book in courses that ar e designed as capston e course s fo r th e mathematic s major , includin g student s with a variety o f interests and backgrounds . Th e study o f measurabl e dynamics ca n b e used t o reinforce an d appl y th e student' s knowledg e of measure theor y an d rea l analysi s whil e introducing som e beautifu l mathematics o f relativel y recen t vintage . Measur e theor y i s devel -oped a s neede d an d applie d t o stud y notion s i n dynamics . Whil e i t has les s emphasis , som e metri c spac e topology , includin g th e Bair e category theorem , i s presente d an d applie d t o topologica l dynamics . Several example s ar e develope d i n detai l t o illustrat e concept s fro m measurable an d topologica l dynamics .
This boo k ca n b e use d a s a special-topic s cours e fo r upper-leve l mathematics students . I t ca n als o b e use d a s a shor t introductio n
vn
vm Preface
to Lebesgu e measur e an d integration , a s a n introductio n t o ergodi c theory, and for independent study . Th e Bibliographical Note s provide some guideline s fo r furthe r reading .
An introductor y cours e coul d star t wit h a shor t revie w o f th e topology on the rea l line and basi c properties o f metric spaces as cov-ered i n Appendix B or with the constructio n o f Lebesgue measure o n the rea l lin e i n Chapte r 2 . Th e reade r wh o want s t o ge t t o ergodi c theory quickl y needs to cover only Section s 2.1 through 2. 4 and coul d then star t wit h Chapte r 3 , perhap s omittin g Section s 3.1 0 throug h 3.12. Topologica l dynamic s i s closel y relate d t o measurabl e dynam -ics, an d th e boo k introduce s som e topic s fro m topologica l dynamics . This i s not necessar y fo r th e mai n developmen t o f th e book , an d th e reader ha s th e optio n o f omitting th e topologica l dynamic s topic s o r of usin g the m t o lear n som e metri c spac e topolog y an d som e elegan t ideas from topologica l dynamics. A more advanced course could cover Chapters 2 and 3 in more detail . Som e of the measur e theor y notion s that ar e covere d includ e th e Caratheodor y extensio n theorem , prod -uct measure s an d L p spaces . Lebesgu e integratio n i s introduce d i n Chapter 4 , an d som e o f thes e notion s ar e use d t o stud y th e eigen -values o f measure-preservin g transformations . Th e chapte r o n th e ergodic theorem , i n additio n t o bein g o f intrinsi c interest , provide s a beautifu l exampl e fo r application s o f various theorem s o f Lebesgu e integration. Th e final chapte r o n mixing uses ideas from al l the othe r chapters.
The boo k contain s bot h simpl e exercises , calle d questions , de -signed t o tes t th e reader' s immediat e gras p o f th e ne w material , an d more challengin g exercise s a t th e en d o f eac h section . Harde r exer -cises are marked wit h a star (*) . Partia l solution s an d hint s fo r som e of the exercise s wil l b e availabl e a t th e book' s webpag e liste d o n th e back cover . Som e section s als o contai n ope n question s designe d t o suggest t o th e reade r som e avenue s o f research . Th e bibliograph y i s not intende d t o b e exhaustive ; i t i s ther e t o provid e suggestion s fo r additional readin g an d t o acknowledg e th e source s I have used .
I a m indebte d t o man y peopl e wh o throug h thei r conversation s and writing s hav e taugh t m e measur e theor y an d ergodi c theory . I firs t learne d analysi s fro m Cesa r Carranza . I wa s introduce d t o
Preface IX
ergodic theor y b y Doroth y Mahara m an d late r wa s influence d b y Shizuo Kakutani an d Joh n Oxtoby . I have also learned muc h from al l my coauthor s an d th e student s I hav e supervise d i n researc h an d i n courses. I a m indebte d t o severa l anonymou s reviewer s an d reader s at variou s stage s o f thi s wor k wh o hav e provide d advic e an d sugges -tions. I n particula r I woul d lik e t o than k m y William s colleague s OUie Beaver, E d Burger , Satya n Devadoss , Fran k Morga n an d Miha i Stoiciu, an d m y editor , Serge i Gelfand . I als o than k Blair e Mador e and Kari n Reinhold , wh o use d a n earl y versio n wit h thei r student s and sen t m e helpfu l suggestions .
I hav e use d earl y version s o f thi s boo k i n courses , tutorial s an d summer SMAL L REU projects an d would like to thank th e many stu -dents wh o correcte d errors , discovere d typo s an d mad e suggestions , in particula r Katherin e Acton , Al i Al-Sabah , Ami e Bowles , Joh n Bryk, Joh n Chatlos , Tega n Cheslack-Postava , Alexandr a Constantin , Jon Crabtree , Michae l Daub , Sara h Day , Chri s Dodd , Jaso n Enelow , Lukasz Fidkowski , Thoma s Fleming , Artu r Fridman , Ily a Grigoriev , Brian Grivna , Kat e Gruher , Fre d Hines , Sara h lams , Catali n lor -dan, Nat e Ince , O n Jesakul , Ann e Jirapattanakul , Jef f Kaye , Eri c Katerman, Bria n Katz , Mi n Kim , Jame s Kingsbery , Dav e Klein -schmidt, Thoma s Koberda , Ros s Kravitz , Gar y Lapon , Ale x Levin , Amos Lubin , Am y Marinello , Earl e McCartney , Ab e Menon , Eric h Muehlegger, Kar l Naden, Nar a Narasimhan , Deepa m Patel , Rav i Pu -rushotham, And y Raich , Hyeji n Rho , Beck y Robinson , Richar d Ro -driguez, Davi d Roth , Charle s Samuels , Bria n Simanek , Pete r Speh , Anita Spielman , Joh n Spivack , Noa h Stein , Josep h Stember , An -drea Stier , Bria n Street , Danie l Sussman , Mik e Touloumtzis , Pau l Vichyanond, Rober t Waelder , Kirste n Wickelgren , Ale x Wolfe , an d Wenhuan Zhao . Thank s ar e du e mos t especiall y t o Darre n Creutz , Daniel Kane, Jennife r James , Kathry n Lindsey , an d Anatol y Preygel .
Finally, I woul d lik e t o dedicat e thi s boo k t o m y wif e an d tw o daughters an d th e memor y o f my parents .
Cesar E . Silv a
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Bibliographical Note s
Chap te r 2
For furthe r informatio n abou t thes e topic s th e reade r i s referre d to [67 ] an d [56] .
Our definitio n o f Lebesgu e measurabl e se t i s a s i n [70 ] an d ou r presentation follow s [56 ] an d [70] . Ou r definitio n o f semi-ring s i s a s in [68] . Fo r the sections on <r~algebras, measure spaces and the mono-tone class theorem w e have followed [28 ] and [68] . Fo r Section 2. 8 we have followe d [68] , [6] , [18 ] an d [28] . Th e quotatio n i n Sectio n 2. 7 on Littlewood' s Thre e Principle s i s from [61] . Fo r a shor t accoun t o f the histor y o f measur e an d integratio n wit h extensiv e reference s se e [18].
Chap t e r 3
For furthe r informatio n abou t thes e topic s th e reade r i s referre d to [24] , [58 ] an d [69] . A n introductio n t o severa l topic s i n dynamic s can b e foun d i n [31 ] an d the n i n [10 ] (includin g a n applicatio n o f ergodic theory to Interne t searches) . Fo r entropy theory , whic h i s not covered here , th e reade r ma y star t wit h th e entrop y chapte r i n [50 ] and the n consul t [39] , [58 ] and [69] .
The baker' s transformatio n i s a well-know n exampl e i n ergodi c theory. Fo r a higher-dimensiona l versio n o f irrationa l rotation s see
251
252 Bibliographical Note s
[15] an d [57] . Fo r th e topologica l dynamic s example s w e hav e fol -lowed [69] . Fo r application s o f th e Bair e Categor y metho d t o dy -namics see [3 ] an d [56] . Fo r a discussio n o f nonmeasurabl e set s se e [56]. Th e proof o f the Baire category theorem i s standard (see , for ex-ample, [56]) . Fo r measure-presevin g transformation s an d recurrenc e we have followe d [24] , [58] and [69] . Fo r multipl e recurrenc e refe r t o [24]. Th e notion s o f Poincare sequenc e an d thic k set s ar e fro m [24] . For th e recen t proo f o f Gree n an d Ta o o n arithmeti c progression s i n the primes and th e role of Furstenberg's ergodi c theoretic methods i n this proo f se e [43] . Th e notio n o f ergodicit y goe s bac k t o a pape r o f Birkhoff an d Smit h [7] . Mos t o f the equivalence s o f Lemma 3.7. 2 ar e in, fo r example , [58] . Par t (4 ) of Lemma 3.7.2 i s probably know n bu t I learne d i t fro m Danie l Kane , a s wel l a s Exercis e 6.2.5 . Th e dyadi c odometer i s due to Kakutani an d von Neumann; a description simila r to our s ca n b e foun d i n [21] . Th e Hajian-Kakutan i transformatio n is from [27] , and weakl y wanderin g set s wer e introduced i n [26] . Fo r further result s on infinite measure-preservin g transformation s se e [1]. For factor s an d isomorphis m w e have followed [24 ] and [63] . Th e no-tion o f a Lebesgu e spac e i s due t o Rohlin ; fo r propertie s o f Lebesgu e spaces see [48 ] and [63] . Th e induced transformatio n i s due to Kaku -tani [37] . Fo r furthe r propertie s o f symbolic systems refer t o [48 ] and [46]. Furstenberg' s questio n ma y b e foun d i n [64] .
Chapter 4
Our developmen t o f Lebesgu e integratio n follow s [11] . Fo r th e Gauss transformation se e [48] . Othe r example s o f invariant measure s may b e foun d i n [8] . Fo r othe r numbe r theoreti c example s se e [16] . For th e Lebesgu e L p space s w e have followe d [11 ] an d [70] . Th e re -sults on eigenvalues ar e standard; see , for example , [69] . Fo r produc t measure w e have followe d [6 ] and [68] .
Chapter 5
The Birkhof f ergodi c theore m ha s ha d a lon g histor y o f differen t proofs. Fo r ou r firs t proo f o f th e ergodi c theore m w e hav e followe d [66], whic h wa s influence d b y [54 ] an d [35] ; these argument s ca n b e extended t o th e rati o subadditiv e ergodi c theore m [66] . Ou r secon d proof o f th e ergodi c theore m follow s [29] . Fo r anothe r proo f o f th e
Bibliographical Note s 253
maximal ergodi c theorem du e to Garsi a see , e.g., [57] . A more recen t proof o f the ergodic theorem with references t o other proofs i s in [41] . For a comprehensiv e surve y o f development s relate d t o th e ergodi c theorem see [45 ] an d [60] . Fo r th e proo f o f th e L 2 ergodi c theore m we followe d [57] . Th e proo f o f Lemm a 5.4. 1 i s fro m [67] . A proo f of Theore m 5.4. 5 ca n b e foun d i n [69] . Th e proo f o f Lemma 5.4. 3 i s from [71] . Fo r additiona l materia l o n equidistributio n se e [57] . Fo r other exposition s o f the ergodi c theore m se e [39] , [58] , [65] , [69].
Chap te r 6
Many o f th e characterization s o f wea k mixin g g o back t o Koop -man an d vo n Neumann an d alread y appea r i n [29] . W e have followe d [19], [24] , [58] , [69] . Th e notio n o f doubl e ergodicit y an d it s equiv -alence t o wea k mixin g appear s i n [24] ; i t wa s generalize d t o infinit e measure-preserving transformation s i n [9] . Fo r a proo f o f th e exis -tence o f sequences o f density on e for a weakly mixin g transformatio n see [20] , [69 ] an d [58] . Fo r Sectio n 6. 4 w e hav e followe d [58 ] an d [63]. Chacon' s transformatio n appear s i n [19] , an d appeare d i n a modified for m i n [13] . Th e transformatio n i n [13 ] i s lightl y mixin g (hence weakl y mixing ) bu t no t mixin g [22] . Th e transformatio n i n [13] is weakly mixing (an d mildl y mixing [22] ) bu t no t lightl y mixin g [19], [22] . Th e proo f o f Theore m 6.5. 2 follow s [13 ] an d [19] . Th e proof o f Lemm a 6.5. 4 (doubl e approximation ) follow s a proo f i n [2 ] and [17] . Th e proof of Theorem 6.5.5 follows a proof in [2 ] of a similar result i n infinit e measure . Anothe r proo f o f wea k mixin g i s i n [63] . Other example s of weakly mixing and no t mixin g transformations ar e in [40 ] and [38] . Th e canonica l Chaco n transformatio n wa s shown t o be prime (n o nontrivial factors ) an d t o commute only with it s power s in [36] . Th e notio n o f mild mixin g an d rigidit y i s from [25] . Fo r th e proof o f Lemma 6.7. 3 we followed [2] , where i t i s in the mor e genera l context o f nonsingula r transformations . Fo r th e proo f o f Proposi -tion 6.7. 5 w e followed [32] . Sectio n 6. 8 i s from [51] . Exercis e 6.8. 5 i s from [47] .
Appendix A
For basi c propertie s o f set s se e [44] . Fo r a n introductio n t o se t theory se e [30] . Fo r propertie s o f th e rea l number s th e reade r ma y
254 Bibliographical Note s
consult [44] , [49] , [62] . Fo r measurabl e dynamic s o n othe r metri c completions o f the field Q se e [12 ] an d [42] .
Appendix B
The proo f o f th e Heine-Bore l Theore m (Theore m B.1.5 ) follow s Borel's proo f a s in [56 , p. 4] . Fo r th e se t M w e followed [51 ] an d a n early versio n o f [51] . Fo r th e othe r topic s se e [44] , [49] , [62].
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Index
(1 - S)-Ml, 98 L2 inner product, 162 L°°-norm, 162 Z^-norm, 193 T-invariant, 9 2 T-invariant mo d /i , 9 2 a-algebra, 26 , 2 7 cr-algebra generate d by , 3 4 <x-finite, 2 9 cr-ideal, 8 2 d-dimensional Lebesgu e oute r
measure, 5 1 d-volume, 5 1 p-norm, 160 , 19 3 rn -odometer , 10 8
a.e., 9 1 absolutely continuous , 15 1 absolutely normal , 19 0 accumulation point , 24 3 algebra, 3 6 almost disjoint , 2 5 almost everywhere , 9 1 atom, 3 0
Baire Categor y Theorem , 7 9 Bernstein set , 7 5 Boole's transformation , 8 5 Borel measurable , 12 6 Borel measurabl e function , 13 4
Borel sets , 3 5 Borel-Canteili, 3 3 bounded above , 23 8 bounded below , 23 7 bounded interval , 23 7
canonical atomi c spaces , 2 9 canonical Chaco n transformation ,
219 canonical Lebesgu e measur e space ,
30 canonical nonatomi c Lebesgu e
measure space , 2 9 canonical representatio n o f a
simple function , 14 2 Cantor middle-third s set , 1 0 Cantor set , 1 4 Cartesian product , 5 1 Cauchy sequence , 23 9 Cesaro convergenc e o f sequences ,
202 characteristic function , 7 7 closed set , 24 3 closure, 24 3 column, 10 4 compact, 24 6 complete, 24 6 complete measur e space , 2 9 compressible, 8 8 conjugate, 16 0
259
260 Index
conservative, 8 7 continued fractio n expansion , 15 4 continuous, 7 1 continuous spectrum , 16 9 converge i n density , 20 4 convergence o f sequences , 23 9 converges, 24 3 copy, 22 0 countable basis , 16 6 countably subadditive , 4 8 counting measure , 3 0 cutting an d stacking , 11 4
dense, 24 3 dense algebra , 4 4 dense ring , 4 4 density one , 20 3 doubling map , 7 6 doubly ergodic , 20 7 dyadic interval , 9 dynamical property , 11 8
eigenfunction, 16 6 eigenvalue, 16 6 eigenvalue group , 16 8 eigenvector, 16 6 element, 23 5 empty set , 23 5 equal almos t everywhere , 13 7 equivalent mo d 1 , 6 7 equivariance, 11 8 ergodic, 9 6 essential supremum , 16 2 exhaustive, 11 2
factor, 11 9 fiber, 17 1 finite measur e space , 2 9 finite measure-preserving , 8 3 finitely additive , 4 7 first category , 7 8 first retur n tim e ,12 1 full measure , 11 6
Gauss map , 15 2 Gelfand's question , 7 2 generate mo d 0 , 4 4
height, 10 4
homeomorphism, 13 0
improper d-algebra , 2 7 incompressible, 8 8 indicator function , 7 7 induced transformation , 12 1 infimum, 23 8 infinite measure-preserving , 8 3 integrable, 15 6 interior point , 24 8 interval, 23 7 invariant, 9 2 invariant measure , 6 9 inverse image , 6 9 invertible measurabl e
transformation, 6 9 invertible measurabl e
transformation mod^t , 9 4 invertible measure-preserving , 7 0 invertible transformation , 6 7 isolated point , 24 3 isomorphic, 116 , 11 8 isomorphism, 11 8
least period , 6 8 Lebesgue integrable , 145 , 15 6 Lebesgue integra l o f a
characteristic function , 14 1 Lebesgue integra l o f a measurabl e
function, 15 6 Lebesgue integra l o f a nonnegativ e
function, 14 5 Lebesgue integra l o f a simpl e
function, 14 2 Lebesgue measurable , 1 7 Lebesgue measurabl e function , 13 4 Lebesgue measure , 2 3 Lebesgue oute r measure , 6 Lebesgue space , 11 7 length, 23 7 level, 10 4 lightly mixing , 22 5 Liouville number , 1 6 lower bound , 23 7
meager, 7 8 mean convergence , 19 3 measurable function , 13 4 measurable rectangle , 17 1
Index 261
measurable transformation , 6 9 measure, 2 8 measure space , 2 9 measure-preserving, 6 9 measure-preserving dynamica l
system, 83 , 11 8 measure-preserving isomorphis m
mod 0 , 11 6 mesh, 13 1 metric, 24 2 metric space , 24 2 mildly mixing , 22 9 minimal, 7 2 mixing sequence , 21 4 monotone class , 3 6 monotone clas s generate d by , 3 6 monotone se t function , 4 8 multiply recurrent , 8 9
negatively nonsingular , 15 2 nonsingular, 15 2 norm, 16 1 normal t o bas e 2 , 18 9 normed linea r space , 16 1 nowhere dense , 14 , 7 8 null set , 9
odometer map , 12 7 open, 24 2 open ball , 24 2 open bounde d interval , 23 7 open set , 24 1 orbit, full , 6 8 orbit, positive , 6 8 orthogonal collection , 16 4 orthogonal complement , 19 4 orthogonal functions , 16 4 orthonormal collection , 16 4 outer measure , 4 8
pair wise disjoint , 23 6 partially rigid , 22 8 partition, 13 1 perfect set , 24 3 period, 6 8 periodic, 6 8 periodic point , 6 8 Poincare Recurrenc e Theorem , 8 8 Poincare sequence , 9 0
positive density , 20 3 positively invariant , 9 1 power set , 2 7 pre-image, 6 9 probability space , 2 9 probability-preserving, 8 3 product measure , 17 1 projection, 19 5
rational eigenvalue , 16 8 real an d imaginar y par t o f a
function, 16 0 recurrent, 8 6 residual, 8 2 Riemann integrable , 13 2 rigid, 22 7 ring, 4 4 ring generate d by , 4 6 rotation b y a , 6 7
semi-ring, 3 9 separable metri c space , 24 6 set function , 4 7 sets restricte d t o Y , 2 8 shift, 12 7 simple eigenvalue , 16 7 simple function , 14 1 simply norma l number , 7 7 simply norma l t o bas e 2 , 18 8 spacers, 110 , 11 4 square integrable , 15 9 strictly invarian t mo d /i , 9 2 strictly invariant , 9 2 strictly periodic , 6 8 strong Cesar o convergenc e o f
sequences, 20 2 sublevels, 22 0 subset, 23 5 sufficient ring , 20 9 sufficient semi-ring , 4 1 supremum, 23 8 sweeps out , 9 7 symbolic binar y representation , 7 6 symmetric difference , 23 5
tent map , 7 8 thick set , 9 0 topologically transitive , 7 8 totally disconnected , 1 4
262 Index
totally ergodic , 10 1 tower, 10 4 transformation, 6 7 triadic interva l o f orde r n , 22 2 trivial cr-algebra , 2 7
unbounded interval , 23 7 uniformly distributed , 19 1 upper bound , 23 7
wandering, 90 , 11 2 weakly mixing , 20 5 weakly wandering , 11 2
zero density , 20 3
