Ergodic Theory via Joinings

http://dx.doi.org/10.1090/surv/101

Mathematical Surveys

and Monographs

Volume 101

Ergodic Theory via Joinings

Eli Glasner

American Mathematical Society

EDITORIAL COMMITTEE Peter S. Landweber Tudor Stefan Ratiu Michael P. Loss, Chair J. T. Stafford

2000 Mathematics Subject Classification. Primary 37Axx, 28Dxx, 37Bxx, 54H20, 20Cxx.

Library of Congress Cataloging-in-Publication D a t a Glasner, Eli, 1945-

Ergodic theory via joinings / Eli Glasner. p. cm. — (Mathematical surveys and monographs ; v. 101)

Includes bibliographical references and indexes. ISBN 0-8218-3372-3 (alk. paper) 1. Ergodic theory. 2. Topological dynamics. 3. Measure theory. I. Title. II. Mathematical

surveys and monographs ; no. 101.

QA611.5.G53 2003 515/.42—dc21 2002043617

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10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03

To Hillel Furstenberg

Contents

Introduction

Part 1. General Group Actions 11

Chapter 1. Topological Dynamics 13 1. Topological transitivity, minimality 13 2. Equicontinuity and distality 18 3. Proximality and weak mixing 22 4. The enveloping semigroup 28 5. Pointed systems and their corresponding algebras 32 6. Ellis' joint continuity theorem 33 7. Furstenberg's distal structure theorem 34 8. Almost equicontinuity 34 9. Weak almost periodicity 38 10. The unique invariant mean on WAP functions 42 11. Van der Waerden's theorem 46 12. Notes 47

Chapter 2. Dynamical Systems on Lebesgue Spaces 49 1. Lebesgue spaces 49 2. Dynamical systems and their factors 53 3. The automorphism group and some basic constructions 56 4. Poincare's recurrence theorem 58 5. Notes 59

Chapter 3. Ergodicity and Mixing Properties 61 1. Unitary representations 62 2. The Koopman representation 67 3. Rohlin's skew-product theorem 69 4. The ergodic decomposition 71 5. Group and homogeneous skew-products 72 6. Amenable groups 79 7. Ergodicity and weak mixing for Z-systems 80 8. The pointwise ergodic theorem 83 9. Mixing and the Kolmogorov property for Z-systems 86 10. Stationary stochastic processes and dynamical systems 89 11. Gaussian dynamical systems 90 12. Weak mixing of Gaussian systems 91 13. Notes 93

viii CONTENTS

Chapter 4. Invariant Measures on Topological Systems 95 1. Invariant probability measures 96 2. Generic points 98 3. Unique ergodicity 99 4. Examples of strictly ergodic systems 101 5. Minimal Heisenberg nil-systems are strictly ergodic 103 6. The geodesic and horocycle flows 105 7. ^-systems 110 8. Notes 114

Chapter 5. Spectral Theory 115 1. The spectral theorem for a unitary operator 115 2. The spectral invariants of a dynamical system 118 3. The spectral type of a if-system 120 4. Irreducible Koopman representations 121 5. Notes 123

Chapter 6. Joinings 125 1. Joinings of two systems 125 2. Composition of joinings and the semigroup of Markov operators 129 3. Group extensions and Veech's theorem 133 4. A joining characterization of homogeneous skew-products 136 5. Finite type joinings 139 6. Disjointness and the relative independence theorem 140 7. Joinings and spectrum 144 8. Notes 144

Chapter 7. Some Applications of Joinings 147 1. The Halmos-von Neumann theorem 147 2. A joining characterization of mixing 148 3. Mixing of all orders of horocycle flows 150 4. a-weak mixing 153 5. Rudolph's counterexamples machine 156 6. Notes 157

Chapter 8. Quasifactors 159 1. Factors and quasifactors 160 2. A proof of the ergodic decomposition theorem 163 3. The order of orthogonality of a quasifactor 164 4. The de Finetti-Hewitt-Savage theorem 166 5. Quasifactors and infinite order symmetric self joinings 168 6. Joining quasifactors 170 7. The symmetric product quasifactors 172 8. A weakly mixing system with a non-weakly mixing quasifactor 175 9. Notes 176

Chapter 9. Isometric and Weakly Mixing Extensions 177 1. £2(X) as a direct integral of the Hilbert bundle i)(Y) 177 2. Generalized eigenfunctions and isometric extensions 181 3. The Hilbert-Schmidt bundle 184

CONTENTS ix

4. The Y-eigenfunctions of a relative product 190 5. Weakly mixing extensions 192 6. Notes 193

Chapter 10. The Furstenberg-Zimmer Structure Theorem 195 1. /-extensions 196 2. Separating sieves, distal and /-extensions 197 3. The structure theorem 201 4. Factors and quasifactors of distal extensions 203 5. Notes 204

Chapter 11. Host's Theorem 205 1. Pairwise independent joinings 205 2. Mandrekar-Nadkarni's theorem 207 3. Proof of the purity theorem 209 4. Mixing systems of singular type are mixing of all orders 212 5. Notes 213

Chapter 12. Simple Systems and Their Self-Joinings 215 1. Group systems 216 2. Factors of simple systems 216 3. Joinings of simple systems I 218 4. JQFs of simple systems 220 5. Joinings of simple systems II 221 6. Pairwise independent joinings of simple Z-systems 223 7. Simplicity of higher orders 226 8. About 2-simple but not 3-simple systems 228 9. Notes 229

Chapter 13. Kazhdan's Property and the Geometry of Mr(X) 231 1. Strong ergodicity and property T 232 2. A theorem of Bekka and Valette 235 3. A topological characterization of property T 240 4. The Bauer Poulsen dichotomy 241 5. A characterization of the Haagerup property 243 6. Notes 244

Part 2. Entropy Theory for Z-systems 245

Chapter 14. Entropy 247 1. Topological entropy 249 2. Measure entropy 254 3. Applications of the martingale convergence theorem 260 4. Kolmogorov-Sinai theorem 263 5. Shannon-McMillan-Breiman theorem 264 6. Examples 266 7. Notes 267

Chapter 15. Symbolic Representations 1. Symbolic systems

269 269

CONTENTS

Kakutani, Rohlin and K-R towers 271 Partitions and symbolic representations 273 (a, e, 7V)-generic points 277 An ergodic theorem for towers 279 A SMB theorem for towers 281 The J-metric 283 The Jewett-Krieger theorem 291 Notes 296

Chapter 16. Constructions 299 1. Rank one systems 299 2. Chacon's transformation 301 3. A rank one a-weakly mixing system 302 4. Notes 305

Chapter 17. The Relation Between Measure and Topological Entropy 307 1. The variational principle 307 2. A combinatorial lemma 310 3. A variational principle for open covers 312 4. An application to expansive systems 315 5. Entropy capacity 316 6. Notes 317

Chapter 18. The Pinsker Algebra, CPE and Zero Entropy Systems 319 1. The Pinsker algebra 319 2. The Rohlin-Sinai theorem 322 3. Zero entropy 325 4. Notes 327

Chapter 19. Entropy Pairs 329 1. Topological entropy pairs 330 2. Measure entropy pairs 332 3. A measure entropy pair is an entropy pair 334 4. A characterization of E^ 336 5. A measure /i with E^ — Ex 337 6. Entropy pairs and the ergodic decomposition 338 7. Measure entropy pairs and factors 340 8. Topological Pinsker factors 341 9. The entropy pairs of a product system 342 10. An application to the proximal relation 344 11. Notes 345

Chapter 20. Krieger's and Ornstein's Theorems 347 1. Ornstein's fundamental lemma 347 2. Krieger's finite generator theorem 352 3. Finitely determined processes 354 4. Bernoulli processes are finitely determined 357 5. Sinai's factor theorem and Ornstein's isomorphism theorem 360 6. Notes 361

CONTENTS xi

Appendix A. Prerequisite Background and Theorems 363

Bibliography 369

Index of Symbols 379

Index of Terms 381

APPENDIX A

Prerequisite Background and Theorems

A.l. THEOREM (The subadditivity lemma). Let {an}^L1 be a sequence of non-negative numbers which is subadditive; i.e. for every m and n, a m + n < am + an, then the limit l imn^0 0 an/n exists and equals infn an/n.

(See e.g. [257], Theorem 4.9.)

Call a commutative Banach algebra B over R, with unit and satisfying | | / 2 | | = | | / | | 2 ,V/ G B, a real commutative C* algebra.

A.2. THEOREM. 1. There is a 1-1 correspondence between real commuta­tive C* algebras B and compact Hausdorff spaces X. Under this correspon­dence B is isometrically isomorphic with C(X) (the Banach algebra of real valued continuous functions on X with sup norm: \\f\\ — supxeX\f(x)\). The space X is the Gelfand space or the spectrum of B; i.e. the set of all continuous multiplicative homomorphisms of the algebra B into R. The topology on X can be described as the weak* topology induced on X when it is considered as a subset of the dual Banach space B*. The algebra B is separable iff X is metrizable.

2. There is a 1-1 correspondence between closed subalgebras containing the unit 1 G D C B, and continuous onto maps (j) : X —» Y from the compact space X, the spectrum of B, onto the compact space Y, the spectrum of D, where for a functional x G X, (j){x) G Y is the restriction of x to the subalgebra D.

3. There is a 1-1 correspondence between self homeomorphisms T of the com­pact space X and automorphisms of C(X) = B (also denoted by T) given by: Tf(x) = / ( T - ^ ) , (x G XJ G C{X)).

(See e.g. [53], Section IV.5.)

A.3. THEOREM (The Stone-Cech compactification). Let X be a completely reg­ular (Hausdorff) topological space and let f3X denote the Gelfand space of the real commutative C* algebra Cb(X) consisting of the bounded continuous real valued functions on X with supremum norm.

1. The map j : X —> /3X sending the point x G X to the multiplicative homomorphism j(x) G j3X defined by j(x)(F) = F(x),F G Cb(X), is a homeomorphism of X onto a dense subset of (3X.

2. The compactification j : X —> j3X (which is called the Stone-Cech com­pactification of X) has the following universal property. Every continuous map f : X —>• Z into a compact Hausdorff space Z has a unique extension to a continuous function f : /3X —> Z.

363

364 A. PREREQUISITE BACKGROUND AND THEOREMS

3. The Stone-Cech compactification of X is unique in the following sense. If i : X —> Z is a homeomorphism of X into a compact Hausdorff space Z with dense image then there exists a homeomorphism onto <p : f3X —> Z which makes the diagram

px ^z commutative.

(See e.g. [53], Section IV.6, and [62], Sections 3.5 and 3.6.)

Let X be a separable metrizable space. Let Cb(X) (or just C(X) when X is compact) be the algebra of bounded continuous real valued functions on X, and M(X) the set of probability Borel measures on X. We equip M(X) with the smallest topology which makes all maps fi i-» /x(/), / G Cb(X), of M(X) into R, continuous. Let d be a compatible metric on X such that the corresponding completion (X,d) is compact (there always is such a metric). X is a dense subset of the compact metric space X. It is a G$ subset of X iff X is Polish. Denote by Ud(X) the space of d-uniformly continuous, real valued functions on X. (Thus, Ud(X) is isometrically isomorphic with the Banach algebra C(X)). Let {fn}n>i be a dense sequence in Ud(X) and let:

£>(M,i/) = V 2 \-^-JM/n)-K/n)l ^ llf II

By the Riesz representation theorem we can identify M(X) with the convex weak* compact subset of C(X)*, the dual Banach space, consisting of all positive func­t ional \i with /i(l) = 1. It is easy to see that the map /i i—> /i, defined by (i(A) = fi(AnX) is an embedding of M(X) into M(X) with image {p, £ M(X) : fi(X) = 1} and that this image is a dense subset of M(X).

A.4. THEOREM. Let X be a separable metrizable space, then 1. M{X) is a separable metrizable space and D is a compatible metric. 2. X is compact iff M(X) is compact. 3. X is Polish iff M(X) is Polish. 4. The Borel a-algebra *B(M(X)) is the smallest a-algebra with respect to

which the maps /i ^ /x(^4), A G *B(X) are measurable and also the smallest a-algebra with respect to which the maps \i ^ / i( /) , / a bounded Borel func­tion on X, are measurable. In particular the Borel structure on M(X) for a standard Borel space X does not depend on a particular Polish topology.

(See e.g. [152].)

A.5. THEOREM. Let X be a separable metrizable space, / i n , / / G M(X), then the following are equivalent:

1. limn^oo/in = \i. 2. For every f G Cb(X), WvUn^oo J f d[in — J f d[i (or if this holds for every

f in a dense subset ofUd(X)).

A. PREREQUISITE BACKGROUND AND THEOREMS 365

3. l imsup n ^ n (F) < id(F) for every closed subset F of X. 4. limmfn jdn(U) > /d(U) for every open subset U of X. 5. limn^(X) /dn(A) = fd(A) for every Borel subset A of X with /d(dA) — 0.

(See [152], Theorem 17.20.)

A.6. THEOREM (The Kolmogorov consistency theorem). Let {Xn : n G N} be a sequence of standard Borel spaces and suppose that for every finite subset F c N there is a probability Borel measure /dp defined on Xp — Y\jeF -^j in such a way that whenever F\ C F2 then the image of /dp2 under the projection map TTF2,F! '• Xp2 —> Xp1 is LIF17 then there is a unique probability Borel measure /d on the standard Borel space X = YijeN^J w^h 7I"F(M) = HF for every finite F c N , where np : X —» Xp is the projection map.

(See e.g. [186], page 82.)

A.7. THEOREM (Measure disintegration). Let X,Y be standard Borel spaces, and IT : X —> Y a Borel map. Let /d G M(X) and let v — 7r(/i) = / i o TT-1 be its image in M(Y), then there is a Borel map y 1—>• /iy, from Y into M(X) such that:

1. For v almost every y GY, /dy(7r~1(y)) = 1. 2.

/d= Liydis(y).

(The latter means that fd(A) — J /dy(A) du(y) for every Borel subset A of X, or equivalently that j f d/d — f f f(x) d/dy(x) dv(y), for every bounded Borel function f : X —> R.J

Moreover such a map is unique in the following sense: If y 1—> /dy is another such map, then fdy = /dyl v-a.e.

(See e.g. [35] and [77], Theorem 5.8.)

Let X =(X, X,/i) and Y =(Y", y,^) be separable probability measure spaces and let cj> : X —• Y be a measurable measure-preserving map. This map defines an embedding of L1(Y) into ^ 1 (X) and we shall thus consider i^1(Y) as a closed subspace of L1(X).

A.8. THEOREM (Conditional expectation). The conditional expectation opera­tor E Y : i^1(X) —> L1(Y) is characterized by the following properties:

1. E Y is a linear operator. 2. E Y / > 0 for f > 0. 3. EYg = g forge LX{Y). 4. EY(gf) - gEYf forge L\Y) and f G L^X). 5. V / G L ^ X ) , fEYfdv = ffd/d.

In addition we have the following properties: 6. Let ip be a convex nonnegative function of a real variable. If f and I/J o f

are in L1 (X) then V>(EY/)<EYW>o/).

7. The conditional expectation operator maps each LP(X) to LP(Y), 1 < p < oo; with | | E Y / | | P < | | / | | p .

366 A. PREREQUISITE BACKGROUND AND THEOREMS

8. On L2(X), E Y is the orthogonal projection of L2(X.) onto L2(Y). 9. When X and Y are standard Borel spaces then — letting \i — J nydv(y)

be the disintegration of \i over v — for every f G L1(X), / is in L1(/jJy) for v-a.e. y and

EY/(2/) = J fdfJLy.

(See e.g. [186].)

A.9. THEOREM. Suppose that (X, X, /x) is a standard Lebesgue space and A C L°°(X, X,//) a conjugation invariant subalgebra. Let 23 be the smallest a-algebra of Borel sets in X with respect to which all functions in A are measurable. Then

els,4 = L 2 (X,2) = {f e L2(X,X,^) : / is ^-measurable }.

(See [267], Theorem 1.2.)

A.10. THEOREM. Let (X, X,/i) and (Y^,v) be probability spaces and let <3> : (y,^) —» (X,/i) be a homomorphism of measure algebras, then the map VIB — 1$(J3) has a unique extension to a linear operator V : L2[y) —» L2(/i) satisfying

1. y 25 an isometry: (Vf,Vg) = (/,#) (Tience | | y / | | = | | / | | ; . 2. V sends bounded functions to bounded functions. 3. V{fg) = V(f)V(g) for bounded functions.

If moreover & is an isomorphism (i. e. onto), then V is a unitary isomorphism of L2(is) onto L2(n). Conversely, ifV : L2{y) —> L2(/i) is a linear operator satisfying the conditions 1 — 3 above, then for every B G y there is unique (mod 0) A G X such that VIB = 1A a^d the map &(B) — A is a homomorphism of measure algebras; an isomorphism when V is a unitary isomorphism.

(See [257], page 56.)

A.11. THEOREM. Let (T, A) be the circle equipped with Lebesgue's measure and V : L2(A) —> L2(X) a unitary operator which is also positive (i.e. Vf > 0 for f > 0). Then V has the form: Vf{t) = J~^(t)f(Tt), where T : T -> T is an invertible measurable map.

For a proof refer to the proof of Lamperti's Theorem in [215], Chapter 15, Theorem 24, together with the following observation: Royden proves a version of Theorem A. 11 for any surjective isometry of LP(X) which applies to all 1 < p < oo but p = 2. Of course this more general statement is false for a general unitary operator V : L2(X) —> L2(X). However, when in addition one assumes that V is positive, then one gets that for two disjoint sets A, B C X the non-negative functions / = VIA and g = VIB satisfy

f f-gdX= (V1A, V1B) = (1A, 1 B ) = [ lA-lBdX = 0

and it follows that (Vl^) • (V\B) = 0 a-e- With this information one can complete the proof of Theorem A. 11 in exactly the same way it is done in Royden's book.

A.12. THEOREM (Eberlein-Smulian's theorem). Let A be a subset of a Banach space X, then for the weak topology on X the following are equivalent

A. PREREQUISITE BACKGROUND AND THEOREMS 367

1. A is conditionally compact; i.e. weak-clsA is weakly compact. 2. A is conditionally sequentially compact; i.e. every sequence of elements of

A has a weakly convergent subsequence. 3. A is conditionally countably compact; i.e. every sequence of elements of A

has a cluster point in X.

(See [53], [47], or [158].)

For the next theorem recall the following definitions. Let E be a locally convex topological vector space and P C E a closed convex cone such tha t P — P = E. A compact convex Q C P is a base for the cone P if for every y G P there is a unique x G X and a > 0 such tha t y — ax. When P admits a base then clearly P D (—P) = {0} and it follows tha t the partial order defined on E by x > y <=^ x — y £ P satisfies: x > y and y > x implies x = y. The pair (£", P) is a lattice if for every x, y G X there exists a greatest lower bound, i.e. a lower bound z such tha t for every lower bound w we have w < z (we write z = x A y). Finally the convex compact subset Q C E is a simplex if it is the base of a cone P as above for which (E, P) is a lattice; it is a Bauer simplex if the set ext (Q) of extreme points of Q is closed.

A. 13. T H E O R E M (Choquet 's theorem). Let Q be a metrizable compact convex subset of a locally convex topological vector space E.

1. The subset ext (Q) of extreme points of Q is a Gs subset of Q. 2. For every XQ G Q there exists a probability measure K on Q such that

tt(ext (Q)) = 1 and such that

/ f(x) dn(x) = f(x0) -'ext (Q)

for every bounded upper semi-continuous affine function / : Q —> M. (Such a measure n is said to represent the point XQ.)

3. The point XQ is in ext (Q) iff the point mass K, = SXQ is the only probability measure on ext (Q) that represents XQ.

4. Q is a simplex iff for every xo G Q there is just one representing measure. 5. Q is a Bauer simplex iff it is affinelly homeomorphic to the simplex M(X),

where X — ext (Q) and M(X) is the space of Borel probability measures on X with the weak* topology.

(See [206] for details.)

A topological group G is called monothetic if there exists an element go G G (called a topological generator) such tha t the set {#Q : n G Z} is dense in G. It is easy to check tha t every monothetic group is abelian and tha t , when G is second countable, the set of topological generators is a dense Gs subset of G.

A. 14. T H E O R E M . Let K be a compact, abelian, second countable, topological group. Then K is monothetic iff its dual group K is isomorphic (as an abstract group) to a countable subgroup of the circle S1 — {z G C : \z\ = 1}. When K is monothetic every such isomorphism is implemented by a choice of a topological generator a G K and then by mapping \ £ K ^n^° x{°) £ S1.

(See [119], page 390.)

368 A. PREREQUISITE BACKGROUND AND THEOREMS

A.15. THEOREM (Peter-Weyl's theorem). Let K be a compact, second count­able, topological group with normalized Haar measure m. Let E denote the countable set of all equivalence classes (with respect to the relation of unitary equivalence) of irreducible unitary representations of G on (necessarily) finite dimensional Hilbert spaces. For every a G XI let U^ : G —> U(S)a) be a representative element of a with dim-S)a — da and {£i, ("2,.. •, Cda} & fixed orthonormal basis. Let

be the unitary matrix of the operator Uxa\ (x G G) in this basis. The set of

functions 4 / 2 « S \ (<re£, l<j,k<da),

is an orthonormal basis for L2(G, m).

(See [120], page 24.)

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Index of Symbols

A(K), H(M) (Galois correspondence for group extensions), 57, 133, 135

(A) (icer generated by A), 341 A n , A_p (ergodic sums), 79, 83 AP(T) (algebra of almost periodic

functions), 38 Aut (X, r ) (group of automorphisms of a to­

pological dynamical system), 13 Aut (X) (group of automorphisms of a mea­

sure dynamical system), 56 a~, a T , 258 a C /?, 349 B(p) = B(Pl, ...,pt) (Bernoulli shifts), 82 B(X) , 67 BV, B V - , 235 B(T), Bn (spaces of matrix coefficients), 38 /3X, /3T (Stone-Cech compactification), 30,

363 c, |c| (a column and its carrier), 271 c(K), Cfj,(K) (entropy capacity), 316 co(r) (space of functions vanishing at infin­

ity), 38 Co, 178 Ci, 186 A x = {(x,x) : x e X} (diagonal), 13 d (metric), 283 dN (metric), 285 dent, 275 dham, 271 draeas, 270, 274, 278, 285, 349, 354 d p a r t , 274, 275 dist(a), 357 E, E3" (expectation, conditional expectation

with respect to the cr-algebra 3r), 365 E(X) , 67 E(X,T), E(X,T) (enveloping

semigroup), 28 Efj, (/i-entropy pairs), 332 ^(X,T)> Ex (topological entropy pairs), 332 £ ( X / Y ) (closure of Y-eigenfunctions), 181 EQ(X) (set of equicontinuity points), 34 ext (Q), ext (K) (set of extreme points), 367,

97, 168, 231

G(X) , 67 gr(/x,0) (graph joining), 126 0 (Hilbert-Schmidt bundle), 185 H(U) (topological entropy of the cover IX),

249 h(U) (topological T-entropy of the cover II),

249 htop(X,T) (topological entropy of the sys­

tem (X,T)) , 249 H(a) (entropy of the partition a ) , 254 H(a\3r) (conditional entropy of the partition

a with respect to the cr-algebra JF), 254 h>n(ct) ~ hn(ct,T) (T,/i-entropy of the par­

tition a ) , 258 7iM(a|5F) (conditional T, /i-entropy of the par­

tition a with respect to the cr-algebra 9), 258

/i(X) (entropy of the dynamical system X) , 258

/ i(X|Y) (conditional entropy of the dynam­ical system X relative to the factor Y ) , 258

H(}$), (Galois correspondence for group ex­tensions), 133, 135

S) (Hilbert space), 62 h (Hilbert bundle), 178 1 (at), I3" (pt) ((conditional) information func­

tion), 254 I (X) , 67 3, Inv(L 2 (X)) , 63, 67, 181 Iso (fj) (group of isometries of a Hilbert

space), 235 J ( X ) , J e (X) , J ( X , Y ) , J e ( X , Y ) (set of (er­

godic) joinings), 125 J p ( X ) , JuXu(X), 127 J Z ( X , Y) (joinings over a common factor),

127 j ( f c)(X), 148 K', ^ k \ R^k\ n^°°\ K(°°^ (joinings associ­

ated with a quasifactor), 164 £ = { 1 , 2 , . . . , ^ } (an alphabet), 16, 269, 310 £(io) (The Banach algebra of bounded linear

operators on a Hilbert space), 62

380 INDEX OF SYMBOLS

A* (adjoint joining), 129 A(a A/3), 274 m (invariant mean on WAP functions), 43 Mr(X), MT(X) (set of T-invariant proba­

bility measures), 96 Mp r g (X) , M^ r g (X) (set of ergodic r-invari-

ant probability measures), 97, 147 Mmax(X) (set of measures of maximal en­

tropy), 310 M ( X ) , M ( X , Y ) (semigroup (set) of com­

muting Markov operators), 132 p(aA(3) ( = < a r t ( a , / ? ) ) , 274 N(A, B) = {7 6 T : 7 A H B / 0}, 13 N(x,B) = {7 e r :jx e £ } , 13 N(U) (minimal cardinality of a subcover of

the cover U), 249 0rx (r-orbit of x), 13 Qrx (r-orbit closure of x), 13 P = P ( X , r ) , P 7 r , p ( n ) (proximal relations),

22 P<r,P3 (orthogonal projections), 67 II(X) (Pinsker algebra), 319 Q(/x), Q(X) (set of quasifactors), 160 Q J M , Q J ( X ) , Q J ( X ) (set of (ergodic) join­

ing quasifactors), 171 Q(7r): Qj(n)> e ^ c (quasifactors over a fac­

tor), 160, 171 Q = Q(X,T),Q7T (regionally proximal rela­

tions), 22 QG{x) (set of measures for which x is quasi-

generic), 98 RTV = {(x,xl) : 7r(x) = K(X')} (relation de­

termined by a homomorphism), 15 S(p) (property), 333 s(/i), s2(/x) (support), 336 SFT (subshift of finite type), 17

spn(e), srn(e), sp(e,T), sr(e,T), sp(T:d), sr(T,d) (Bowen's entropy), 251

sym (symmetrization map), 58 S (group of permutations), 168 CTX (spectral measure), 116 t, |t| (a tower and its carrier), 271 T (Kazhdan's property), 231 Tw (Weak Kazhdan property), 243 r o A (composition of joinings), 130 u (minimal idempotent), 30 u (minimal idempotent in E) , 63, 67 C/7, UT (Koopman representation), 67 U(S)) (The group of unitary operators), 38 Wo(r) (space of flight functions), 44 WAP(T) (algebra of weakly almost periodic

functions), 38 X = (X,X, /x , r ) , X = (X,X,/x,T) (mea­

sure-preserving dynamical systems), 53, 55

X^7/ (r-symmetric product system), 58 (X,T), (X,Z), (X,T) (topological dynami­

cal systems), 13 ( X n , T ) (n-fold product system), 13 Xd, Xeq (maximal distal (equicontinuous)

factor), 18 X p , Xp(fi) (topological Pinsker

factors), 341 Y = (Y,y , i / , r ) , Y = (Y,y,i/ ,T) (measure-

preserving dynamical systems), 53, 55 Y x (U, p) (skew-product with cocycle a),

a 57

Z = (Z,Z,77,r), Z = (Z,Z,77,T) (measure-preserving dynamical systems), 53, 55

Z(x) (cyclic subspace generated by x), 116 l r -< 7T, l r C 7r, 235

Index of Terms

AE (almost equicontinuous), 34 affine actions (on a Hilbert space), 235 affine property (of the entropy function),

260, 276, 310 affine transformation, 204 amenable group, 79, 86, 96, 121, 141, 167 analytic set, 51 aperiodic

Z-system, 296 matrix, 82

asymptotically invariant sets, 232

Banach's problem, 120 barycenter (equation), 160 Bernoulli shifts, 16, 82, 87, 266, 360 Birkhoff (pointwise ergodic theorem), 83

for towers, 279 Borel

action, 55 set, 51 space, 51 cr-algebra, 51

Breiman (lemma), 264 Bruhat (decomposition), 105

Cartan (decomposition), 105 Chacon (system), 27, 101, 301 chaos (first chaos of a Gauss system), 91, 92 Choquet (theorem), 367, 72, 95, 276 Chung (lemma), 261 coboundary, 57, 71, 235 cocycle

(associated with a dynamical system), 57, 71

(associated with a unitary representation), 235

cohomologous (cocycles), 71, 235 copying (names on towers), 273 CPE (complete positive entropy)

measure, 323 topological, 330

cyclic subspace (for a representation in a Hilbert space), 116

de Finetti-Hewitt-Savage (theorem), 167

deterministic (process, system), 319 discrete spectrum, 63, 67, 147

generalized, 196 relative (discrete spectrum over Y) , 181

disjointness (of dynamical systems) topological, 13 measure, 140 weak, 13

distal measure, 197 topological, 18, 34

Eberlein-Smulian (theorem) 366, 38 eigenfunction

generalized, or Y-eigenfunction, 179, 181 measure, 103, 119 topological, 19

Ellis (joint continuity theorem), 33 empirical distribution, 270 entropy capacity, 316 entropy pair

measure, 332 topological, 330

enveloping semigroup, 28 equicontinuous, 18

almost (AE), 34 ergodic decomposition, 72, 85, 164 ergodic theorem

Birkhoff or pointwise, 83, for amenable groups, 86 for towers, 279 von Neumann or mean, 79

ergodic topological system, 110 ergodicity (of a dynamical system), 67

intrinsic, 310 strict, 99 strong, 232 unique, 99

ergodicity (of a representation), 63 E-system, 110, 111, 113 expansive (dynamical system), 315 extension (or factor)

almost 1-1, 26 distal

382 INDEX OF TERMS

measure, 197, 201, 202, 203 topological, 22

group measure, 57, 76, 77, 133 topological, 15

homogeneous measure, 58, 137, 183

I, 15, 198 isometric, 182, 196, 183, 190, 198 proximal, 22 with separating sieve, 197, 198 weakly mixing

measure, 192, 201, 202 topological, 25

extremal cr-algebra, 324, 325

finitely determined, 354, 358 fixed point property, 96 flight function, 44, 93 flow (real), 105, 110 F0mer sequence, 79

tempered, 85 full measure, 97, 111 Furstenberg (structure theorem for distal

systems), 34 Furstenberg-Zimmer (structure theorem),

202, 327

Galois correspondence for group extensions, 135

Gauss system, 90, 92, 242 Gelfand (space), 363, 32, 38 generator, strong generator, 263 generic point, 98, 100, 78, 110, 137

(i/, e, 7V)-generic, or (e, A/")-generic (sequence), 270

geodesic flow, 105, 106 golden mean (subshift), 17, 251 Grothendieck (theorem), 38

Haagerup property, 243 Hahn-Banach (theorem), 45, 99 Halmos-von Neumann (theorem), 147 Heisenberg (group, nil-system), 20, 104 Herglotz (theorem), 116 horocycle flow, 26, 105, 101, 109 Host (singular spectrum theorem), 205

icer (invariant closed equivalence relation), 15, 341

incontractible, 26 intrinsically ergodic system, 310 inverse limit (system), 127 ZP-set, 15 isometric (system)

extension, 182 (measure), 66, 69 system, 67 (topological), 18

Iwasawa (decomposition), 105

Jankov-von Neumann (theorem), 52 Jewett-Krieger (theorem), 291 joining, 125, 136, 142, 221, 168, 347, 349

composition of, 130 finite type, 139 graph, 126 minimal self-joinings, 126, 215, 217, 228 off-diagonal, 126, 215 orthogonal, 130 relatively independent, 126 topological, 13

JQF (joining quasifactor), 171 Jordan decomposition, 99

Kazhdan property, 231 K-system (Kolmogorov property), 87, 88 ,

120, 323, 326 Kolmogorov (consistency theorem), 365, 90,

127 Kolmogorov representation, 89 Kolmogorov-Sinai (theorem), 263 Koopman representation, 67 Krieger (finite generator theorem), 352 Kronecker (system)

measure 58, 119 topological, 18

Kronecker-Weyl (theorem), 81 Krylov-Bogolubov (theorem), 97

Lebesgue space, 52 Lusin (theorem), 52

M-system, 14, 110, 111 Mackey range (of a cocycle), 72 Mackey-Weil (theorem), 207, 136, 218 Mandrekar-Nadkarni (theorem), 207 Markov-Kakutani (fixed point theorem), 43,

45 Markov operators (commuting), 132 Markov systems, 82, 266 martingale (convergence theorem), 261, 262,

335 matrix coefficients, 38, 169 Mautner (lemma), 106 mean (left or right invariant), 43 mean ergodic theorem (von Neumann), 79 measure of maximal entropy, 310 mild mixing, 169, 169 mincenter, 14 minimal cocycle, 72 minimal (dynamical system), 14 minimal point, 14 minimal self-joinings (MSJ) , 126, 215, 217,

228 min-max point, 235 mixing

measure, 86, 149 of order /c, of all orders, 86, 205, 228 representation, 63 topological, 23

INDEX OF TERMS 383

uniform, 86 model (for a dynamical system, topological,

Cantor, Polish), 55, 291, 53 module (over Y , or Y-module), 179 monothetic group, 81, 367, 19, 147 Morse (system), 27, 101 /i-measurable set, 51

nil-system, 20, 101, 103 non-periodic (ergodic dynamical system),

272

order of orthogonality (of a quasifactor), 166, 220

Ornstein (fundamental lemma), 349 (isomorphism theorem), 360

orthogonal (joinings), 130

P-system, 14, 110, 111 Peter-Weyl (theorem), 368, 40, 66, 69, 183 Pinsker algebra (II(X)), 319 Pinsker factor

measure, 319 /i topological, 341 topological, 341

Pinsker's formula, 321 Pinsker's inequality, 357 Pinsker conjecture, 325 Poincare (recurrence theorem), 58 point transitive (dynamical system), 13 pointed dynamical system, 13 pointwise minimal (dynamical system), 14 Polish space, 51 POOD (product of off-diagonals), 215, 220,

221 positive definite (function), 90, 115, 92, 242 prime (dynamical system), 297 process (associated with a partition), 273 proximal relation, 22

quasi-discrete spectrum, 204 quasifactor, 160, 168, 203, 220, 326

ergodically embedded, 126 finite type, 172 joining, 171 mixingly embedded, 172 order of orthogonality, 166, 220 points, 160 trivial, 160 weak mixingly embedded, 172

quasi-generic point, 98 quasi-invariant measure, 207

Rajchman (measure), 119 rank one (system), 299 regionally proximal relation, 22 regular points, 110 regular (topological dynamical system), 26 replete partition, 332 return time function, 271

rigid system, 63, 67, 68, 81, 119, 153, 224, 227

{nfc}-rigid system, 81, 169 Rohlin (skew-product theorem), 69 Rohlin's lemma, 272, 296 Rohlin's problem, 87, 205, 228 Rohlin-Sinai (theorem), 323 Rohlin tower (or column), 271 Rudolph (counter-examples), 156

semigroup enveloping, 28 semitopological, 41, 63, 132

separating open cover, 315 partition, 332 sieve, 197

SMB (The Shannon-McMillan-Breiman the­orem), 265

for towers, 281 simple (dynamical system), 215,

of order 2, of all orders, 126 of order k, 215

simple spectrum, 117, 120, 300 simplex, 231, 163, 367

Bauer, 231, 367, 167, 240 Poulsen, 231, 242

Sinai (factor theorem), 360 skew-product, 57

group, 58, 72, 77 homogeneous, 58, 72, 77, 183 Rohlin's skew product theorem, 69

spectral measure, multiplicity, sequence, type, 117 theorem for unitary operators, 117

spectrum (of a unitary operator) absolutely continuous, discrete, homoge­

neous, Lebesgue, infinite Lebesgue, sim­ple, singular, 117

standard Borel space, 51 standard cover, 330 stochastic process (stationary), 273 Stone-Cech compactification, 30, 363 strong ergodicity 232 subshift, 16, 82

of finite type, 17 substitution (system), 27 subsystem, 13 sweeping set, 271 symbolic representation

(of a process), 273 measure, 273

syndetic, 15 syndetically transitive, 24

thick, 15 thickly syndetic, 44 Toeplitz (sequence, system), 28

384 INDEX OF TERMS

totally ergodic (dynamical system), 119, 291 tower (height, roof, floor)

Kakutani, 271 Kakutani-Rohlin, K-R, 271, 291 Rohlin, 271, 349

transitive (topologically transitive system), 13

typical sequence, (is, e, N, 2n + 1)-typical, 270

uniformly (or well) distributed sequences, 102

universally measurable set, 51 UPE (uniform positive entropy), 330, 297 upper semi-continuity, 15, 270, 310, 367

van der Waerden (theorem), 47 Veech (theorem), 136, 216 von Neumann (mean ergodic theorem), 79

WAP (weakly almost periodic) function, 38 system, 40, 63, 132

weak containment, 235 weak mixing (of a representation), 63 weak mixing (of a dynamical system), 67

a, 153, 156, 302 measure, 67 topological, 14

weak Pinsker conjecture, 325 weakly isomorphic systems, 157

Weyl (theorem), 102

Titles in This Series

99 Phi l ip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guil lemin, Viktor Ginzburg, and Yael Karshon, Moment maps,

cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Mart in Markl, Steve Shnider, and J im Stasheff, Operads in algebra, topology and

physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D . Neuse l and Larry Smith , Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2:

Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1:

Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery , A tour of subriemannian geometries, their geodesies and

applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant

magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi , Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. NLaz'ya., and J. Rossmann, Spectral problems associated with

corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant

differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential

operators, and layer potentials, 2000 80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module

theory, 2000 79 Joseph A. Cima and Wil l iam T. Ross , The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt , KP or mKP: Noncommutative mathematics of Lagrangian,

Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and D e n e s Petz , The semicircle law, free random variables and entropy,

2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W . Stroock, An introduction to the analysis of paths on a Riemannian manifold,

2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems

and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor) , Surgery on compact manifolds, second edition,

1999 68 David A. Cox and Sheldon Katz , Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N . Wallach, Continuous cohomology, discrete subgroups, and

representations of reductive groups, second edition, 2000

TITLES IN THIS SERIES

66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,

1999 64 Rene A. Carmona and Boris Rozovskii , Editors, Stochastic partial differential

equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W . Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic

algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Wil l iams, Morita equivalence and continuous-trace

C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B . Frenkel, and Alexander A. Kirillov, Jr., Lectures on

representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum

groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in

analysis, 1997 53 Andreas Kriegl and Peter W . Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in

domains with point singularities, 1997 51 Jan Maly and Wil l iam P. Ziemer, Fine regularity of solutions of elliptic partial

differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential

equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert , Integer-valued polynomials, 1997 47 A. D . Elmendorf, I. Kriz, M. A. Mandell , and J. P. May (with an appendix by

M. Cole) , Rings, modules, and algebras in stable homotopy theory, 1997 46 S tephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and A d a m O. Hausknecht, Cogroups and co-rings in

categories of associative rings, 1996 44 J. Amoros , M. Burger, K. Corlette , D . Kotschick, and D . Toledo, Fundamental

groups of compact Kahler manifolds, 1996 43 James E. Humphreys , Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B . Nat ion , Free lattices, 1995 41 Hal L. Smith , Monotone dynamical systems: an introduction to the theory of

competitive and cooperative systems, 1995 40.5 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 5, 2002 40.4 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 4, 1999 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 3, 1998 40.2 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 2, 1995

TITLES IN THIS SERIES

40.1 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups, number 1, 1994

39 Sigurdur Helgason, Geometric analysis on symmetric spaces, 1994 38 Guy David and Stephen Semmes , Analysis of and on uniformly rectifiable sets, 1993 37 Leonard Lewin, Editor, Structural properties of polylogarithms, 1991 36 John B. Conway, The theory of subnormal operators, 1991 35 Shreeram S. Abhyankar, Algebraic geometry for scientists and engineers, 1990 34 Victor Isakov, Inverse source problems, 1990 33 Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean

fields, 1990 32 Howard Jacobowitz , An introduction to CR structures, 1990 31 Paul J. Sally, Jr. and David A. Vogan, Jr., Editors, Representation theory and

harmonic analysis on semisimple Lie groups, 1989 30 Thomas W . Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, 1989 29 Alan L. T. Paterson, Amenability, 1988 28 Richard Beals , Percy Deift , and Carlos Tomei, Direct and inverse scattering on the

line, 1988 27 Nathan J. Fine, Basic hypergeometric series and applications, 1988 26 Hari Bercovici , Operator theory and arithmetic in if00, 1988 25 Jack K. Hale, Asymptotic behavior of dissipative systems, 1988 24 Lance W . Small, Editor, Noetherian rings and their applications, 1987 23 E. H. Rothe , Introduction to various aspects of degree theory in Banach spaces, 1986 22 Michael E. Taylor, Noncommutative harmonic analysis, 1986 21 Albert Baernstein, David Drasin, Peter Duren, and Albert Marden, Editors,

The Bieberbach conjecture: Proceedings of the symposium on the occasion of the proof, 1986

20 Kenneth R. Goodearl , Partially ordered abelian groups with interpolation, 1986 19 Gregory V. Chudnovsky, Contributions to the theory of transcendental numbers, 1984 18 Frank B. Knight, Essentials of Brownian motion and diffusion, 1981 17 Le Baron O. Ferguson, Approximation by polynomials with integral coefficients, 1980 16 O. T imothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl , Jr., Vector measures, 1977 14 V. Guil lemin and S. Sternberg, Geometric asymptotics, 1977 13 C. Pearcy, Editor, Topics in operator theory, 1974 12 J. R. Isbell, Uniform spaces. 1964 11 J. Cronin, Fixed points and topological degree in nonlinear analysis, 1964 10 R. Ayoub, An introduction to the analytic theory of numbers, 1963 9 Arthur Sard, Linear approximation, 1963 8 J. Lehner, Discontinuous groups and automorphic functions, 1964

7.2 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume II, 1961 7.1 A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Volume I, 1961

6 C. C. Chevalley, Introduction to the theory of algebraic functions of one variable, 1951 5 S. Bergman, The kernel function and conformal mapping, 1950 4 O. F. G. Schilling, The theory of valuations, 1950 3 M. Marden, Geometry of polynomials, 1949 2 N . Jacobson, The theory of rings, 1943 1 J. A. Shohat and J. D . Tamarkin, The problem of moments, 1943