http://dx.doi.org/10.1090/surv/101
Ergodic Theory via Joinings 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 Data 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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© 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 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 269 1. Symbolic systems 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, am+n < am + an, then the limit limn^00 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 commuta tive C* algebras B and compact Hausdorff spaces X. Under this correspon dence 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 com pact 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 reg ular (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 com pactification 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
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/) = \-^-JM/n)-K/n)V2 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 func tional \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 func tion 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, /in,// 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. limsupn^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 FcN 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 w the standard Borel space X = YijeN^J ^h 7I"F(M) = HF for every finite FcN, 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) =/io 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 1. For v almost every y GY, /dy(7r~ (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 opera tor EY : i^1(X) —> L1(Y) is characterized by the following properties: 1. EY is a linear operator. 2. EY/ > 0 for f > 0. 3. EYg = g forge LX{Y). 4. EY(gf) - gEYf forge L\Y) and f G L^X). 5. V/GL^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/)
8. On L2(X), EY 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) 1 1 be the disintegration of \i over v — for every f G L (X), / is in L (/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 = L2(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 — 2 2 1$(J3) has a unique extension to a linear operator V : L [y) —» L (/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, 1B) = [ 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 that 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 that y — ax. When P admits a base then clearly P D (—P) = {0} and it follows that 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 that 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. THEOREM (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 that the set {#Q : n G Z} is dense in G. It is easy to check that every monothetic group is abelian and that, when G is second countable, the set of topological generators is a dense Gs subset of G. A. 14. THEOREM. 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 count able, 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
a be the unitary matrix of the operator Ux \ (x G G) in this basis. The set of functions /2 4 «S\ ( [1] J. Aaronson, An introduction to infinite ergodic theory, Math. Surveys and Monographs 50, Amer. Math. Soc. 1997. [2] L. M. Abramov, Metric automorphisms with quasi-discrete spectrum, Izv. Akad. Nauk, SSR Ser. Mat. 26, (1962), 513-530. Amer. Math. Soc. Transl. 39, (1964), 37-56. [3] S. Adams, Another proof of Moore's ergodicity theorem for SL(2,R), Topological dy namics and applications, Contemporary Mathematics 215, a volume in honor of R. Ellis, 1988, pp. 183-187. [4] R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114, (1965), 309-319. [5] R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. USA 57, (1967), 1573-1576. [6] , Similarity of automorphisms of the torus, Memoirs of the Amer. Math. Soc. 98, 1970. [7] E. Akin, The general topology of dynamical systems, Amer. Math. Soc. , Providence, RI, 1993. [8] , Recurrence in topological dynamics: Furstenberg families and Ellis actions, Plenum Press, New York, 1997. [9] E. Akin, J. Auslander, and K. Berg, When is a transitive map chaotic , Convergence in Ergodic Theory and Probability, Walter de Gruyter & Co. 1996, pp. 25-40. [10] , Almost equicontinuity and the enveloping semigroup, Topological dynamics and applications, Contemporary Mathematics 215, a volume in honor of R. Ellis, 1998, pp. 75-81. [11] J. Auslander, Minimal Flows and their Extensions, Mathematics Studies 153, Notas de Matematica, 1988. [12] L. Auslander, L. Green and F. Hahn Flows on homogeneous spaces, Annals of mathe matics studies 53, Princeton University Press, Princeton, New Jersey, 1963. [13] J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's Definition of Chaos, Amer. Math. Monthlly 99, (1992),332-334. [14] J. R. Baxter, A class of ergodic transformations having simple spectrum, Proc. Amer. Math. Soc. 27, (1971), 275-279. [15] M. B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Math. Soc. Lecture Note Series 269, Cambridge University Press, 2000. [16] M. B. Bekka and A. Valette, Kazhdan's property (T) and amenable representations, Math. Zeitschrift 212, (1993), 293-299. [17] F. Beleznay and M. Foreman, The collection of distal flows is not Borel, Amer. J. Math. 117, (1995), 203-239. [18] K. R. Berg, Entropy of torus automorphisms, Topological dynamics, an international symposium, J. Auslander and W.A. Gottschalk, eds., W. A. Benjamin, New York, 1968. [19] V. Bergelson, H. Furstenberg, N. Hindman and Y. Katznelson, An algebraic proof of van der Waerden's theorem, L'Enseignement Math. 35, (1989), 209-215. [20] V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Sze- meredi's theorems, Journal of Amer. Math. Soc. 9, (1996), 725-753 [21] V. Bergelson and J. Rosenblatt, Mixing actions of groups, 111. J. of Math. 32, (1988), 65-80. [22] P. Billingsley, Ergodic theory and information, Wiley, New York, 1965. 369 370 BIBLIOGRAPHY [23] G. D. Birkhoff, Recurrence theorem on strongly transitive systems, Proc. of the Academy of Science USA 17, (1931), 650-655. [24] , Proof of the ergodic theorem, Proc. of the Academy of Science USA 17, (1931), 656-660. [25] F. Blanchard, Fully positive topological entropy and topological mixing, Contemp. Math. Symbolic Dynamics and Applications 135, Amer. Math. Soc. , Providence, 1992, pp. 95-105. [26] , A disjointness theorem involving topological entropy, Bull, de la Soc. Math, de France 121, (1993), 465-478. [27] F. Blanchard, E. Glasner and B. Host, A variation on the variational principle and applications to entropy pairs, Ergod. Th. Dynam. Sys. 17, (1997), 29-43. [28] F. Blanchard, B. Host, A. Maass, S. Martinez and D. Rudolph Entropy pairs for a measure, Ergod. Th. and Dynam. Sys. 15, (1995), 621-632. [29] F. Blanchard, B. Host and A. Maass Topological complexity, Ergod. Th. Dynam. Sys. 20, (2000), 641-662. [30] F. Blanchard and Y. Lacroix Zero-entropy factors of topological flows, Proc. Amer. Math. Soc. 119, (1993), 985-992. [31] J. Bourgain, On the spectral type of Ornstein's class one transformations, Israel J. Math. 84, (1993), 53-63. [32] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153, (1971), 401-414. [33] , On axiom A diffeomorphisms, Regional Conference Series in Mathematics, Amer. Math. Soc. 35, 1978. [34] M. Boyle Algebraic aspects of symbolic dynamics, Topics in symbolic dynamics and applications, Editors: F. Blanchard, A. Maass and A. Nogueira, LMS Lecture note Series 279, Cambridge University Press, 2000, 57-88. [35] L. Breiman, Probability, Addison-Wesley, Reading, Massachusetts, 1968. [36] I. U. Bronstein, Extensions of minimal transformation groups, Sijthoff & Noordhoff, 1979. [37] R.B. Burckel, Weakly almost periodic functions on semigroups, Notes on mathematics and its applications, Gordon and Breech, New York, 1970. [38] R. Burton, M. Keane and J. Serafin, Residuality of dynamical morphisms, Colloq. Math. 84/85, Dedicated to the memory of Anzelm Iwanik, part 2, (2000), 307-317. [39] R. Burton and A. Rothstein, Isomorphism theorems in ergodic theory, Technical report, Oregon State University, 1977. [40] R. V. Chacon, Weakly mixing transformations which are not strongly mixing, Proc. Amer. Math. Soc. 22, (1969), 559-562. [41] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg and A. Valette, Groups with the Haagerup property (Gromov's a-T-menability), Progress in Mathematics, 197, Birkhauser Verlag, Basel, 2001. [42] A.Connes and B. Weiss, Property T and almost invariant sequences, Israel J. Math. 37, (1980), 209-210. [43] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic theory, Springer-Verlag, New York, 1982. [44] A. I. Danilenko, Entropy theory from orbital point of view, Monatsh. Math. 134, (2001), 121-141. [45] A. I. Danilenko and K. K. Park, Generators and bernoullian factors for amenable actions and cocycles on their orbits, to appear, Ergod. Th. Dynam. Sys. [46] M. Denker, C. Grillenberger and K. Sigmund, Ergodic theory on compact spaces, Lecture Notes in Math. 527, Springer-Verlag, 1976. [47] J. Diestel, Sequences and series in Banach spaces, Graduate texts in mathematics 92, Springer-Verlag, 1984. [48] J. L. Doob, Stochastic processes, Wiley, New York, 1953. [49] A. H. Dooley and V.Ya. Golodets, The spectrum of completely positive entropy actions of countable amenable groups, preprint, (2001). [50] T. Downarowicz, The Choquet simplex of invariant measures for minimal flows, Israel J. of Math. 74, (1991), 241-256. BIBLIOGRAPHY 371 [51] , Weakly almost periodic flows and hidden eigenvalues, Topological dynamics and applications, Contemporary Mathematics 215, a volume in honor of R. Ellis, 1998, pp. 101-120. [52] L. Dubins, Bernstein-like polynomial approximation in higher dimensions, Pacific J. of Math. 109, (1983), 305-311. [53] N. Dunford and J. Schwartz, Linear operators, Part I, Interscience, New York, third printing 1966. [54] , Linear operators, Part II, Interscience, New York, third printing 1967. [55] W. F. Eberlein, abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67, (1949), 217-240. [56] R. Ellis, Locally compact transformation groups, Duke Math. J. 24, (1957), 119-126. [57] , Distal transformation groups, Pacific J. Math. 8, (1957), 401-405. [58] , Lectures on Topological Dynamics, W. A. Benjamin, Inc. , New York, 1969. [59] R. Ellis and W. H. Gottschalk, Homomorphisms of transformation groups, Trans. Amer. Math. Soc. 94, (1960), 258-271. [60] R. Ellis and M. Nerurkar, Weakly almost periodic flows, Trans. Amer. Math. Soc. 313, (1989), 103-119. [61] R. Ellis and W. Perrizo, Unique ergodicity of flows on homogeneous spaces, Israel J. of Math. 29, (1978), 276-284. [62] R. Engelking, General topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. [63] J. Feldman, Representations of invariant measures, (dittoed notes, 17 pp.), 1963. [64] J. M. G. Fell, An extension of Mackey's method to Banach* algebraic bundles, Memoires of the Amer. Math. Soc. 90, 1969. [65] W. Feller, An introduction to probability theory and its applications, Vol. I , second eddition, Wiley, New York, 1957. [66] S. Ferenczi, Systems of finite rank, Colloquium. Math. 73, (1997), 35-65. [67] A. Fieldsteel, Stability of the weak Pinsker property for flows, Ergod. Th. Dynam. Sys. 4, (1984), 381-390. [68] M. Foreman and B. Weiss An anti-classification theorem for ergodic measure-preserving transformations, in preparation. [69] M. Frank, Ergodic theory in the 1930's: a study in international mathematical activity. Mathematics Unbound: The evolution of an international mathematical community, 1800-1945, Eds.: K. Parshall & A. Rice [70] N. Friedman, Introduction to ergodic theory, van Nostrand Reinhold, New York, 1970. [71] H. Furstenberg, Stationary processes and prediction theory, Princeton University Press, Princeton, New Jersey, 1960. [72] , Strict ergodicity and transformation of the torus, American J. of Math. 83, (1961), 573-601. [73] , The structure of distal flows, American J. of Math. 85, (1963), 477-515. [74] , Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. System Theory 1, (1967), 1-49. [75] , The unique ergodicity of the horocycle flow, Recent advances in Topological dynamics (A. Beck, ed.) Springer-Verlag, Berlin, 1972, pp. 95-115. [76] , Ergodic behavior of diagonal measures and a theorem of Szemeredi on arith metic progressions, J. d'Analyse Math. 31, (1977), 204-256. [77] , Recurrence in ergodic theory and combinatorial number theory, Princeton uni versity press, Princeton, N.J., 1981. [78] H. Furstenberg and E. Glasner, On the existence of isometric extensions, Amer. J. of Math. 100, (1978), 1185-1212. [79] , m-systems, preprint, (1998). [80] H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. d'Analyse Math. 57, (1991), 64-119. [81] H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. d'Analyse Math. 34, (1978), 61-85. [82] , The finite multipliers of infinite transformation, Spriger Verlag Lecture Notes in Math. 688, (1978), 127-132. [83] F. R. Gantmacher, Applications of the theory of matrices, Interscience, New York, 1959. 372 BIBLIOGRAPHY [84] A. M. Garsia, A simple proof of E. Hopf's maximal ergodic theorem, J. Math. Mech. 14, (1956), 381-382. E. Glasner, Proximal flows, Lecture Notes in Math. 517, Springer-Verlag, 1976. , Quasifactors in ergodic theory, Israel J. of Math. 45, (1983), 198-208. , A simple characterization of the set of ji-entropy pairs and applications, Israel J. of Math. 102, (1997), 13-27. , Topological ergodic decompositions and applications to products of powers of minimal transformations, J. d'Analyse Math. 64, (1994), 241-262. , Structure theory as a tool in topological dynamics, Descriptive set theory and dynamical systems, LMS Lecture note Series 277, Cambridge University Press, 2000, 173-209. , Quasifactors of minimal systems, Topol. Meth. in Nonlinear Anal. 16, (2000), 351-370. E. Glasner, B. Host and D. Rudolph, Simple systems and their higher order self-joinings, Israel J. of Math. 78, (1992), 131-142. E. Glasner and D. Maon, Rigidity in topological dynamics, Ergod. Th. Dynam. Sys. 9, (1989), 309-320. E. Glasner J.-P. Thouvenot and B. Weiss, Entropy theory without a past, Ergodic theory and Dynam. Sys. 20, (2000), 1355-1370. E. Glasner and B. Weiss, Minimal transformations with no common factor need not be disjoint, Israel J. of Math. 45, (1983), 1-8. , Processes disjoint from weak mixing, Trans. Amer. Math. Soc. 316, (1989), 689-703. , On the construction of minimal skew-products, Israel J. of Math. , 34, (1979), 321-336. , Sensitive dependence on initial conditions, Nonlinearity 6, (1993), 1067-1075. , Strictly ergodic, uniform positive entropy models, Bull. Soc. Math. Prance 122, (1994), 399-412. , A simple weakly mixing transformation with nonunique prime factor, Amer. J. Math. 116, (1994), 361-375. , Quasifactors of zero-entropy systems, J. of Amer. Math. Soc. 8, (1995), 665-686. , Topological entropy of extensions, Proceedings of the 1993 Alexandria conferece, Ergodic theory and its connections with harmonic analysis, Editors: K. E. Petersen and I. A. Salama, LMS Lecture note Series 205, Cambridge University Press, 1995, 299-307. , Kazhdan's property T and the geometry of the collection of invariant measures, GAFA, Geom. funct. anal. 7, (1997), 917-935. , Quasifactors of ergodic systems with positive entropy, to appear, Israel J. of Math. T. N. T. Goodman, Relating topological entropy with measure theoretic entropy, Bull. London. Math. Soc. 3, (1971), 176-180. G. R. Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems, J. Dynam. Control Systems 5, (1999), 173-226. W. H. Gottschalk and G. A. Hedlund, Topological Dynamics, AMS Colloquium Publi cations, Vol. 36, 1955. F. P. Greenleaf, Invariant means on topological groups, Van Nostrand, New York, 1969 U. Grenander, Stochastic processes and statistical inference, Archiv for Matematik 1, (1950), 195-277. A. Grothendieck, Criteres de compacite dans les espaces fonctionnels generaux, Amer. J. of Math. 74, (1952), 168-186. F. Hahn and Y. Katznelson, On the entropy of uniquely ergodic transformations, Trans. Amer. Math. Soc. 126, (1967), 335-360. F. Hahn and W. Parry, Some characteristic properties of dynamical systems with quasi- discrete spectrum, Math. Systems Theory 2, (1968), 179-190. P. R. Halmos, Measure theory, Van Nostrand, Princeton, 1950. , Lectures on ergodic theory, Chelsea Publishing Company, New York, 1956. J. Hansel and J. P. Raoult, Ergodicity, uniformity and unique ergodicity, Indiana. Univ. Math. J. 23, (1974), 221-237. BIBLIOGRAPHY 373 [115] P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 2000. [116] P. de la Harpe et A. Valette, La propriete (T) de Kazhdan pour les group localement compacts, Asterisque, 1989. [117] G. A. Hedlund, Fuchsian groups and transitive horocycle, Duke Jour, of Math. 2, (1936), 530-542. [118] , Dynamics of geodesic flows, Bull. Am. Math. Soc. 45, (1939), 241-260. [119] E. Hewitt and K. Ross, Abstract harmonic analysis, Volume I, Springer-Verlag, Berlin, 1963. [120] , Abstract harmonic analysis, Volume II, Springer-Verlag, Berlin, 1970. [121] E. Hewitt and L.J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc. 80, (1955), 470-501. [122] G. Hjorth, On invariants for measure preserving transformations, Fund. Math. 169, (2001), 51-84. [123] B. Host, Mixing of all orders and independent joinings of systems with singular spec trum, Israel J. Math. 76, (1991), 289-298. [124] , Substitution subshifts and Bratteli diagrams, Topics in symbolic dynamics and applications, Editors: F. Blanchard, A. Maass and A. Nogueira, LMS Lecture note Series 279, Cambridge University Press, 2000, 35-55. [125] R. Howe and C.C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal. 32, (1979), 299-314. [126] K. ltd, On the ergodicity of a certain stationary process, Proc. Imper. Acad. Tokyo 20, (1944), 54-55. [127] K. Jacobs, Lecture notes on ergodic theory, Aarhus University, 1962. [128] K. Jacobs and M. Keane, 0-1-sequences of Toeplitz type, Z. Wahrsch. 13, (1969), 123-131. [129] R. I. Jewett, The prevalence of uniquely ergodic systems, J. Math. Mech. 19, (1970), 717-729. [130] V. F. R. Jones and K. Schmidt, Asymptotically invariant sequences and approximate finiteness, Amer. J. of Math. 109, (1987), 91-114. [131] A. del Junco, Transformations with discrete spectrum are stacking transformations, Canad. J. of Math. 24, (1976), 836-839. [132] , A simple map with no prime factors, Israel. J. of Math. 104, (1998), 301-320. [133] A. del Junco and M. Keane, On generic points in the cartesian square of Chacon's transformation, Ergodic theory and Dynam. Sys. 5, (1985), 59-69. [134] A. del Junco and M. Lemahczyk, Generic spectral properties of measure-preserving maps and applications, Proc. Amer. Math. Soc. 115, (1992), 725-736. [135] , Simple systems are disjoint with Gaussian systems, Studia Math. 133, (1999), 249-256. [136] A. del Junco, M. Lemahczyk and M. K. Mentzen, Semisimplicity, joinings and group extensions, Studia Math. 112, (1995), 141-164. [137] A. del Junco, M. Rahe and L. Swanson Chacon's automorphism has minimal self- joinings, J. d'Analyse Math. 37, (1980), 276-284. [138] A. del Junco and D.J. Rudolph, On ergodic actions whose self-joinings are graphs, Ergod. Th. Dynam. Sys. 7, (1987), 531-557. [139] , A rank-one rigid simple prime map, Ergod. Th. Dynam. Sys. 7, (1987), 229-247. [140] S. Kalikow, T,T_1 transformation is not loosely Bernoulli, Annals of Math. 115, (1982), 393-409. [141] , Two fold mixing implies three fold mixing for rank one transformations, Ergod. Th. Dynam. Sys. 4, (1984), 237-259. [142] R. Kallman, Certain quatint spaces are countably separated, III, J. Funct. Anal. 22, (1976), 225-241. [143] J. W. Kammeyer and D. Rudolph, Restricted orbit equivalence for actions of discrete amenable groups, preprint, (181 pages, 1995). [144] A. Katok, Smooth non-Bernoulli K-automorphisms, Invent. Math. 61, (1980), 291-300. [145] A. Katok (assisted by E. A. Robinson), Constructions in Ergodic Theory, unpublished notes, 1986. [146] A. Katok and B. Hasselblatt, Modern theory of dynamical systems, Encyclopedia of mathematics and its applications, Vol. 54, Cambridge University Press, 1995. 374 BIBLIOGRAPHY Y. Katznelson and B. weiss, When all points are recurrent/generic , Ergodic theory and dynamical systems I, Proceedings, Special year, Maryland 1979-80, Birkhauser, Boston, 1981. D. A. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal, and its Appl. 1, (1967), 63-65. M. Keane, Ergodic theory and subshifts of finite type, Ergodic theory, symbolic dynamics and hyperbolic spaces, Editors: T. Bedford, M. Keane and C. Series, Oxford University Press, 1991, 35-70. M. Keane and J. Serafin, On the countable generator theorem, Fund. Math. 157, (1998), 255-259. M. Keane and M. Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. Math. 109, (1979), 397-406. A. S. Kechris, Classical descriptive set theory, Springer-Verlag, Graduate texts in math ematics 156, 1991. H. B. Keynes and J. B. Robertson, Eigenvalue theorems in topological transformation groups, Trans. Amer. Math. Soc. 139, (1969), 359-369. J. King, The commutant is the weak closure of the powers for rank one transformations, Ergod. Th. Dynam. Sys. 6, (1986), 363-384. , Ergodic properties where order 4 implies infinite order, Israel J. of Math. 80, (1992), 65-86. B. Kitchens, Symbolic dynamics. One-sided, two-sided and countable Markov shifts, Springer-Verlag, 1998. A. N. Kolmogorov, A new metric invariant of transitive dynamical systems and Lebesgue space automorphisms, Dokl. Acad. Sci. SSSR 119, (1958), 861-864. S. Kremp, An elementary proof of the Eberlein-Smulian's theorem and the double limit criterion, Arc. Math. 47, (1986), 66-69. U. Krengel, Ergodic theorems, W. de Gruyter, Berlin, 1985. W. Krieger, On entropy and generators of measure-preserving transformations, Trans. Amer. Math. Soc. 149, (1970), 453r464. , On unique ergodicity, Proc. Sixth Berkeley Simposium on Math. Stat, and Prob., (1970), 327-346. F. Ledrappier, Un champ markovien peut etre d'entropie nulle et meelangeant, C. R. Acad. Sc. Paris, Ser. A 287, (1987), 561-562. K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105, (1961), 63-97. E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. of Math. 57, (1987), 239-255. M. Lemaficzyk, Introduction to ergodic theory from the point of view of spectral the ory, Lecture notes, Edited by Geon Ho Choe, Korea Advanced Institute of Science and Technology, Mathematics Research center, Korea, 1995. M. Lemanczyk and E. Lesigne Ergodicity of Rohlin cocycles, J. d'Analyse Math. 85, (2001), 43-86. M. Lemanczyk, F. Parreau and J.-P. Thouvenot, Gaussian automorphisms whose ergodic self-joinings are Gaussian, Fund. Math. 164, (2000), 253-293. M. Lemanczyk, J.-P. Thouvenot and B. Weiss, Relative discrete spectrum and joinings, Monatsh. Math. 137, (2000), 57-75. D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge University Press, 1995. E. Lindenstrauss, Lowering topological entropy, J. d'Analyse Math. 67, (1995), 231-267. , Measurable distal and topological distal systems, Ergod. Th. Dynam. Sys. 19, (1999), 1063-1076. , Pointwise theorems for amenable groups, Invent. Math. 146, (2001), 259-295. J. Lindenstrauss, G. H. Olsen and Y. Sternfeld, The Poulsen simplex, Ann. Inst. Fourier (Grenoble) 28, (1978), 91-114. G. W. Mackey, Borel structures in groups and their duals, Trans. Amer. Math. Soc. 85, (1957), 134-165. , Ergodic theory and virtual groups, Math. Ann. 166, (1966), 187-297. , Point realizations of transformation groups, Illinois J. Math. 6, (1962), 327-335. BIBLIOGRAPHY 375 [177] V. Mandrekar and M. Nadkarni, On ergodic quasi-invariant measures on the circle group, J. Funct. Anal. 3, (1969), 157-163. [178] B. Marcus, Unique ergodicity of some flows related to axiom A diffeomorphiss, Israel. J. of Math. 21, (1975), 111-132. [179] , The horocycle flow is mixing of all degrees, Invent. Math. 46, (1978), 201-209. [180] N. G. Markley and M. E. Paul, Almost automorphic minimal sets without unique ergod icity, Israel. J. of Math. 34, (1979), 259-272. [181] D. C. McMahon, Relativized weak disjointness and relative invariant measures, Trans. Amer. Math. Soc. 236, (1978), 225-137. [182] M. Misiurewicz, A short proof of the variational principle for a Z™ action on a compact space, Int. Conf. Dyn. Systems in Math. Physics, Societe Mathematique de France, Asterisque 40 (1976), 147-158. [183] S. Mozes, Mixing of all orders of Lie group actions, Invent. Math. 107 (1992), 235-241. Erratum, Invent. Math. 119 (1995), 399. [184] M. G. Nadkarni, Spectral theory of dynamical systems, Birkhauser advanced texts, 1998. [185] I. Namioka, Separate continuity and joint continuity, Pacific J. of Math. 51 (1974), 515-531. [186] J. Neveu, Mathemaical foundations of the calculus of probability, Holden-Day, Inc., San Francisco, 1965. [187] , Processus aleatoires gaussiens, Presses de TUniversite de Montreal, 1971. [188] P. J. Nicholls, The ergodic theory of discrete groups, London Math. Soc. Lecture Note Series 143, Cambridge University Press, 1989. [189] D. S. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4, (1970), 337-352. [190] , On the root problem in ergodic theory, Proc. sixth Berkeley symposium Math. Statist. Probab. Univ. of California Press, 1970, 347-356. [191] , A K-automorphism with no square root and Pinsker's conjecture, Advances in Math. 10, (1973), 89-102. [192] , Ergodic theory, randomness and dynamical systems, Yale Mathematical Mono graphs 5, Yale University Press, New Haven, CT, 1974. [193] D.S. Ornstein and P. Shields, An uncountable family of K-auotomorphisms, Advances in Math. 10, (1973), 63-88. [194] D.S. Ornstein and B. Weiss, Geodesic flows are Bernoullian, Israel. J. of Math. 14, (1973), 184-198. [195] , Ergodic theory of amenable group actions I. The Rohlin lemma, Bull. Amer. Math. Soc. 2, (1980), 161-164. [196] , The Sannon-McMillan-Breiman theorem for amenable groups, Israel. J. of Math. 44, (1983), 53-60. [197] , Entropy and isomorphism theorems for actions of amenable groups, J. d'Analyse Math. 48, (1987), 1-141. [198] , On the Bernoulli nature of systems with some hyperbolic structure, Ergod. Th. Dynam. Sys. 18, (1998), 441-456. [199] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112, (1964), 55-66. [200] , Zero entropy of distal and related transformations, in Topological dynamics, J. Auslader and W. Gottschalk, eds., Benjamin, New York, 1967. [201] , Entropy and generators in ergodic theory, W. A. Benjamine, INC, New York, 1969. [202] , Dynamical systems on nilmanifolds, London Math. Soc. 2, (1970), 37-40. [203] , Topics in ergodic theory, Cambridge University Press, Cambridge, 1981. [204] K. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc. 24, (1970), 278-280. [205] , Ergodic theory, Cambridge University Press, Cambridge, 1983. [206] R. R. Phelps, Lectures on ChoqueVs theorem, Van Nostrand mathematical studies 7, new York, 1966. [207] S. Polit, Weakly isomorphic maps need not be isomorphic, Ph.D. Dissertation, Stanford university, 1974. [208] M. Queffelec, Substitution dynamical systems, Lecture Notes in Math. 1294, Springer- Verlag, 1987. 376 BIBLIOGRAPHY [209] A. Ramsay, Virtual groups and group actions, Advances in Math. 6, (1971), 253-322. [210] M. Ratner, Horocycle flows, joinings and rigidity of products, Anna, of Math. 118, (1983), 277-313. [211] V. A. Rohlin, Exact endomorphisms of Lebesgue spaces, Izv. Acad. Si. USSR, Ser. Mat. 25, (1961), 499-530. [212] , Lectures on the entropy theory of measure-preserving transformations, Russian Math. Surveys 22, (1967), no. 5, 1-52. [213] V. A. Rohlin and Ja. G. Sinai, Construction and properties of invariant measurable partitions, Dokl. Acad. Nauk. SSSR. 141, (1961), 1038-1041. Sov. Math. 2 (6) (1961), 1611-1614. [214] A. Rosental, Strictly ergodic models and amenable group actions, preprint, (1988). [215] H. L. Royden, Real Analysis, third edition, Macmillan Publishing Co., New York, 1988. [216] W. Rudin, Fourier analysis on groups, Interscience Publishers, New York, 1960. [217] D. J. Rudolph, An example of a measure-preserving transformation with minimal self- joinings and applications, J. d'Analyse Math. 35, (1979), 97-122. [218] , k-fold mixing lifts to weakly mixing isometric extensions, Ergod. Th. Dynam. Sys. 5, (1985), 445-447. [219] , Zn andRn cocycle extensions and complementary algebras, Ergod. Th. Dynam. Sys. 6, (1986), 583-599. [220] , Fundamentals of Measurable Dynamics, Clarendon Press, Oxford, 1990. [221] D. J. Rudolph and B. Weiss, Entropy and mixing for amenable group actions, Ann. of Math. 151, (2000), 1119-1150. [222] V.V. Ryzhikov, Joinings and multiple mixing of finite rank actions, Funktsional. Anali. i Prilozhen. 27, (1993), 63-78; translation in Funct. Anal. Appl. 27, (1993), 128-140. [223] , Joinings, intertwining operators, factors and mixing properties of dynamical systems, Russian Acad. Izv. Math. 42, (1994), 91-114. [224] , Around simple dynamical systems. Induced joinings and multiple mixing, J. Dynam. Control Systems 3, (1997), 111-127. [225] J. Serafin Finitary codes and isomorphisms, Ph.D. Thesis, Technische Universiteit Delft, 1996. [226] K. Schmidt, Cocycles of ergodic transformation groups, Lecture Notes in Math. Vol. 1, MacMillan Co. of India, 1977. [227] , Asymptotically invariant sequences and an action of SL(2,R) on the 2-sphere, Israel J. of Math. 37, (1980), 193-208. [228] , Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group actions, Ergod. Th. and Dynam. Sys. 1, (1981), 223-236. [229] , Asymptotic properties of unitary representations and mixing, Proc. London Math. Soc. 48, (1984), 445-460. [230] , Algebraic ideas in ergodic theory, AMS-CBMS Reg. Conf. 76, Providence, 1990. [231] , Dynamical systems of algebraic origin, Birkhauser, 1995. [232] , Mildly mixing actions of locally compact groups, Proc. London Math. Soc. 45, (1982), 506-518. [233] L. Shapiro, Proximality in minimal transformation groups, Proc. Amer. Math. Soc. 26, (1970), 521-525. [234] P. Shields, The theory of Benoulli shifts, University of Chicago Press, 1973. [235] , The ergodic theory of discrete sample paths, Graduate studies in mathematics 13, Amer. Math. Soc. 1996. [236] M. Shub and B. Weiss, Can one always lower topological entropy?, Ergod. Th. and Dynam. Sys. 11 (1991), 535-546. [237] Ja. G. Sinai, A weak isomorphism of transformations with an invariant measure, Dokl. Acad. Nauk. S.S.S.R. 147, (1962), 797-800. [238] M. Smorodinsky, Ergodic theory, entropy, Lecture Notes in Math. 214, Springer-Verlag, New Yourk, 1971. [239] , Information, entropy and Bernoulli systems, Development of mathematics 1950-2000, 993-1012, Birkhuser, Basel, 2000. [240] M. Smorodinsky and J.-P. Thouvenot, Bernoulli factors that span a transformation, Israel J. of Math. 32, (1979), 39-43. BIBLIOGRAPHY 377 [241] A. M. Stepin, Spectral properties of generic dynamical systems, Math. U.S.S.R. Izv. 50, (1986), 159-192. [242] E. Szemeredi's, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27, (1975), 199-245. [243] A. Tempelman, Ergodic theorems for group actions, Kluwer Acad. Publ. , Dordrecht, 1992. [244] J.-P. Thouvenot, Quelques proprietes des systemes dynamiques qui se decomposent en un produit de deux systmes dont I'un est un schema de Bernoulli, Israel J. of Math. 21, (1975), 177-207. [245] , On the stability of the weak Pinsker property, Israel J. of Math. 27, (1977), 150-162. [246] , Some properties and applications of joinings in ergodic theory, Proceedings of the 1993 Alexandria conferece, Ergodic theory and its connections with harmonic analysis, Editors: K. E. Petersen and I. A. Salama, LMS Lecture note Series 205, Cambridge University Press, 1995, 207-238. [247] V. S. Varadarajan, Groups of automorphisms of B or el spaces, Trans. Amer. Math. Soc. 109, (1963), 191-220. [248] , Geometry of quantum theory, Vol II, Van Nostrand, Princeton, 1970. [249] , Lie groups, Lie algebras, and their representations, Graduate texts in mathe matics 102, Springer-Verlag, 1984. [250] W. A. Veech, The equicontinuous structure relation for minimal Abelian transformation groups, Amer. J. of Math. 90, (1968), 723-732. [251] , A fixed point theorem-free approach to weak almost periodicity, Trans. Amer. Math. Soc. 177, (1973), 353-362. [252] , Topological dynamics, Bull. Amer. Math. Soc. 83, (1977), 775-830. [253] , Unique ergodicity of horospherical flows, Amer. J. of Math. 99, (1977), 827-859. [254] , A criterion for a process to be prime, Monatsh. Math. 94, (1982), 335-341. [255] J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Nat. Acad. Sci. USA 18, (1932), 70-82. [256] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publishers, 1993. [257] P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982. [258] A. Weil, L,integration dans les groupes topologiques et ses applications, Actualites Sci. et Ind. 869, 1145. Paris: Hermann & Cie. 1941 and 1951. [259] B. Weiss, Topological transitivity and ergodic measures, Math. Systems Theory 5, (1971), 71-75. [260] , Ergodic theory, Unpublished lecture notes, 1972. [261] , Strictly ergodic models for dynamical systems, Bull, of the Amer. Math. Soc. 13, (1985), 143-146. [262] , Countable generators in dynamics — Universal minimal models, Contemporary Mathematics 94, (1989), 321-326. [263] , Multiple recurrence and doubly minimal systems, Contemporary Mathematics 215, (1998), 189-196. [264] , Single orbit dynamics, CBMS, Regional Conference Series in Math. 95, Amer. Math. Soc. Providence RI, 2000. [265] H. Weyl, Uber die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77, (1916), 313- 352. [266] S. Williams, Toeplitz minimal flows which are not uniquely ergodic, Z. Wahrsch. 67, (1984), 95-107. [267] R. J. Zimmer, Extensions of ergodic group actions, Illinois J. Math. 20, (1976), 373-409. [268] , Ergodic actions with generalized discrete spectrum, Illinois J. Math. 20, (1976), 555-588. [269] , Ergodic theory and semisimple groups, Birkhauser, 1984. Index of Symbols A(K), H(M) (Galois correspondence for G(X), 67 group extensions), 57, 133, 135 gr(/x,0) (graph joining), 126 (A) (icer generated by A), 341 0 (Hilbert-Schmidt bundle), 185 An, A_p (ergodic sums), 79, 83 H(U) (topological entropy of the cover IX), AP(T) (algebra of almost periodic 249 functions), 38 h(U) (topological T-entropy of the cover II), Aut (X, r) (group of automorphisms of a to 249 pological dynamical system), 13 htop(X,T) (topological entropy of the sys Aut (X) (group of automorphisms of a mea tem (X,T)), 249 sure dynamical system), 56 H(a) (entropy of the partition a), 254 T a~, a , 258 H(a\3r) (conditional entropy of the partition a C /?, 349 a with respect to the cr-algebra JF), 254 B(p) = B(Pl, ...,pt) (Bernoulli shifts), 82 h>n(ct) ~ hn(ct,T) (T,/i-entropy of the par B(X), 67 tition a), 258 BV, BV-, 235 7iM(a|5F) (conditional T, /i-entropy of the par B(T), Bn (spaces of matrix coefficients), 38 tition a with respect to the cr-algebra /3X, /3T (Stone-Cech compactification), 30, 9), 258 363 /i(X) (entropy of the dynamical system X), c, |c| (a column and its carrier), 271 258 c(K), Cfj,(K) (entropy capacity), 316 /i(X|Y) (conditional entropy of the dynam co(r) (space of functions vanishing at infin ical system X relative to the factor Y), ity), 38 258 Co, 178 H(}$), (Galois correspondence for group ex Ci, 186 tensions), 133, 135 Ax = {(x,x) : x e X} (diagonal), 13 S) (Hilbert space), 62 d (metric), 283 h (Hilbert bundle), 178 dN (metric), 285 1 (at), I3" (pt) ((conditional) information func dent, 275 tion), 254 dham, 271 I(X), 67 2 draeas, 270, 274, 278, 285, 349, 354 3, Inv(L (X)), 63, 67, 181 dpart, 274, 275 Iso (fj) (group of isometries of a Hilbert dist(a), 357 space), 235 3 E, E " (expectation, conditional expectation J(X), Je(X), J(X,Y), Je(X,Y) (set of (er with respect to the cr-algebra 3r), 365 godic) joinings), 125 E(X), 67 Jp(X), JuXu(X), 127 E(X,T), E(X,T) (enveloping JZ(X, Y) (joinings over a common factor), semigroup), 28 127 Efj, (/i-entropy pairs), 332 j(fc)(X), 148 ^(X,T)> Ex (topological entropy pairs), 332 K', ^k\ R^k\ n^°°\ K(°°^ (joinings associ £(X/Y) (closure of Y-eigenfunctions), 181 ated with a quasifactor), 164 EQ(X) (set of equicontinuity points), 34 £= {1,2,...,^} (an alphabet), 16, 269, 310 ext (Q), ext (K) (set of extreme points), 367, £(io) (The Banach algebra of bounded linear 97, 168, 231 operators on a Hilbert space), 62 380 INDEX OF SYMBOLS A* (adjoint joining), 129 spn(e), srn(e), sp(e,T), sr(e,T), sp(T:d), A(a A/3), 274 sr(T,d) (Bowen's entropy), 251 m (invariant mean on WAP functions), 43 sym (symmetrization map), 58 Mr(X), MT(X) (set of T-invariant proba S (group of permutations), 168 bility measures), 96 CTX (spectral measure), 116 Mprg(X), M^rg(X) (set of ergodic r-invari- t, |t| (a tower and its carrier), 271 ant probability measures), 97, 147 T (Kazhdan's property), 231 Mmax(X) (set of measures of maximal en Tw (Weak Kazhdan property), 243 tropy), 310 r o A (composition of joinings), 130 M(X), M(X,Y) (semigroup (set) of com u (minimal idempotent), 30 muting Markov operators), 132 u (minimal idempotent in E), 63, 67 p(aA(3) (= 0rx (r-orbit of x), 13 X = (X,X,/x,r), X = (X,X,/x,T) (mea Qrx (r-orbit closure of x), 13 sure-preserving dynamical systems), 53, n P = P(X,r),P7r,p( ) (proximal relations), 55 7 22 X^ / (r-symmetric product system), 58 P Q = Q(X,T),Q7T (regionally proximal rela Y = (Y,y,i/,r), Y = (Y,y,i/,T) (measure- tions), 22 preserving dynamical systems), 53, 55 QG{x) (set of measures for which x is quasi- Y x (U, p) (skew-product with cocycle a), generic), 98 a RTV = {(x,xl) : 7r(x) = K(X')} (relation de 57 termined by a homomorphism), 15 Z = (Z,Z,77,r), Z = (Z,Z,77,T) (measure- S(p) (property), 333 preserving dynamical systems), 53, 55 s(/i), s2(/x) (support), 336 Z(x) (cyclic subspace generated by x), 116 SFT (subshift of finite type), 17 lr -< 7T, lr C 7r, 235 Index of Terms AE (almost equicontinuous), 34 deterministic (process, system), 319 affine actions (on a Hilbert space), 235 discrete spectrum, 63, 67, 147 affine property (of the entropy function), generalized, 196 260, 276, 310 relative (discrete spectrum over Y), 181 affine transformation, 204 disjointness (of dynamical systems) amenable group, 79, 86, 96, 121, 141, 167 topological, 13 analytic set, 51 measure, 140 aperiodic weak, 13 Z-system, 296 distal matrix, 82 measure, 197 asymptotically invariant sets, 232 topological, 18, 34 Banach's problem, 120 Eberlein-Smulian (theorem) 366, 38 barycenter (equation), 160 eigenfunction Bernoulli shifts, 16, 82, 87, 266, 360 generalized, or Y-eigenfunction, 179, 181 Birkhoff (pointwise ergodic theorem), 83 measure, 103, 119 for towers, 279 topological, 19 Borel Ellis (joint continuity theorem), 33 action, 55 empirical distribution, 270 set, 51 entropy capacity, 316 space, 51 entropy pair cr-algebra, 51 measure, 332 Breiman (lemma), 264 topological, 330 Bruhat (decomposition), 105 enveloping semigroup, 28 equicontinuous, 18 Cartan (decomposition), 105 almost (AE), 34 Chacon (system), 27, 101, 301 ergodic decomposition, 72, 85, 164 chaos (first chaos of a Gauss system), 91, 92 ergodic theorem Choquet (theorem), 367, 72, 95, 276 Birkhoff or pointwise, 83, Chung (lemma), 261 for amenable groups, 86 coboundary, 57, 71, 235 for towers, 279 cocycle von Neumann or mean, 79 (associated with a dynamical system), 57, ergodic topological system, 110 71 ergodicity (of a dynamical system), 67 (associated with a unitary representation), intrinsic, 310 235 strict, 99 cohomologous (cocycles), 71, 235 strong, 232 copying (names on towers), 273 unique, 99 CPE (complete positive entropy) ergodicity (of a representation), 63 measure, 323 E-system, 110, 111, 113 topological, 330 expansive (dynamical system), 315 cyclic subspace (for a representation in a extension (or factor) Hilbert space), 116 almost 1-1, 26 de Finetti-Hewitt-Savage (theorem), 167 distal 382 INDEX OF TERMS measure, 197, 201, 202, 203 Jankov-von Neumann (theorem), 52 topological, 22 Jewett-Krieger (theorem), 291 group joining, 125, 136, 142, 221, 168, 347, 349 measure, 57, 76, 77, 133 composition of, 130 topological, 15 finite type, 139 homogeneous graph, 126 measure, 58, 137, 183 minimal self-joinings, 126, 215, 217, 228 I, 15, 198 off-diagonal, 126, 215 isometric, 182, 196, 183, 190, 198 orthogonal, 130 proximal, 22 relatively independent, 126 with separating sieve, 197, 198 topological, 13 weakly mixing JQF (joining quasifactor), 171 measure, 192, 201, 202 Jordan decomposition, 99 topological, 25 Kazhdan property, 231 extremal cr-algebra, 324, 325 K-system (Kolmogorov property), 87, 88 , finitely determined, 354, 358 120, 323, 326 fixed point property, 96 Kolmogorov (consistency theorem), 365, 90, flight function, 44, 93 127 flow (real), 105, 110 Kolmogorov representation, 89 F0mer sequence, 79 Kolmogorov-Sinai (theorem), 263 tempered, 85 Koopman representation, 67 full measure, 97, 111 Krieger (finite generator theorem), 352 Furstenberg (structure theorem for distal Kronecker (system) systems), 34 measure 58, 119 Furstenberg-Zimmer (structure theorem), topological, 18 202, 327 Kronecker-Weyl (theorem), 81 Krylov-Bogolubov (theorem), 97 Galois correspondence for group extensions, 135 Lebesgue space, 52 Gauss system, 90, 92, 242 Lusin (theorem), 52 Gelfand (space), 363, 32, 38 M-system, 14, 110, 111 generator, strong generator, 263 Mackey range (of a cocycle), 72 generic point, 98, 100, 78, 110, 137 Mackey-Weil (theorem), 207, 136, 218 (i/, e, 7V)-generic, or (e, A/")-generic Mandrekar-Nadkarni (theorem), 207 (sequence), 270 Markov-Kakutani (fixed point theorem), 43, geodesic flow, 105, 106 45 golden mean (subshift), 17, 251 Markov operators (commuting), 132 Grothendieck (theorem), 38 Markov systems, 82, 266 martingale (convergence theorem), 261, 262, Haagerup property, 243 335 Hahn-Banach (theorem), 45, 99 matrix coefficients, 38, 169 Halmos-von Neumann (theorem), 147 Mautner (lemma), 106 Heisenberg (group, nil-system), 20, 104 mean (left or right invariant), 43 Herglotz (theorem), 116 mean ergodic theorem (von Neumann), 79 horocycle flow, 26, 105, 101, 109 measure of maximal entropy, 310 Host (singular spectrum theorem), 205 mild mixing, 169, 169 icer (invariant closed equivalence relation), mincenter, 14 15, 341 minimal cocycle, 72 incontractible, 26 minimal (dynamical system), 14 intrinsically ergodic system, 310 minimal point, 14 inverse limit (system), 127 minimal self-joinings (MSJ) , 126, 215, 217, ZP-set, 15 228 isometric (system) min-max point, 235 extension, 182 mixing (measure), 66, 69 measure, 86, 149 system, 67 of order /c, of all orders, 86, 205, 228 (topological), 18 representation, 63 Iwasawa (decomposition), 105 topological, 23 INDEX OF TERMS 383 uniform, 86 rigid system, 63, 67, 68, 81, 119, 153, 224, model (for a dynamical system, topological, 227 Cantor, Polish), 55, 291, 53 {nfc}-rigid system, 81, 169 module (over Y, or Y-module), 179 Rohlin (skew-product theorem), 69 monothetic group, 81, 367, 19, 147 Rohlin's lemma, 272, 296 Morse (system), 27, 101 Rohlin's problem, 87, 205, 228 /i-measurable set, 51 Rohlin-Sinai (theorem), 323 Rohlin tower (or column), 271 nil-system, 20, 101, 103 Rudolph (counter-examples), 156 non-periodic (ergodic dynamical system), 272 semigroup order of orthogonality (of a quasifactor), enveloping, 28 166, 220 semitopological, 41, 63, 132 Ornstein separating (fundamental lemma), 349 open cover, 315 (isomorphism theorem), 360 partition, 332 orthogonal (joinings), 130 sieve, 197 SMB (The Shannon-McMillan-Breiman the P-system, 14, 110, 111 orem), 265 Peter-Weyl (theorem), 368, 40, 66, 69, 183 for towers, 281 Pinsker algebra (II(X)), 319 simple (dynamical system), 215, Pinsker factor of order 2, of all orders, 126 measure, 319 of order k, 215 /i topological, 341 simple spectrum, 117, 120, 300 topological, 341 simplex, 231, 163, 367 Pinsker's formula, 321 Bauer, 231, 367, 167, 240 Pinsker's inequality, 357 Poulsen, 231, 242 Pinsker conjecture, 325 Sinai (factor theorem), 360 Poincare (recurrence theorem), 58 skew-product, 57 point transitive (dynamical system), 13 group, 58, 72, 77 pointed dynamical system, 13 homogeneous, 58, 72, 77, 183 pointwise minimal (dynamical system), 14 Rohlin's skew product theorem, 69 Polish space, 51 spectral POOD (product of off-diagonals), 215, 220, measure, 221 multiplicity, sequence, type, 117 positive definite (function), 90, 115, 92, 242 theorem for unitary operators, 117 prime (dynamical system), 297 spectrum (of a unitary operator) process (associated with a partition), 273 absolutely continuous, discrete, homoge proximal relation, 22 neous, Lebesgue, infinite Lebesgue, sim quasi-discrete spectrum, 204 ple, singular, 117 quasifactor, 160, 168, 203, 220, 326 standard Borel space, 51 ergodically embedded, 126 standard cover, 330 finite type, 172 stochastic process (stationary), 273 joining, 171 Stone-Cech compactification, 30, 363 mixingly embedded, 172 strong ergodicity 232 order of orthogonality, 166, 220 subshift, 16, 82 points, 160 of finite type, 17 trivial, 160 substitution (system), 27 weak mixingly embedded, 172 subsystem, 13 quasi-generic point, 98 sweeping set, 271 quasi-invariant measure, 207 symbolic representation (of a process), 273 Rajchman (measure), 119 measure, 273 rank one (system), 299 syndetic, 15 regionally proximal relation, 22 syndetically transitive, 24 regular points, 110 regular (topological dynamical system), 26 thick, 15 replete partition, 332 thickly syndetic, 44 return time function, 271 Toeplitz (sequence, system), 28 384 INDEX OF TERMS totally ergodic (dynamical system), 119, 291 van der Waerden (theorem), 47 tower (height, roof, floor) Veech (theorem), 136, 216 Kakutani, 271 von Neumann (mean ergodic theorem), 79 Kakutani-Rohlin, K-R, 271, 291 WAP (weakly almost periodic) Rohlin, 271, 349 function, 38 transitive (topologically transitive system), system, 40, 63, 132 13 weak containment, 235 typical sequence, (is, e, N, 2n + 1)-typical, weak mixing (of a representation), 63 270 weak mixing (of a dynamical system), 67 uniformly (or well) distributed sequences, a, 153, 156, 302 102 measure, 67 universally measurable set, 51 topological, 14 UPE (uniform positive entropy), 330, 297 weak Pinsker conjecture, 325 upper semi-continuity, 15, 270, 310, 367 weakly isomorphic systems, 157 Weyl (theorem), 102 Titles in This Series 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel 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 William 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 Denes 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. Williams, 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 William 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 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam 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. Nation, 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. Timothy O'Meara, Symplectic groups, 1978 15 J. Diestel and J. J. Uhl, Jr., Vector measures, 1977 14 V. Guillemin 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