2.2.3 Pontryagin duality, Bochner's theorem, and the Bohr compact- ification. ..... mathematics related to locally compact topological groups. Renling Jin ...
ERGODIC AVERAGES, CORRELATION SEQUENCES, AND SUMSETS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University
By John Thomas Griesmer, B.A., M.S. *****
The Ohio State University 2009 Dissertation Committee:
Approved by
Professor Vitaly Bergelson, Advisor Professor Neil Falkner Professor Alexander Leibman
Advisor Graduate Program in Mathematics
ABSTRACT
The study of multiple ergodic averages and multiple correlation sequences was initiated by Furstenberg to give an ergodic-theoretic proof of Szemer´edi’s theorem on arithmetic progressions. In this dissertation we generalize some of the recent results on multiple ergodic averages and correlation sequences. We also apply some of the techniques from the study of multiple ergodic averages as a means of generalizing some of the recent results on sumsets in infinite groups.
ii
ACKNOWLEDGMENTS
Foremost, I would like to thank my advisor, Professor Vitaly Bergelson, for introducing me to many interesting ideas and methods in ergodic theory, and providing me with many interesting problems for research. I would also like to thank the other members of my dissertation committee, Professors Neil Falkner and Alexander Leibman, as well as Professor Manfred Einsiedler, for many helpful discussions during my time as a student of mathematics. Thanks are also due to the National Science Foundation, whose VIGRE grant funded much of my study at The Ohio State University. For moral support, encouragement, and interesting discussions, my gratitude is due to Professors Chris Miller and Tim Carlson. Finally, I would like to thank Michael Bj¨orklund and Alexander Fish, who introduced me to a problem that inspired the work on sumsets in this dissertation.
iii
VITA 2002-Present . . . . . . . . . . . . . . . . . . . . . . . . . .
Graduate Teaching Associate, The Ohio State University
2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M.S. in Mathematics, The Ohio State University
2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.S. in Mathematics, Miami University
PUBLICATIONS
Research Papers • (With Zoltan T. Balogh.) On the multiplicity of jigsawed bases in compact and countably compact spaces. In Topology and its Applications.
FIELDS OF STUDY
Major Field: Mathematics Specialization: Ergodic Theory and Combinatorial Number Theory
iv
TABLE OF CONTENTS Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
CHAPTER 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 1.2 1.2.1 1.3 1.4
2
PAGE 1
Sumsets in discrete groups. . . . . . . . . . . . . . . . . . Characteristic factors for some commuting actions Zd . . . Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation Sequences. . . . . . . . . . . . . . . . . . . . Convergence of cubic ergodic averages for commuting actions of amenable groups. . . . . . . . . . . . . . . . . . .
1 2 7 7
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1 2.1.1 2.1.2 2.1.3 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.4 2.4.1
Review of measure theory and measure preserving systems. Conditional expectation. . . . . . . . . . . . . . . . . . . . . . Disintegration of measures. . . . . . . . . . . . . . . . . . . . . The σ-algebra of invariant sets. . . . . . . . . . . . . . . . . . Locally compact topological groups. . . . . . . . . . . . . Haar measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . Unitary representations of a discrete group. . . . . . . . . . . Pontryagin duality, Bochner’s theorem, and the Bohr compactification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Bohr compactification. . . . . . . . . . . . . . . . . . . . Topological systems. . . . . . . . . . . . . . . . . . . . . . Measure preserving systems. . . . . . . . . . . . . . . . . Joinings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
8
10 11 12 13 13 14 15 15 17 18 18 20
2.4.2 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.6 2.7 2.7.1 2.8 3
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21 22 23 26 26 27 29 30 32 34 36 38 38
Sumsets of Dense Sets and Sparse Sets . . . . . . . . . . . . . . . . . .
41
3.1 3.2 3.2.1 3.3 3.3.1 3.3.2 3.4 3.4.1 3.5 4
Ergodicity and ergodic decomposition. . . . . . . . . . . . Amenable groups. . . . . . . . . . . . . . . . . . . . . The mean ergodic theorem. . . . . . . . . . . . . . . . . . The von Neumann-Koopman representation. . . . . . . . . Isometric systems and group extensions. . . . . . . . . . . Cocycles and skew product extensions. . . . . . . . . . . . The Mackey group. . . . . . . . . . . . . . . . . . . . . . . Isometric extensions. . . . . . . . . . . . . . . . . . . . . . Shift spaces and the Furstenberg correspondence principle. Product sets in compact topological groups. . . . . . . Large sets and Bohr sets in groups. . . . . . . . . . . Characterizations of Bohr sets when Γ is abelian. . . . . . van der Corput lemmas. . . . . . . . . . . . . . . . . .
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41 43 46 50 50 52 53 57 60
Characteristic factors for some commuting actions of Zd . . . . . . . . .
62
4.0.1 4.1 4.1.1 4.1.2 4.1.3 4.2 4.3 4.3.1 4.3.2 4.3.3 4.3.4
Introduction. . . . . . . . . . . . . . . . . . . . Ergodic averaging schemes for abelian groups. . Examples of ergodic averaging schemes. . . . . . . . Proof of Theorem 3.1.6, Part 1 . . . . . . . . . Proof of Theorem 3.1.7. . . . . . . . . . . . . . . . Proof of theorem 3.1.6, Part 2. . . . . . . . . . . . . Proof of Proposition 3.3.1. . . . . . . . . . . . Identifying US(D). . . . . . . . . . . . . . . . . . . Remarks. . . . . . . . . . . . . . . . . . . . . .
Notation. . . . . . . . . . . . . . . . . . . Introduction. . . . . . . . . . . . . . . Nilsystems. . . . . . . . . . . . . . . . . . Background. . . . . . . . . . . . . . . . . . Cocycles and group extensions. . . . . . . Outline of the proof of Theorem 4.1.2. Definition of the measures µ[k] . . . . . The cube Vk and symmetries. . . . . . . . The measures µ[k] . . . . . . . . . . . . . . The case k = 1. . . . . . . . . . . . . . . . The side actions. . . . . . . . . . . . . . . vi
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62 62 62 64 64 67 69 69 70 72 74
4.3.5 4.3.6 4.4 4.4.1 4.4.2 4.5 4.5.1 4.5.2 4.6 4.6.1 4.7 4.7.1 4.7.2 4.8 4.8.1 4.8.2 4.8.3 4.8.4 4.9 4.9.1 4.9.2 4.9.3 4.9.4 4.10 4.10.1 4.10.2 4.10.3 4.11 4.12 4.12.1 4.12.2 4.13 4.14 4.14.1 4.15 4.15.1 4.15.2 4.15.3
Symmetries. . . . . . . . . . . . . . . . . . . . . . . . . Gowers-Host-Kra seminorms. . . . . . . . . . . . . . . Definition of the characteristic factors. . . . . . . . ∗ ∗ ∗ The marginal (X [k] , X[k] , µ[k] ) . . . . . . . . . . . . . Systems of order k. . . . . . . . . . . . . . . . . . . . . A group associated to each ergodic system. . . . . General properties. . . . . . . . . . . . . . . . . . . . . Faces and commutators. . . . . . . . . . . . . . . . . . Relations between consecutive factors. . . . . . . . Description of the extension. . . . . . . . . . . . . . . . Cocycles of type k and systems of order k. . . . . Cocycles of type k and automorphisms. . . . . . . . . . Cocycles of type k and group extensions. . . . . . . . . Systems of order 2. . . . . . . . . . . . . . . . . . Systems of order 1 . . . . . . . . . . . . . . . . . . . . The Conze-Lesigne Equation . . . . . . . . . . . . . . . Systems of order 2. . . . . . . . . . . . . . . . . . . . . The group of a Zd -system of order 2 . . . . . . . . . . . The main induction. . . . . . . . . . . . . . . . . . The systems Xs . . . . . . . . . . . . . . . . . . . . . . The factors Zk (Xs ). . . . . . . . . . . . . . . . . . . . . Connectivity. . . . . . . . . . . . . . . . . . . . . . . . Countability. . . . . . . . . . . . . . . . . . . . . . . . Systems of order k and nilmanifolds. . . . . . . . . Reduction to toral systems. . . . . . . . . . . . . . . . Building nilmanifolds. . . . . . . . . . . . . . . . . . . End of the proof. . . . . . . . . . . . . . . . . . . . . . Characteristic factors. . . . . . . . . . . . . . . . . The Zk are characteristic factors. . . . . . . . . . . Proof that Zk is an inverse limit of nilsystems. . . . . . Proof that Zk is characteristic. . . . . . . . . . . . . . . Necessity of the conditions on the endomorphisms. Appendix, part 1. . . . . . . . . . . . . . . . . . . Lie groups. . . . . . . . . . . . . . . . . . . . . . . . . Appendix, part 2. . . . . . . . . . . . . . . . . . . Measurability properties. . . . . . . . . . . . . . . . . . Cocycles and groups of automorphisms. . . . . . . . . . Variations of Lemma 4.15.5. . . . . . . . . . . . . . . . vii
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77 79 81 81 87 88 89 94 96 101 111 113 114 119 119 119 122 123 132 132 135 138 140 143 143 145 154 156 160 164 165 166 168 168 169 170 176 180
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Multiparameter correlation sequences . . . . . . . . . . . . . . . . . . . 189 5.1 5.2 5.3
6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 189 Reduction to nilsystems. . . . . . . . . . . . . . . . . . . 192 Nilmanifolds and orbits of diagonal measures. . . . . . . . 195
Cubic ergodic averages . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 6.1 6.1.1 6.2 6.3 6.3.1 6.3.2 6.3.3 6.3.4 6.4 6.4.1 6.4.2 6.5 6.6 6.6.1
Introduction. . . . . . . . . . . . . . . . . . . . . Remark. . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary lemmas and notation. . . . . . . . . The case d = 2 . . . . . . . . . . . . . . . . . . . Reduction to partial characteristic factors. . . . . . . An invariant measure on X × X × X. . . . . . . . . . Evaluation of the limit and the minimal characteristic An example. . . . . . . . . . . . . . . . . . . . . . . . The box joinings and seminorms. . . . . . . . . . Box measures. . . . . . . . . . . . . . . . . . . . . . . Box seminorms. . . . . . . . . . . . . . . . . . . . . . The Magic Extension . . . . . . . . . . . . . . . Proofs of convergence. . . . . . . . . . . . . . . . Large limits and combinatorial corollaries. . . . . . .
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198 200 200 202 204 205 208 209 210 210 212 214 216 218
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
viii
CHAPTER 1 INTRODUCTION
In this chapter we introduce the main results of this dissertation and explain their relation to recent mathematical developments.
1.1
Sumsets in discrete groups.
If G is a locally compact group, with two subsets A and B of positive Haar measure, then {ab : a ∈ A, b ∈ B} contains an open set. This well-known fact, sometimes referred to as the Steinhaus Lemma, has been fruitfully exploited in many areas of mathematics related to locally compact topological groups. Renling Jin established an analogue of the Steinhaus Lemma for sets in Z. To state this result, we recall the notions of upper Banach density and syndeticity for sets in Z. A set A ⊂ Z has positive upper Banach density if there is a sequence of intervals In with lengths |In | → ∞, such that lim supn→∞
|A∩In | |In |
> 0. A set A ⊂ Z is syndetic
if A has bounded gaps, that is, if A ∩ I 6= ∅ for every interval I of some large enough length L. We say S ⊂ Z is piecewise syndetic if there exists a sequence of intervals In S with lengths tending to infinity and a syndetic set A such that S = A ∩ n In . Note that such a set must have positive Banach density. R. Jin established the following result using nonstandard analysis. 1
Theorem 1.1.1. ([38], Corollary 3) Let A, B ⊂ Z have positive upper Banach density. Then A + B is piecewise syndetic. Theorem 1.1.1 was generalized in [7] and [6], and we discuss these results in Chapter 3. The main theorem of Chapter 3 is a new generalization of Theorem 1.1.1 that weakens the hypothesis that d∗ (A) > 0. As a consequence, we obtain the following corollary. Theorem 1.1.2. Let α ∈ R>0 \ Z. Let C ⊂ Z with lim supn→∞
|C∩[1,...,n]| n
> 0. If
B ⊂ Z has positive Banach density, then {bnα c + b : n ∈ C, b ∈ B} is piecewise syndetic. The conclusion we establish in Chapter 3 is stronger; we will show that A + B is piecewise Bohr, a notion introduced by Bergelson, Furstenberg, and Weiss in [7]. Let us postpone the definition of Bohr sets and piecewise Bohr sets to Section 2.7 of Chapter 2.
1.2
Characteristic factors for some commuting actions Zd .
In this section we assume some familiarity with ergodic theory; the relevant definitions and terminology are given in Chapter 2. In order to give an ergodic theoretic proof of Szemer´edi’s theorem on arithmetic progressions, Furstenberg [22] analyzed the behavior of averages of the form AN,M
N −1 Z X 1 f · f ◦ T n · f ◦ T 2n · · · · · f ◦ T kn dµ(x), := N − M n=M
2
(1.1)
where T is a measure preserving transformation of a probability space (X, X, µ). One of the main results in [22] is that lim inf N −M →∞ AN,M > 0 whenever f is a nonnegative R function with f dµ > 0. The investigations in [22] raised the question of whether the limit N −1 X 1 f ◦ T n · f ◦ T 2n · · · · · f ◦ T kn lim N −M →∞ N − M n=M
(1.2)
exists (in the sense of L2 (µ)), and if so, what is the limit? In fact the limit in (1.2) exists, and for ergodic systems the limit can be evaluated through understanding an appropriate characteristic factor of (X, X, µ, T ), which in many cases is isomorphic to an inverse limit of flows on nilmanifolds. Let us define these terms. Definition 1.2.1. (cf. [25]) Let (X, X, µ, T ) be a probability measure preserving Zd system, and let p1 , . . . , pk : Zd → Zd . We say that a factor (Y, Y, ν, S) (interpreted as a sub σ-algebra Y ⊆ X) is a characteristic factor for the scheme (T p1 (~n) , T p2 (~n) , · · · , T pk (~n) ) if 1 lim N −M →∞ (N − M )d
X
k Y
fi ◦ T
~ n∈[M,N −1]d i=1
pi (~ n)
−
k Y
E(fi |Y) ◦ T pi (~n) = 0
i=1
for all bounded fi . (Here E(f |Y) denotes the conditional expectation of f on the σ-algebra Y. See Chapter 2 for an overview of conditional expectation and related results.) Definition 1.2.2. If G is a k-step nilpotent Lie group, and H ⊆ G is a discrete subgroup of G such that G/H is compact, we say that G/H is a k-step nilmanifold. A k-step Zd -nilsystem is a dynamical system (G/H, X, µ, T ), where µ is the natural 3
projection of Haar measure from G to the quotient, X is the Borel σ-algebra, and T (n1 ,...,nd ) x is given by tn1 1 tn2 2 · · · tnd d x for commuting t1 , . . . , td ∈ Λ. We will drop the prefix Zd from “Zd - nilsystem” when the value of d is clear. Note that a 1-step d-nilsystem is a Kronecker system, that is, a system of the form (Z, Z, mZ , R), where Z is a compact abelian group, mZ is Haar measure on Z, and Rn1 ,...,nd z = tn1 1 · · · tnd d z for some t1 , . . . , td ∈ Z (using multiplicative notation). The proof of convergence of the averages in (1.1) for the case k = 2 is essentially done in [22], where the Kronocker factor of (X, X, µ, T ) is shown to be characteristic for the scheme (T n , T 2n ) (assuming T is ergodic). Convergence in the case k = 3 is proved in [16] and [17]. The characteristic factor for the scheme (T n , T 2n , T 3n ) is a special abelian group extension of the Kronecker factor, which is shown to be an inverse limit of 2-step nilsystems. Convergence for all k ≥ 1 is proved by Host and Kra in [36], and independently by T. Ziegler in [68], and the characteristic factor for the scheme (T n , T 2n , . . . , T kn ) is shown to be an inverse limit of k − 1-step nilsystems. In Chapter 4 we investigate multiparameter averages such as N −1 X 1 f1 ◦ T n S m · f2 ◦ T 3n−m S n+m · f3 ◦ T 2m S 4n , (N − M )2 n,m=M
(1.3)
where T and S are commuting transformations. In order to understand the averages in (1.3), we show that a characteristic factor for the scheme (T n S m , T 3n−m S n+m , T 2m S 4n ) is an inverse limit of 2-step 2-nilsystems. In contrast, an example we present in Chapter 6, due to A. Leibman, shows that there are systems (X, X, µ, T, S) where there
4
is no characteristic factor for the scheme (T n , S m , T n S m ) which is an inverse limit of nilsystems. We consider schemes of the form L (n1 ,...,nd )
(T1 11
L (n1 ,...,nd )
T2 12
L
· · · Td 1d
(n1 ,...,nd )
L
, . . . , T1 k1
(n1 ,...,nd )
L
· · · Td kd
(n1 ,...,nd )
),
where the Ti commute and generate an ergodic measure preserving action of Zd , and the Lij : Zd → Z are linear functions. We give a necessary and sufficient condition on the Lij to guarantee that such schemes have characteristic factors which are inverse limits of nilsystems. The model for the main result of Chapter 4 is the following result from [36] and [68] Theorem 1.2.3. Let X = (X, X, µ, T ) be an invertible ergodic measure preserving system, and k ∈ N. Then there is a system Zk = (Zk , Zk , νk , T ) which is a factor of X (we view Zk ⊆ X as a T -invariant sub-σ algebra) such that (1) For bounded functions fi : X → C, N k k Y 1 XY fi ◦ T in − E(fi |Zk ) ◦ T in = 0 N →∞ N n=1 i=1 i=1
lim
(2) The system Zk is an inverse limit of k − 1-step nilsystems. In Chapter 4 we provide a straightforward generalization of Theorem 1.2.3 to the setting where Zd acts by measure preserving transformations on a space (X, X, µ). We consider averages of the form k 1 XY fi ◦ T Mi a , |FN | a∈F i=1 N
5
(1.4)
where FN is a Følner sequence in Zd and the Mi are endomorphisms of Zd . (See Chapter 2 for the definition of Følner sequence). The following is the main result of Chapter 4. Theorem 1.2.4. Let (X, X, µ, T ) be an ergodic Zd system, and let Mi be endormorphisms of Zd such that the images of each of Mi , Mi − Mj , i 6= j have finite index in Zd . There is a factor (Z, Z, ν, T ), isomorphic to an inverse limit of k-step nilsystems, such that k k Y 1 XY fi ◦ T Mi a − E(fi |Z) ◦ T Mi a = 0, N →∞ |FN | i=1 a∈F i=1
lim
N
for all Følner sequences FN , and the convergence takes place in L2 (µ). We will also show that the hypothesis that the images of the Mi , Mi − Mj each have finite index in Zd is necessary to conclude that there is a characteristic factor for the scheme (T M1 a , . . . , T Mk a ) which is an inverse limit of nilsystems. One application of the characteristic factors developed in [36] is the description in [21] of characteristic factors for averages such as N k 1 XY fi ◦ Tin , N n=1 i=1
where the Ti are commuting transformations such that each Ti , Ti Tj−1 , i 6= j is ergodic. In Section 4.12 of Chapter 4, we generalize the results of [21] to the case where we consider commuting Zd -actions, rather than commuting Z-actions.
6
1.2.1
Remarks.
If one were interested only in whether the averages in Theorem 1.2.4 converge, one could simply cite the recent result of T. Austin ([2]), where convergence of averages of the form k 1 XY fi ◦ Tia , |FN | a∈F i=1
(1.5)
N
where the Ti are commuting measure preserving actions of a countable abelian group A on a probability space (X, X, µ). The remarkably elegant proof in [2], however, gives no description of the limit, and seemingly does not give precise information about the correlation sequences Z I(a) :=
f0 ·
k Y
fi ◦ Tia dµ.
i=1
The problem of finding a precise description of the correlation sequence I(a) = R Q f0 · ki=1 f ◦ Tin dµ where the Ti are commuting measure preserving transformations remains open. Austin’s result generalizes Tao’s result in [61], which demonstrates convergence in the case A = Z and FN = [1, . . . , N ]. Towsner, in [63] has given an interesting version of Tao’s argument, using nonstandard analysis to prove the same result.
1.3
Correlation Sequences.
An application of Theorem 1.2.3 is a description of correlation sequences such as R Ik (n) = f · f ◦ T n · f ◦ T 2n · · · · · f ◦ T kn dµ. The following description is given in [8]: 7
Theorem 1.3.1. Let (X, X, µ, T ) be an ergodic system. Let f be a bounded function R Qk in on X. Let Ik (n) denote dµ. Then i=0 f ◦ T Ik (n) = a0 (n) + ak (n) where limN −M →∞
1 N −M
PN −1
n=M
|a0 (n)| = 0, and ak is a k-step nilsequence, that is, for
all ε > 0 there exists a k-step nilmanifold Z = G/H, t, x ∈ G, and φ ∈ C(Z) so that |φ(tn xH) − ak (n)| < ε. In analogy with Theorem 1.3.1, we will show in Chapter 5 that, under the hypotheses of Theorem 1.2.4, the correlation sequences Z k Y Jk (n1 , . . . , nd ) := f0 · fi ◦ T Mi (n1 ,...,nd ) dµ
(1.6)
i=1
can be decomposed as Jk (n1 , . . . , nd ) = a0 (n1 , . . . , nd ) + ak (n1 , . . . , nd ), where limN →∞
1 (2N +1)d
P
~ n∈[−N,N ]d
|a0 (~n)| = 0, and ak (n1 , . . . , nd ) can be uniformly
approximated by a function of the form φ(tn1 1 · · · tnd d xH), for φ ∈ C(G/H), for some k-step nilmanifold G/H, and commuting ti ∈ G.
1.4
Convergence of cubic ergodic averages for commuting actions of amenable groups.
In Chapter 6, we find characteristic factors for an averaging scheme not covered by the results of Chapter 4. We investigate the averages N −1 X 1 f1 ◦ T n · f2 ◦ S m · f3 ◦ T n S m . (N − M )2 n,m=M
8
(1.7)
One of the main results in Chapter 6 is that the limit of the averages in (1.7) exists as N − M → ∞. We provide a formula for the limit and describe the minimal characteristic factors for these averages. Our methods here are a combination of methods from [22] and [11]. In fact, we will work in a more general setting, where T and S are replaced by commuting actions of an amenable group. We will adapt the methods of Host ([35]) and Chu ([15]), who recently demonstrated convergence of higher dimensional analogues of the averages in (1.7), and generalize their results to the setting of commuting actions of an amenable group. Remark 1.4.1. While Austin proved the existence of limits of the form (1.7) in [2], it is not immediately obvious how to generalize his proof to the setting where T and S are replaced by commuting actions of an amenable group. The methods of [15] and [35], together with an argument in Chapter 6 for the base case, accomplish this generalization.
9
CHAPTER 2 PRELIMINARIES
2.1
Review of measure theory and measure preserving systems.
In this section we present some of the basic facts we need from measure theory. See [20] or [59] for more details. A measure space is a triple (X, X, µ) where X is a set, X is a σ-algebra, and µ : X → R is a nonnegative, countably additive function. That is, if {Ai : i ∈ N} is a countable collection of mutually disjoint sets with Ai ∈ X for all i, then [ X µ( Ai ) = µ(Ai ). i
i
A measure space (X, X, µ) is separable if there is countable collection of sets B ⊂ X such that for all A ∈ X and all ε > 0, there exists B ∈ B such that µ(A4B) < ε. If X is a compact Hausdorff space and X is the Borel σ-algebra of subsets of X, then we say that a measure µ on X is a Radon measure if (i) µ(K) is finite whenever K ⊂ X is compact. (ii) µ(U ) = sup{µ(K) : K ⊂ U, K is compact} whenever U ⊂ X is open. (iii) µ(K) = inf{µ(U ) : U ⊃ K, U is open} whenever K ⊂ X is compact. 10
See [20], Chapter 7 for more on Radon measures. We will mainly be concerned with measure spaces whose underlying set is a σ-compact metric space, and whose measure is a Radon measure. We call a measure space (X, X, µ) standard if it is measure-theoretically isomorphic to a measure space (Y, Y, ν), where Y is a compact metric space, Y is the Borel σalgebra of Y, and ν is a Radon measure on Y. (Here a measure-theoretic isomorphism is a map φ : X → Y such that φ−1 (Y) ⊂ X, φ(X) ⊂ Y, and ν(φ(D)) = µ(D) for all D ∈ X.) A standard probability space is measure-theoretically isomorphic to a measure space ∞ (Y, B, m), where Y = [0, l] ∪ {ai }∞ i=1 , [0, l] ⊂ R is an interval of length l ≤ 1, {ai }i=1
is a countable discrete set disjoint from [0, l], B is the Borel σ-algebra of Y, and P l + i m({ai }) = 1. See [59], Chapter 14 for a proof. 2.1.1
Conditional expectation.
We continue to consider a probability measure space (X, X, µ). If V ⊂ L2 (µ) is a closed subspace, we can consider the orthogonal projection PV : L2 (µ) → V onto V. Of special interest is the case where V consists of functions measurable with respect to some sub σ-algebra Y ⊂ X. Letting L2 (Y) denote the space of Y-measurable functions in L2 (µ), we write Eµ (f |Y) for the orthogonal projection of f onto L2 (Y). We use this
11
notation because the map f 7→ Eµ (f |Y) agrees with the conditional expectation1 of f on Y when f ∈ L1 (µ). See [31] for a standard development of conditional expectation. R R Note that Eµ (f |Y) dµ = f dµ for all f ∈ L1 (µ). 2.1.2
Disintegration of measures.
If (X, X, µ) is a standard probability measure space, A ⊂ X is a countably generated sub σ-algebra, the disintegration of µ over A is a A-measurable function from X to the space of probability measures on (X, X), denoted by x 7→ µA x , with the following properties: (i) For all f ∈ L1 (µ), f ∈ L1 (µx ) for µ-almost every x, and
R
f dµA x = E(f |A)(x),
for µ-almost every x. (ii) The map x 7→ µA x is A-measurable. When there is no ambiguity, we will use x 7→ µx to denote the disintegration of µ over A. See [23], Theorem 5.8 for a proof of the existence of disintegration of measures.. If A is realized as π −1 (Y) for a measurable map π : X → Y, where (Y, Y, ν) is a probability space, then we can consider the map x 7→ µA x as a Y-measurable map y 7→ µy from Y to the space of probability measures on (X, X). This is the convention used in [23], [26], and others. 1
If f ∈ L1R(µ), we can define Eµ (f |Y) in the following way: define a measure µf on (X, Y) by µf (A) = R1A · f dµ0 ,R where µ0 = µ|Y . By the Radon-Nikodym theorem , there exists h ∈ L1 (µ0 ) such that g dµf = gh dµ0 for all g ∈ L1 (µ0 ). We call h the conditional expectation of f on Y with respect to µ.
12
If A ⊂ X is a σ-algebra, we say that A ∈ A is an atom of A if for all B ∈ A, either A ⊂ B or A ∩ B = ∅. When A is countably generated, one can view the measures µA x as being supported on the atoms of A. 2.1.3
The σ-algebra of invariant sets.
One σ-algebra we encounter will usually not be countably generated; in this situation we will use a countably generated substitute. If (X, X, µ) is a separable probability measure space, and Γ is a countable group of transformations of X preserving X and µ, let I0 denote the σ-algebra of Γ-invariant sets. In the future, when we refer to “the σ-algebra of Γ-invariant sets,” we will mean I0 , but the σ-algebra I generated by the sets {x ∈ X : p < f (x) < q, f ∈ R}, where R is a countable dense subset of the space of Γ-invariant functions in L2 (µ). Although there is some ambiguity in this definition, it will cause no problems.
2.2
Locally compact topological groups.
In this section we summarize some of the standard facts about locally compact topological groups and Haar measure. These can be found in [19] or [34]. Notation: if G is a group, we denote the identity of G by idG , or simply id if there is no ambiguity.
13
2.2.1
Haar measure.
Every locally compact topological group admits a left invariant measure. To be precise, we state the following theorem from [19]. Theorem 2.2.1. (Existence of Haar measure) If G is a locally compact group with Borel σ-algebra B, there exists a nonzero Radon measure mG on (G, B) such that mG (tA) = mG (A) for all t ∈ G, A ∈ B, such that mG (U ) > 0 if U is open. The measure mG is unique up to a multiplicative constant. That is, if mG , m0G are two such measures, then mG = cm0G for some c > 0. Furthermore, if G is a locally compact group, we have the following implications. · If A, B ∈ B with mG (A), mG (B) < ∞, then each of the functions G → R, t 7→ mG (A ∩ tB), t 7→ mG (A ∩ Bt) is continuous. · If G is compact, mG is right invariant as well as left invariant, and mG (A−1 ) = mG (A) for all Borel sets A ⊂ G. If G is compact and H ⊂ G is a closed subgroup, then G acts on G/H by left multiplication: g(aH) := (ga)H. One can define a measure mG/H on G/H by mG/H (AH) = mG (AH), and we will refer to mG/H as the Haar measure of G/H. Note that mG/H is invariant under the action of G on G/H. If G is compact, then mG is finite and nonzero, and we assume that mG is normalized with mG (G) = 1.
14
2.2.2
Unitary representations of a discrete group.
The theory of unitary representations of locally compact groups is presented in [19]. Here we are mainly concerned with representations of countable discrete groups. If H is a Hilbert space with inner product h·, ·i, we say that T : H → H is unitary if T is a linear map satisfying hT x, T yi = hx, yi for all x, y ∈ H. If Γ is a group and H is a Hilbert space, then an action U of Γ on H is a unitary representation if for all γ ∈ Γ, Uγ is a unitary transformation of H. Two unitary actions U, V on Hilbert spaces H, H0 are unitarily equivalent if there is a surjective isometry θ : H → H0 satisfying θ(Uγ x) = Vγ θ(x) for all γ ∈ Γ, x ∈ H. If U = {Uγ : γ ∈ Γ} is a family of maps such that γ 7→ Uγ −1 is a unitary representation, then we call U a unitary anti-representation of Γ. A unitary representation U of Γ is called ergodic if the only vector x ∈ H satisfying Uγ x = x for all γ ∈ Γ is x = 0.
2.2.3
Pontryagin duality, Bochner’s theorem, and the Bohr compactification.
The material in this subsection can be found in [34]. Let G be a locally abelian compact group, and let T denote {z ∈ C : |z| = 1}. The dual b is the group of continuous homomorphisms χ : G → T, where of G, denoted by G, b is endowed with the topology the group operation is pointwise multiplication. If G b is again locally compact. If G is of uniform convergence on compact subsets, then G
15
discrete, then the topology of uniform convergence on compact subsets is the topology b is compact. of pointwise convergence, so G b are called characters of G. The elements of G One of the fundamental results concerning locally compact abelian groups is that every such group is canonically isomorphic to the dual of its dual. This is the content of the following theorem, often referred to as “Pontryagin duality,” or the Pontryaginvan Kampen duality theorem. b be its dual. Theorem 2.2.2. Let G be a locally compact abelian group, and let G bb For every g ∈ G, define a map eg ∈ G by eg (χ) = χ(g). Then the map g 7→ eg is a d b homeomorphism and isomorphism from G onto (G). For a proof, see [19], Theorem 4.31, or [34], Theorem 24.8. A useful consequence of Theorem 2.2.2 is the following structure theorem for locally compact abelian groups. Theorem 2.2.3. (cf. [34], Theorem 9.14) Every locally compact abelian group has an open subgroup isomorphic to Rn × K, where K is a compact abelian group and n ∈ N ∪ {0}. b is an element of L2 (mG ). If G is a compact abelian group, then each character χ ∈ G It is well known that when G is compact abelian, the characters of G form an orthonormal basis of L2 (mG ), and span a dense subset of C(G). The following theorem, known as the Bochner-Herglotz theorem or Bochner’s theorem, is fundamental to understanding unitary representations of a locally compact abelian group G. 16
Theorem 2.2.4. Let G be a locally compact abelian group, and let g 7→ Ug be a unitary representation of G on a Hilbert space H. For all x ∈ H, there is a probability b such that measure νx on G Z hUγ x, xi =
χ(g) dνx (χ).
See [19], Theorem 4.18 for a proof and references. Corollary 2.2.5. Let G be a locally compact abelian group, and let g 7→ Uγ be a unitary representation of G on a Hilbert space H. Let x ∈ H, and denote by Hx the closure of the span of {Uγ x : g ∈ G}. Let U (x) be the representation of G on Hx (x)
given by g 7→ Uγ |Hx . Let νx be the measure associated to the map g 7→ hUg x, xi by the Bochner-Herglotz theorem. Then the representation U (x) is equivalent to the representation V on L2 (νx ) given by Vg f = eg f. Furthermore, U (x) is ergodic if and only if νx ({idGˆ }) = 0. 2.2.4
The Bohr compactification.
As is shown in Section 4.7 of [19], every locally compact abelian group G can be embedded into a compact abelian group bG with the following property: if K is a compact group and ρ : G → K is a continuous homomorphism, then ρ extends to b a continuous homomorphism from bG to K. In particular, every character χ ∈ G c extends to a continuous homomorphism χ˜ ∈ bG.
17
2.3
Topological systems.
Although topological dynamics is closely related to measure preserving dynamics, we will use only the notions of minimality and equicontinuity in this dissertation. More on these topics is contained in [23] and [26]. Let Γ be a countable discrete group. If (X, d) is a compact metric space and T is an action of Γ on X by homeormorphisms, we say that (X, T ) is a topological system. The topological system (X, T ) is called minimal if for all x ∈ X, {Tγ x : γ ∈ Γ} is dense in X. The system (X, T ) is called equicontinuous if the collection {Tγ : γ ∈ Γ} is an equicontinuous family of functions. In this case, there is a metric d0 on X equivalent to the original metric such that every Tγ is an isometry with respect to d0 . Thus, we will also call an equicontinuous system an isometric system.
2.4
Measure preserving systems.
In this section we summarize some of the standard facts about measure preserving dynamics. The material in this section appears in [26]. Introductory material on measure preserving dynamics can be found in [65], [57], [60], and [23]. In this section, we fix a countable group Γ. First let us fix notation for actions of Γ. If T is an action of Γ on a set X, we let Tγ be the element of T corresponding to Γ, so that Tγ : X → X for each γ, and Tγ Tγ 0 = Tγγ 0 for all γ, γ 0 ∈ Γ. If T and S are actions of Γ, then T × S denotes the action of Γ on X × X given by γ 7→ Tγ × Sγ .
18
If Γ is abelian, as it will often be in Chapters 4 and 5, we may use the notation γ 7→ T γ rather than γ 7→ Tγ . If (X, X, µ) is a measure space, we say that a map S : X → X is µ-measure preserving (or simply measure preserving if there is no confusion) if for all A ∈ X, S −1 (A) ∈ X and µ(S −1 A) = µ(A). If T is an action of Γ on X such that for all γ ∈ Γ, Tγ : X → X is measure preserving, then we call (X, X, µ, T ) a measure preserving Γ-system. We henceforth assume all measure preserving Γ-systems have separable probability spaces as their underlying measure spaces, and that the underlying σ-algebras are countably generated. If X = (X, X, µ, T ) and Y = (Y, Y, ν, S) are two measure preserving Γ-systems, then we say that Y is a factor of X if there is a map φ : X → Y such that φ−1 (Y) ⊆ X, µ(φ−1 (A)) = ν(A) for all A ∈ Y, and φ(Tγ x) = Sγ φ(x) for all γ ∈ Γ, for µalmost every x ∈ X. We call φ a factor map. If, in addition to the above conditions, φ : X → Y is a measure-theoretic isomorphism from (X, X, µ) to (Y, Y, ν), we say that X and Y are isomorphic, and that φ is an isomorphism. If Y is a factor of X, we also say that X is an extension of Y. Given a factor Y of X with factor map φ : X → Y, the σ-algebra φ−1 (Y) is T invariant, and we sometimes abuse notation and identify Y with φ−1 (Y). Conversely, if D ⊂ X is a countably generated T -invariant sub σ-algebra, then there is a factor (Y, Y, ν, S) of X, with factor map ψ, such that D = ψ −1 (Y).
19
2.4.1
Joinings.
Fix k ∈ N. If Xi = (Xi , Xi , µi , T (i) ), 1 ≤ i ≤ k are Γ-systems, then a joining X of (Xi )ki=1 is a Γ-system (X1 × · · · × Xk , X1 ⊗ · · · ⊗ Xk , λ, T (1) × · · · × T (k) ), where λ is a T (1) × · · · × T (k) -invariant measure satisfying the following marginal condition: for each i, λ
i−1 Y
X ×A×
j=1
k Y
X = µi (A)
j=i+1
for all A ∈ Xi . The set of joinings of a collection (Xi )ki=1 is always nonempty, as it Q includes the product system ki=1 Xi .
Relatively independent joinings. A generalization of the product system is given by the following construction. Let X = (X, X, µ, T ) be a Γ-system, and let Y = (Y, Y, ν, S) be a factor of X. Let x 7→ µx denote the disintegration of µ over Y. The relatively independent joining of X R over Y is the system (X × X, X ⊗ X, µ ×Y µ, T × T ), where µ ×Y µ = µx × µx dµ(x). R That is, µ ×Y µ(C) = µx × µx (C) dµ(x) for all C ∈ X ⊗ X. Assuming that (X, X, µ) is a regular Borel probability space, the measure µ ×Y µ can also be defined in the following way: define a linear functional L on the space of functions spanned by linear combinations of functions of the form f ⊗ g, where R f, g : X → C are bounded and X-measurable, by L(f ⊗ g) = Eµ (f |Y)Eµ (g|Y) dµ. Then L can be extended to a positive linear functional on the space of all bounded,
20
X ⊗ X-measurable functions. By the Caratheodory, there is a regular Borel measure ν R R on(X × X, X ⊗ X) such that ψ dν = L(ψ) for all ψ ∈ C(X × X). Since f ⊗ g dν = R f ⊗ g dµ ×Y µ for all f, g ∈ C(X × X), we conclude ν = µ ×Y µ. In general, if X and Y are Γ-systems with factors X0 = (X 0 , X0 , µ0 , T 0 ) and Y0 = (Y 0 , Y0 , ν 0 , S 0 ), we say that a joining λ of X and Y is relatively independent over a joining λ0 of X0 and Y0 if Z
Z f ⊗ g dλ =
2.4.2
Eµ (f |X0 ) Eν (g|Y0 ) dλ0 .
Ergodicity and ergodic decomposition.
If Γ is a countable group, and X = (X, X, µ, T ) is a Γ-system, we say that X is ergodic if for all A ∈ X, µ(A4Tγ A) = 0 for all γ ∈ Γ implies µ(A) = 0 or µ(A) = 1. Write MTerg (X) for the set of measures ν such that (X, X, ν, T ) is ergodic. For an arbitrary Γ-system, X, one can write µ as a convex combination of ergodic measures. Theorem 2.4.1. (cf. [26], Theorem 8.7) Let X = (X, X, µ, T ) be a Γ-system. and let I ⊂ X be the σ-algebra of T -invariant sets. Let Y = (X, X, ν, S) be the factor determined by I with factor map π, so that I = π −1 (Y). Let y 7→ µy be the disintegration of µ over I, and for every y ∈ Y, denote Xy = π −1 (y) and let Xy = {A ∩ Xy : A ∈ X}. Then 1. S acts as the identity on Y. 2. For ν-almost every y ∈ Y, the system (Xy , Xy , µy , S) is ergodic. 21
3. This decomposition is unique in the following sense. If (Z, Z, η) is a standard probability space and ψ : z → µ ˜z is a measurable map from Z into MTerg (X) such R that µ = Z µ ˜z dη(z), then there exists a measurable map ψ : (Z, Z, η) → (Y, Y, ν) such that for η-almost every z, µφ(z) = µ ˜z .
2.5
Amenable groups.
Here we consider a class of groups which admit translation invariant notions of density. A countable group Γ is called amenable if there exists a left Følner sequence for Γ, that is, a sequence (Φn )n∈N of finite subsets of Γ, such that for all |Φn ∩ γΦn | = 1. n→∞ |Φn |
γ ∈ Γ, lim If instead one has limn→∞
|Φn γ∩Φn | |Φn |
= 1 for all γ ∈ Γ, then (Φn )n∈N is called a right
Følner sequence. If (Φn )n∈N is both a left- and a right Følner sequence, then we say it is a two sided Følner sequence. Most of the material in this section is contained in [30], [41], and [55]. The theory of measure preserving actions of amenable groups was studied extensively in [52]. Recurrence phenomena for amenable groups was studied in [9], [11], and [10]. A left Følner sequence (Φn )n∈N for a group Γ can be used to define a translationinvariant notion of density as follows: for E ⊂ Γ, let dΦ (E) = lim sup n→∞
|E ∩ Φn | . |Φn |
It is easy to verify that dΦ (E) = dΦ (gE) whenever (Φn )n∈N is a left Følner sequence.
22
2.5.1
The mean ergodic theorem.
The mean ergodic theorem can be stated for amenable groups as follows. Theorem 2.5.1. Suppose that Γ is a countable amenable group, and (Φn )n∈N is a left Følner sequence for Γ. If U is a unitary anti-representation of Γ on a Hilbert space H, then for all x ∈ H, 1 X Uγ x = P x, n→∞ |Φn | γ∈Φ lim
(2.1)
n
in the norm topology of L2 (µ), where P is the orthogonal projection onto the closed space of U -invariant vectors. The proof is a straightforward generalization of the usual proof of the mean ergodic theorem for powers of a single transformation (e.g. Theorem 1.2 of Chapter 2 of [57]). Proof. Consider the subspaces H0 := {x ∈ H : Uγ x = x for all γ ∈ Γ} H1 := span{Uγ x − x : f ∈ H, γ ∈ Γ}. We claim that H = H0 ⊕ H1 , that is, H is the orthogonal direct sum of H0 and H1 . First we verify that H0 ⊥ H1 . It suffices to check that hx, Uγ y − yi = 0 for all x ∈ H0 , y ∈ H, and γ ∈ Γ. We have hx, Uγ y − yi = hx, Uγ yi − hx, yi = hUγ x, Uγ yi − hx, yi = 0, 23
for all γ ∈ Γ, since each Uγ is unitary. To show that H0 ⊕ H1 is dense in H, it suffices to show that x ∈ H0 if x ⊥ H1 . But if x ⊥ H1 , then hx, Uγ x − xi = 0 for all γ ∈ Γ, and so hx, Uγ xi = hx, xi for all γ ∈ Γ. Since kUγ xk = kxk for all γ ∈ Γ, this implies that Uγ x = x for all γ ∈ Γ, as desired. Thanks to the decomposition H = H0 ⊕ H1 , we will prove the theorem if we prove it in the special cases x ∈ H0 and x ∈ H1 . In the case x ∈ H0 , the sequence of averages in (2.1) is constant, and every term is equal to P x. For the case x ∈ H1 , x can be approximated in norm by linear combinations of vectors of the form Uβ y − y, so we consider only vectors of this form. Computing the averages in this case, we find 1 X 1 X 1 X Uγ (Uβ y − y) = Uβγ y − Uγ y |Φn | γ∈Φ |Φn | γ∈Φ |Φn | γ∈Φ n
n
=
1 |Φn |
n
X
Uγ y −
γ∈βΦn \Φn
1 |Φn |
X
Uγ y.
γ∈Φn \βΦn
1 kyk |Φn |
· |βΦn 4Φn |. Since (Φn )n∈N is a
left Følner sequence, the norm tends to 0 as n → ∞.
The norm of the last average is bounded by
Corollary 2.5.2. If Γ is a countable group, (Φn )n∈N is a right Følner sequence for Γ, and U is unitary anti-representation of Γ on a Hilbert space H, then for all x ∈ H, 1 X Uγ x = P x n→∞ |Φn | γ∈Φ lim
n
in the weak topology of L2 (µ), where P x is the orthogonal projection of x on the space of U -invariant vectors. Proof. We need to show that for all x, y ∈ H, D 1 X E lim Uγ x, y = hP x, yi. n→∞ |Φn | γ∈Φ n
24
Noting that hUγ x, yi = hx, Uγ −1 yi for all γ ∈ Γ, we obtain D 1 X E D E 1 X Uγ x, y = x, Uγ −1 y |Φn | γ∈Φ |Φn | γ∈Φ n n E D X 1 Uγ y . = x, |Φn | −1
(2.2)
γ∈Φn
Since (Φn )n∈N is a right Følner sequence, (Φ−1 n )n∈N is a left Følner sequence, so Theorem 2.5.1 implies that 1 X Uγ y = P y. n→∞ |Φn | −1 lim
(2.3)
γ∈Φn
in the norm topology of L2 (µ). Equations (2.2) and (2.3) imply that D 1 X E Uγ x, y = hx, P yi, n→∞ |Φn | γ∈Φ lim
n
and hx, P yi = hP x, yi.
Commuting representations. If Γ and Λ are discrete groups, and U and V are unitary actions of Γ and Λ on a Hilbert space H, we say that U and V commute if Uγ Vλ = Vλ Uγ for all γ ∈ Γ, λ ∈ Λ. If IU ⊂ H and IV ⊂ H denote the spaces of U -invariant and V -invariant vectors, respectively, and let PU , PV denote the orhtogonal projections onto IU and IV , respectively. then Vλ IU ⊆ IU and Uγ IV ⊆ IV for all λ ∈ Λ, γ ∈ Γ, whenever U and V commute. As a consequence of the mean ergodic theorem, if Γ and Λ are amenable, then PU commutes with PV whenever U and V commute. Furthermore, if J ⊂ H is the space of vectors which are simultaneously U - and V -invariant, then PJ = PU PV = PV PU .
25
2.5.2
The von Neumann-Koopman representation.
If (X, X, µ, T ) is a probability measure preserving Γ-system, then there is a unitary representation, called the von Neumann-Koopman representation, V (T ) on L2 (µ) (T )
given by Vγ f = f ◦ Tγ−1 for γ ∈ Γ. For convenience, we will usually consider instead the anti-representation given by Uγ f = f ◦ Tγ , and we call this the von Neumann-Koopman anti-representation. For the special case of the von Neumann-Koopman anti-representation, the mean ergodic theorem can be stated as follows. Theorem 2.5.3. Let Γ be a countable group and let (Φn )n∈N be a left Følner sequence for Γ. If (X, X, µ, T ) is a probability measure preserving Γ-system, then for all f ∈ L2 (µ) 1 X f ◦ Tγ = Eµ (f |IT ), n→∞ |Φn | γ∈Φ lim
n
2
in the norm topology of L (µ), where Eµ (f |IT ) is the orthogonal projection of f on the space of T -invariant functions. If instead (Φn )n∈N is a right Følner sequence, then the limit exists in the weak topology of L2 (µ).
2.5.3
Isometric systems and group extensions.
A probability measure preserving Γ-system (X, X, µ, T ) is called isometric if X is a compact metric space, X is the Borel σ-algebra of X, µ is a regular Borel probability measure of full support on X and Tγ : X → X is an isometry for each γ ∈ Γ. The following theorem of G. Mackey (see [47]) classifies ergodic isometric Γ-systems. 26
Theorem 2.5.4. If (X, X, µ, T ) is an ergodic isometric Γ-system, then there is a compact group G, a closed subgroup L, and a homomorphism ψ : Γ → G, such that (X, X, µ, T ) is isomorphic to (G/L, B, mG/L , R), where Rγ (xL) = ψ(γ)xL, for γ ∈ Γ and B is the Borel σ-algebra of G/L. In the case Γ = Z, one can conclude that G is abelian. In this case, Theorem 2.5.4 is known as the Halmos - von Neumann Theorem (see [32] or [26] for a proof). As a special case of the Furstenberg-Zimmer structure theorem ([22], [70],[71]) in the case Γ = Z), every ergodic Γ-system (X, X, µ, T ) has a maximal isometric factor (Z, Z, m, R). This factor is useful for identifying the T × T -invariant functions on (X × X, X ⊗ X, µ × µ). Proposition 2.5.5. Let (X, X, µ, T ) be an ergodic Γ-system, and let (Z, Z, m, R) be its maximal isometric factor. Then every T × T -invariant function on (X × X, X ⊗ X, µ × µ) is measurable with respect to the σ-algebra Z ⊗ Z. For a proof, see [22] for the case Γ = Z, [70] for the general case. As a consequence, if f ∈ L2 (µ) and Eµ (f |Z) = 0, then for all g ∈ L2 (µ), Eµ×µ (f ⊗ g|IT ×T ) = 0.
2.5.4
Cocycles and skew product extensions.
Here we consider a general way of constructing extensions of measure preserving Γ-systems.
27
If (X, X, µ, T ) is a Γ-system and G is a group, then a function α : Γ×X → G (defined µ-almost everywhere) is called a cocycle if α(γγ 0 , x) = α(γ, Tγ 0 x)α(γ 0 , x) for all x ∈ X, γ ∈ Γ. Two cocycles α and β are said to be cohomologous if there is a function F : X → G such that β(γ, x) = F (Tγ x)−1 α(γ, x)F (x) for µ-almost all x ∈ X and all γ ∈ Γ. A cocycle α : Γ × X → G is called a coboundary if α(γ, x) = f (Tγ x)−1 f (x) for some f : X → G. So α is a coboundary if and only if it is cohomologous to the map (γ, x) 7→ idG . Given a Γ-system X and a compact group G, a cocycle α : Γ × X → G can be used to define the group extension X ×α G = (X × G, X ⊗ B, µ × mG , T (α) ), where (α)
Tγ (x, g) = (Tγ x, α(γ, x)g), and B is the Borel σ-algebra of G. If β is cohomologous to α so that β(γ, x) = F (Tγ x)−1 α(γ, x)F (x), then the system X ×β G is isomorphic to X ×α G, with the isomorphism given by (x, g) 7→ (x, F (x)−1 g). A Γ-system X is a homogeneous extension of a system Y if there is a compact group G with closed subgroup H such that X = Y × G/H, µ = ν × mG , and there is a cocycle α : Γ × Y → G such that Tγ (y, gH) = (Sγ y, α(γ, y)gH) for (y, gH) ∈ X. Measurability. If X is a Γ-system and S is the circle group {z ∈ C : |z| = 1}, let C(X, S) denote the group of cocycles ρ : Γ × X → S under pointwise addition. Given the topology of convergence in measure and the group operation of pointwise multiplication, it is a Polish group, due to completeness and separability of L2 (µ). 28
In Chapter 4, we will consider maps ω → ρω , from a probability space (Ω, P ) to C(X, S). Because each ρω is defined µ almost everywhere, the map (ω, γ, x) 7→ ρω (γ, x) is not a priori defined for all ω. Fortunately, there exists a map R : Ω × Γ × X → S, defined P × µ-almost everywhere, such that for every ω ∈ Ω, ρω (γ, x) = R(ω, γ, x). (See, e.g., [64], p. 65 for a proof.)
2.5.5
The Mackey group.
While a group extension X of Y may not be ergodic, the ergodic components have a convenient description. Fix an ergodic Γ-system Y = (Y, Y, ν, S), a compact group G, and form a group extension X of Y by a cocycle α : Γ × Y → G. Then G acts on X = Y × G by Mg (y, h) = (y, hg). The following fact is a consequence of Theorem 3.25 of [26]. It was first proved explicitly in [70], but is stated in [48]. Theorem 2.5.6. There is a closed subgroup K ⊂ G and a cocycle β : Γ × Y → K cohomologous to α such that the group extension Y ×β K is ergodic, and Z ν × mG =
Mg (ν × mK ) dmH\G (g). K\G
The subgroup K is unique up to conjugacy, and X is ergodic if and only K = G. We abuse notation and call the conjugacy class of the group K in Theorem 2.5.6 the Mackey group of the cocycle α.
29
2.5.6
Isometric extensions.
If X is a G-system and Y is a factor of X, the maximal isometric extension of Y is the factor of X generated by linear combinations of functions of the form Z x 7→
H(x, t)ψ(t) dµYx (t),
where H ∈ L2 (µ ×Y µ) is T × T -invariant. If X is ergodic, then the maximal isometric extension of Y is actually a homogeneous extension of Y. (See [70], or [22] for the case Γ = Z.) Zimmer gave several characterizations of isometric extensions in [70], and Furstenberg independently gave these characterizations in the case Γ = Z in [22]. These characterizations will be needed in Chapter 4. We summarize the relevant consequences, as presented in [26], Chapter 9. Fix an ergodic probability measure preserving Γ-system X = (X, X, µ, T ) and a factor Y = (Y, Y, ν, S). Let y 7→ µy be the disintegration of µ over Y. We say that a subspace M ⊂ L2 (µ) is a Y-module if for all f ∈ M, g ∈ L∞ (Y), g · f ∈ M. The rank of M is the least number r so that there exist k functions f1 , . . . , fr ∈ M P such that every f ∈ M can be written as a combination ri=1 gi ·fi , where gi ∈ L∞ (Y). Such a collection {f1 , . . . , fr } is called a Y-basis. If Y is ergodic, then there is a Y R basis {f1 , . . . , fr } such that fi · f¯j dµy = δij for ν-almost all y. We call such a basis an orthonormal Y-basis. If {h1 , . . . , hr } is an orthonormal Y-basis, then for each γ ∈ Γ, hi (Tγ−1 x)
=
r X
λij (γ, y)hj (x),
j=1
30
where the λij (γ, y) are the entries of a unitary matrix Λ(γ, y). A function f ∈ L2 (µ) is called a generalized eigenfunction if {f ◦ Tγ : γ ∈ Γ} belongs to some Γ-invariant finite rank Y-module. Let E(X/Y) be the closure of the span of the generalized Y-eigenfunctions in L2 (µ). We say that X has relatively discrete spectrum over Y if L2 (µ) = E(X/Y). The following, Theorem 9.14 from [26], characterizes isometric extensions as compact group extensions. Theorem 2.5.7. If X is an ergodic extension of Y, it is an isometric extension if and only if it is a homogeneous skew product extension. The next theorem describes the eigenfunctions of a relative product system. In particular, it shows that the invariant functions of a relative product are measurable with respect to the σ-algebra generated by functions of the form φ ⊗ ψ, where φ and ψ are relative eigenfunctions over Y. Theorem 2.5.8. ([26], Theorem 9.21) Let X1 , X2 be two (not necessarily ergodic) dynamical systems, with a common factor Y, and corresponding factor maps πi : Xi → Y for i = 1, 2. Let ρi : Zi → Y be the maximal intermediate isometric extensions of Y within Xi for each i, and let ρ : Z → Y be the maximal isometric extension of Y within X1 ×Y X2 → Y. Then Z = Z1 ×Y Z2 , in the sense that E(X1 ×Y X2 /Y) is spanned by functions of the form φ ⊗ ψ, where ψ ∈ E(X1 /Y), φ ∈ E(X2 /Y). In particular, every invariant function on X1 ×Y X2 is measurable with respect to Z1 ⊗ Z2 . 31
If X are ergodic, Theorem 2.5.8 implies that the invariant functions on X × X are (relative to µ × µ) measurable with respect to Z ⊗ Z, where Z ⊂ X is the σ-algebra corresponding to the maximal isometric factor of X. This implies that whenever Eµ (f |Z) = 0 and g ∈ L∞ (µ), then Eµ×µ (f ⊗ g|IT ×T ) = 0, where IT ×T is the algebra of T × T -invarant sets in X ⊗ X. 2.5.7
Shift spaces and the Furstenberg correspondence principle.
Let G be a countable group. If Ω = {0, 1}Γ , we consider points ξ ∈ Ω as functions ξ : Γ → {0, 1}. If we give Ω the product topology, it is a compact metrizable space, and a sequence (ξn )n∈N converges to a point ξ ∈ Ω if and only if for all finite F ⊂ G, there exists N ∈ N such that ξn |F = ξ|F for all n ≥ N. The space Ω admits two natural actions of Γ. We will mainly be interested in the right shift, σ (r) , given by (σγ(r) (ξ))(γ 0 ) = ξ(γ 0 γ). The next proposition is a version of Furstenberg’s correspondence principle from [22], adapted to the setting actions of amenable groups. It is proved, though not explicitly stated, as Proposition 6.2 in [11]. Proposition 2.5.9. Let Γ be a countable amenable group, and let (ΦN )N ∈N be a left Følner sequence for Γ. Suppose that E ⊂ Γ with dΦ (E) > 0. Then there is a measure preserving Γ-system (X, B, µ, T ), where X is a compact metric space with
32
Borel σ-algebra B, and a clopen set A ∈ B with µ(A) = dΦ (E), such that for all γ1 , . . . , γk ∈ Γ, A) ≤ dΦ (E ∩ γ1−1 E ∩ · · · ∩ γk−1 E). A ∩ · · · ∩ Tγ−1 µ(A ∩ Tγ−1 1 k
(2.4)
Proof. Passing to a subsequence (Φ0N )N ∈N of (ΦN )N ∈N , we assume that dΦ (E) = limN →∞
|E∩Φ0N | . |Φ0N |
Let Ω = {0, 1}Γ , be the shift space with the product topology, and
define an action of Γ on Ω by (Tγ ξ)(h) = ξ(hγ). Consider 1E as an element of Ω, and let X be the orbit closure {Tγ 1E : γ ∈ Γ}. Let A = {ξ ∈ X : ξ(e) = 1}. Note that γ ∈ E if and only if Tγ 1E ∈ A. Let the measure µ on X be a weak∗ limit of the measures µN :=
1 X δTγ ξ . |Φ0N | 0 γ∈ΦN
Then µ(A) = dΦ (E), and one can check from the definition of µ and A that inequality (2.4) holds. To see that µ is T -invariant, note that Z f ◦ Tγ 0 dµN =
1 X f (Tγ 0 Tγ 1E ), |ΦN | γ∈Φ N
P f dµN = |Φ1N | γ∈ΦN f (Tγ 1E ). Since (ΦN )N ∈N is a left Følner sequence, the R R limits of f ◦ Tγ 0 dµN and f dµN are the same.
while
R
As a consequence of Proposition 2.5.9 and ergodic decomposition, whenever Γ is a countable amenable group, and E ⊂ Γ with dΦ (E) > 0 with respect to some left Følner sequence (Φn )n∈N , there exists an ergodic measure µ on the orbit closure of 1E in the shift space {0, 1}Γ .
33
2.6
Product sets in compact topological groups.
In this section, we prove a classical result, sometimes referred to as the Steinhaus Lemma, and a variation thereof. We only consider the special case of compact groups. A proof in the general case may be found by combining Theorem E, Section 59 of [31], and Theorem A, Section 61 of the same volume. Lemma 2.6.1. If G is a compact topological group, and A, B ⊂ G with mG (A), mG (B) > 0, then the set A−1 B := {a−1 b : a ∈ A, b ∈ B} contains a nonempty open set. Proof. Consider the group G×G, and its diagonal subgroup ∆ = {(g, g) : g ∈ G}. Let Y denote the σ-algebra on G × G generated by sets of the form {(g, h) : g −1 h ∈ D}, where D is a Borel set. Then Y is countably generated, and the atoms of Y are the cosets of ∆. The disintegration of mG × mG over Y, denoted by (g, h) 7→ mY(g,h) can be realized as the map taking (g, h) to Haar measure µg−1 h of the coset (g −1 h, e)∆. R For each t ∈ G, µt is a measure on G × G, and µt (A × B) = 1A (x)1B (xt) dmG (x). Hence the map t 7→ µt (A × B) is continuous. The measures µt reveal information about A−1 B as follows: observing that t ∈ A−1 B if µt (A × B) > 0, we see that it suffices to show that µt (A × B) > 0 for a nonempty open set of t ∈ G. By continuity of the map t 7→ µt (A × B), it suffices to show that µt (A × B) > 0 for at least one t. R Since µt dmG (t) = mG , we have Z µt (A × B) dmt (G) = mG (A × B) > 0, hence there exists at least one t with µt (A × B) > 0.
34
From the proof of Lemma 2.6.1, we see that the set of t with µt (A × B) > 0 is an open, nonempty set. In Chapter 3, we will need the following variations of Lemma 2.6.1. Lemma 2.6.2. Let G be a compact group, and let A, B ⊂ G with mG (A), mG (B) > 0. Let S ⊂ G be a Borel set with the following property: for all a ∈ A, mG (S ∩ a−1 B) = mG (a−1 B). Then there exists a nonempty open set U ⊂ G such that mG (U \ S) = 0. If, in addition, we assume that H ⊂ G is a closed subgroup and BH = B, then we can make the same conclusion with the additional condition that U H = U. Proof. Note that the hypothesis of the lemma is equivalent to the assertion that for all a ∈ A, there exists Borel set Ba ⊂ B with mG (Ba ) = mG (B), and a−1 Ba ⊂ S. Let E = {(a, b) : a ∈ A, b ∈ Ba }. One can verify that E is measurable by writing E as the set {(a, b) : a ∈ A, b ∈ B, a−1 b ∈ S}. Let t 7→ µt be as in the proof of Lemma 2.6.1. Note that t ∈ S if µt (E) > 0. Since mG ×mG (E \A×B) = 0, Fubini’s theorem implies µt (E) = µt (A×B), for mG -almost every t. It follows that {t : µt (E) > 0} agrees with {t : µt (A × B) > 0}, except for a set of t of mG -measure 0. Hence we can take U = {t : µt (A × B) > 0}. R Since µt (A × B) = 1A (x)1B (xt)dmG (x), the assumption that Bh = B for all h ∈ H means that µth (A × B) > 0 whenever µt (A × B) > 0 and h ∈ H. Hence we can take U H = U if BH = B.
Corollary 2.6.3. Let G be a compact group and H a closed subgroup. Let m be Haar measure on G/H. Let A ⊂ G with mG (A) > 0, and let B ⊂ G/H with m(B) > 0. If
35
S ⊂ G/H is a Borel set such that for all a ∈ A, m(aB \ S) = 0, then there exists an open set U ⊂ G/H such that U ⊂ S, up to m-measure 0. Proof. Let θ : G → G/H be the quotient map. We consider B as a union of left cosets of H, denoted by B 0 ⊂ G, and similarly we consider S as a union of left cosets of H, denoted by S 0 ⊂ G. By hypothesis, for all a ∈ A, mG (aB 0 \ S 0 ) = 0. By Lemma 2.6.2, there exists an open set U 0 ⊂ G with U 0 H = U 0 , such that mG (U 0 \ S 0 ) = 0. The set U = θ(U 0 ) is the desired open set.
2.7
Large sets and Bohr sets in groups.
In preparation for Chapter 3, we summarize some of the notions of largeness for countable discrete groups. Definition 2.7.1. Let Γ be a countable discrete group, and let S ⊂ Γ. We say that S S is left (right-) syndetic if there is a finite set F ⊂ Γ such that γ∈F γS = Γ S ( γ∈F Sγ = Γ). We call a set S ⊂ Γ left (right-) thick if for all finite F ⊂ Γ, there exists γ ∈ Γ such that γF ⊂ S (F γ ⊂ S). Note that the intersection of a left thick set and a left syndetic set is nonempty, and in fact, the intersection of a right thick set and a left syndetic set is nonempty. We say that E ⊂ Γ is piecewise syndetic if E is the intersection of a thick set and a syndetic set. A set B ⊂ Γ is a left Bohr set if there is a compact metric space X, a minimal action
36
T of Γ on X by isometries, an open set U and an x ∈ X such that B ⊃ {γ : Tγ x ∈ U }. Since the action T is minimal, {γ : Tγ x ∈ U } is left syndetic. We say that E ⊂ Γ is piecewise left Bohr if E is the intersection of a thick set and left Bohr set. The notion of “piecewise Bohr” was introduced in [7] for Γ = Z, and for general groups in [6]. Lemma 2.7.2. Let B ⊂ Γ be a left Bohr set, and let E ⊂ Γ. If for all finite F ⊂ Γ, there exists γ ∈ Γ such that (F ∩ B)γ ⊂ E, then E is piecewise left Bohr. Proof. Since B is a left Bohr set, there exists a minimal isometric Γ-system (X, T ), an open set U ⊂ X and x ∈ X such that {g : Tγ x ∈ U } ⊂ B. For convenience, S assume that B = {γ : Tγ x ∈ U }. Write Γ as an increasing union j∈N Fj of finite sets Fj . For each j, let hj ∈ Γ such that E ⊃ (Fj ∩ B)hj . The set Fj ∩ B can be written as {γ ∈ Fj : Tγ x ∈ U }, so (Fj ∩ B)hj = {γ ∈ Fj hj : Tγh−1 x ∈ U }. j By compactness of X, there exists a subsequence h0j of hj such that Th−1 0 x converges j
to a point z ∈ X. Since (X, T ) is isometric, there is a neighborhood V of z and an open set U 0 ⊂ U such that Tγ V ⊂ U whenever Tγ z ∈ U 0 . Thus, for large enough j, the set {γ ∈ Fj h0j : 0 0 Tγ (Th−1 0 x) ∈ U } contains {γ ∈ Fj hj : Tγ z ∈ U }. This shows that E is piecewise left j
Bohr.
37
2.7.1
Characterizations of Bohr sets when Γ is abelian.
In the case where Γ is abelian, there are several well known characterizations of Bohr sets. We do not need them for our proofs, but we state them here for reference. Lemma 2.7.3. Let Γ be a countable abelian group and A ⊂ G. The following are equivalent. (i) There exists a linear combination of characters φ =
P
i
b such that χ i , χi ∈ Γ
{γ : Re φ(γ) > 0} ⊂ A. (ii) There exists a compact abelian group Z, a homomorphism R : Γ → Z, x ∈ Z and a continuous f : Z → R such that {γ : f (R(γ) + x) > 0} ⊂ A. (iii) A is a Bohr set.
2.8
van der Corput lemmas.
Many proofs of results concerning multiple ergodic averages use some variation of the van der Corput lemma. The following variation appeared as Lemma 4.2 in [11]. Lemma 2.8.1. Let H be an amenable group, and let {ug : g ∈ H} be a bounded set in a Hilbert space. Let (ΦN )N ∈N , (ΨN )N ∈N be left Følner sequences for H. If X 1 1 lim sup M →∞ |ΨM |2 N →∞ |ΦN | h,k∈Ψ lim
M
1 P
Then limN →∞ |ΦN | g∈ΦN ug = 0.
38
X g∈ΦN
huhg , ukg i = 0
The proof of Lemma 2.8.1 uses the following fact: if {xi : i ∈ F } is a finite set of P P elements in a Hilbert space H, then k |F1 | i∈F xi k2 ≤ |F1 | i∈F kxi k2 . For the proof of Lemma 2.8.1, we state an observation of independent interest. Observation 2.8.2. Let (ΦN )N ∈N be a left Følner sequence in Γ. For all M ∈ N and all δ > 0, there exists N0 such that for all N ≥ N0 , and all sets {ug : g ∈ Γ} of vectors in a Hilbert space with kug k ≤ 1 for all g ∈ Γ,
1 X X 1 1
u − u
g hg ≤ δ |ΦN | g∈Φ |Φm | |ΦN | g∈Φ ,h∈Φ N
N
m
whenever m ≤ M. Consequently
1 X
1 X
1 X
ug ≤ uhg + δ
|ΦN | g∈Φ |ΦN | g∈Φ |Φm | h∈Φ N
m
N
for all N ≥ N0 , m ≤ M. Proof of Lemma 2.8.1. Let (ΦN )N ∈N , (ΨN )N ∈N , and (ug )g∈Γ be as in the statement of the Lemma, and let ε > 0. Assume additionally that kug k ≤ 1 for all g. Choose M0 so that for all M ≥ M0 , lim sup
X 1 1 |ΨM |2 |ΦN | h,k∈Ψ
huhg , ukg i < ε2 .
(2.5)
1 X
X 1 1
u − u
g hg ≤ ε |ΦN | g∈Φ |Ψm | |ΦN | g∈Φ ,h∈Ψ
(2.6)
N →∞
M
X g∈ΦN
Choose N0 so that for all N ≥ N0 ,
N
N
whenever m ≤ M0 .
39
M
We can estimate the norm of
1 1 |Ψm | |ΦN |
P
g∈ΦN ,h∈Ψm
uhg :
1
2
X 1 1 X
1 X uhg ≤ uhg
|Ψm | |ΦN | g∈Φ ,h∈Ψ |ΦN | g∈Φ |Ψm | h∈Ψ N
m
N
m
X 1 X 1 = huhg , ukg i |ΦN | g∈Φ |Ψm |2 h,k∈Ψ m
N
For large enough N, inequalities (2.5) and (2.6) imply k |Φ1N |
40
P
g∈ΦN
ug k < ε.
CHAPTER 3 SUMSETS OF DENSE SETS AND SPARSE SETS
3.1
Introduction.
In this chapter we study relationships between notions of density and sumsets in groups. A classical notion of density for sets in Z is upper Banach density: the upper Banach density of a set A ⊂ Z is d∗ (A) = limk→∞ supn∈Z words, d∗ (A) = lim sup|I|→∞
|A∩I| , |I|
|A∩[n,...,n+k−1]| . k
In other
where I ranges over intervals in Z.
One of the themes of additive number theory is that sumsets, that is, sets of the form {a + b : a ∈ A, b ∈ B} for sets A, B ⊂ Z, have more structure than arbitrary sets. One manifestation of this theme is the following theorem of Renling Jin. Theorem 3.1.1. ([38], Corollary 3) Let A, B ⊂ Z with d∗ (A), d∗ (B) > 0. Then A+B is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to the following. Theorem 3.1.2. ([7], Theorem I) Let A, B ⊂ Z with d∗ (A), d∗ (B) > 0. Then A + B is piecewise Bohr. In [6], the preceding theorem was then generalized to the setting where Z is replaced by an amenable group. 41
Theorem 3.1.3. Let Γ be a countable amenable group with left Følner sequences (Φn )n∈N , (Ψn )n∈N , and let A, B ⊂ Γ with dΦ (A) > 0, dΨ (B) > 0. Then {ab : a ∈ A, b ∈ B} is piecewise Bohr. The main result of this chapter is a generalization of Theorem 3.1.3. We weaken the hypothesis dΦ (A) > 0 and establish the same conclusion. Definition 3.1.4. Let Γ be a countable group, and let (νn )n∈N be a sequence of probability measures on Γ. We say that (νn )n∈N is an ergodic averaging scheme if for all probability measure preserving Γ-systems (X, X, µ, T ), and all f ∈ L2 (µ), the averages Z f ◦ Tγ dνj (γ) converge (in the weak topology of L2 (µ)) to Eµ (f |IT ), the orthogonal projection of f onto the space of T -invariant functions. Definition 3.1.5. If (νn )n∈N is a sequence of probability measures on Γ and C ⊂ Γ, then the upper density of C with respect to (νn )n∈N is dν (C) := lim sup νn (C). n→∞
With these definitions we can state the main result of this chapter. Theorem 3.1.6. Let Γ be a countable amenable group, and suppose that (νn )n∈N is an ergodic averaging scheme for Γ. Let B ⊂ Γ have dΦ > 0 with respect to some left Følner sequence (Φn )n∈N , and let A ⊂ Γ. Then 1. If dν (A) > 0, then {ab : a ∈ A, b ∈ B} is piecewise Bohr. 42
2. If dν (A) = 1, then {ab : a ∈ A, b ∈ B} is thick. This is equivalent to the following theorem, as the condition that (νn )n∈N be an ergodic averaging scheme is equivalent to the condition that (νn0 )n∈N be such a scheme, where νn0 (A) = νn (A−1 ). Theorem 3.1.7. Let Γ be a countable amenable group, and suppose that (νn )n∈N is an ergodic averaging scheme for Γ. Let B ⊂ Γ have dΦ > 0 with respect to some left Følner sequence (Φn )n∈N , and let A ⊂ Γ. Then 1. If dν (A) > 0, then {a−1 b : a ∈ A, b ∈ B} is piecewise Bohr. 2. If dν (A) = 1, then {a−1 b : a ∈ A, b ∈ B} is thick.
3.2
Ergodic averaging schemes for abelian groups.
When Γ is abelian, ergodic averaging schemes can be described in terms of the Pontryagin dual of Γ. This is explicitly done in [13] for the case Γ = Z, and has been used implicitly since the earliest work in ergodic theory. Proposition 3.2.1. Let Γ be a countable abelian group, and let (νn )n∈N be a sequence of probability measures on Γ. The following conditions are equivalent. (i) (νn )n∈N is an ergodic averaging scheme. b limn→∞ (ii) For every nontrivial character χ ∈ Γ,
R
χ(γ) dνn (γ) = 0.
(iii) For every probability measure preserving Γ-system (X, X, µ, T ), and every funcR tion f ∈ L2 (µ), limn→∞ f ◦ Tγ dνn (γ) = Eµ (f |IT ) in the norm topology of L2 (µ). 43
Proof. For the implication (i) =⇒ (ii) we consider the system (bΓ, B, m, T ), where bΓ is the Bohr compactification of Γ, B is the Borel σ-algebra of bΓ, m is Haar measure on bΓ, and T is the action of Γ on bΓ given by Tγ h = γ + h for all γ ∈ Γ, h ∈ bΓ. Let χ be a nontrivial character of Γ. Then there is a nontrivial character χ˜ of bΓ such that χ| ˜ Γ = χ. We have Z ˜m= hχ˜ ◦ Tγ , χi
χ(γ ˜ + h)χ(h) ˜ dm(h) Z
=
χ(γ) ˜ χ(h) ˜ χ(h) ˜ dm(h)
= χ(γ) ˜ = χ(γ). Since (νn )n∈N is an ergodic averaging pair, and
R
χ˜ dm = 0, we have
Z 0 = lim
n→∞
χ(γ) dνn (γ).
For the implication (ii) =⇒ (iii), we apply the Bochner-Herglotz theorem. Let (X, X, µ, T ) be a measure preserving Γ-system, and let f ∈ L2 (µ). Then there is b such that a measure λ on Γ Z hf ◦ Tγ , f ◦ Th iµ =
χ(γ)χ(h) dλ(χ)
for all γ, h ∈ Γ. Then for all m, n ∈ N, Z
Z
f ◦ Tγ dνn (γ) − f ◦ Tγ dνm (γ) 2 L (µ) Z
Z
= χ(γ) dνn (γ) − χ(γ) dνm (γ)
L2 (λ)
44
(3.1) .
R b (3.1) implies that the sequence f ◦ χ(g) dνn (g) exists for all χ ∈ Γ, R Tγ dνn (γ) converges in L2 (µ). The limit is Eµ (f |IT ), since the limit of χ(γ) dµn (γ)
Since limn→∞
R
is 1 if χ is trivial, and 0 otherwise. The implication (iii) =⇒ (i) follows from the definition of the weak topology of L2 (µ). As a consequence of Proposition 3.2.1, we can use ergodic averaging schemes on Γ to find ergodic averaging schemes on Γ × Γ. Lemma 3.2.2. Suppose that Γ and H are countable abelian groups, and (νn )n∈N and (ηn )n∈N are ergodic averaging schemes for Γ and H, respectively. Then the sequence (νn × ηn )n∈N is an ergodic averaging scheme for Γ × H. Proof. Let (X, X, µ, T ) be a probability measure preserving Γ × H system. Let T (Γ) be the action of Γ on (X, X, µ) given by γ 7→ Tγ for γ ∈ Γ, and similarly let T (H) be the action of H on (X, X, µ) given by h 7→ Th for h ∈ H. We want to show that Z lim hf ◦ Tk , ψiµ dνn × ηn (k) = hEµ (f |IT ), ψiµ n→∞
for all f, ψ ∈ L2 (µ). We have Z Z (H) lim hf ◦ Tk , ψiµ dνn × ηn (k) = lim hf ◦ Tγ(Γ) , ψ ◦ Th−1 iµ dνn (γ) dηn (h) n→∞ n→∞ Z Z (H) (Γ) = h lim f ◦ Tγ dνn (γ), ψ ◦ Th dηn (h)iµ n→∞
= hEµ (f |IT (Γ) ), Eµ (ψ|IT (H) )iµ . The last inner product is equal to hEµ (Eµ (f |IT (Γ) )|IT (H) ), ψiµ , 45
and Eµ (Eµ (f |IT (Γ) )|IT (H) ) = Eµ (f |IT ). 3.2.1
Examples of ergodic averaging schemes.
In this subsection we discuss the classical examples of ergodic averaging schemes for amenable groups, and present some examples for Γ = Z due to Boshernitzan, Kolesnik, Quas, and Wierdl. Other ergodic averaging schemes are discussed in [58]. The classical examples of ergodic averaging schemes are given by Følner sequences: if Γ is a countable group and (Φn )n∈N is a left (or right) Følner sequence, and νn denotes normalized counting measure on Φn , then (νn )n∈N is an ergodic averaging scheme, by Theorem 2.5.1 and its corollary. Thus, a special case of Theorem 3.1.6 is the following theorem of Beiglbock, Bergelson, and Fish in [6]. Corollary 3.2.3. Let Γ be a countable amenable group, let (Ψn )n∈N be a left Følner sequence for Γ, and suppose that B ⊂ Γ with dΨ (B) > 0. If A ⊂ Γ and there is a left Følner sequence or a right Følner sequence (Φn )n∈N such that dΦ (A) > 0, then AB is piecewise Bohr. We now turn to ergodic averaging schemes (νn )n∈N where the union of the supports of the νn has upper Banach density 0. These schemes are provided by the results in [13], and we first present some of the definitions from that article.
Germs of functions. A germ of a function f : (0, ∞) → R is an equivalence class of pairs ((a, ∞), g), where a ≥ 0, g : (a, ∞) → R. Let us define this equivalence relation:
46
If a1 , a2 > 0, and g1 : (a1 , ∞) → R, g2 : (a2 , ∞) → R, then ((a1 , ∞), g1 ) is equivalent to ((a2 , ∞), g2 ) if there exists c ≥ max{a1 , a2 } with g1 |(c,∞) = g2 |(c,∞) . If [f ] and [g] are germs, we define [f ] + [g] by taking representatives ((a, ∞), f0 ), ((b, ∞), g0 ) for g, and letting [f ] + [g] be the equivalence class of (c, f0 + g0 ), where c = max{a, b}. The product [f ] · [g] is defined similarly. If f is a function defined on some interval (a, ∞), then the germ of f is the equivalence class of ((a, ∞), f ) under the above equivalence relation. Definition 3.2.4. Let B be the collection of germs of functions f : R>0 → R. A subfield of B which is closed under differentiation is called a Hardy field. Let U denote the union of all Hardy fields. The collection U contains many interesting examples of functions, including the logarithmico-exponential functions introduced by Hardy in [33]. These are all functions that can be obtained by addition, multiplication, division, and composition of the functions x 7→ x, x 7→ exp(x), and x 7→ log x. If f and g are two functions belonging to the same Hardy field, we write f ≺ g if limx→∞ f (x)/g(x) = 0. Definition 3.2.5. A function a ∈ U is subpolynomial if a(x) ≺ xk for some k. A function r ∈ U is nonpolynomial if it is subpolynomial and for each k ∈ Z+ , either r(x) ≺ xk or xk ≺ r(x). The symbol CQ[x] will denote the set of all real multiples of polynomials with rational coefficients, and the symbol CA[x] will denote the set of all real multiples of polynomials with algebraic coefficients. 47
Every subpolynomial a ∈ U can be uniquely expressed as p+r, where p is a polynomial and r is a nonpolynomial satisfying r(x) ≺ xs for each nonzero term cs xs of p. The expression p + r is called the canonical decomposition of a. We say that a : Z → Z is ergodic if for every probability space (X, X, µ), every measure preserving transformation T of (X, X, µ), and every f ∈ L2 (µ), N 1 X f ◦ T a(n) = Eµ (f |IT ). N →∞ N n=1
lim
If the function a : Z → R is one-to-one, then a is ergodic if and only if the sequence (νn )n∈N , where νn is normalized counting measure on {a(k) : 1 ≤ k ≤ n}, is an ergodic averaging scheme. The following facts are contained in Section 3 of [13]. Proposition 3.2.6. Let a ∈ U be subpolynomial with canonical decomposition p + r. Then bac is ergodic if one of the following conditions holds. 1. For all q ∈ CQ[x], |a(x) − q(x)| log x. 2. There exists ε > 0 such that r(x) xε . 3. The degree of p = n ≥ 2, and for some ε > 0, r(x) (log x)2
n+1 −1+ε
.
4. The polynomial p has the property that the ratio of two of its non-constant coefficients is badly approximable by rationals. Combining these facts with Theorem 3.1.6, we obtain the following corollaries. Recall ¯ d(C) := lim supn→∞
|C∩{1,...,n}d | nd
for C ⊂ Zd .
48
Corollary 3.2.7. Let a : R>0 → R satisfy one of the conditions 1-4 in Proposition ¯ 3.2.6. Then for all B ⊂ Z with d∗ (B) > 0, and all C ⊂ Zd with d(C) > 0, the set {ba(n)c + b : n ∈ C, b ∈ B} ¯ is piecewise Bohr. For all C ⊂ Z with d(C) = 1, the set {ba(n)c + b : n ∈ C, b ∈ B} is thick. Combining Proposition 3.2.6 with Proposition 3.2.2, we obtain a corollary for sumsets in Zd . Corollary 3.2.8. Let d ∈ N, and let a1 , . . . , ad : R>0 → R satisfy one of the conditions 1-4 in Proposition 3.2.6. Then for all B ⊂ Zd with d∗ (B) > 0, and all C ⊂ Z ¯ d ) > 0, the set with d(C {(ba1 (n1 )c, . . . , bad (nd )c) + b : ni ∈ C, b ∈ B} ¯ is piecewise Bohr. For all C ⊂ Z with d(C) = 1, the set {(ba1 (n1 )c, . . . , bad (nd )c) + b : ni ∈ C, b ∈ B} is thick. Remark 3.2.9. The fact that a(n) = bnα c is ergodic when α ∈ R>0 \ Z was already proved in Mate Wierdl’s PhD dissertation, [66].
49
3.3
Proof of Theorem 3.1.6, Part 1
We will actually prove Theorem 3.1.7, which is equivalent to Theorem 3.1.6. The main tool in the proof of Theorem 3.1.7 is the following proposition. Proposition 3.3.1. Let Γ be a countable group, and let (X, X, µ, T ) be an ergodic Γ-system with maximal isometric factor (Z, m, R), where π : X → Z denotes the factor map. If (νn )n∈N is an ergodic averaging scheme, A ⊂ Γ with dν (A) > 0, and if D ⊂ X with µ(D) > 0, then V :=
[
Tγ−1 D
γ∈A
contains, up to µ-measure 0, a set of the form π −1 U, where U ⊂ Z is open and nonempty. We will first prove Theorem 3.1.7, assuming the validity of Proposition 3.3.1.
3.3.1
Proof of Theorem 3.1.7.
For the remainder of this section, we fix a countable group Γ, a set B ⊂ Γ, and we let X be the orbit closure of 1B in the topological system ({0, 1}Γ , T ), where T = σ (r) is S the right shift. Let O = {ξ ∈ X : ξ(idΓ ) = 1}, and let V = a∈A Ta−1 O. Lemma 3.3.2. (a) Suppose that there is a point ξ ∈ X such that {γ : Tγ ξ ∈ V } is a left Bohr set. Then A−1 B is piecewise left Bohr. (b) If there exists ξ ∈ X such that Tγ ξ ∈ V for all γ ∈ Γ, then A−1 B is right thick. Proof. (a) By Lemma 2.7.2, we need only show that there is a Bohr set S ⊂ Γ such that for all finite F ⊂ S, there exists h ∈ Γ such that F h ⊂ A−1 B. 50
Let ξ be such a point in X. Write ξ = 1E , where E = {γ ∈ Γ : ξ(γ) = 1}. We then have γ ∈ A−1 E if and only if there exists a ∈ A such that aγ ∈ E, which is so if and S only if ξ ∈ a∈A Ta−1 O. Hence A−1 E is a Bohr set. We will show that A−1 B contains a right translate of every finite subset of A−1 E. S Since ξ is in the orbit closure of 1B , we can write E = k (Fk γk ∩ B)γk−1 , for some S increasing sequence (Fk )k∈N of finite subsets of Γ with k Fk = Γ and some γk ∈ Γ. S Hence A−1 E = k A−1 (Fk γk ∩ B)γk−1 , and A−1 E =
[
A−1 (Fk γk ∩ B)γk−1 .
k
If F ⊂ Γ is finite, then F ∩ A−1 E = F ∩ A−1 (Fk γk ∩ B)γk−1 for some k (and we can take k to be arbitrarily large). Hence (F ∩ A−1 E)γk = F ∩ A−1 Fk γk ∩ A−1 B. In particular, A−1 B contains (F ∩ A−1 E)γk . By Lemma 2.7.2, this implies that A−1 B is piecewise Bohr. (b) Suppose that there exists ξ ∈ X such that Tγ ξ ∈ V for all γ ∈ Γ. By the same argument as in part (a), we can conclude that there exists a set E ⊂ Γ such that 1E ∈ X and A−1 E = Γ. Letting Fk be as in part (a), we have that for all finite F ⊂ Γ, there exists k with F ⊂ Fk and F = F ∩ A−1 (Fk γk ∩ B)γk−1 , so F γk = F ∩ A−1 Fk γk ∩ A−1 B. In particular, F γk ⊂ A−1 B.
Proof of Theorem 3.1.7, Part 1. By the hypothesis of the theorem and Proposition 2.5.9, there is an ergodic invariant measure µ on (X, T ), where T = σ (r) is the right 51
shift, and X is the orbit closure of 1B under T. Let (Z, m, R) denote the maximal isometric factor of (X, X, µ, T ), with factor map π : X → Z. S By Proposition 3.3.1, the set V := γ∈A Tγ−1 O contains (up to µ-measure 0) a set of the form π −1 (U ), where U ⊂ Z is open. Let N0 = π −1 (U \ V ), so that µ(N0 ) = 0, S and let N = γ∈Γ Tγ−1 N. Let ξ ∈ X have the following properties: (i) π(Tγ ξ) = Rγ π(ξ) for all γ ∈ Γ. (ii) ξ ∈ / N. Such a ξ exists, since the set of ξ ∈ X satisfying each of (i) and (ii) separately has full measure. By the definition of N, we have Tγ ξ ∈ V if Rγ π(ξ) ∈ U. Hence, {γ : Tγ ξ ∈ V } contains the Bohr set {γ : Rγ π(ξ) ∈ U }. By Lemma 3.3.2, A−1 B is a left piecewise Bohr set. 3.3.2
Proof of theorem 3.1.6, Part 2.
Lemma 3.3.3. If (X, X, µ, T ) is an ergodic Γ-system, (νn )n∈N is an ergodic averaging S scheme, and A ⊂ Γ with dν (A) = 1, then γ∈A Tγ−1 D = X, up to µ-measure 0, whenever µ(D) > 0. Proof. Let E ∈ X with µ(E) > 0. We will show that µ(E ∩
S
γ∈A
we have Z lim
n→∞
µ(E ∩ Tγ−1 D) dνn (γ) = µ(E)µ(D),
52
Tγ−1 D) > 0. In fact,
since (νn )n∈N is an ergodic averaging scheme. In particular, there exists γ ∈ A such that µ(E ∩ Tγ−1 D) > 0.
To prove Part 2 of Theorem 3.1.6, we apply Part (b) of Lemma 3.3.2. Let B ⊂ Γ have dΦ (B) > 0 with respect to some Følner sequence (Φn )n∈N , and let A ⊂ Γ with dν (A) = 1 for some ergodic averaging scheme (νn )n∈N . Let (X, X, µ, T ) and O ⊂ X S be as in the proof of Part 1 of Theorem 3.1.6. Then µ(O) > 0, so V := γ∈A Tγ−1 O has µ-measure 1. In particular, there exists ξ ∈ X such that Tγ ξ ∈ V for all γ ∈ Γ. By Part (b) of Lemma 3.3.2, AB is right thick.
3.4
Proof of Proposition 3.3.1.
Before we prove Proposition 3.3.1, we need two preliminary facts about ergodic averaging schemes. Lemma 3.4.1. Let Γ be a countable group, let (Z, Z, m, R) be an isometric Γ-system, and let (νn )n∈N be an ergodic averaging scheme. For all continuous functions f : Z → C, the averages Z lim
n→∞
f ◦ Tγ dνn (γ)
converge uniformly to Em (f |IR ). In particular, if (Z, Z, m, R) is an ergodic rotation on a quotient Z = G/H, where G is a compact group, H is a closed subgroup, and Rγ xH = φ(γ)xH for some R homomorphism φ : Γ → G, then for all f ∈ C(Z), limn→∞ f (φ(γ)H) dνn (γ) = R f dm whenever (νn )n∈N is an ergodic averaging scheme. 53
Proof. Let f : Z → C be continuous. Since every Rγ is an isometry, the collection R of functions M := { f ◦ Rγ dνn (γ) : n ∈ N} is equicontinuous, and therefore is precompact in the uniform topology and has an uniform accumulation point f ∗ . R Choose a subsequence (νn0 )n∈N of (νn )n∈N such that f ◦ Rγ dνn (γ) → f ∗ uniformly. R By hypothesis, f ◦ Rγ dνn (γ) → Em (f |IR ) weakly, so f ∗ = Em (f |IR ), m-almost everywhere. Since m has full support, this implies that f ∗ is the only accumulation point of M in the uniform topology. This fact, together with the precompactness of M, implies the conclusion.
The following standard fact is used in proofs of multiple recurrence results. Lemma 3.4.2. Let Γ be a countable discrete group, (X, X, µ, T ) an ergodic Γ-system, and (νn )n∈N an ergodic averaging scheme. Let (Z, Z, m, R) be the maximal isometric factor of (X, X, µ, T ). If f ∈ L2 (µ) and Eµ (f |Z) = 0, then for all g ∈ L2 (µ), and all ε > 0, n Z o dν γ : f ◦ Tγ · g dµ > ε = 0. In the proof we use the fact, presented in Chapter 2, that the T ×T -invariant functions on (X × X, X ⊗ X, µ × µ) are measurable with respect to Z ⊗ Z. Proof. The conclusion is equivalent to the assertion that Z Z 2 lim f ◦ Tγ · g dµ dνn (γ) = 0, n→∞
and this is what we will prove. R Write | f ◦ Tγ · g dµ|2 as Z 2 Z f ◦ Tγ · g dµ = f ⊗ f¯ ◦ (Tγ × Tγ ) · g ⊗ g¯ dµ × µ. 54
Since (νn )n∈N Z lim n→∞
is an ergodic averaging scheme, Z Z 2 ¯ dµ × µ, f ◦ Tγ · g dµ dνn (γ) = Eµ×µ (f ⊗ f¯|IT ×T ) · h ⊗ h
which is zero, since f ⊗ f¯ is orthogonal to L2 (IT ×T ).
We now fix an ergodic Γ-system (X, X, µ, T ) with maximal isometric factor (Z, Z, m, R) and factor map π : X → Z. If D ⊂ X is a Borel set, we denote by hull(D) the set {x ∈ X : E(f |Z) > 0}. Since x 7→ µx is Z-measurable, we can identify hull(D) with a subset of Z. We will prove Proposition 3.3.1 in two steps. The first step is to show, essentially, that S S −1 −1 γ∈A Tγ hull(D). This is not really true, but what we do show will γ∈A Tγ D = S have the desired consequences. The second step is to show that γ∈A Tγ−1 hull(D), or rather a substitute for this set, contains, up to µ-measure 0, a set of the form π −1 (U ), where U ⊂ Z is open and nonempty. Definition 3.4.3. For D ∈ X, let J(D) := {E ∈ X : there exists δ > 0 such that dν ({γ ∈ A : µ(E ∩Tγ−1 D) > δ}) > 0}. Let S(D) := {E ∈ X : for all H ⊂ E with µ(H) > 0, H ∈ J(D)}. Lemma 3.4.4. For all D ∈ X, J(D) = J(hull(D)). Hence S(D) = S(hull(D)). Proof. Since D ⊂ hull(D), it is easy to see that J(hull(D)) ⊂ J(D). To prove the reverse inclusion, write 1D = f0 + f1 , where f1 = Eµ (1D |Z), and Eµ (f0 |Z) = 0. We can then decompose µ(E ∩ Tγ−1 D) as Z Z −1 µ(E ∩ Tγ D) = 1E · f0 ◦ Tγ dµ + 1E · f1 ◦ Tγ dµ. 55
Lemma 3.4.2 implies that
R
1E · f0 ◦ Tγ dµ tends to 0 in density, in the sense that
n Z o dν γ : 1E · f0 ◦ Tγ dµ > λ =0 for all λ > 0. Hence, for all λ > 0, dν ({γ ∈ A : µ(E ∩
Tγ−1 D)
Z n o > λ}) = dν γ ∈ A : 1E f0 ◦ Tγ dµ > λ .
(3.2)
To prove the lemma, it suffices to establish the following claim. Claim. If dν ({γ ∈ A : µ(E ∩ Tγ−1 hull(D)) > λ}) > 0 for some λ > 0, then Z dν ({γ ∈ A :
1E · f1 ◦ Tγ dµ > δ})
for some δ > 0. Proof of Claim. Let η > 0 such that µ({x : f0 (x) > η}) > µ(hull(D)) − λ2 , and let f (x) = f1 (x) if f1 (x) > η, and f (x) = 0 otherwise. Then Z
Z 1E · f1 ◦ Tγ dµ ≥
1E · f ◦ Tγ dµ 1 ≥ λµ E ∩ Tγ−1 hull(D) , 2
for all γ ∈ Γ, so by (3.2), we can take δ = ηλ/2. This proves the claim, completing the proof of the lemma.
We want S(D) to serve as a substitute for {Tγ−1 D : γ ∈ A}. The following claim shows that this is reasonable. 56
Claim. For all E ∈ S(D), E ⊂ Proof. If µ(E \
S
γ∈A
S
γ∈A
Tγ−1 D, up to µ-measure 0.
Tγ−1 D) > 0, then H := E \
S
γ∈A
Tγ−1 D is a subset of E with
µ(H) > 0 and H ∈ / J(D). Hence E ∈ / S(D). We would like to use
S
S(D) as a substitute for
S
γ∈A
Tγ−1 D, but S(D) may not be
countable, so this may not be desirable. Instead, we make the following construction. Let M := sup{µ(E) : E is a union of countably many elements of S(D)}. For each n > 0, let En be a countable union of elements of S(D) such that µ(En ) > S M − n1 . Let US(D) = n∈N En , so that US(D) is a countable union of elements of S(D) S with µ(US(D)) = M. By the above Claim, US(D) ⊂ g∈A Tg−1 D, up to µ-measure 0.
3.4.1
Identifying US(D).
We now identify US(D). First, since US(D) = US(hull(D)), we can assume that D ∈ Z, so we consider D as a subset of Z = G/H. Definition 3.4.5. Let (Y, ρ) be a compact metric space. If b : Γ → Y and (νn )n∈N is a sequence of probability measures on Γ, we define dclν {b(γ) : γ ∈ C} by dclν {b(γ) : γ ∈ C} := {y ∈ Y : for all ε > 0, dν {γ ∈ C : ρ(b(γ), y) < ε} > 0}. So dclν {b(γ) : γ ∈ C} is the set of points in Y that are frequently close to bi (g), where “frequently” is defined in terms of (νn )n∈N .
57
It will be useful to know that K = dclν {b(γ) : γ ∈ C} is compact and nonempty whenever dν (C) > 0. Compactness is easy to verify; we now verify that K is nonempty. Note that if Q ⊂ Y with dν ({γ ∈ C : b(γ) ∈ Q} > 0, and Q is covered by finitely many sets F1 , . . . , Fr , there exists Fi such that dν ({γ ∈ C : b(γ) ∈ Q} > 0. Hence one can construct a sequence of closed sets Qn with diameter tending to 0 such that T dν ({γ ∈ C : b(γ) ∈ Qn }) > 0 for all n. Then n Qn is contained in dclν {b(γ) : γ ∈ C}. Lemma 3.4.6. Let G be a compact metric group with Haar measure mG , and let φ : Γ → G be a homomorphism. If (νn )n∈N is an ergodic averaging scheme, A ⊂ Γ, and K = dclν {φ(γ) : γ ∈ A}, then mG (K) ≥ dν (C). Proof. Let ε > 0. Since mG is a regular Borel measure, there exists a continuous R function f : G → [0, 1] satisfying f |K = 1K , and f dmG < mG (K) + ε. Let η < 1, and let Lη = {γ ∈ A : f (φ(γ)) ≤ η}. Claim. dν (Lη ) = 0. Proof of Claim. Since {γ ∈ Γ : f (γ) ≤ η} is compact, dclν {R(γ) : γ ∈ Lη } will be nonempty if dν (Lη ) > 0. On the one hand, f (γ) ≤ η for all γ ∈ dclν {φ(γ) : γ ∈ Lη }, and on the other hand dclν {φ(γ) : γ ∈ Lη } ⊂ K, and f (g) = 1 for all g ∈ K. Hence dν (Lη ) = 0.
The above claim implies that dν (A \ Lη ) = dν (A).
58
Since (νn )n∈N is an ergodic averaging scheme, we have Z
Z f dmG = lim
n→∞
f (φ(γ))dνn (γ)
≥ η lim sup νn (A \ Lη ) n→∞
= ηdν (A). By the definition of f, we then have Z ηdν (A) ≤
f dmG ≤ mG (K) + ε.
Letting η → 1 and ε → 0, we find mG (K) ≥ dν (A).
Lemma 3.4.7. Suppose that (Z, Z, m, R) is an ergodic, isometric Γ-system, with Z = G/H for some compact group G and closed subgroup H ⊂ G. If (νn )n∈N is an ergodic averaging scheme, A ⊂ Γ with dν (A) > 0, and D ⊂ Z with m(D) > 0, write K = dclν {Rγ idG : γ ∈ A} ⊂ G. Then for all k ∈ K, US(D) ⊃ k −1 D, up to m-measure 0. Proof. By the definition of US(D), it suffices to show that k −1 D ∈ S(D) for all k ∈ K. From the definition of S(D), we must show that for all E ⊂ D with m(E) > 0, k −1 E ∈ J(D). Fix some k ∈ K, and let E ⊂ D with m(E) > 0. By continuity of the map t 7→ m(t−1 D ∩k −1 E), there exists a neighborhood W of k with m(w−1 D ∩k −1 E) ≥ 12 m(E) for all w ∈ W. Since k ∈ dclν {Rγ idG : γ ∈ A}, we have dν {γ ∈ A : Rγ idG ∈ W } > 0, and hence dν {γ ∈ A : m(Rγ D ∩ kE) > 21 m(E)} > 0. It follows that k −1 E ∈ J(D).
59
By the Corollary to Lemma 2.6.2, Lemma 3.4.7 shows that US(D) contains an open S set, up to m-measure 0. Since US(D) ⊂ γ∈A Tγ−1 D, up to m-measure 0, this implies S that γ∈A Tγ−1 D contains an open set, up to m-measure 0. This completes the proof of Proposition 3.3.1.
3.5
Remarks.
(i) The condition that dν (A) > 0 for an ergodic averaging sequence can be considered a “pseudo-randomness” condition. In the finitary setting, a subset A of a finite group Z may be called ε-pseudo-random (or ε-uniform) if its characteristic function has small R Fourier coefficients. That is, we say A ⊂ Z is ε-uniform if | 1A (z)χ(z) dmZ (z)| < ε ˆ Such sets are well studied in combinatorial for every nontrivial character χ ∈ Z. number theory, and in particular are known to contain many three-term arithmetic progressions, if ε is small and mZ (A) is bounded away from 0. In our setting, one R can view the assertion that limn→∞ χ(g) dνn (g) → 0 in the definition of the ergodic averaging schemes as the assertion that νn has small Fourier coefficients. See [62] for an extensive exposition and bibliography of results on sumsets in finite groups. (ii) As an extreme example, R. Pavlov has constructed a set A ⊂ Z such that A + B is thick whenever B is infinite. The construction appears in Section 3 of [56]. (iii) Theorem 3.1.6 raises some questions. Question 3.5.1. If A = {n2 : n ∈ N} and B ⊂ Z with d∗ (B) > 0, is A+B necessarily piecewise Bohr? 60
The Proof of Theorem 3.1.6 cannot easily be adapted to answer Question 3.5.1 in the affirmative, since there are ergodic compact abelian group rotations (Z, m, R) such that the closure {Rn2 z : n ∈ N} has Haar measure 0 for all z ∈ Z. For instance, let Q Zp denote the p-adic integers for a prime p, and let Z = p prime Zp . Let α be the element whose coordinates are (12 , 13 , 15 , . . . ), where 1p is the multiplicative identity in Zp . This raises the next question. Question 3.5.2. Are there any compact abelian groups G with compact subsets K such that mG (K) = 0, but K + D contains an open set whenever mG (D) > 0.?
61
CHAPTER 4 CHARACTERISTIC FACTORS FOR SOME COMMUTING ACTIONS OF Zd
4.0.1
Notation.
In this chapter, we use the notation g 7→ T g to denote an element of a group action, rather than g 7→ Tg . This is to avoid confusing double subscripts. Given a measure preserving system X = (X, X, µ, T ), we will abuse notation and refer to “the system X,” rather than X, unless there is some chance of confusion. We will use the symbol T to denote R/Z, and the symbol S to denote the set {z ∈ C : |z| = 1} as a multiplicative subgroup of C.
4.1
Introduction.
Before we state the main result of this chapter, we give the definition of a certain class of dynamical systems.
4.1.1
Nilsystems.
A Lie group is a group G which is also a C ∞ manifold, such that the group operations (g, h) 7→ gh, g 7→ g −1 are C ∞ . See [14], [40] for the general theory of Lie groups.
62
We will be concerned with nilpotent Lie groups and their compact quotients. The general theory in this area was developed by Mal’cev in [49]. Definition 4.1.1. Let k and d be integers, and let G be a k-step nilpotent Lie group. Suppose that H ⊂ G is a closed subgroup of G such that the quotient G/H is compact, and t1 , . . . , td ∈ Λ such that ti tj xH = tj ti xH for all x ∈ G and all i 6= j. We can define a Zd -system (X, X, µ, T ) where X = G/H, X is the Borel σ-algebra of X, µ is Haar measure on G/H, and T (n1 ,...,nd ) xH = tn1 1 tn2 2 · · · tnd d xH for (n1 , . . . , nd ) ∈ Zd . Such a system is called a k-step d-nilsystem, or simply k-step nilsystem if there is no ambiguity about d. Nilsystems have been studied extensively, in particular in [1], [53], [54], [45], [42], and [43]. The following is the main theorem of this chapter. Theorem 4.1.2. Let (X, X, µ, T ) be an ergodic Zd -system, and let Mi , i = 1, . . . , k be endormorphisms of Zd such that the images of each of Mi , Mi − Mj , i 6= j have finite index in Zd . There is a factor (Z, Z, ν, T ), isomorphic to an inverse limit of k-step nilsystems, such that k k Y 1 XY Mi a lim fi ◦ T − E(fi |Z) ◦ T Mi a = 0, N →∞ |FN | i=1 a∈F i=1 N
for all Følner sequences FN , and the convergence takes place in L2 (µ).
63
We will also show that the hypothesis that the images of the Mi , Mi − Mj each have finite index in Zd is necessary to conclude that there is a characteristic factor for the scheme (T M1 a , . . . , T Mk a ) which is an inverse limit of nilsystems. In Section 4.12, we give an application of Theorem 4.1.2 where we replace the actions a 7→ T Mi a with commuting actions of Zd satisfying some ergodicity hypotheses. In the case d = 1, this theorem was proved independently in [36] and [68].
4.1.2
Background.
The various steps in our proof of Theorem 4.1.2 will be done in strict analogy to those in [36]. In fact, proofs of facts which make no reference to cocycles or group extensions will be virtually identical to the proofs of the corresponding facts in [36], but with the word “action” (or “Zd -action”) replacing the word “transformation.” To explain the differences in the cases where group extensions and cocycles are involved, we define these notions now.
4.1.3
Cocycles and group extensions.
In this subsection, we repeat and expand some of the discussion on cocycles from section 2.5.4 in Chapter 2. Let Γ be a countable discrete group and let (X, X, µ) a standard probability space. Let T be an action of Γ on (X, B, µ) by measure preserving transformations, and let G be a compact group. We say that ρ : Γ × X → G is a cocycle if ρ(ab, x) = ρ(a, T b x)ρ(b, x) for all a, b ∈ Γ, x ∈ X
64
(4.1)
Note that ρ is defined only almost everywhere. If equation (4.1) is satisfied, then the transformations from X to itself defined by Tρa (x, g) = (T a x, ρ(a, x)g) constitute a Γ action. In the case Γ = Z, the function ρ is Q determined by the values ρ(1, x), and one can deduce that ρ(n, x) = ni=1 ρ(1, T i−1 x). This fact is implicitly exploited in many of the proofs in [36], due to the consequence that if there are only countably many possibilities for the function x 7→ ρ(1, x), then there are only countably many possibilities for ρ : Z × X → G. For any group action, the values of ρ are determined by the values ρ(a, x), where a ranges over a generating set, and x ∈ X. In particular, for finitely generated groups, if ρ(a, ·) is allowed to range over only countably many functions for a given a, then ρ itself can only have countably many values. We say that a map ρ : Γ × X → G is a coboundary if it is of the form ρ(a, x) = F (T a x)(F (x))−1 for some F : X → G. Note that every such ρ is a cocycle.
Zd -actions. We now specialize to the case Γ = Zd for some d, and use additive notation for the group operation. If a cocycle ρ(a, x) is a constant function of x for each a, then ρ(a+b, x) = ρ(a, x)ρ(b, x), for all a, b ∈ Zd and almost all x ∈ X, and we identify the set of such cocycles with the set of homomorphisms hom(Zd , G) from Zd into G. If G is an abelian Lie group, then hom(Zd , G) is also an abelian Lie group with pointwise addition. To be precise, we identify hom(Zd , G) with Gd by φ ↔ (φ(ei ))di=1 , where {ei }di=1 is a generating set for Zd . 65
Let (Y, Y, ν, S) be a factor of (X, X, µ, T ), with factor map π. Let ρ : Zd × Y → U be a cocycle of (Y, Y, ν, S). Then the cocycle ρ ◦ π : Zd × X → U defined by ρ ◦ π(a, x) = ρ(a, π(x)) is a cocycle of (X, X, µ, T ). We will also use this notation when π is an automorphism of (X, X, µT ). We call a homomorphism φ : Zd → C an eigenvector of the system (X, X, µ, T ) if there exists a nonzero f ∈ L2 (µ) such that f ◦ T a = φ(a)f for all a ∈ Zd . Eigenspaces corresponding to different eigenvalues are mutually orthogonal. We will assume throughout that (X, X, µ) is a standard probability space, so that L2 (µ) is separable and (X, X, µ, T ) can have only countably many eigenvalues. Throughout this chapter, we fix d ∈ N, a regular probabilty measure space (X, X, µ), and an ergodic action T of Zd on X. Let T a denote the element of T corresponding to a ∈ Zd . A Følner sequence in Zd is a sequence of sets FN ⊆ Zd satisfying limN →∞
|(FN −a)∩FN | |FN |
=
1 for all a ∈ Zd . The mean ergodic theorem for Zd -systems may be stated as follows. Theorem 4.1.3. Let (X, X, µ, T ) be a measure preserving Zd system, and let V be the closed subspace of L2 (µ) consisting of the T -invariant functions. Let FN be a Følner sequence in Zd . Then 1 X f ◦ Ta = Pf N →∞ |FN | a∈F lim
N
where P f is the orthogonal projection of f on V. The convergence is in the sense of L2 (µ).
66
We will sometimes use, without comment, the following fact in the course of the proof of Theorem 4.1.2. It is a special case of Theorem 25.31 of [34]. Lemma 4.1.4. Let K be a compact abelian group, and suppose that T is a closed subgroup of K with quotient Z = K/T. Then K is isomorphic to Z × T. b and Z \ Proof. We produce an isomorphism between K × T. By Pontryagin duality, this implies that K and Z × T are isomorphic. b Since the b such that χ|T generates T. First we claim that there is a character χ ∈ K characters of K span a dense subset of C(K), the functions of the form χ|T , where b span a dense subset of C(T). But this is possible only if every character of T χ ∈ K, b is of the form χ|T for some χ ∈ K. Let χ be as above, and let π denote the projection map π : K → Z. Define a b to K b by (ψ, χn ) 7→ ψ ◦ π · χn . This map is injective, since map B from Zb × T ψ ◦ π · χn = ψ 0 ◦ π · χm implies that χn−m = (ψ 0 ψ) ◦ π, so that χn−m is defined on K/T, and therefore is constant on T. This means that n = m, so that ψ 0 = ψ. The map B is clearly a homomorphism. To see that B is surjective, note that the functions of the form ψ 0 ◦ π · χm separate points: if x 6= y, and π(x) = π(y), then there exists m such that χm (x − y) 6= 0. If π(x) 6= π(y), there exists ψ ∈ Zb such that ψ(π(x)) 6= ψ(π(y)).
4.2
Outline of the proof of Theorem 4.1.2.
The proof of Theorem 4.1.2 will follow [36] closely. Although this chapter is quite long, the only novelties are in the proofs of Proposition 4.8.9 and Proposition 4.10.12. 67
Let us fix an ergodic measure preserving Zd -system (X, X, µ, T ) and outline the definition of the factors Zk . k
For each k, we will define a joining µ[k] on X 2 . We will use µ[k] to define a seminorm ||| · |||k on L∞ (X), and the σ-algebras Zk . We will see that E(f |Zk−1 ) = 0 if and only if |||f |||k = 0, and this relationship will help establish part (1) of Theorem 4.1.2. Part (2) of Theorem 4.1.2 will be accomplished by constructing a group G = G(X, X, µ, T ), defined in Section 4.5, associated to X. This group will have the following useful properties: (1) T a ∈ G for each a ∈ Zd . (2) G(Zk ) is nilpotent. (3) G acts transitively on Zk whenever Zk is toral. (See Section 4.8 for a definition of “toral system”.) From these properties, it is more or less clear that when Zk is toral, the space Zk can be realized as a finite-volume quotient of G, and this is essentially part (2) of Theorem 4.1.2. The main difficulty is in showing that G(Z) acts transitively on toral systems Z, that G(Z) is a Lie group, and showing that Zk is actually an inverse limit of toral systems. In order to describe the Zk , we show in Section 4.6 that Zk+1 is an abelian group extension of Zk . That is, there exists a compact abelian group U and a cocycle ρ : Zd × Zk → U such that Zk+1 is isomorphic to the extension Zk ×ρ U. We then show that the cocycle ρ has additional structure, and this additional structure will 68
permit a useful description of G. More precisely, we will define an invariant of cocycles, called the type. This invariant will behave like the degree of a polynomial, in the sense that various “differentiations” reduce the type by 1. This is the content of Section 4.7 In Sections 4.8 and 4.9, we establish some “rigidity” results for cocycles, specifically Propositions 4.8.9 and Theorem 4.9.6. These are essential for showing that the group G acts transitively on Z. There is an obstruction to applying Theorem 4.9.6 as is done for the case d = 1 in [36], and we will use some tools from [68] to overcome this. The necessary material from [68] is presented at the end of Section 4.15. In many cases, the proofs we give will follow the corresponding proofs from [36] closely, sometimes reproducing arguments verbatim.
4.3 4.3.1
Definition of the measures µ[k] . The cube Vk and symmetries.
Following the notation of [36], let Vk = {0, 1}k , and write an element ε ∈ Vk as ε1 . . . εk , where each εi = 0 or 1. k
If X is a set, then X [k] is an abbreviation of X ({0,1} ) , and we write points in X [k] as (xε )ε∈Vk . If T : X → X is a transformation, we write T [k] for the transformation of X [k] given by T [k] (xε )ε∈Vk = (T xε )ε∈Vk . If T is an action of a group G on X, we write T [k] for the action generated by the (T a )[k] , a ∈ G and may denote the element (T a )[k] by (T [k] )a . For l ≤ k, we call a subset W of Vk a face of dimension k − l if there are η1 , . . . , ηl ∈
69
{0, 1}, n1 < n2 < · · · < nl so that W = {ε : εn1 = η1 , . . . , εnl = ηl }. A face of dimension k − 1 is called a side. Associated to each face W of dimension r, there is a natural projection X [k] → X [r] given by (xε )ε∈Vk 7→ (xε )ε∈W . The set X [k] has a group of symmetries induced by the group of symmetries of Vk . Definition 4.3.1. We call σ : Vk → Vk a digit permutation if there is a permutation τ of {1, . . . , k} such that σ(ε1 . . . εk ) = εσ(1) . . . εσ(k) for ε = ε1 . . . εk ∈ Vk . Call ρ : Vk → Vk a reflection if there exists i ∈ {1, . . . , k} such that ρ(ε1 . . . εi . . . εk ) = ε1 . . . (1 − εi ) . . . εk for ε1 . . . εk ∈ Vk . The group Sk is the group generated by the digit permutations and reflections. The group Sk acts on X [k] as follows: for σ ∈ Sk , x = (xε )ε∈Vk ∈ X [k] let σ∗ (x) = (xσ(ε) )ε∈Vk . 4.3.2
The measures µ[k] .
We now define, as in [36], for each integer k ≥ 0, measures µ[k] on X [k] , and Zd -actions T [k] : X [k] → X [k] which preserve µ[k] . Let X [0] = X, T [0] = T, and µ[0] = µ. Assuming µ[k] is defined, let I[k] denote the T [k] -invariant σ-algebra of (X [k] , µ[k] , T [k] ). We then define µ[k+1] on X [k+1] by ! ! Z Z O O O fε dµ[k+1] = E fη0 |I[k] E fη1 |I[k] dµ[k] , X [k+1] ε∈V k+1
X [k]
η∈Vk
η∈Vk
so that µ[k+1] is the relatively independent joining of µ[k] over I[k] .
70
(4.2)
As in [36], we describe the relationship between the ergodic decompositions of the µ[k] . Let µ
[k]
Z
µ[k] ω dPk (ω)
=
(4.3)
Ωk
denote the ergodic decomposition of µ[k] under T [k] . Then by definition µ
[k+1]
Z
[k] µ[k] ω × µω dPk (ω).
=
(4.4)
Ωk
In complete analogy with [36], Lemma 3.1, we have Lemma 4.3.2. (cf. [36], Lemma 3.1) Let k, l ≥ 1 be integers. Then µ
[k+l]
Z =
[l] (µ[k] ω ) dPk (ω).
Ωk
The proof is virtually identical to the proof for the case d = 1 in [36]. We give it for completeness. [k]
[k]
Proof. By definition, µω is a measure on X [k] and so (µω )[l] is a measure on (X [k] )[l] , which we identify with X [k+l] . For l = 1 the formula is equation (4.4). By induction assume that it holds for some l ≥ 1. Let Jω denote the invariant σ-algebra of the [k]
[k]
system ((X [k] )[l] , (µω )[l] , (T [k] )[l] ) = (X [k+l] , (µω )[l] , T [k+l] ). Let f and g be two bounded functions on X [k+l] . By the pointwise ergodic theorem1 , [k]
[k+l]
applied to both the system (X [k+l] , X[k+l] , (µω )[l] , T [k+l] ) and (X [k+l] , X [k+l] , µω 1
, T [k+l] ),
Here one may apply the L2 ergodic theorem and pass to a subsequence to get pointwise convergence, rather than using the full strength of the pointwise ergodic theorem for Zd .
71
[k]
for almost every ω the conditional expectation of f on Ik+l (for µ[k+l] ) is equal (µω )[l] [k]
almost everywhere to the conditional expectation of f on Jω (for (µω )[l] ). As the same holds for g, we have Z f ⊗ g dµ
[k+l+1]
Z
E(f |I[k+l] ) · E(g|I[k+l] ) dµ[k+l] Z Z [k+l] [k+l] [k] [l] dPk (ω) E(f |I ) · E(g|I )d(µω ) = Ωk X [k+l] Z Z [k] [l] = E(f |Jω ) · E(g|Jω )d(µω ) dPk (ω) Ωk X [k+l] Z Z [k] [l+1] = f ⊗ gd(µω ) dPk (ω)
=
X [k+l]
Ωk
X [k+l]
[k]
where the last identity uses the definition of (µω )[l+1] . This means that µ[k+l+1] = R [k] (µω )[l+1] dPk (ω). Ω
4.3.3
The case k = 1.
We repeat the discussion in [36], section 3.2. Let Z1 = (Z1 , µ1 , t1 , . . . , td ) denote the Kronecker factor of (X, X, µ, T ). This is the factor generated by the eigenfunctions of X, and the Halmos-von Neumann theorem says that such a system is isomorphic to an ergodic group system (Z1 , µ1 , t1 , . . . , td ), ti ∈ Z1 where T acts by translations: T (n1 ,...,nd ) z = tn1 1 · · · tnd d z for z ∈ Z1 . Let π1 denote the factor map X → Z1 . For s ∈ Z1 , let µ1,s denote the image of the measure µ1 under the map z 7→ (z, sz) from Z1 to Z12 , so that µ1,s is Haar measure of the coset (0, s) + ∆ of the diagonal
72
in Z × Z. This measure is invariant under T [1] = T × T and is a self-joining of the system Z1 . Let µs denote the measure on Z × Z given by Z Z f (x0 )g(x1 ) dµs (x0 , x1 ) = E(f |Z1 )(z)E(g|Z1 )(sz) dµ1 (z). Z×Z
(4.5)
Z
The invariant σ-algebra I[1] of (X × X, µ × µ, T × T ) is the collection of sets of the form {(x, y) ∈ X × X : π1 (x) − π1 (y) ∈ A}, where A ⊂ Z1 . That is, T [1] -invariant functions on X × X are functions of cosets of the diagonal in Z × Z. The ergodic decomposition of µ × µ under T × T can be written as Z µ×µ=
µs dµ1 (s).
(4.6)
Z1
By Lemma 4.3.2, for an integer l > 0 we have Z [l+1] µ = (µs )[l] dµ1 (s).
(4.7)
Z1
Formula (4.4) becomes µ
[2]
Z µs × µs dµ1 (s).
= Z1
When fε , ε ∈ V2 , are four bounded functions on X, writing f˜ε = E(fε |Z1 ) and viewing these functions as defined on Z1 , by Equation (4.5) we have Z O fε dµ[2] = X 4 ε∈V 2
Z Z Z
f˜00 (z)f˜01 (z + s1 )f˜10 (z + s2 )f˜11 (z + s1 + s2 ) dµ1 (z) dµ1 (s1 ) dµ1 (s2 ).
Z13
(4.8) 73
[2]
[2]
[2]
The projection under π1 of µ[2] on Z1 is the Haar measure µ1 of the closed subgroup {(z, z + s1 , z + s2 , z + s1 + s2 ) : z, s1 , s2 ∈ Z1 } [2]
of Z1 = Z14 . 4.3.4
The side actions.
In this subsection we define Zd -actions on X [k] in analogy with the transformations defined in section 3.3 of [36]. [k]
Definition 4.3.3. If α is a face of Vk with k ≥ 1, and a ∈ Zd , let (Tα )a denote the transformation of X [k] given by
((Tα[k] )a x)ε =
T a (xε ) for ε ∈ α
xε
otherwise [k]
This is called a face transformation. When α is a side of Vk , we call Tα a side [k]
[k]
transformation. We can also write (T a )α for (Tα )a . [k]
[k]
Let Tk−1 be the group generated by the transformations (Tα )a , where α is a side of [k]
[k]
Vk (a face of dimension k − 1), and a ∈ Zd . Let T∗ be the subgroup of Tk−1 generated [k]
by those (Tα )a where α is a side not containing 0. Analogously to Lemma 3.3 of [36], we have Lemma 4.3.4. (cf. [36], Lemma 3.3) For k ≥ 1, the measure µ[k] is invariant under [k]
the group Tk−1 of side transformations.
74
The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. [k]
Proof. We proceed by induction. For k = 1 the group Tk−1 is generated by, id ×T a and T a × id, a ∈ Zd , and µ[1] = µ × µ is invariant under each of these transformations. Assume that the result holds for some k ≥ 1, and let a ∈ Zd . We consider first the side α = {ε ∈ Vk+1 : εk+1 = 0}. Identifying X [k+1] with the X [k] × X [k] , we have [k+1]
(T a )α
= (T [k] )a ×id[k] . Since (T [k] )a leaves each set in I[k] invariant, by the definition
(4.2) of µ[k+1] , this measure is invariant under ((T a )[k+1] )α . The same method gives [k+1]
the invariance under (T a )α
, where α0 is the side opposite from α.
Any other side β of Vk+1 can be written as γ × {0, 1} for some side γ of Vk . Under [k+1]
the identification of X [k+1] with X [k] × X [k] , we have (T a )β
[k]
[k]
= (T a )γ × (T a )γ . By
[k]
the inductive hypothesis, the transformation (T a )γ leaves the measure µ[k] invariant. Furthermore, it commutes with each (T b )[k] , b ∈ Zd , and so commutes with the conditional expectation on I[k] . By the definition (4.2) of µ[k+1] , this measure is invariant [k+1]
under (T a )β
.
Notation. Let J[k] (X) denote the σ-algebra of subsets of X [k] that are invariant under [k]
the group T∗ . If there is no confusion, we will write J[k] for J[k] (X). The following is a version of Proposition 3.4 of [36]. Proposition 4.3.5. (cf. [36], Proposition 3.4) On (X [k] , X[k] , µ[k] ), the σ-algebra J[k] coincides with the σ-algebra of sets depending only on the coordinate 0. The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. 75
[k]
Proof. If α is a side not containing 0, then ((Tα )a x)0 = x0 for every x ∈ X [k] . Thus a subset of X [k] depending only on the coordinate 0 is obviously invariant under the [k]
group T∗ and so belongs to J[k] . [k]
We prove the reverse inclusion by induction. For k = 1, X [1] = X 2 , the group T∗ contains id ×T a for all a ∈ Zd , and the result is obvious.
Assume the result holds for some k ≥ 1. Let F be a bounded function on X [k+1] that is measurable with respect to the σ-algebra J[k+1] . Write x = (x0 , x00 ) for a point of X [k+1] , where x0 , x00 ∈ X [k] . Since (X [k+1] , µ[k+1] , T [k+1] ) is a self-joining of (X [k] , X[k] , µ[k] , T [k] ), the function F (x) = F (x0 , x00 ) on X [k+1] can be approximated in L2 (µ[k+1] ) by finite sums of the form X
Fi (x0 )Gi (x00 ),
i
where Fi and Gi are bounded functions on X [k] . Since id ×(T [k] )a is one of the side transformations of X [k+1] for all a ∈ Zd , it leaves F invariant and by passing to ergodic averages, we can assume that each of the functions Gi is invariant under T [k] . Thus, by the construction of µ[k+1] , for all i, Fi (x0 ) = Gi (x00 ) for µ[k+1] -almost every (x0 , x00 ). Therefore the above sum is equal µ[k+1] -almost everywhere to a function depending only on x0 . Passing to the limit, there exists a bounded function H on X [k] such that F (x) = H(x0 ) µ[k+1] -almost everywhere. Under the natural embedding of Vk in Vk+1 given by the first side, each side of Vk [k+1]
is the intersection of a side of Vk+1 with Vk since F is invariant under T∗
, H is
[k]
also invariant under T∗ and thus is measurable with respect to J[k] . By the induction hypothesis, H depends only on the 0 coordinate. 76
Corollary 4.3.6. (cf. [36], Corollary 3.5) (X [k] , µ[k] ) is ergodic for the group of side [k]
transformations Tk−1 . [k]
Proof. A subset A of X [k] invariant under the group Tk−1 is also invariant under the [k]
group T∗ . Thus its characteristic function is equal almost everywhere to a function depending only on the 0 coordinate. Since A is invariant under T [k] , this last function is invariant under T and so is constant.
Corollary 4.3.7. (cf. [36], Corollary 3.6) (Ωk , Pk ) is ergodic under the action induced [k]
by T∗ . To see that Corollary 4.3.7 follows from Corollary 4.3.6, note that T [k] induces the trivial action on (Ωk , Pk ).
4.3.5
Symmetries.
Following Proposition 3.7 in [36], we have Proposition 4.3.8. (cf. [36], Proposition 3.7) The measure µ[k] is invariant under the transformation σ∗ for every σ ∈ Sk . This is a consequence of Lemma 6.4.1 in Chapter 6, but here we present the proof from [36]. Proof. First we show by induction that µ[k] is invariant under reflections. For k = 1 the map (x0 , x1 ) 7→ (x1 , x0 ) is the unique reflection and it leaves the measure µ[1] = µ × µ invariant.
77
Assume that for some integer k ≥ 1, the measure µ[k] is invariant under reflections. For 1 ≤ j ≤ k + 1, let Rj be the reflection of X [k+1] corresponding to the digit j. If j < k + 1, Rj can be written Sj × Sj , where Sj is the reflection of X [k] for the digit j. Since µ[k] is invariant under Sj , by construction µ[k+1] is invariant under Rj . The reflection Rk+1 simply exchanges the two sides of X [k+1] and by construction of the measures, it leaves the measures µ[k+1] invariant. Next we show that µ[k] is invariant under digit permutations. For k = 1, there is no nontrivial digit permutation and so nothing to prove. For k = 2, there is one nontrivial digit permutation, the map (x00 , x01 , x10 , x11 ) 7→ (x00 , x10 , x01 , x11 ). By Formula (4.8), µ[2] is invariant under this map. Assume that for some integer k ≥ 2, the measure µ[k] is invariant under all digit permutations. The group of permutations of {1, . . . , k, k + 1} is spanned by the permutations leaving k + 1 fixed and the transposition (k, k + 1) exchanging k and k + 1. Consider first the case of a permutation of {1, . . . , k, k + 1} leaving k + 1 fixed. The corresponding transformation R of X [k+1] = X [k] × X [k] can be written as S × S, where S is a digit permutation of X [k] and so leaves µ[k] invariant. By construction, µ[k+1] is invariant under R. Next consider the case of the transformation R of X [k+1] associated to the permutation (k, k + 1). By the ergodic decomposition of Formula (4.3) of µ[k−1] and Equation (4.4) [k−1] [2]
for k − 1, the measure (µω
)
(as a measure on (X [k−1] )[2] ) is invariant by the
transposition of the two digits. Thus, when we consider the same measure as a
78
measure on X [k+1] , it is invariant under R. The integral, µ[k+1] , is invariant under R and therefore µ[k+1] is invariant under all digit permutations.
Corollary 4.3.9. (cf. [36], Corollary 3.8) The image of µ[k] under any side projection X [k] → X [k−1] is µ[k−1] . Proof. By construction of µ[k] , the result holds for the side projection associated to the side {ε ∈ Vk : εk = 0} of Vk . The result for the other side projections follows immediately from Proposition 4.3.8.
4.3.6
Gowers-Host-Kra seminorms.
We define the seminorms |||·|||k as in [36]: for k ≥ 0 and bounded, real-valued functions f, let |||f |||k :=
Z O
f dµ[k]
1/2k
.
(4.9)
ε∈Vk
Note that |||f |||1 = kf kL2 (µ) . Note that ||| · |||0 is not actually a seminorm, since |||f |||0 is just
R
f dµ.
Since their introduction in [27] (for finite systems (X, T )) and [36] (for general systems), these seminorms have been applied to many problems in recurrence and combinatorial number theory. See, for instance, [8], [28], [29], and [37]. Lemma 4.3.10. (cf. [36], Lemma 3.9) (1) When fε , ε ∈ Vk , are bounded functions on X, Z Y O [k] f dµ |||fε |||k ≤ ε ε∈Vk
ε∈Vk
79
(2) For each k ∈ N, ||| · |||k is a seminorm on L∞ (µ). (3) For a bounded function f, |||f |||k ≤ |||f |||k+1 , for all k. We repeat the proof from [36] for completeness. Proof. (1) Using the definition of µ[k] , the Cauchy-Schwarz inequality and again using definition of µ[k] ,
Z O
[k]
fε dµ
2
O
[k−1] fη0 |I ) ≤ E(
ε∈Vk
L2 (µ[k−1] )
η∈Vk−1
=
Z O
O
[k−1] fη1 |I ) · E(
L2 (µ[k−1] )
η∈Vk−1
Z O gε dµ · hε dµ[k] [k]
ε∈Vk
ε∈Vk
(4.10) where the functions gε and hε are defined for η ∈ Vk−1 by gη0 = gη1 = fη0 and hη0 = hη1 = fη1 . For each of the two integrals in (4.10), we permute the digits k − 1 RN [k] 4 and k and then use the same method. Thus ( ε∈Vk fε dµ ) is bounded by the product of four integrals. Iterating this procedure k times, we have the statement. (2) The only nontrivial property is the subadditivity of ||| · |||k . Let f and g be bounded k
functions on X. Expanding |||f + g|||2k , we get the sum of 2k integrals. Using part (1) to bound each of them, we have the subadditivity. (3) For a bounded function f on X,
2k+1
|||f |||k+1 = kE(
O
f |I[k] )k2L2 (µ[k] ) ≥
η∈Vk
Z O
!2 f dµ[k]
k+1
= |||f |||k2
.
η∈Vk
Part (1) of Lemma 4.3.10 allows one to estimate the L2 -norm of expectations on I[k] : 80
Corollary 4.3.11. (cf. [36], Corollary 3.10) If fε , ε ∈ Vk are bounded functions, then kE(
O
fε |I[k] )kL2 (µ[k] ) ≤
ε∈Vk
4.4
Y
|||fε |||k+1 .
ε∈Vk
Definition of the characteristic factors. ∗
∗
∗
The marginal (X [k] , X[k] , µ[k] )
4.4.1
∗
For x ∈ X [k] , write x = (x0 , x˜), where x0 ∈ X and x˜ = (xε ; ε ∈ Vk∗ ) ∈ X [k] . Let µ[k]
∗
∗
denote the measure on X [k] given by Z O
[k]∗
fε dµ
Z 1X (x0 ) ·
=
ε∈Vk∗
O
fε (xε ) dµ[k] .
ε∈Vk∗
∗
∗
So µ[k] is the image of µ[k] under the natural projection x → x˜ from X [k] onto X [k] . ∗
∗
∗
We consider (X [k] , X[k] , µ[k] ) as a factor of (X [k] , X[k] , X[k] , µ[k] ), so that (X [k] , X[k] , µ[k] ) ∗
∗
∗
is a joining of (X, X, µ) and (X [k] , X[k] , µ[k] ). ∗
∗
∗
∗
Let I[k] denote the σ-algebra of (T [k] )∗ -invariant sets of (X [k] , X[k] , µ[k] ) and J[k]
∗
∗
denote the σ-algebra of subsets of X [k] which are invariant under the action of T∗k . Definition 4.4.1. For k ≥ 1, Zk−1 (X) is the σ-algebra of subsets B of X for which ∗
there exists a subset A of X [k] such that 1B (x0 ) = 1A (˜ x) for µ[k] -almost every x = (x0 , x˜). Let Zk (X) be the factor of X corresponding to the σ-algebra Zk (X). By ergodicity of T, Z0 (X) is the trivial factor. Analogous to Lemma 4.2 of [36], we have 81
Lemma 4.4.2. For an integer k ≥ 1, (X [k] , X[k] , µ[k] ) is the relatively independent ∗
∗
∗
∗
joining of (X, X, µ) and (X [k] , X[k] , µ[k] ) over Zk−1 when identified with J[k] . The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. ∗
Proof. Let f be a bounded function on X and let g be a bounded function on X [k] . [k]
Since µ[k] is invariant under the group Tk−1 , we have Z
[k]
Z
f (x0 )g(˜ x) dµ (x) =
f (x0 )g(S x˜) dµ[k] (x),
[k]
[k]
whenever S ∈ T∗ . Thus by averaging over a Følner sequence in T∗ and taking the limit, Z
Z
[k]
f (x0 )g(˜ x) dµ (x) =
∗
f (x0 )E(g|J[k] )(˜ x) dµ[k] (x)
Z =
E(f |Zk−1 )(x0 )E(g|J
[k] ∗
(4.11) [k]
)(˜ x) dµ (x).
Corresponding to [36], Lemma 4.3, we have Lemma 4.4.3. Let f be a bounded function on X. Then E(f |Zk−1 ) = 0 ⇐⇒ |||f |||k = 0. The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. Assume that E(f |Zk−1 ) = 0. By Equation (4.11) applied to g(˜ x) = we have |||f |||k = 0 by definition (4.9) of the seminorm. 82
Q
ε∈Vk∗
f (xε ),
Conversely, assume that |||f |||k = 0. By Lemma 3.9, for every choice of fε , ε ∈ Vk∗ , Z f (x0 ) X [k]
Y
fε (xε ) dµ[k] (x) = 0.
ε∈Vk∗
By density, the function x 7→ f (x0 ) is orthogonal in L2 (µ[k] ) to every function defined ∗
∗
on X [k] , and in particular to every function measurable with respect to J[k] . But this means that f is orthogonal in L2 (µ) to every Zk−1 -measurable function and so E(f |Zk−1 ) = 0.
As a consequence of of Lemma 4.3.10, we have Corollary 4.4.4. (cf. [36], Corollary 4.4) The factors Zk (X), k ≥ 1, form an increasing sequence of factors of X. If p : (X, X, µ, T ) → (Y, Y, ν, T ) is a factor map, let p[k] denote the map defined by p[k] (x)ε = (p(xε ))ε∈V k . Lemma 4.4.5. (cf. [36], Lemma 4.5) Let p : (X, X, µ, T ) → (Y, Y, ν, T ) be a factor map and let k ≥ 1 be an integer. (1) The map p[k] : (X [k] , X[k] , µ[k] , T [k] ) → (Y [k] , Y[k] , ν [k] , T [k] ) is a factor map. (2) For a bounded function f on Y, |||f |||k = |||f ◦ p|||k . Proof. (1) It is clear that p[k] is a factor map since p[k] commutes with T [k] . To show that the image of µ[k] under p[k] is ν [k] , note that this is true for k = 0, and assume by way of induction that the result is true for k. Let fε , ε ∈ Vk be bounded functions on Y.
83
Since p[k] is a factor map, it commutes with the operators of conditional expectation on the invariant σ-algebras and we have E (
O
O fε ) ◦ p[k] |I[k] (X) = E fε |I[k] (Y ) ◦ p[k] .
ε∈Vk
ε∈Vk
Let gε , ε ∈ Vk+1 be bounded functions on Y. Computing Z O gε ◦ p[k+1] dµ[k+1]
RN
ε∈Vk+1
gε ◦ p[k] dµ[k+1] :
ε∈Vk+1
Z =
E
O
O gη0 ◦ p[k] |I[k] (X) E gη1 ◦ p[k] |I[k] (X) dµ[k]
η∈Vk
Z =
E Z
=
E
O
η∈Vk
O gη1 |I[k] (Y ) ◦ p[k] dµ[k] gη0 |I[k] (Y ) ◦ p[k] E
η∈Vk
η∈Vk
O
O
gη0 |I[k] (Y ) ◦ p[k] E
η∈Vk
Z =
O
gη1 |I[k] (Y ) dν [k]
η∈Vk
gε dν [k+1] .
ε∈Vk+1
Statement (2) follows from (1) and the definition of ||| · |||k .
Proposition 4.4.6. (cf. [36], proposition 4.6) Let p : (X, X, µ, T ) → (Y, Y, ν, T ) be a factor map and let k ≥ 1 be an integer. Then p−1 (Zk−1 (Y )) = Zk−1 (X) ∩ p−1 (Y). ∗
∗
Proof. For k = 1 there is nothing to prove. Let k ≥ 2 and let p[k] : X [k] → Y [k]
∗
denote the natural map. By Lemma 4.4.5, it is a factor map. Let f be a bounded function on X that is measurable with respect to p−1 (Zk−1 (Y )). Then f = g ◦ p for some function g on Y which is measurable with respect to Zk−1 (Y ). There exists ∗
∗
a function F on Y [k] , measurable with respect to J[k] , so that g(y0 ) = F (˜ y ) for ∗
ν [k] -almost every y = (y0 , y˜) ∈ Y [k] . Thus g ◦ p(x0 ) = F ◦ p[k] (˜ x) for µ[k] -almost every 84
x = (x0 , x˜) ∈ X [k] and the function f = g ◦ p is measurable with respect to Zk−1 (X). We have p−1 (Zk−1 (Y )) ⊂ Zk−1 (X) ∩ p−1 (Y). Conversely, assume that f is a bounded function on X, measurable with respect to Zk−1 (X) ∩ p−1 (Y). Then f = g ◦ p for some g on Y. Write g = g 0 + g 00 , where g 0 is measurable with respect to Zk−1 (Y ) and E(g 00 |Zk−1 (Y )) = 0. By the first part, g 0 ◦p is measurable with respect to Zk−1 (X). By Lemma 4.4.3 and Part (2) of Lemma 4.4.5, |||g 00 |||k = 0 and so |||g 00 ◦ p|||k = 0 and E(g 00 ◦ p|Zk−1 (X)) = 0. Since f = g 0 ◦ p + g 00 ◦ p is measurable with respect to Zk−1 (X), we have g 00 ◦ p = 0. Thus g 00 = 0 and g is measurable with respect to Zk−1 (Y ).
Proposition 4.4.7. (cf. [36], Proposition 4.7) Let k ≥ 1 be an integer. (1) As a joining of 2k copies of (X, X, µ), (X [k] , X[k] , µ[k] ) is a relatively independent [k]
[k]
joining over the joining (Zk−1 , µk−1 ) of 2k copies of (Zk−1 , µk−1 ). (2) Zk is the smallest factor Y of X so that the σ-algebra I[k] is measurable with respect to Y [k] . The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. (1) The statement is equivalent to showing that whenever fε , ε ∈ Vk , are bounded functions on X, Z
O
X [k] ε∈V k
[k]
fε dµ
Z
O
= [k] Zk−1
85
ε∈Vk
[k]
E(fε |Zk−1 ) dµk−1 .
(4.12)
It suffices to show that Z O
fε dµ[k] = 0
(4.13)
ε∈Vk
whenever E(fη |Zk−1 ) = 0 for some η ∈ Vk . By Lemma 4.4.3, if E(fη |Zk−1 ) = 0, we have that |||fη |||k = 0. Lemma 4.3.10 implies equality (4.13). (2) Let fε , ε ∈ Vk , be bounded functions on X. We claim that E
O
O fε |I[k] = E E(fε |Zk )|I[k] .
ε∈Vk
(4.14)
ε∈Vk
As above, it suffices to show this holds when E(fη |Zk ) = 0 for some η ∈ Vk . By Lemma 4.4.3, this condition implies that |||Fη |||k+1 = 0. By Corollary 4.3.11, the left hand side of Equation (4.14) is equal to zero and the claim follows. Every bounded function on X [k] which is measurable with respect to I[k] can be N approximated in L2 (µ[k] ) by finite sums of functions of the form E( ε∈Vk fε |I[k] ) where fε , ε ∈ Vk are bounded functions on X. By Equation (4.14), one can assume N that these functions are measurable with respect to Zk . In this case, ε∈Vk fε is [k]
[k]
[k]
measurable with respect to Zk (recall that πk : X [k] → Zk is a factor map by Part N (1) of Lemma 4.4.5). Since this σ-algebra is invariant under T [k] , E( ε∈Vk fε |I[k] ) is [k]
[k]
also measurable with respect to Zk . Therefore I[k] is measurable with respect to Zk . We use induction to show that Zk is the smallest factor of X with this property. For k = 0, I[0] and Z0 are both the trivial factor of X and there is nothing to prove. Let k ≥ 1 and assume that the result holds for k − 1. Let Y be a factor of X so that I[k] is measurable with respect to Y[k] . For any bounded function f on X with E(f |Y) = 0, we have to show that E(f |Zk ) = 0. 86
By projecting on the first 2k−1 coordinates, I[k−1] is measurable with respect to Y[k−1] . By the induction hypothesis, Y ⊇ Zk−1 . since µ[k] is a relatively independent joining [k]
over Zk−1 , it is a relatively independent joining over Y [k] . This impliesthat when fε , ε ∈ Vk are bounded functions on X, E(
O
fε |Y[k] ) =
ε∈Vk
O
E(fε |Y).
ε∈Vk
We apply this with fε = f for all ε. The function x 7→
Q
ε∈Vk
f (xε ) has zero conditional
expectation with respect to Y[k] . By hypothesis, it has zero conditional expectation with respect to I[k] . By the definition (4.9) of the seminorm, |||f |||k+1 = 0 and by Lemma 4.4.3, E(f |Zk ) = 0.
4.4.2
Systems of order k.
By Corollary 4.4.4, the factors Zk (X) form an increasing sequence of factors of X. Definition 4.4.8. An ergodic system (X, X, µ, T ) is of order k for an integer k ≥ 0 if X = Zk (X). Proposition 4.4.9. (cf. [36], Proposition 4.11) (1) A factor of a system of order k is a system of order k. (2) Let X be an ergodic system and Y be a factor of X. If Y is a system of order k, then it is a factor of Zk (X). (3) An inverse limit of a sequence of systems of order k is of order k.
87
Proof. Parts (1) and (2) follow from Proposition 4.4.6. To prove Part (3), let X be an inverse limit of a collection of systems Xi of order k. If f is measurable with respect ∗
to Xj for some j, then by the definition of Zk there exists a function F on X [k] such that f (x0 ) = F (˜ x) µ[k] -almost everywhere. The functions that are measurable with respect to some Xj are dense in L2 (µ), so the result follows.
As a consequence of Proposition 4.4.9, we have the following. Corollary 4.4.10. (cf. [36], Corollary 4.12) An ergodic system (X, X, µ, T ) is of order k if and only if |||f |||k+1 6= 0 for every nonzero bounded function f on X.
4.5
A group associated to each ergodic system.
Definition 4.5.1. (cf. [36], Definition 5.1) Let (X, X, µ, T ) be an ergodic system. We write G(X) or G for the group of measure preserving transformations x 7→ g · x which satisfy for every integer l > 0 the property: (Pl ) The transformation g [l] of X [l] leaves the measure µ[l] invariant and acts trivially on the invariant σ-algebra I[l] (X). We give G(X) the topology of convergence in probability. We recapitulate the remarks of [36], section 5: (i) T a ∈ G(X) for each a ∈ Zd . Property Pk implies Pl for l ≤ k, and factor maps p : X → Y induce maps from G(X) to G(Y ). (ii) G(X) is a Polish group.
88
(iii) If p : (X, X, µ, T ) → (Y, Y, ν, S) is a factor map and g ∈ G(X) such that g maps Y to itself, then the transformation h of Y induced by g is in G(Y ). (iv) Property (Pl ) implies property (Pk ) for k < l. The proofs of (i)-(iv) are entirely analogous to the proofs of the corresponding facts for the case d = 1, as presented in [36].
4.5.1
General properties.
Lemma 4.5.2. (cf. [36], Lemma 5.2) Let (X, X, µ, T ) be an ergodic system. Then for all k ≥ 0, every g ∈ G(X) maps the σ-algebra Zk = Zk (X) to itself and thus induces a measure preserving transformation of Zk , belonging to G(Zk ). The proof for the case d = 1 in [36] suffices. We reproduce it for completeness. Proof. Let g ∈ G and k ≥ 0 be an integer. Let f be a bounded function on X with E(f |Zk ) = 0. By Lemma 4.4.3 and the definition of the seminorm, 0=
k+1 |||f |||2k+1
Z =
O
[k+1]
f dµ
X [k+1 ε∈V k+1
Z =
O
f ◦ g dµ[k+1] .
X [k+1] ε∈V k+1
Since g [k+1] leaves the measure µ[k+1] invariant, we have |||f ◦ g|||k+1 = 0 and E(f ◦ g|Zk ) = 0. By using the same argument with g −1 substituted for g, we have that E(f ◦ g|Zk ) = 0 implies E(f |Zk ) = 0. We deduce that g · Zk = Zk . Thus g induces a transformation of Zk . By Remark (iii) above, this transformation pk g belongs to G(Zk ).
Notation. (From [36], Section 5.1) Let G be a group. Let k ≥ 1 be an integer and let
89
α be a face of Vk . Analogous to the definition of the side transformations, for g ∈ G [k]
we also write gα for the element of G[k] given by gα[k]
ε
= g if ε ∈ α ; gα[k]
ε
= idG otherwise.
[k]
When G acts on a space X, we also write gα for the transformation of X [k] associated to this element of G[k] : For x ∈ X [k] , gα[k] · x ε =
g · xε if ε ∈ α
xε otherwise
Lemma 4.5.3. (cf. [36], Lemma 5.3) Let (X, X, µ, T ) be an ergodic system and let 0 ≤ l < k be integers. For a measure preserving transformation g : x 7→ g · x of X, the following are equivalent: [k]
(1) For any l-face α of Vk , the transformation gα of X [k] leaves the measure µ[k] invariant and maps the σ-algebra I[k] to itself. [k+1]
(2) For any (l +1)-face β of Vk+1 the transformation gβ
leaves the measure µ[k+1]
invariant. [k]
(3) For any (l + 1)-face γ of Vk the transformation gγ leaves the measure µ[k] invariant and acts trivially on the σ-algebra I[k] . The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. We note first that if any one of these properties holds for a face of Vk , then by permuting the coordinates, it holds for any face of Vk of the same dimension. 90
[k]
(1) =⇒ (2). Let α be an l-face of Vk . The transformation gα preserves the measure µ[k] and the σ-algebra I[k] , and thus commutes with the conditional expectation on [k]
this σ-algebra. For any bounded function F on X [k] , we have E(F |I[k] ) ◦ gα = [k]
E(F ◦ gα |I[k] ). So, for bounded functions F 0 , F 00 on X [k] , Z X [k+1]
(F 0 ⊗ F 00 ) ◦ (gα[k] ×gα[k] ) dµ[k+1] Z = E(F 0 ◦ gα[k] |I[k] ) · E(F 00 ◦ gα[k] |I[k] ) dµ[k] [k] ZX = E(F 0 |I[k] ) ◦ gα[k] · E(F 00 |I[k] ) ◦ gα[k] dµ[k] [k] ZX = E(F 0 |I[k] ) · E(F 00 |I[k] ) dµ[k] [k] ZX = F 0 ⊗ F 00 dµ[k+1] X [k+1] [k]
[k]
[k+1]
and the measure µ[k+1] is invariant under gα × gα . But this transformation is gβ for some (l + 1)-face β of Vk+1 and so Property (2) follows.
(2) =⇒ (3). Let γ be an (l + 1)-face of Vk . Under the bijection between Vk and the first k-face of Vk+1 , γ corresponds to an (l + 1)-face β of Vk+1 . Under the usual [k+1]
identification of X [k+1] with X [k] ×X [k] , we have gβ [k+1]
µ[k+1] is invariant under gβ
[k]
= gγ ×id[k] . Since the measure
and each of its projections on X [k] is equal to µ[k] , this [k]
last measure is invariant under gγ . For a bounded function F on X [k] , measurable
91
with respect to I[k] , we have kF k2L2 (µ[k] )
Z =
F ⊗ F dµ[k+1]
Z Z = Z = [k]
[k+1]
(F ⊗ F ) ◦ gβ
=
dµ[k+1]
(F ◦ gγ[k] ) ⊗ F dµ[k+1] E(F ◦ gγ[k] |I[k] ) · F dµ[k] .
[k]
Thus E(F ◦ gγ |I[k] ) = F and F ◦ gγ = F. Property (3) is proved. [k]
(3) =⇒ (1). Let α be an l-face of Vk and let γ be an (l + 1)-face of Vk . Since gγ
acts trivially on I[k] , by using the definition of the conditional expectation we have [k]
E(F ◦gγ |I[k] ) = E(F |I[k] ) for any bounded function F on X [k] . By the definition of the [k]
measure µ[k+1] , this measure is invariant under gγ × id[k] . But this transformation is [k+1]
equal to Gβ
for some (l +1)-face β of Vk+1 . By permuting coordinates, the measure [k+1]
µ[k+1] is invariant under gβ [k]
for every (l + 1)-face β of Vk+1 . As the transformation
[k]
gα × gα is a transformation of this kind, it leaves the measure µ[k+1] invariant. By [k]
projection, the measure µ[k] is invariant under gα .
92
Let F be a bounded function on X [k] , measurable with respect to I[k] . Then Z [k] [k] 2 kE(F ◦ gα |I )kL2 (µ[k] ) = (F ◦ gα[k] ) ⊗ (F ◦ gα[k] ) dµ[k+1] Z = (F ⊗ F ) ◦ (gα[k] × gα[k] ) dµ[k+1] Z = F ⊗ F dµ[k+1] = kE(F |I[k] )k2L2 (µ[k] ) = kF k2L2 (µ[k] ) = kF ◦ gα[k] k2L2 (µ[k] ) , [k]
and this means that F ◦ gα is measurable with respect to I[k] .
Corollary 4.5.4. (cf. [36], Corollary 5.4.) Let (X, X, µ, T ) be an ergodic system and g : x 7→ g · x a measure preserving transformation of X. The following are equivalent: (1) For every integer k > 0 and every side α of Vk the measure µ[k] is invariant [k]
under gα . (2) For every integer k > 0 and every side α of Vk , the measure µ[k] is invariant [k]
under gα and this transformation maps the σ-algebra I[k] to itself. (3) g ∈ G(X). By an automorphism of the system (X, X, µ, T ), we mean a measure preserving transformation of X that commutes with T a for all a ∈ Zd . Lemma 4.5.5. (cf. [36], Lemma 5.5) Let (X, X, µ, T ) be an ergodic system. Then every automorphism of X belongs to G(X). 93
Moreover, if g : x 7→ g · x is an automorphism of X acting trivially on Zl (X) for some integer l ≥ 0, then for every integer k > 0 the measure µ[l+k] is invariant under [l+k]
gα
for every (k − 1)-face α of Vl+k .
The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. Let g be an automorphism of X as in the second part of the lemma. We use the formula (4.3) for µ[l+1] and the expression given by Lemma 4.3.2 for µ[l+k] : µ
[l+1]
Z =
µ[l+1] ω
dPl+1 (ω) and µ
[l+k]
Ωl+1
Z =
(µ[l+1] )[k−1] dPl+1 (ω). ω
Ωl+1 [l+1]
As µ[l+1] is relatively independent over Zl [l+1]
µ[l+1] is invariant under gε
and g acts trivially on Zl , the measure [l+1]
for any vertex ε ∈ Vl+1 . As the transformation gα
commutes with T [l+1] , it induces a measure preserving transformation h of Ωl+1 . [l+1]
Moreover, for Pl+1 -almost every ω ∈ Ωl+1 , the image of µω
[l+1]
under gα
[l+1]
follows that the measure µ[l+k] is invariant under the transformation gα [l+k]
(2k−1 times). But this transformation is gα
[l+1]
is µh·ω . It [l+1]
×· · ·×gα
, for some (k − 1)-face α of Vk+1 .
The second part of the lemma follows by permutation of coordinates. The first part of the lemma follows from the second part with l = 0 and Corollary 4.5.4.
4.5.2
Faces and commutators. [k]
Definition 4.5.6. Let G be a Polish group. Write Gl for the closed subgroup of G[k] generated by {gα[k] : g ∈ G and α is an l-face of Vk }. 94
[k]
[k]
We call Gk−1 the side subgroup and G1 the edge subgroup of G[k] . For j ≥ 0, let G(j) denote the closed j th iterated commutator subgroup of G. Lemma 4.5.7. (cf. [36], Lemma 5.7) Let G be a Polish group. For integers 0 ≤ j < k, [k]
[k]
the j th iterated commutator subgroup of Gk−1 contains (G(j) )k−j−1 . The proof is identical to that in [36]. We reproduce it for completeness. Proof. For g, h ∈ G and faces α, β of Vk , an immediate computation gives [k]
[k]
[gα[k] ; hβ ] = [g; h]α∩β .
(4.15)
For j = 0 the statement of the lemma is trivial. For j > 0 the statement is proved by induction. Every (k − j − 1)-face γ of Vk can be written as the intersection of a side α and a (k − j)-face β. By using Equation (4.15) we get the result.
Corollary 4.5.8. (cf. [36], Corollary 5.8) Let (X, X, µ, T ) be an ergodic system and G = G(X). Then, for integers 0 ≤ j < k, and g ∈ G(j) and any (k − j − 1)-face α of [k]
Vk , the map gα leaves the measure µ[k] invariant and maps the σ-algebra I[k] to itself. The proof is identical to that in [36]. We reproduce it for completeness. Proof. Let k ≥ 1 and let H be the subgroup of G[k] consisting of the transformations g = (gε : ε ∈ Vk ) of X [k] that leave the measure µ[k] invariant and map the σ-algebra [k]
I[k] to itself. By Corollary 4.5.4, H contains the side group Gk−1 . By Lemma 4.5.7, [k]
H contains (G(j) )k−j−1 for 0 ≤ j < k.
Corollary 4.5.9. (cf. [36], Corollary 5.9) If (X, X, µ, T ) is a system of order k, then the group G(X) is k-step nilpotent. 95
The proof is identical to that in [36]. We reproduce it for completeness. Proof. Let g ∈ G(k) . By Corollary 4.5.8, for any vertex ε ∈ Vk+1 , the measure µ[k+1] [k+1]
is invariant under gε
. Let f be a bounded function on X. Then Z Y 2k+1 (f (g · xε ) − f (xε )) dµ[k+1] . |||f ◦ g − f |||k+1 = ε∈Vk+1
All 2k+1 integrals obtained by expanding the right side of this equality are equal up to sign and so this expression is zero. By Corollary 4.12, f = f ◦ g so that g acts trivially on X, thus is the identity element of G. The group G(k) is trivial.
Corollary 4.5.10. (cf. [36], Corollary 5.10) Let (X, X, µ, T ) be a system of order k and u an automorphism of X inducing the trivial transformation on Zk−1 (X). Then u belongs to the center of G(X). The proof is identical to that in [36]. We reproduce it for completeness. Proof. By Lemma 4.5.5, u belongs to G(X). Let g ∈ G. Let ε be a vertex of Vk+1 . We choose an edge α and a side β of Vk+1 with ε = α ∩ β. By Lemma 4.5.5, µ[k+1] is [k+1]
invariant under uα
[k+1]
. By Corollary 5.4 this measure is invariant under gβ [k+1]
this measure is invariant under [uα
[k+1]
; gβ
[k+1]
] = [u; g]ε
. Thus
. We conclude as in the proof
of the preceding corollary that [u; g] is the identity.
4.6
Relations between consecutive factors.
In this section, we use the following fact, as stated in Section 2.5.6 of Chapter 2. Fact. Let (X, X, µ, T ) be a joining of Zd systems (Xi , Xi µi , T ), i = 1, . . . , k which is relatively independent over a common factor (Y, Y, µ, T ). Then the algebra I of 96
T -invariant subsets of X is contained in the σ-algebra generated by
Nk
i=1
bi , where Y
bi is the maximal isometric extension of Y as a factor of Xi . Y We also use the facts about isometric extensions summarized at the beginning of Section 6 of [36]. In particular, if Y is an ergodic system, and X = Y ×ρ G is an ergodic group extension of Y by the cocycle ρ, then every factor of X which extends Y is an isometric extension of the form X ×ρ G/K for some closed subgroup K ⊆ G. We present a proof of this fact here for completeness. Claim. If Y ×ρ G is an ergodic group extension of Y, then every factor of Y × G containing Y is of the form Y ×ρ G/K, for some closed subgroup K of G. Proof. Let W be a factor of X, corresponding to an invariant σ-algebra W ⊃ Y. Let K ⊆ G be the closed subgroup of G consisting of those k with the property that Ak = A µ-almost everywhere, for all A ∈ W. Clearly the elements of W are unions of cosets of K, so we may consider W to be a factor of Y × G/K. To show that W is the Borel σ-algebra of Y × G/K, it is enough to show that W separates almost every pair of points of Y × G/K. To this end, let h ∈ G \ K. Then there exists A ∈ W such that A is not equal to Ah µ-almost everywhere. In particular, B := {(y, g) ∈ A : (y, gh) ∈ / A} has positive measure. Then for all a ∈ Zd and (y, g) ∈ B, (T a y, ρ(a, y)g) ∈ T a B, while (T a y, ρ(a, y)gh) ∈ / T a B. By the ergodicity of T , we have that for almost all (y, g) ∈ Y × G, there exists C ∈ W such that (y, g) ∈ C and (y, gh) ∈ / C. This implies that for all h ∈ G and almost all (y, gK) ∈ Y × G/K, with gK 6= ghK, there exists C ∈ W such that (y, gK) ∈ C and (y, ghK) ∈ / C - take
97
C to be one of the T a B. Thus, W separates almost every pair of points of Y × G/K. Lemma 4.6.1. (cf. [36], Lemma 6.1) Let W = Y × G/H be an ergodic isometric extension of Y so that the corresponding extension Y × G is ergodic. Then, for every g ∈ G, g [1] = g × g acts trivially on the invariant σ-algebra I[1] (W ) of W × W. Proof. Let T denote the action of Zd on W. Consider the factor K of W generated by Y and the Kronecker factor Z1 (W ) of W. Then K is an extension of Y by a compact abelian group.2 Therefore, K = Y × G/L for some closed subgroup L of G containing H and containing the commutator subgroup G0 of G.3 Thus, for all g ∈ G, the action of g on K commutes with T and induces an automorphism of the Kronecker factor Z1 (K) = Z1 (W ). But an automorphism of an ergodic rotation is itself a rotation. Recalling that the invariant subsets of W × W are of the form {(x, y) : π(x) − π(y) ∈ A}, where A ⊂ Z1 (W ), and π : W → Z1 (W ) is the factor map, we see that g acts trivially.
2
To see this, note the Kronecker factor of Z1 (Y ) is a homomorphic image of the group Z1 (W ), with a homomorphism π induced by the factor map from W to Y. The elements of ker π act trivially on Z1 (Y ), and so induce automorphisms of K. The only subsets of W invariant under the action of ker π by these automorphisms are subsets of Y, and so K is an extension of Y by ker π.
3
To see this, let f ∈ L2 (Y ), and let ψ be an eigenfunction of W. For h ∈ G, φ ∈ L2 (Y × G), let φh be given by φh (y, g) = φ(y, gh). Then φh is an eigenvector of Y × G with eigenvalue λ if and only if φ is. By ergodicity of Y × G, we have φh = λ(h)φ for some constant λ whenever φ is an eigenvector of Y × G. We then have (f ψ)h1 h2 = f ψh1 h2 = λ(h1 )λ(h2 )f ψ = f ψh2 h1 . This means that f ψ does not distinguish between h1 h2 and h2 h1 , so the subgroup L must contain the commutator.
98
Lemma 4.6.2. (cf. [36], Lemma 6.2) Let (X, X, µ, T ) be an ergodic system and let k ≥ 2 be an integer. Then Zk is an isometric extension of Zk−1 . Proof. Let Y be the maximal isometric extension of Zk−1 which is a factor of X. ∗
∗
∗
∗
We consider (X [k] , X[k] , µ[k] , T [k] ) as a joining of (X, X, µ, T ) and (X [k] , X[k] , µ[k] , T [k] ), and recall that this joining is relatively independent with respect to the common fac∗
tor Zk−1 = I[k] of these two systems. Recall that if X1 = (X1 , X1 , µ1 , T1 ) and X2 = (X2 , X2 , µ2 , T2 ) are systems with a common factor W, then the invariant sets b ⊗ W, b where in the relative product of X1 and X2 are measurable with respect to W W is the maximal isometric extension of W. In particular, the invariant σ-algebra I[k] ∗
of (X [k] , X[k] , µ[k] , T [k] ) is measurable with respect to Y ⊗ X[k] . Let f be a bounded function on X with E(f |Y) = 0. Write F for the function Q [k] x 7→ ε∈Vk f (xε ) on X [k] . Since µ[k] is relatively independent with respect to Zk−1 ∗
and Y ⊃ Zk−1 , F has zero conditional expectation on the σ-algebra Y ⊗ X[k] and so R has zero conditional expectation on I[k] . Thus, X [k+1] F (x0 )F (x00 ) dµ[k+1] (x0 , x00 ) = 0. That is, |||f |||k+1 = 0 by definition of ||| · |||k+1 . Thus, E(f |Zk ) = 0, by Lemma 4.4.3. Therefore Zk ⊂ Y.
Proposition 4.6.3. (cf. [36], Proposition 6.3) Let (X, X, µ, T ) be a system of order k ≥ 2. (1) X is a compact abelian group extension of Zk−1 , written X = Zk−1 × U, where U is a compact abelian group. [k]
(2) For every u ∈ U and every edge α of Vk , the transformation uα acts trivially on I[k] . 99
The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. By Lemma 4.6.2, X is an isometric extension of Zk−1 and so we can write X = Zk−1 × (G/H), where G is a compact group and H a closed subgroup. As in Section 6.1 we write ρ : Zd × Zk−1 → G for the cocycle defining this extension and let λ denote the Haar measure G/H. [k]
Since µ[k] is relatively independent with respect to Zk−1 , this measure is invariant [k]
under the map gε for any g ∈ G and any ε ∈ Vk . Consequently, it is invariant under [k]
gα for any g ∈ G and any edge α of Vk . [k]
Claim. For any g ∈ G and any edge α of Vk , the transformation gα acts trivially on I[k] . [k−1]
Consider the dergodic decompositions of µ[k−1] and µk−1 as in Section 3.1. Since [k−1]
I[k−1] is measurable with respect to Zk−1 , these decompositions can be written as Z Z [k−1] [k−1] [k−1] [k−1] µ = µω dPk−1 (ω) and µk−1 = µk−1,ω dPk−1 (ω), Ωk−1
Ωk−1
[k−1]
[k−1]
where µk−1,ω is the projection of µω
[k−1]
on Zk−1 .
By Part (1) of Proposition 4.4.7, (X [k−1] , X[k−1] , µ[k−1] , T [k−1] ) is the relatively inde[k−1]
pendent joining of 2k−1 copies of (X, X, µ, T ) over Zk−1 . Thus we can identify X [k−1] [k−1]
[k−1]
with Zk−1 × (G[k−1] /H [k−1] ). The measure µ[k−1] is the product of µk−1 by the 2k−1 power λ⊗[k−1] of λ, which is the Haar measure of G[k−1] /H [k−1] and X [k−1] is the [k−1]
[k−1]
isometric extension of Zk−1 given by the cocycle ρ[k−1] : Zd × Zk−1 → G[k−1] . [k−1]
So for almost every ω ∈ Ωk−1 , the system (X [k−1] , X[k−1] µω [k−1]
[k−1]
[k−1]
, T [k−1] ) is an isometric
extension of (Zk−1 , Zk−1 , µk−1,ω , T [k−1] ), with fiber G[k−1] /H [k−1] . 100
[k−1]
Let g ∈ G and let ε ∈ Vk−1 . Since gε [k−1]
transformation gε
[k−1]
× gε
belongs to G[k−1] , by Lemma 4.6.1 the
of X [k] = X [k−1] × X [k−1] acts trivially on the T [k] = [k−1]
T [k−1] × T [k−1] -invariant σ-algebra of (X [k] , µω
[k−1]
× µω
, T [k] ).
We recall (see Formula (4.4)) that µ
[k]
Z
µ[k−1] × µ[k−1] dP (ω). ω ω
= Ωk−1
[k−1]
Thus gε
[k−1]
× gε
[k−1]
acts trivially on the invariant σ-algebra I[k] . But gε
[k−1]
× gε
is
[k]
equal to gα for some edge α of Vk . The claim follows by permuting the coordinates. Claim. G is abelian. Let g, h ∈ G, and let ε be a vertex of Vk+1 . Choose two edges α and β of Vk+1 with [k]
[k]
[k]
α ∩ β = ε. By Equation (4.15), [gα ; hβ ] = [g; h]ε . By the first step and Lemma [k+1]
4.5.3, the transformations gα [k+1]
transformation [g; h]ε
[k+1]
and hβ
preserve the measure µ[k+1] , thus also the
. As this holds for every vertex ε, we conclude as in the proof
of Corollary 4.5.9 that [g; h] acts trivially on X. This means that [g; h] = 1 and so G is abelian. Since G is abelian, we can take U = G.
4.6.1
Description of the extension.
Here we recapitulate Section 6.3 of [36] with some modifications. Notation. For k ≥ 1 and ε ∈ Vk , we write |ε| =
X
εi and s(ε) = (−1)|ε| .
101
Let X be a set, U an abelian group written with additive notation and f : X → U a map. For every k ≥ 1, we define a map ∆k f : X [k] → U by ∆k f (x) =
X
s(ε)f (xε ).
ε∈Vk
Proposition 4.6.4. (cf. [36], Proposition 6.4) Let (X, X, µ, T ) be a Zd -system of order k ≥ 2, By proposition 4.6.3, X is an extension of Zk−1 by a compact abelian group U for some cocycle ρ : Zd × Zk−1 → U. Then [k]
[k]
[k]
(1) ∆k ρ : Zd × Zk−1 → U is a coboundary of the system (Zk−1 , µk−1 , T [k] ), meaning [k]
that there exists F : Zk−1 → U with [k]
[k]
∆k ρ(a, x) = F ◦ (T [k] )a (x) − F (x) for all a ∈ Zd , µk−1 , x ∈ Zk−1 .
(4.16)
(2) The σ-algebra I[k] (X) is spanned by the σ-algebra I[k] (Zk−1 ) and the map Φ : X [k] → U given by Φ(y, u) = F (y) −
X
s(ε)uε
(4.17)
ε∈Vk [k]
for y ∈ Zk−1 and u ∈ U [k] where X is identified with Zk−1 × U and X [k] with [k]
Zk−1 × U [k] . The proof of is essentially identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. Here we consider characters of U as homomorphisms from U to the circle group S, written with multiplicative notation. 102
b . Define the function ψ on X = Zk−1 × U by ψ(y, u) = χ(u) and the (1) Let χ ∈ U [k]
function Ψ on X [k] = Zk−1 × U [k] by Ψ(y, u) = χ
X
s(ε)uε for y ∈ Y [k] and u ∈ U [k] .
ε∈Vk
Since X is of order k, |||ψ|||k+1 6= 0 by Corollary 4.4.10 and E(Ψ|I[k] ) 6= 0 by Lemma 4.3. [k]
Let J be the linear map from L2 (µk−1 ) to L2 (µ[k] ) given by Jf (y, u) = f (y)Ψ(y, u) [k]
[k]
for f ∈ L2 (µk−1 ), y ∈ Zk−1 and u ∈ U [k] . J is an isometry and its range Hχ is a closed [k]
subspace of L2 (µ[k] ). Furthermore, for f ∈ L2 (µk−1 ), and a ∈ Zd write ga for the map x 7→ χ(∆k ρ(a, x) · f ((T [k] )a x). Then Jga = (Jf ) ◦ (T [k] )a and so the space Hχ is invariant under T [k] . Since the function Ψ belongs to Hχ , the function E(Ψ|I[k] ) also belongs to this space. We get that there exists a nonidentically [k]
zero function f on Zk−1 with [k]
χ(∆k ρ(a, y)) · f ◦ (T [k] )a (y) = f (y) µk−1 -a.e. [k]
(4.18)
Let A = {y ∈ Zk−1 : f (y) 6= 0}. Then µk−1 (A) 6= 0 and A is T [k] -invariant by R [k] [k] Equation 4.18. We use the ergodic decomposition of µk−1 as Ωk µk−1,ω dP (ω). Since A is invariant, it corresponds to a subset B of Ωk , with Pk (B) 6= 0. Define [k]
[k]
C = {ω ∈ Ωk : χ ◦ ∆k ρ is a coboundary of (Zk−1 , µk−1,ω , T [k] )}. 103
By Lemma 4.15.3 C is measurable in Ωk and it contains B by Equation (22) and the definition of B. Thus Pk (C) > 0. We show now that C is invariant under the [k]
group Tk−1 of side transformations. Let ω ∈ Ωk , a ∈ Zd , and let α be a side of Vk not [k]
[k]
containing 0 so that (Tα )a ω ∈ C. Let φ : Zk−1 → T be chosen with its coboundary for T [k] equal to χ ◦ ∆k ρ almost everywhere for the measure µ [k]
[k] [k]
k−1,(Tα )a ω
. The coboundary
[k]
of φ ◦ (Tα )a for T [k] is equal to χ ◦ (∆k ρ) ◦ Tα almost everywhere for the measure [k]
[k]
[k]
µk−1,ω . But the map ((∆k ρ) ◦ (Tα )a ) ◦ (Tα )b − ∆k ρ (where the symbol b is a variable) from Zd × Y [k] to U is the coboundary for T [k] of the map y 7→
X
s(ε)ρ(yε ).
ε∈α [k]
[k]
Therefore χ ◦ ∆k ρ is a coboundary of the system (Zk−1 , µk−1,ω , T [k] ) and ω ∈ C. Thus [k]
[k]
the set C is invariant under Tα . By Corollary 4.3.7, the action of the group T∗ on Ωk is ergodic. As P (C) > 0, we have P (C) = 1. Therefore, for Pk -almost every ω ∈ Ωk , χ ◦ ∆k ρ is a coboundary of the system [k]
[k]
[k]
[k]
(Zk−1 , µk−1,ω , T [k] ). By Corollary 4.15.4, χ ◦ ∆k ρ is a coboundary of (Zk−1 , µk−1 , T [k] ). b , ∆k ρ is a coboundary of this system by Lemma 4.15.1 As this holds for every χ ∈ U and the first part of the proposition is proved. b [k] . For θ = (θε : ε ∈ Vk ) ∈ U b [k] and We identify the dual group of U [k] with U u = (uε : ε ∈ Vk ) ∈ U [k] , θ(u) =
Y
θε (uε ).
ε∈Vk
Let H be the subspace of L2 (µ[k] ) consisting of functions invariant under T [k] . For
104
b [k] , we write Lθ for the subspace of L2 (µ[k] ) consisting of those functions of the θ∈U form (y, u) 7→ f (y)θ(u)
(4.19)
[k]
for some f ∈ L2 (µk−1 ). Then Lθ is a closed subspace of L2 (µ[k] ), invariant under T [k] . [k]
Since the measure µ[k] is relatively independent over µk−1 , using the Fourier transform b [k] . Therefore, the we see that L2 (µ[k] ) is the Hilbert sum of the spaces Lθ for θ ∈ U invariant subspace H of L2 (µ[k] ) is the Hilbert sum of the invariant subspaces H ∩ Lθ of Lθ . b [k] and assume that H ∩ Lθ contains a nonidentically zero function φ. Let Let θ ∈ U [k]
α = (ε, η) be an edge of Vk and let u ∈ U. By Equation (4.19) we have φ ◦ uα = [k]
φ · θε (u)θη (u). But by part (2) of Proposition 4.6.3, φ ◦ uα = φ and we get that θε (u)θη (u) = 1. Since this holds for every u ∈ U, θε θη = 1. As it holds for every edge b with θε = χs(ε) for every ε ∈ Vk . Finally, φ is a function α = (ε, η), there exists χ ∈ U of the form φ(y, bu) = f (y) · χ
X
s(ε)uε
ε∈Vk [k]
for some f ∈ L2 (µk−1 ), and φ(y, u) = χ(−Φ(y, u)) · χ(F (y))f (y)
(4.20)
where Φ is the map defined by equation (4.20). Since Φ and φ are invariant under T [k] , the function χ ◦ F · f is also invariant under this action and is measurable with respect to I[k] (Zk−1 ). We conclude that φ is measurable with respect to the σ-algebra spanned by Φ and I[k] (Zk−1 ). 105
Since the invariant space H of L2 (µ[k] ) is the Hilbert sum of the spaces H ∩ Lθ , every function in H is measurable with respect to this σ-algebra and the second part of the proposition is proved.
Proposition 4.6.5. (cf. [36], Proposition 6.5) Let (X, X, µ, T ) be a Zd -system of order [l]−1
k. Then for l > k the invariant σ-algebra I[l] is spanned by the σ-algebras ξα
(I[k] ),
where α is a k-face of Vl . The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Let (X, X, µ, T ) be a system of order k. We use the notation of Proposition 4.6.4 and the maps F and Φ defined in Equations (4.16) and (4.20). Let l > k. [l]
[l]
[l]
[l]
We identify X [l] with Zk−1 × U [l] . As the projection of µ[l] on Zk−1 is µk−1 , for µk−1 [l]
almost every y ∈ Zk−1 there exists a measure λy on U [l] such that [l]
Z
µ = [l] Zk−1
[l]
δy × λy dµk−1 (y).
For every u ∈ U, the corresponding vertical rotation is an automorphism of X and acts trivially on Zk−1 . By Lemma 4.5.5, for every (l − k)-face β of Vl the measure µ[l] [l]
is invariant under uβ . It follows that the measure λy is invariant under this transfor[l]
mation for µk−1 -almost every y. By separability, for almost every y the measure λy [l]
is invariant under translation by any element of the group Ul−k . We identify U [l] with U [l−1] × U [l−1] and we write u = (u0 , u00 ) for an element of U [l] ; [l]
[l−1]
[l−1]
we write also y = (y0 , y00 ) for a point of Zk−1 = Zk−1 × Zk−1 ; and x = (y0 , u0 , y00 , u00 ) [l]
for a point of X [l] , with y = (y0 , y00 ) ∈ Zk−1 and u = (u0 , u00 ) ∈ U [l] . 106
[l−1]
: X [l−1] → U is T [l−1] -invariant, it
[l−1]
(x0 ) = Φk ◦ ξγ
Let γ be a k-face of Vl−1 . As the map Φk ◦ ξγ
follows from the construction of µ[l] that Φk ◦ ξγ
[l−1]
[l]
(x00 ) for µk -almost
every x; that is, X
s(ε)u0ε −
X
s(ε)u00ε = F (ξγ[l−1] y0 ) − F (ξγ[l−1] y00 ) µ[l] -a.e.
ε∈γ
ε∈γ [l]
[l]
For µk−1 -almost every y = (y0 , y00 ) ∈ Zk−1 , this identity is true for λy -almost every u = (u0 , u00 ) ∈ U [l] and the measure λy is concentrated on a coset of the group {(u0 , u00 ) ∈ U [l] :
X
u0ε −
ε∈γ
X
s(ε)u00ε = 0}.
ε∈γ
We write δ for the (k + 1)-face γ × {0, 1} of Vl , and we notice that this group is equal to {u ∈ U [l] :
X
[l] −1
s(ε)uε = 0} = ξδ
[k+1]
(U1
).
ε∈δ
By permutation of coordinates, the same property holds for any k + 1-face δ of Vl , and λy is concentrated on a coset of the intersection {u ∈ U [l] :
X
s(ε)uε = 0 for every (k + 1)-face δ of Vl }
ε∈δ
107
(4.21)
of the corresponding subgroups of U [l] . By an elementary algebraic computation4 , [l]
we see that this group is equal to Ul−k . [l]
Finally, λy is invariant under translation by Ul−k and is concentrated on a coset of this group. Thus this measure is the image of the Haar measure of this group by [l]
some translation. Moreover, for almost every y ∈ Zk−1 and all a ∈ Zd , the measure λ(T [l] )a y is the image of the measure λy by the translation ρ[l] (y). We conclude that: [l]
[l]
[l]
The system (X [l] , X[l] , µ[l] , T [l] ) is an extension of (Zk−1 , Zk−1 , µk−1 , T [l] ) by the com[l]
pact abelian group Ul−k . It follows from the description of µ[l] just above that this Hilbert space L2 (µ[l] ) can be decomposed as in the proof of Proposition 4.6.4: L2 (µ[l] ) is the Hilbert sum for d [l] θ ∈ Ul−k of the subspaces [l]
Lθ = {f : f (u, x) = θ(u)f (x) µ[l] -a.e. for every u ∈ Ul−k }. Each space Lθ is invariant under the action T [l] and thus the T [l] -invariant subspace H of L2 (µ[l] ) is the Hilbert sum of the spaces Hθ = H ∩ Lθ . On the other hand, by Lemmas 4.5.5 and 4.5.3, each function in H is invariant under 4
[l]
Write Wk+1,l for the group in display (4.21). It is easy to see that Ul−k ⊆ Wk+1,l , for the intersection of a face of dimension l − k and a face of dimension k + 1 is a face of dimension P [l] at least 1, and Ul−k is generated by elements u such that ε∈δ s(ε)uε = 0 for every face δ of dimension of at least 1. We will show by induction on l that the reverse inclusion is true. The [l−1] statement is easy to check for l = 1, so let l > 1 and assume that Wk+1,l = Ul−1−k holds for all k. Let u ∈ Wk+1 , and write u = (u0 , u00 ), where u0 , u00 ∈ U [l−1] . Note that u0 ∈ Wk,l−1 , so we [l−1] [l] can write u0 as an element of Ul−1−k . But then (u0 , u0 ) ∈ Ul−k , and so (u0 , u0 ) ∈ Wk+1,l . This [l]
implies that (0, u0 − u00 ) ∈ Wk+l,l − Ul−k . In fact u0 − u00 ∈PWk−1,l−1 , since every k − 1-face δ of Vl−1 can be extended to a k-face δ × δ of Vl , and the sum ε∈δ×δ s(ε)(0, u00 − u0 )ε is unaffected by the indices ε of δ × δ appearing in the first side of Vl = Vl−1 × Vl−1 . The induction hypothesis [l] [l] now implies that (0, u00 − u0 ) ∈ Ul−k , so (u0 , u00 ) ∈ Ul−k .
108
[l]
the map x 7→ u·x for any u ∈ Ul−k+1 . Therefore Hθ is trivial except if θ belongs to the [l]
[l]
annihilator of Ul−k+1 in the dual group of Ul−k . By the same algebraic computation as above, we get that [l]
Ul−k+1 = {u ∈ U [l] :
X
s(ε)uε = 0 for every k-face α of Vl }.
ε∈α [l] b )[l] }, where σ : U [l] → It follows that the annihilator of Ul−k+1 in Uc[l] is {χ ◦ σ : χ ∈ (U k [l]
U [l] is the map (xε )ε∈Vl 7→ (s(ε)xε )ε∈Vl . Therefore, the subspace H of L2 (µk ) is the closed linear span of the family of invariant functions of the type [l]
[l]
b . φ(y, u) = ψ(y) · θ ◦ σ(u), where ψ ∈ L2 (µk−1 ) and θ ∈ U k b [l] is spanned by the elements We consider an invariant function φ of this type. As U k [l] b and α is a k-face of Vl , there exist k-faces α1 , . . . , αm of of the form χα , where χ ∈ U
b with Vl and characters χ1 , . . . , χm ∈ U θ(u) =
m Y Y
χj (uε )
j=1 ε∈αj [l]
for u ∈ U [l] . For each j the function χj ◦ Φk ◦ ξαj is invariant, and thus so is the function φ·
m Y
χj ◦ Φk ◦ ξα[l]j .
j=1 [l]
But clearly this function factors through Zk−1 and is measurable with respect to I[l] (Zk−1 ). Therefore, the function φ is measurable with respect to the σ-algebra as[l] −1
panned by I[l] (Zk−1 ) and ξαm
(I[k] (X)), 1 ≤ j ≤ m. We get: [l] −1
The σ-algebra I[l] (X) spanned by the σ-algebras I[l] (Zk−1 ) and the σ-algebras ξα for α a k-face of Vl . 109
(I[k] (X)),
We now proceed by induction on k. For k = 0 the system X is trivial and there is nothing to prove. We take k > 0 and assume that the assertion holds for every system of order k − 1. Let X be a system of order k and let l > k. By the inductive [l] −1
hypothesis I[l] (Zk−1 ) is spanned by the σ-algebras ξα [l] −1
Vl . But, for each α, ξα
[l] −1
(I[k] (Zk−1 )) ⊆ ξα
(I[k] (Zk−1 )) for α a k-face of
(I[k] (X)) and the result follows from the
preceding paragraph.
Corollary 4.6.6. Let (X, X, µ, T ) be a system of order k and let x 7→ g · x be a measure preserving transformation of X satisfying the property (Pk ) of Definition 4.5.1. Then g ∈ G(X). The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. We have to show that property (Pl ) holds for every l. For l = k there is nothing to prove. For l < k, (Pl ) follows immediately from (Pk ) by projection. For l > k we proceed by induction. Let l > k and assume that Pl−1 holds. By Lemma [l]
4.5.3, the measure µ[l] is invariant under gβ for any (l − 1)-face β of Vl and it follows immediately that it is invariant under g [l] . By hypothesis, g [k] acts trivially on I[k] and it follows that for every k-face α of Vl the transformation g [l] acts trivially on the [l] −1
σ-algebra ξα
(I[k] ). By Proposition 4.6.5, g [l] acts trivially on I[l] .
110
4.7
Cocycles of type k and systems of order k.
Notation. Let (X, X, µ) be a probability space and U a compact abelian group. We write C(X, U ) for the group of measurable cocycles from Zd × X → U. We write B(X, U ) for the space of all measurable maps from X to U. Definition 4.7.1. Let k ≥ 1 be an integer, (X, X, µ, T ) and ergodic system, U a compact abelian group, (written additively) and ρ : Zd × X → U a cocycle. We say that ρ is a cocycle of type k if the cocycle ∆k ρ : Zd × X [k] → U is a coboundary of (X [k] , X[k] , µ[k] , T [k] ). In the previous section it was shown that Zk is an extension of Zk−1 by a cocycle of type k. Remark 4.7.2. A cocycle cohomologous to a cocycle of type k is also of type k. By Lemma 4.15.1 we get: Remark 4.7.3. ρ : Zd × X → U is of type k if and only if χ ◦ ρ : Zd × X → T is of type k for every character χ of type U. It follows that for any closed subgroup V of U, a V -valued cocycle is of type k if and only if it is of type k as a U -valued cocycle. Notation. Let (X, X, µ, T ) be an ergodic system, k ≥ 1 be an integer, and U a compact abelian group. Let Dk (X, U ) denote the set of cocycles ρ : Zd × X → U with ∆k ρ = 0. Lemma 4.7.4. (cf. [36], Lemma 7.4) Let (X, X, µ, T ) be an ergodic system, k ≥ 1 be an integer, and U be a compact abelian group. Then Dk (X, U ) is a closed subgroup 111
of C(X, U ). Furthermore, if U is a compact abelian Lie group, then Dk (X, U ) admits the group hom(Zd , U ) of homomorphisms as an open subgroup. The proof follows the proof of Lemma 7.4 from [36], with minor modifications. Proof. The first assertion is obvious. We prove the second statement by induction on k. By definition, D1 (X) = hom(Zd , U ) Assume the conclusion holds for some k ≥ 1. Our induction hypothesis is that for all compact abelian Lie groups U, the Dk (X, U ) is a closed subgroup of C(X, U ), and the group hom(Zd , U ) is an open subgroup of Dk (X, U ). Let B ⊂ Zd be a finite generating set. R [k] [k] We use Formula (4.4) to write µ[k+1] = Ωk µω × µω dPk (ω). Also, ρ belongs to [k]
[k]
Dk+1 (X, U ) if and only if ∆(∆k ρ) = 0, µω × µω -almost everywhere for Pk -almost all ω ∈ Ωk . This condition means that for Pk -almost all ω ∈ Ωk , ∆k ρ is in hom(Zd , U ), [k]
µω -almost everywhere. Thus, ∆k ρ(a, (T [k] )b x) = ∆k ρ(a, x) µ[k] -almost everywhere, for each b ∈ Zd . As (∆k ρ) ◦ (T [k] )b = ∆k (ρ ◦ T b ), for b ∈ Zd this condition is equivalent to ∆k (ρ ◦ T b − ρ) = 0 for each b ∈ Zd . Thus, ρ ◦ T b − ρ ∈ Dk (X, U ), for each b ∈ Zd . Consider the map ∂ from Dk+1 (X, U ) to Dk (X, U B ) given by ∂ρ = (ρ◦T b −ρ)b∈B . The kernel K of ∂ consists of those cocycles ρ so that ρ(a, T b x) = ρ(a, x) for all b ∈ Zd , which by ergodicity of T means that K = hom(Zd , U ). By the induction hypothesis, hom(Zd , U B ) is an open subgroup of Dk (X, U B ), and therefore has countable index in Dk (X, U B ). Claim. The image of ∂ meets hom(Zd , U B ) in a countable set. To prove this claim, suppose that ∂ρ ∈ hom(Zd , U B ). Then for each b ∈ Zd , we have b , we have χ(ρ(a, T b x)) = ρ(a, T b x) − ρ(a, x) = ca,b for some constant ca,b ∈ U. If χ ∈ U 112
χ(ca,b )χ(ρ(a, x)). This means that χ ◦ ρ(a, ·) is an eigenvector of (X, X, µ, T ), for all b . Since an ergodic system has only countably many eigenvectors, there are only χ∈U b . Since U is a Lie group, U b countably many possible values of χ(ca,b ) for each χ ∈ U is finitely generated, so the values of χ(ca,b ) are determined on a finite set of a, b, and χ. Hence, there are only countably many possible values for ca,b , and the image of ∂ meets hom(Zd , U B ) in a countable set, as claimed. Since hom(Zd , U Z ) has countable index in Dk (X, U Z ), the claim above implies that d
d
the image of ∂ is countable. Thus the kernel of ∂ has countable index, and is therefore an open subgroup of Dk+1 (X, U ).
4.7.1
Cocycles of type k and automorphisms.
Corollary 4.7.5. (cf. [36], Corollary 7.5) Let (X, X, µ, T ) be an ergodic system, ρ : X → U a cocycle and k an integer. Then (1) If ρ is of type k ≥ 1, then for any automorphism S of X the cocycle ρ ◦ S − ρ is of type k − 1. (2) If X is of order k ≥ 2 and ρ is of type k, then for any vertical rotation x 7→ u · x of X over Zk−1 the cocycle ρ ◦ u − ρ is a coboundary. (3) If X is of order k ≥ 1 and ρ is of type k + 1, then for any vertical rotation x 7→ u · x of X over Zk−1 the cocycle ρ ◦ u − ρ is of type 1. The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. 113
Proof. (1) Let F : X [k] → U be a map with F ((T [k] )a x) − F (x) = ∆k ρ(a, x). Let α [k]
be the first side of Vk . By Lemma 4.5.5, the measure µ[k] is invariant under Sα . As this transformation commutes with (T [k] )a for all a ∈ Zd , by the definition of F we have (∆k−1 ρ)(a, ξα[k] S [k] x) − ρ(a, ξα[k] x) = (F ◦ Sα[k] − F ) ◦ (T [k] )a (x) − (F ◦ Sα[k] − F )(x). By Lemma 4.15.7, ∆k−1 (ρ ◦ S − ρ) is a coboundary on X [k−1] and ρ ◦ S − ρ is of type k − 1. (2) By Proposition 4.6.3, X = Zk−1 × W for some compact abelian group W. The [k]
measure µ[k] is conditionally independent over Zk−1 and thus invariant under the [k]
vertical rotation by wε for every ε ∈ Vk and every w ∈ W. The same computation [k]
as above shows that (ρ ◦ w − ρ) ◦ ξε is a coboundary on X [k] and so ρ ◦ w − ρ is a coboundary on X. (3) Let W be as in Part (2). Let w ∈ W. For any ε ∈ Vk , the measure µ[k] is invariant [k]
uner wε . This transformation commutes with (T [k] )a for all a ∈ Zd and thus maps the σ-algebra I[k] to itself. By Lemma 4.5.5, for any edge α of Vk−1 the measure µ[k+1] [k+1]
is invariant under wα
4.7.2
. We conclude as in part (2).
Cocycles of type k and group extensions.
Let Y be an ergodic extension of a system X by a compact abelian group U. For u ∈ U, let pk u denote the automorphism of Zk (Y ) induced by the vertical rotation by u on Y.
114
Proposition 4.7.6. Let (X, X, µ, T ) be an ergodic system, U a compact abelian group, ρ : Zd ×X → U an ergodic cocycle and (Y, Y, ν, S) = (X ×U, X⊗B(U ), µ×mU , Tρ ) the extension it defines. Let k ≥ 1 be an integer. For u ∈ U, let pk u be the automorphism of Zk (Y ) defined just above. Let W = {u ∈ U : pk u = id}. Then (1) W is a closed subgroup of U. b is the subgroup Γk := {χ ∈ U b : χ◦ρ is of type k}. (2) The annihilator W ⊥ of W in U (3) The cocycle ρ mod W : X → U/W is of type k. (4) Zk (Y ) is an extension of Zk (X) by the compact abelian group U/W, given by a cocycle ρ0 : Zk (X) → U/W of type k. Moreover, the cocycle ρ0 ◦ πX,k is cohomologous to ρ mod W : X → U/W. The proof is identical to that in [36], with obvious modifications. We reproduce it for completeness. Proof. (1) is obvious. For every u ∈ U, let u denote its image in U/W. If we think of factors as invariant σ-algebras, then X is the collection of sets in Y with are invariant under the vertical rotation associated to any u ∈ U. By Proposition 4.4.6 we have Zk (X) = Zk (Y ) ∩ X. Thus Zk (X) consists of those sets of Zk (Y ) which are invariant under pk u for every u ∈ U. Therefore, as an extension of Zk (X), Zk (Y ) is isomorphic to an extension by the compact abelian group U/W. We identify Zk (Y ) with Zk (X) × U/W and Y with X × U and study the factor map πY,k : X ×U → Zk (X)×U/W. By construction, for (x, u) ∈ X ×U, the transformation pk v is given by pk v(z, u) = (z, v + u). That is, it is the vertical rotation by v of Zk (Y ) 115
over X. Since πY,k ◦ v = pk v ◦ πY,k , it follows that there exists φ : X → U/W such that πY,k (x, u) = (πX,k (x), u + φ(x)). Let ρ0 : Zd × Zk (X) → U/W be a cocycle defining the extension Zk (X) × U/W of Zk (X). Since πY,k : X × U → Zk (X) × U/W is a factor map, we get ρ0 ◦ πX,k (a, x) = ρ(a, x) + φ(T a x) − φ(x) and ρ0 ◦ πX,k is cohomologous to ρ = ρ mod W. [ = W ⊥ . Here we consider χ as taking values in the circle group S. We Let χ ∈ U/W define a map ψ on Zk (Y ) = Zk (X) × U/W by ψ(x, u) = χ(u) and define a function Ψ on Zk (Y )[k] = Zk (X)[k] × (U/W )[k] by Ψ(x, u) = χ(
X
s(ε)uε ) for x ∈ Zk (X)[k] and u ∈ (U/W )[k]
ε∈Vk
and continue exactly as in the proof of the first part of Proposition 6.4. Then χ ◦ ρ0 is of type k. [ , the cocycle ρ0 is of type k and Part (4) of Proposition As this holds for every χ ∈ U/W 4.7.6 is proved. Part (3) follows immediately, as does the inclusion W ⊥ ⊆ Γk . We now prove the opposite inclusion. Let χ ∈ Γk . Then χ ◦ ρ is a cocycle of type k. We consider χ as taking values in T. Let F : X [k] → T be a map with F ((T [k] )a x) − F (x) = ∆k (χ ◦ ρ) µ[k] -almost everywhere, for all a ∈ Zd . We define a map Φ from Y [k] = X [k] × U [k] to T by Φ(x, u) = F (x) −
X
s(ε)χ(uε ) for x ∈ X [k] and u ∈ U [k] .
ε∈Vk
The projection of ν [k] on X [k] is µ[k] and each of the one-dimensional marginals of ν [k] is ν. From these remarks and the definition of F we get that Φ ◦ S [k] = Φ ν [k] -almost everywhere. The map Φ is measurable with respect to I[k] (Y ). 116
Let w ∈ W and ε ∈ Vk . The measure ν [k] is relatively independent with respect to Zk−1 (Y ) and thus with respect to Zk (Y ). Since the vertical rotation w acts trivially [k]
on Zk (Y ), the measure ν [k] is invariant under wε . Moreover this transformation acts [k]
[k]
trivially on Zk (Y ), thus also on I[k] (Y ), and Φ ◦ wε = Φ ν [k] -almost everywhere. [k]
But Φ ◦ wε − Φ is equal to the constants s(ε)χ(w) and we get that χ(w) = 1. As this holds for every w ∈ W, we have χ ∈ W ⊥ and so Γk ⊂ W ⊥ . Combining these two inclusions, we have the statement of Part (2).
Corollary 4.7.7. (cf. [36], Corollary 7.7) Let k ≥ 1 be an integer, (X, X, µ, T ) a system of order k, U a compact abelian group and ρ : Zd × X → U an ergodic cocycle. Then the extension of X associated to ρ is of order k if and only if ρ is of type k. The proof is identical to that in [36]. We reproduce it for completeness. Proof. If Y is of order k then Zk (Y ) = Y, W is the trivial subgroup of U and ρ is of b ; thus W is trivial, and Zk (Y ) = Y. type k. If ρ is of type k, then Γk = U
Corollary 4.7.8. (cf. [36], Corollary 7.8) Assume that (X, X, µ, T ) and (Y, Y, ν, S) are ergodic systems and that X is of order k for some integer k ≥ 1. Assume that π : X → Y is a factor map and ρ : Zd × Y → U is a cocycle. Then ρ is of type k on Y if and only if ρ ◦ π is of type k on X. Proof. If ρ is of type k, it follows immediately from the definition that ρ ◦ π is of type k. Assume that ρ ◦ π is of type k. It suffices to show that χ ◦ π is of type k for every b . Since χ ◦ (ρ ◦ π) is of type k, without loss of generality, we can assume that χ∈U U = T. 117
The set {φ ∈ hom(Zd , T) : φ + ρ is not ergodic} is either empty or is a coset of the countable subgroup {φ ∈ hom(Zd , T) : nφ is an eigenvalue of (X, X, µ, T ) for some n 6= 0}. Therefore, there exists φ ∈ hom(Zd , T) so that φ + ρ is ergodic. Substituting ρ + φ for ρ, we can assume that ρ is ergodic. By Proposition 4.7.6, the extension of X associated to ρ ◦ π is of order k because ρ ◦ π is of type k. Furthermore, the extension of Y associated to ρ is a factor of this extension and so is of order k as well. Therefore ρ is of type k. Corollary 4.7.9. (cf. [36], Corollary 7.9) Let (X, X, µ, T ) be an ergodic system, U a compact abelian group, and ρ : Zd ×X → U a cocycle of type k for some integer k ≥ 1. Then there exists a cocycle ρ0 : Zd × Zk (X) → U of type k so that ρ is cohomologous to ρ0 ◦ πk . The proof is identical to that in [36]. We reproduce it for completeness. Proof. If ρ is ergodic, the result follows immediately from the preceding proposition, since by Part(2), the subgroup W is trivial. Assume that ρ is not ergodic. There exist a closed subgroup V of U and an ergodic cocycle σ : Zd × X → V so that ρ and σ are cohomologous as U -valued cocycles (this V is the Mackey group, as presented in Chapter 2, Section 2.5.4). Also, σ is of type k as a U -valued cocycle, thus also as a V -valued cocycle. There exists a cocycle ρ0 : Zd × Zk (X) → V of type k so that σ is cohomologous to ρ0 ◦ πk as a V -valued cocycle on Zk (X). Thus, as a U -valued cocycle, ρ0 is of type k and ρ0 ◦ πk is cohomologous to ρ.
118
Corollary 4.7.10. (cf. [36], Corollary 7.10) Let k ≥ 2 be an integer, (X, X, µ, T ) be a system of order k and ρ : X → U a cocycle of type k. Assume that X is an extension of Zk−1 by a compact connected abelian group. Then there exists a cocycle ρ0 : Zd × Zk−1 → U of type k so that ρ is cohomologous to ρ0 ◦ πk−1 . The proof is identical to that in [36]. We reprodce it for completeness. Proof. Write X = Zk−1 × V and assume that V is connected. By Corollary 4.7.5, for every v ∈ V the cocycle ρ ◦ v − ρ is a coboundary. By Lemma 4.15.9, there exists a cocycle ρ0 : Zd × Zk−1 so that ρ0 ◦ πk−1 is cohomologous to ρ. By Corollary 4.7.8, ρ0 is of type k.
4.8 4.8.1
Systems of order 2. Systems of order 1
Here we present the obvious modifications of the remarks in Section 8.1 of [36]. A system (X, X, µ, T ) of order 1 is a Kronecker system, that is X = Z for some compact abelian group Z, µ = Haar measure of Z, and T a z = ta11 · · · tadd z for some t1 , . . . , td ∈ Z. Furthermore G(Z) coincides with Z: every S ∈ G(Z) is given by Sz = sz for some s ∈ Z.
4.8.2
The Conze-Lesigne Equation
Throughout this section, (Z, t) denotes an ergodic Kronecker system. Z is a compact abelian group, endowed with Haar measure m = mZ and with the ergodic Zd action T given by T a z = ta11 · · · tadd z, where t1 , . . . , td are fixed elements of Z. 119
Lemma 4.8.1. (cf. [36], Lemma 8.1) Let (Z, t) be an ergodic rotation, U be a torus and ρ : Zd × Z → U a cocycle of type 2. For every s ∈ Z, there exist f : Z → U and φ ∈ hom(Zd , U ) so that ρ(a, sz) − ρ(a, z) = f (ta11 · · · tadd z) − f (z) + φ(a)
(4.22)
for all a ∈ Zd . Proof. For every s ∈ Z, the map z 7→ sz is an automorphism of Z. By Corollary 4.7.5, the cocycle (a, z) 7→ ρ(a, sz) − ρ(a, z) is of type 1. Since U is a torus, the cocycle is a quasi-coboundary by Lemma 4.15.5. This gives equation (4.22).
Lemma 4.8.2. (cf. [36], Lemma 8.2) Let (Z, t) be an ergodic rotation and ρ : Zd ×Z → T a cocycle of type 2 and assume that there exists an integer n 6= 0 so that nρ is a quasi-coboundary. Then ρ is a quasi-coboundary. Proof. Let s, f and φ be as in equation (4.22). Since nρ is a quasi-coboundary, the map (a, z) 7→ n(ρ(a, sz) − ρ(a, z)) is a coboundary. Substituting into equation (4.22), we find n(ρ(a, sz) − ρ(a, z)) − n(f (T a z) − f (z)) = nφ(a), (using multiplicative notation for T so that φ is itself a coboundary). Writing nφ(a) = F ◦ T a · F , we see that nφ is an eigenvalue of (Z, T ). So for all s, f and φ satisfying equation (4.22), φ belongs to the countable subgroup Γ of hom(Zd , T), where Γ = {φ ∈ hom(Zd , T) : nφ is an eigenvalue of (Z, T )}.
120
Define Z0 = {s ∈ Z : the cocycle (a, z) 7→ ρ(a, sz) − ρ(a, z) is a coboundary}. The group Z0 is a Borel subgroup of Z, since the set of coboundaries is a Borel subset of the set of maps from Zd × Z to T. Let (s, f, φ) and (s0 , f 0 , φ0 ) satisfy equation (4.22). If φ = φ0 , the map (a, z) 7→ ρ(a, s0 z) − ρ(a, sz) is a coboundary. Thus so is the map (a, z) 7→ ρ(a, s0 s−1 z) − ρ(a, z) and s0 s−1 ∈ Z0 . As Γ is countable, Z0 has countable index in Z. As Z0 is Borel, Z0 is an open subgroup of Z. One can check, using the cocycle equation, that Z0 contains ti , i = 1, . . . , d. Since the ti generate a dense subgroup, Z0 = Z and the cocycle (a, z) 7→ ρ(a, sz)−ρ(a, z) is a coboundary for every s ∈ Z. This means that the map (a, z0 , z1 ) 7→ ρ(a, z1 ) − ρ(a, z0 ) is a coboundary of the system (Z × Z, m × m, T × T ). By Lemma 4.15.5, ρ is a quasi-coboundary. Lemma 4.8.3. (cf. [36], Lemma 8.3) Let (Z, t) be an ergodic rotation, U a torus and ρ : Zd × Z → U a cocycle of type 2. Then there exist a closed subgroup Z0 of Z so that Z/Z0 is a compact abelian Lie group and a cocycle ρ : Zd × Z/Z0 → U of type 2 so that ρ is cohomologous to ρ0 ◦ π, where π : Z → Z/Z0 is the natural projection. The proof is essentially identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. By Equation (4.22), for every s ∈ Z the cocycle (a, z) 7→ ρ(z, sz) − ρ(az) is a quasi-coboundary. Applying Lemma 4.15.10 with the action of Z on itself by translations and Corollary 4.7.8, we get the result.
121
4.8.3
Systems of order 2.
Corollary 4.8.4. (cf. [36], Corollary 8.4) For every ergodic system (X, X, µ, T ), Z2 (X) is an extension of Z1 (X) by a compact connected abelian group. The proof is essentially identical to the proof of Corollary 8.4 in [36]. We reproduce it for completeness. Proof. By Proposition 4.6.3, Z2 is an extension of Z1 by a compact abelian group U given by an ergodic cocycle σ : Zd × Z1 → U of type 2. Assume that U is not connected. Then it admits an open subgroup U0 so that U/U0 is isomorphic to Z/nZ for some integer n > 1. Write σ : Zd × Z1 → U/U0 for the reduction of σ modulo U0 , meaning the composition of σ with the quotient map U 7→ U/U0 . It is an ergodic cocycle of type 2. Using the isomorphism from U/U0 to Z/nZ and an embedding of Z/nZ as a finite closed subgroup of T, we get a (nonergodic) cocycle ρ : Zd × Z1 → T of type 2 with nρ = 0. By Lemma 4.8.2, ρ is a quasi-coboundary and thus of type 1. Viewed as a cocycle with values in Z/mZ, ρ is also of type 1 (even if it is not a quasi-coboundary) and σ is of type 1. By Corollary 4.7.7 the extension Tσ associated to σ is a Zd -system of order 1, meaning it is an ergodic rotation. But this extension is obviously a factor of Z2 , which is the extension of Z1 associated to σ and thus also a factor of X. The maximal property (Proposition 4.4.9) of Z1 provides a contradiction.
Definition 4.8.5. (cf. [36], Definition 8.5) A system X of order 2 is toral if its Kronecker factor Z1 is a compact abelian Lie group and X is an extension of Z1 by a torus. 122
Proposition 4.8.6. (cf. [36], Proposition 8.6) Every Zd -system of order 2 is the inverse limit of a sequence of toral systems of order 2. The proof is essentially the proof of Proposition 8.6 in [36].
4.8.4
The group of a Zd -system of order 2
In this section, let (X, X, µ, T ) be a Zd -system of order 2, and assume that X is an extension of its Kronecker factor (Z1 , t) by a torus U, and write ρ : Zd × Z1 → U for the cocycle defining this extension. A map f : Z1 → U is said to be affine if it is the sum of a constant and a continuous group homomorphism from Z1 to U and we write A(Z1 , U ) for the group of affine maps. It is a closed subgroup of the group of maps from Z1 to U, and is the direct sum of the compact group U of constants and the discrete group of continuous group homomorphisms from Z1 to U. For each s ∈ Z1 and f : Z1 → U, let Ss,f denote the measure preserving transformation of Z1 × U given by Ss,f (z, u) = (sz, u + f (z)).
(4.23)
The group of such transformations, under the topology of convergence in probability, is a Polish group. Lemma 4.8.7. (cf. [36], Lemma 8.7) The group G consists of the transformations of X of the type given by equation (4.23), for s ∈ Z1 and f : Z1 → U satisfying equation (4.22) for some φ ∈ hom(Zd , U ). 123
Proof. Let g ∈ G. By Lemma 4.5.2, g induces a measure preserving transformation of Z1 belonging to G(Z1 ) and thus of the form z 7→ sz for some s ∈ Z1 . Moreover, by Corollary 4.5.10, the transformation g commutes with all vertical rotations of X over Z1 and thus is of the form given by Equation (4.23) for some map f : Z1 → U. For each a ∈ Zd , the commutator [g; T a ] induces the trivial transformation of Z1 : [g; T a ](z, u) = (z, u + f (T a z) − f (z) + ρ(a, z) − ρ(a, sz)).
(4.24)
As G is 2-step nilpotent, [g; T a ] belongs to the center of G for each a and thus commutes with T b for all b. It follows that [g; T a ] is a vertical rotation of X over Z1 , given by some φ(a) ∈ U. Since [g; T a ] is in the center of G, we have [g; T a ][g; T b ] = [g; T a+b ] for a, b ∈ Zd . Hence φ(a + b) = φ(a) + φ(b). We see from equation (4.24) that s, f, and φ satisfy equation (4.22). Conversely, let s ∈ Z1 and f : Z1 → U be such that Equation (4.22) is satisfied for some φ ∈ hom(Zd , U ). Let us show that the transformation Ss,f belongs to G. Let α be an edge of V2 . The transformation s given by z 7→ sz of Z1 induced on Z1 by g [2]
[2]
belongs to G(Z1 ) and thus the transformation sα leaves the measure µ1 invariant [2]
and maps the σ-algebra I(Z1 )[2] to itself. We define a map F : Z1 → U and a map Φ : X [2] → U as in Proposition 4.6.4. An immediate computation shows that the [2]
[2]
map Φ ◦ gα − Φ is invariant under T [2] and so Φ ◦ gα is also invariant under the [2]
action T [2] . By Proposition 4.6.4, the transformation gα maps the σ-algebra I(X)[2] to itself. By Lemma 4.5.3 and Corollary 4.6.6, g ∈ G.
The map p given by Ss,f 7→ s is a continuous group homomorphism from G to Z1 and is onto by Lemma 8.1. The kernel of this homomorphism is the group of transformations 124
of the kind S1,f , where f (T a z) − f (z) = φ(a) for some φ ∈ hom(Zd , U ). By ergodicity of the rotation (Z1 , t1 , . . . , td ), a map f : Z1 → U satisfies this condition if and only if it is affine. The map f 7→ S1,f is then an algebraic and topological embedding of A(Z1 , U ) in G with range ker(p). We identify A(Z1 , U ) with ker(p). This identification generalizes the preceding identification of U with the group of vertical rotations. By Corollary A.2 of [36], G is locally compact. Lemma 4.8.8. (cf. [36], Lemma 8.8) Every toral Zd -system of order 2 is isomorphic to a nilsystem. The proof of Lemma 4.8.8 in [36], with obvious modifications, suffices. Proof. Assume that Z1 is a compact abelian Lie group. The kernel A(Z1 , U ) of p is the direct sum of the torus U and a discrete group and thus it is a Lie group also. By Lemma A.3 of [36], G is a Lie group. We recall that G is 2-step nilpotent. Let Γ be the stabilizer of (1, 0) ∈ X for the action of G on this space. Then Γ consists of the transformations associated to (1, f ), where f is a continuous group homomorphism from Z1 to U. Thus Γ is discrete. The map g 7→ g · (1, 0) induces a bijection j from the nilmanifold G/Γ onto X. For any g ∈ G, the transformation j −1 ◦ g ◦ j of G/Γ is the (left) translation by g on the nilmanifold G/Γ. In particular j −1 ◦ T a ◦ j is the (left) translation x 7→ T a · x by T a ∈ G. Moreover, since every g ∈ G is a measure preserving transformation of X, the image of µ under j −1 is invariant under the (left) action of G on G/Γ and this is the Haar measure on this space. The map j is the announced isomorphism.
125
Proposition 4.8.9. (cf. [36], Proposition 8.9) Let (Z, t1 , . . . , tk ) be an ergodic Kronecker system. Up to the addition of a quasi-coboundary, there are only countably many T-valued cocycles of type 2 on Z. We follow the proof in [36], with a minor change. The proof in [36] uses the fact that for d = 1, ergodicity implies that Z is monothetic. We will use Lemma 4.8.3 to reduce to the case where Z is a Lie group, which means that Z has a monothetic group of finite index - namely, the connected component of the identity. We can then use the cocycle identity and results from Section 4.7 to establish the conclusion in the case where Z has a monothetic group of finite index. Proof. We first consider the case that Z is monothetic, then we consider the case that Z has a monothetic subgroup of finite index. At the end of the proof, we will use Lemma 4.8.3 to reduce to this case. As in [36], we make use of explicit distances on some groups of functions. We reproduce those definitions. Let B(Z) denote the space of functions f : Z → T. For u ∈ T, write kuk = | exp(2πiu) − 1|. For f : Z → T, write kf k =
Z
1/2 kf (z)k2 dm(z) ,
The distance between two functions f, g : Z → T is defined to by kf − gk. Let A(Z) now denote the group of affine maps f. For c, c0 ∈ T and γ, γ 0 ∈ Zb we have √ k(c + γ) − (c0 + γ 0 )k ≥ 2 whenever γ 6= γ 0 .
126
Let Q(Z) denote the quotient group Q(Z) = B(Z)/A(Z) and write q : B(Z) → Q(Z) for the quotient map. The quotient distance5 between Φ ∈ Q and 0 ∈ Q is written kΦkQ and the quotient distance between two elements Φ, Ψ of this group is kΦ − ΨkQ . Endowed with this distance, Q(Z) is a Polish group. We also use the group F of continuous maps from Z to Q, endowed with the distance of uniform convergence: If s 7→ Φ(s) is an element of F, write kΦk∞ = sup kΦ(s)kQ . s∈Z
The distance between two elements Φ and Ψ ∈ F is kΦ − Ψk∞ . As Z is compact and Q is a Polish group, F is also a Polish group. We say that a cocycle ρ : Zd × Z → T is weakly mixing if the Kronecker factor of Z ×ρ T is Z. That is, ρ is weakly mixing if every eigenfunction of the system Z ×ρ T b is of the form (z, s) 7→ χ(z), where χ ∈ Z. First Step. Let ρ : Zd × Z → T be a weakly mixing cocycle of type 2. Let X be the extension of Z associated to this cocycle. Then X is of order 2 and Z1 (X) = Z. Let s 7→ Ss,fs be an arbitrary cross section of the map p : G → Z. For every s ∈ Z, fs belongs to B(Z) and satisfies Equation (4.22) for some φ ∈ hom(Zd , T). Define Φρ (s) ∈ Q(Z) to be the image of fs under q. Since the kernel of p : G → Z is A(Z), Φρ (s) does not depend on the choice of fs . In fact, the map s 7→ Φρ (s) from Z to Q(Z) is the reciprocal of the isomorphism G/ ker(p) → Z and thus it is continuous. In other words, the map s 7→ Φρ (s) is an element of F. 5
That is, the number inf g∈A(Z) kΦ − gk.
127
Second Step. We continue assuming that ρ is a weakly mixing cocycle of type 2. Φρ is defined as above. Lemma 4.8.10. (cf. [36], Lemma 8.10) If kΦρ k∞ < 1/20, then ρ is cohomologous to an affine cocycle (that is, a cocycle such that ρ(a, ·) : Z → T is affine for each a ∈ Zd .) Proof. Define a subset K of G by K = {Ss,f ∈ G : There exists c ∈ T with kc + f k ≤ 1/10}. Let s ∈ Z. By hypothesis kΦρ (s)kQ < 1/20 and there exists f : Z → T with Ss,f ∈ G and kf k < 1/20, thus Ss,f ∈ K. The restriction p|K of p : G → Z to K is therefore onto. Claim. K is a subgroup of G. Proof of Claim. Let Ss,f and Ss0 ,f 0 ∈ K. We have Ss0 ,f 0 ◦ Ss,f = Ss0 s,f 00 where f 00 (z) = f (s0 z) + f 0 (z). Choose c, c0 ∈ T with kc + f k ≤ 1/10 and kc0 + f k ≤ 1/10. Then kf 00 + c + c0 k ≤ kf + ck + kf 0 + c0 k ≤ 1/5. On the other hand, there exists an element of K with projection on Z equal to ss0 . This means that there exists g : Z → T with kgk < 1/20 and Ss0 s,g ∈ G. We get that S1,f 00 −g ∈ G and thus f 00 − g ∈ A(Z) and f 00 − g + c + c0 ∈ A(Z). But kf 00 − g + c + c0 k ≤ kf 00 + c + c0 k + kgk ≤
1 4
and so
f 00 − g + c + c0 is equal to a constant d ∈ T. Finally, kf 00 + c + c0 − dk = kgk < 1/20 and Ss0 s,f 00 ∈ K. Clearly, the identity transformation S1,0 belongs to K and the inverse of an element of K belongs to K. The claim is proved.
128
K clearly contains the group T of vertical rotations. If f is an affine map and kc+f k ≤ 1/10 for some constant c, then f is constant. It follows that the kernel of the group homomorphism p|K : K → Z is the group T of vertical rotations. Moreover, K is clearly closed in G and is locally compact. Since the kernel T and the range Z of p|K are compact, K is a compact group. Claim. K is abelian. Proof of claim. We consider the commutator map (g, h) 7→ [g; h]. It is continuous and bilinear because K is 2-step nilpotent. But the commutator group K0 is included in T because K0 is the kernel of the group homomorphism pK ranging in the compact abelian group Z. Thus the commutator map has range in T. Moreover, T is included in the center of K. (This can be seen either by applying Proposition 4.6.3 or by checking directly.) Thus the commutator map is trivial on T × K and K × T. Therefore, it induces a continuous bilinear map from K/T × K/T → T and finally a continuous bilinear map b : Z × Z → T. Let t be such that {tn : n ∈ Z} is dense in Z. Choose m n f : Z → T with St,f ∈ K. For all integers m, n, the transformations St,f and St,f
commute and by definition of b, b(tm , tn ) = 0. Since {tn : n ∈ Z} is dense in Z, the bilinear map b is trivial. Thus the commutator map K × K → K0 is trivial and the second claim is proved. The compact abelian group K admits T as a closed subgroup, with quotient Z. Thus it is isomorphic to T ⊕ Z. This means that the group homomorphism p|K : K → Z admits a cross section Z → K, which is a group homomorphism and is continuous.
129
This cross section has the form s 7→ Ss,fs and the map s 7→ fs is continuous from Z to B(Z) satisfies for all s, s0 ∈ Z fss0 (z) = fs0 (sz) + fs (z) for almost every z ∈ Z. By Lemma 4.15.8, there exists f ∈ B(Z) so that fs (z) = f (sz) − f (z) for every s ∈ Z. Define ρ0 (a, z) = ρ(a, z) − f (ta11 · · · tadd z) + f (z). The cocycle ρ0 is cohomologous to ρ. Moreover, for every s we have Ss,fs ∈ K ⊆ G and this means that s and fs satisfy Equation (4.22) for some φ ∈ hom(Zd , T). Substituting in the definition of ρ0 we have ρ0 (a, sz) − ρ(a, z) = φ(a). As this holds for every s ∈ Z, ρ0 is an affine cocycle. This completes the proof of Lemma 4.8.10.
End of the proof of Proposition 4.8.9. Let W be the family of weakly mixing cocycles of type 2 on Z. To every cocycle ρ ∈ W, we have associated an element Φρ of F. Since F is separable, there exists a countable family {ρi : i ∈ I} in W so that for every ρ ∈ W, there exists i ∈ I with kΦρ − Φρi k < 1/20. Let ρ : Zd × Z → T be a cocycle of type 2. Assume first that ρ is not weakly mixing. We claim that there exists an integer n 6= 0 so that nρ is a quasi-coboundary. If ρ is not weakly mixing, then the Kronecker factor of Z ×ρ T has the form Z ×ρ (T/H), where H is a closed proper subgroup of T. Thus H is finite, so there is an n > 0 so that nH = {0}. By Corollary 4.7.7 The cocycle ρ : Z × Z → (T/H) given by ρ(z) = ρ(a, z) mod n1 Z is of type 1, so ρ is a quasi-coboundary: write ρ(a, z) = F (ta11 · · · tadd z) − F (z) + φ(a) for some F : Z → T/H, φ ∈ hom(Zd , T). Then nρ(a, z) = n(F (ta11 · · · tadd z) − F (z) + φ(a)), so nρ is a quasi-coboundary. By Lemma 4.8.2, ρ is itself a quasi-coboundary. 130
Assume now that ρ is weakly mixing. Choose i ∈ I so that kΦρ − Φρi k < 1/20. If ρ − ρi is not weakly mixing, by the same argument as above this cocycle is a quasicoboundary and ρ is the sum of ρi and a quasi-coboundary. If ρ − ρi is weakly mixing, then Φρ−ρi = Φρ − Φρi . Thus kΦρ−ρi k < 1/20 and by Lemma 8.10 the cocycle ρ − ρi is cohomologous to some affine cocycle. In this case, ρ is the sum of ρi , an affine cocycle, and a quasi-coboundary. This completes the proof of Proposition 4.8.9 in the case where Z is monothetic. Now we consider the case where Z is a Lie group. Since Z is a Lie group, the connected component of the identity, Z0 , is isomorphic to Tm for some finite m ≥ 0, and therefore monothetic. Let Γ ⊂ Zd be the subgroup {(a1 , . . . , ad ) : ta11 · · · tadd ∈ Z0 }. Given a cocycle ρ : Zd × Z → T we consider the extension Z ×ρ T, and the system Y = (Z ×T, µ×mT , Γ), where (a1 , . . . , ad ) ∈ Γ acts on Z ×T by T (a1 ,...,ad ) . The ergodic components of Y are lifts of cosets of Z0 , so by the case where Z is monothetic, we see that there is a countable collection of cocycles σi : Γ × Z → T such that for every cocycle τ : Γ × Z → T, there exists F : Z → T, φ ∈ hom(Zd , T) and i with τ (a, z) = σi (a, z) + F (ta11 · · · tadd z) − F (z) + φ(a) for a ∈ Γ (here we have used the fact that every homomorphism from a subgroup of Zd to T can be extended to a homomorphism defined on all of Zd ). In particular, we have ρ(a, z) = σi (a, z) + F (t1a1 · · · tadd z) − F (z) + φ(a) for some F, i, and φ. Note that the domain of σi can be extended to all of Zd × Z by defining σi (a, z) := ρ(a, z) − F (ta11 · · · tadd z) + F (z) − φ(a) for a ∈ Zd , z ∈ Z. Let ρ0 (a, z) = ρ(a, z) − σi (a, z) − F (ta11 · · · tadd z) + F (z) − φ(a) for (a, z) ∈ Zd × Z. Then 131
ρ0 (a, z) = 0 for a ∈ Γ. It then follows from the cocycle identity that ρ0 (a, tb1 · · · tbd z) = ρ0 (a, z) for a ∈ Zd , (b1 , · · · , bd ) ∈ Γ. Thus ρ0 is constant on cosets of Z0 . We now consider ρ0 as a function on Zd × (Z/Z0 ) and examine the extension W of Z/Z0 by ρ0 . Consider an ergodic component of W. This is an ergodic extension of a finite system by a compact abelian group, and is therefore a Kronecker system. It follows from Corollary 4.7.7 that the cocycle ρ0 : Zd × (Z/Z0 ) → T is type 1, and therefore a quasi-coboundary, by Lemma 4.15.5. Then ρ(a, z) = ρ0 (a, z)+σi (a, z)+F (ta11 · · · tadd z)−F (z)−φ(a). This proves Proposition 4.8.9 in the case where Z is a compact abelian Lie group. To prove the proposition in the general case, we apply Lemma 8.3 to see that every cocycle ρ of type 2 is cohomologous to a cocycle of the form ρ0 ◦ π, where π : Z → W is a quotient map, and W is a Lie group which is a quotient of Z. Since the quotients b there of Z which are Lie groups correspond to the finitely generated subgroups of Z, are only countably many such quotients. Hence, up to quasi-coboundary, there are only countably many cocycles of the form ρ0 ◦ π for such π.
4.9 4.9.1
The main induction. The systems Xs .
In this section, we use the following notation. Let (X, X, µ, T ) be an ergodic Zd system. For every integer k ≥ 2, Zk = Zk (X) is an extension of Zk−1 by a compact abelian group Uk , given by a cocycle ρk : Zk−1 → Uk of type k.
132
The ergodic decomposition of µ × µ is given by Z µ×µ=
µs dµ1 (s),
(4.25)
Z1
where µs is the measure on X ×X given by
R
f ⊗g dµs =
R
E(f |Z1 )(sz)E(g|Z1 )(z) dz.
Notation. For every s ∈ Z1 , let Zs denote the system (X × X, µs , T × T ). We recall that Xs is ergodic for µ1 -almost every s ∈ Z1 . Lemma 4.9.1. (cf. [36], Lemma 9.1) Let (X, X, µ, T ) be an ergodic system, U a compact abelian group, ρ : Zd × X → U a cocycle and k ≥ 0 an integer. Then the set A = {s ∈ Z1 : ∆ρ is a cocycle of type k of Xs } is measurable and µ1 (A) = 0 or 1. Furthermore, the cocycle ρ is of type k + 1 if and only if µ1 (A) = 1. The proof is similar to the proof in [36]. Proof. We recall that ∆ρ is defined on Zd ×X ×X by ∆ρ(a, x0 , x00 ) = ρ(a, x0 )−ρ(a, x00 ). Under the identification of X [k+1] with (X × X)[k] , we can write ∆k+1 ρ = ∆k (∆ρ). R Moreover, by Equation (4.7) we have (µs )[k] dµ1 (s) = µ[k+1] . By the definition of a cocycle of type k + 1 on Zd × X, the definition of a cocycle of type k on Zd × Xs , and Corollary 4.15.4, we get immediately that A is a measurable subset of Z1 and that ρ is of type k + 1 if and only if µ1 (A) = 1. It only remains to show that µ1 (A) = 0 or 1. Let s ∈ Z1 , a ∈ Zd with T a s ∈ A. The map id ×T a is an isomorphism of Xs onto XT a s . Thus ∆ρ◦(id ×T a ) is a cocycle of type k on Xs . By Corollary 4.7.5, ∆ρ◦(id ×T a )−∆ρ is of type k − 1. It follows that ρ is the sum of a cocycle of type k and a cocycle of 133
type k − 1 and therefore is of type k on Xs . Therefore s ∈ A. This shows that A is T -invariant, and therefore has measure 0 or 1.
Let p : (X, X, µ, T ) → (Y, Y, ν, S) be a factor map which induces a factor map p1 from the Kronecker factor Z1 (X) of X to the Kronecker factor Z1 (Y ) of Y. We write νs instead of νp1 (s) and Ys instead of Yp1 (s) . By the ergodic decomposition, for µ1 -almost every s ∈ Z1 , the measure νs is the image of µs under p × p. In other words, p × p is a factor map from Xs to Ys . Lemma 4.9.2. (cf. [36], Lemma 9.2) Let (X, X, µ, T ) be an inverse limit of a sequence {Xn }n of ergodic systems. Then for µ1 -almost every s ∈ Z1 , Xs = lim Xn,s , where ←− Xn,s is the system associated to Xn in the same way that Xs is associated to X. The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. There exists a countable family {fi : i ∈ I} of bounded functions defined everywhere on X, dense in L2 (µ) and so that the linear span of the family {fi ⊗ fj : i, j ∈ I} is dense in L2 (µ) for every probability measure ν on X × X. For every i and every n, we consider E(fi |Xn ) as a function defined everywhere on X. For every i ∈ I, E(fi |Xn ) converges to fi µ-almost everywhere. There exists a subset X0 of X, with µ(X0 ) = 1, so that E(fi |Xn )(x) → fi (x) for all i ∈ I and all x ∈ X0 . For µ1 -almost every s ∈ Z1 , we have µs (X0 × X0 ) = 1. Fix such an s, and consider X × X as endowed with µs . For every i, j ∈ I, E(fi |Xn ) ⊗ E(fj |Xn ) converges to fi ⊗ fj on X0 × X0 , thus µs -almost everywhere. For every n, E(fi |Xn ) ⊗ E(fj |Xn ) is measurable with respect to Xn ⊗ Xn and it follows that 134
fi ⊗ fj is measurable with respect to the inverse limit lim Xn,s of the factors Xn,s of ←− Xs . By density, every function in L2 (µs ) is measurable with respect to lim Xn,s . ←−
4.9.2
The factors Zk (Xs ).
We compute the factors Zk (Xs ) of Xs . For every k and µ1 -almost every s ∈ Z1 , we associate to the system (Zk , Zk , µk , T ) a measure µk,s on Zk × Zk in the same way that µs is associated to (X, X, µ, T ). Let Zk,s denote the system (Zk × Zk , Zk ⊗ Zk , µk,s , T × T ). The measure µs is a relatively independent joining of µ over the joining µ1,s of µ1 . Thus, for every k, µk,s is a relatively independent joining of µk over µ1,s and thus over the joining µk−1,s of µk−1 . Therefore, the system (Zk,s , Zk,s , µk,s , T × T ) is an extension of (Zk−1,s , Zk−1,s , µk−1,s , T × T ) by the group Uk × Uk , given by the cocycle ρk × ρk : (a, x0 , x00 ) 7→ (ρk (a, x0 ), ρk (a, x00 )). Lemma 4.9.3. (cf. [36], Lemma 9.3) Let k ≥ 1 be an integer. Then: (1) For µ1 -almost every s ∈ Z1 , ρk × ρk is a cocycle of type k on Zk−1,s . (2) For µ1 -almost every s ∈ Z1 , Zk,s is a Zd -system of order k. In particular, if X is of order k then Xs is of order k for µ1 -almost every s ∈ Z1 . The proof from [36], with obvious modifications, suffices. We present it for completeness. [k]
[k]
[k+1]
Proof. (1) We identify Zk−1 × Zk−1 and with Zk−1 . We recall that there exists [k]
[k]
Fk : Zk−1 → Uk with ∆k ρk (a, x) = Fk ((T [k] )a x) − Fk (x), µk−1 -almost everywhere. 135
[k]
[k]
Define G : Zk−1 × Zk−1 → Uk × Uk by F (x0 , x00 ) = (Fk (x0 ), Fk (x00 )). As each of the [k+1]
[k]
[k]
two projections of µk−1 on Zk−1 is equal to µk−1 , the equality ∆k (ρk × ρk )(a, x0 , x00 ) = G ◦ (T [k+1] )a (x0 , x00 ) − G(x0 , x00 ) [k+1]
[k+1]
follows for µk−1 -almost every (x0 , x00 ). As µk−1 =
R Z1
(µk−1,s )[k] dµ1 (s), for µ1 -almost
every s, the same relation holds (µk−1,s )[k] -almost everywhere and ρk × ρk is a cocycle of type k of Zk−1,s . (2) This follows by induction on k, applying Corollary 4.7.7 at each step.
Proposition 4.9.4. (cf. [36], Proposition 9.4) For every integer k ≥ 1 and µ1 -almost every s ∈ Z1 , Zk (Xs ) is a factor of Zk+1,s ; it is an extension of Zk,s by Uk+1 , given by the cocycle ∆ρk+1 : (a, x0 , x00 ) 7→ ρk+1 (a, x0 ) − ρk+1 (a, x00 ), when viewed as a cocycle on Zk,s . Furthermore, Zk+1,s is an extension of Zk (Xs ) by Uk+1 , given by the cocycle (x0 , x00 ) 7→ ρk+1 (x00 ). The proof in [36] suffices. We reproduce it for completeness. Proof. By Proposition 4.4.7, the invariant σ-algebra I[k+1] (X) of the system R [k] [k+1] (X [k+1] , µ[k+1] , T [k+1] ) is measurable with respect to Zk+1 . As µ[k+1] = µs dµ1 (s), [k]
by classical arguments for µ1 -almost every s ∈ Z1 , the invariant σ-algebra of Xs = [k]
(X [k+1] , µs , T [k+1] ) is measurable with respect to the same σ-algebra, that is, with respect to (Zk+1 × Zk+1 )[k] . By the minimality property of the factor Zk (Xs ) (Proposition 4.4.7 again), the σ-algebra Zk (Xs ) is measurable with respect to Zk+1 × Zk+1 . In other words, Zk (Xs ) is a factor of Zk+1,s . [ Let χ0 , χ00 ∈ U k+1 and consider here these characters as taking values in T. Write [ [ χ = (χ0 , χ00 ) ∈ U k+1 × Uk+1 , which we identify with the dual group of Uk+1 × Uk+1 . 136
Let σ : Zd × Zk × Zk → Uk+1 be the map given by σ(a, x0 , x00 ) = χ0 (ρk+1 (a, x0 )) + χ00 (ρk+1 (a, x00 )). Define A = {s ∈ Z1 : σ is a cocycle of type k of Zk,s }. By the same method as in the proof of Lemma 4.9.1, we get that A is invariant under T and µ1 (A) = 0 or 1. Let us assume that µ1 (A) = 1. For µ1 -almost every s ∈ Z1 , ∆k σ is a coboundary of [k+1]
the system (Zk
[k+1]
, µk
[k+1]
, T [k+1] ) and there exists a map F : Zk X
F ((T [k+1] )a x) − F (x) =
→ Uk+1 , with
s(ε)χε (ρk+1 (a, xε ))
ε∈Vk+1
where χε =
χ0 if ε1 = 0
−χ00 if ε1 = 1. [k+1]
[k]
[k]
[k+1]
The function Φ, defined on Zk+1 = Zk = Zk × Uk+1 by X
Φ(x, u) = F (x) −
s(ε)χε (uε ),
ε∈Vk+1 [k+1]
is invariant under T [k+1] . By Proposition 4.6.3, it is invariant under uε
for every
edge α = (ε, η) of Vk+1 and every u ∈ Uk+1 . This means that s(ε)χε (u)+s(η)χη (u) = 1 and thus χε (u) = χη (u). As this holds for every u ∈ Uk+1 , χε = χη , which holds for every edge α and so χ00 = −χ0 . In summary, if χ00 6= −χ0 , then µ1 (A) 6= 1, and so µ1 (A) = 0. Then for µ1 -almost every s, the cocycle σ of Zk,s is not of type k. If χ00 = −χ0 , then σ = χ0 ◦ ∆ρk+1 , which is a cocycle of type k on Zk,s for µ1 -almost every s ∈ Z1 by Lemma 4.9.1. 137
We recall that Zk+1,s is the extension of Zk,s associated to the cocycle ρk+1 × ρk+1 with values in Uk+1 × Uk+1 and apply Proposition 4.7.6. The annihilator of the group 0 0 0 [ W appearing in this proposition is {(χ0 , −χ00 ) : χ ∈ U k+1 }. Thus W = {(u , u ) : y ∈
Uk+1 }. The map (u0 , u00 ) 7→ (u0 − u00 , u00 ) is an isomorphism of Uk+1 × Uk+1 on itself. It maps W to {0} × Uk+1 and we can identify (Uk+1 × Uk+1 )/W with Uk+1 . Under this identification, the cocycle ρk+1 × ρk+1 mod W is simply ∆ρk+1 . We get that Zk (Xs ) is the extension of Zk,s associated to the cocycle ∆ρk+1 . Using the identification of the subgroup W with Uk+1 explained above, we have the last statement of the proposition.
4.9.3
Connectivity.
Theorem 4.9.5. (cf. [36], Theorem 9.5) Let k ≥ 1 be an integer. (1) Let (X, X, µ, T ) be a Zd -system of order k, ρ : Zd × X → T a cocycle of type k + 1 and n 6= 0 an integer. If nρ is of type k, then ρ itself is of type k. (2) For every ergodic system, (X, X, µ, T ), Zk+1 (X) is an extension of Zk (X) by a compact connected abelian group. Proof. For k = 1, these results are Lemma 4.8.2 and Corollary 4.8.4. Let k > 1 and assume that the two properties hold for k − 1. Let X, ρ and n be as in the first statement of the theorem. Note that X is an extension of Zk−1 = Zk−1 (X) by a compact abelian group U, which is connected by the inductive hypothesis. As usual, for u ∈ U we also use u to denote the corresponding vertical rotation of X over Zk−1 . 138
Since nρ is of type k, by Corollary 4.7.10 there exists a cocycle σ : Zk−1 → T and a map f : X → T so that nρ(a, x) = σ ◦ πk−1 (x) + f ◦ T a (x) − f (x), a ∈ Zd . Let u ∈ U. By part (3) of Corollary 4.7.5, the cocycle ρ ◦ u − ρ is a quasi-coboundary and so there exist φ : X → T and ψ ∈ hom(Zd , T) with ρ ◦ u(a, x) − ρ(a, x) = φ ◦ T a (x) − φ(x) + ψ(a), a ∈ Zd . Plugging in to the preceding equation, we get that nψ is a coboundary of X. That is, nφ is an eigenvalue (X, X, µ, T ) and ψ belongs to the countable subgroup Γ = {ψ ∈ hom(Zd , T) : nψ is an eigenvalue of (X, X, µ, T )} of hom(Zd , T). For every ψ ∈ Γ, define Uψ = {u ∈ U : ρ ◦ u − ρ − ψ is a coboundary of X}. Each of these sets is a Borel subset of U and their union is U. Thus there exists ψ ∈ Γ such that mU (Uψ ) > 0, where mU is the Haar measure of U. But U0 is clearly a subgroup of U and Uψ is a coset of this subgroup. It follows that mU (U0 ) > 0 and that U0 is an open subgroup of U. Since U is connected, U0 = U. Thus, for every u ∈ U the cocycle ρ ◦ u − ρ is a coboundary. By Lemma 4.15.9, there exists τ : Zk−1 → T and g : X → T with ρ(a, x) = τ ◦ πk−1 + g(T a x) − g(x) for all a ∈ Zd , x ∈ X By considering X as a system of order k + 1, we see that τ is a cocycle of type k + 1 on Zk−1 by Corollary 4.7.8 and nτ is a cocycle of type k. 139
By Lemma 4.9.3, Zk−1,s is a system of order k − 1 for almost every s ∈ Z1 . By Lemma 4.9.1, for almost every s, the cocycle ∆τ of the system Zk−1,s is of type k and the cocycle n∆τ of this system is of type k − 1. By the inductive assumption, ∆τ is a cocycle of type k − 1 of this system. Using Lemma 4.9.1 again, τ is a cocycle of type k on the system Zk−1 and by Corollary 7.8 ρ is a cocycle of type k on X. The second assertion for k is deduced exactly as in the proof of Corollary 4.8.4.
4.9.4
Countability.
Notation. We let Ck (X) denote the group of functions f : Zd × X → T which are cocycles of type k. Theorem 4.9.6. (cf. [36], Theorem 9.6) Let k ≥ 2 be an integer, (X, X, µ, T ) be an ergodic system. Let (Ω, P ) be a (standard) probabilty space and ω 7→ ρω a measurable map from Ω → Ck (X). Then there exists a subset Ω0 of Ω, with P (Ω0 ) > 0, so that ρω − ρω0 ∈ C1 (X) for every (ω, ω 0 ) ∈ Ω0 × Ω0 . The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. We proceed by induction on k. By Corollary 4.7.9, Theorem 4.9.5 and Corollary 4.7.10, for every cocycle ρ of type 2 on X there exists a cocycle ρ0 of type 2 on Z1 so that ρ is cohomologous to ρ0 ◦ π1 . By Proposition 4.8.9, C1 (Z1 ) has countable index in C2 (Z1 ) and so C1 (X) has countable index in C2 (X). The statement of the theorem follows immediately for k = 2. Fix an integer k ≥ 2 and assume that the theorem holds for k. Let (X, X, µ, T ), (Ω, P ) 140
be as in the statement of the theorem and let ω 7→ ρω be a measurable map from Ω to Ck+1 (X). We use the usual ergodic decomposition (formula (4.6)) of µ×µ for T ×T and formula (4.7)) for µ[k+1] . The map ω 7→ ∆ρω from Ω to Ck (X × X) is measurable. By Lemma 4.15.3 the subset A = {(ω, s) ∈ Ω × Z1 : ∆ρω is a cocycle of type k on Xs } of Ω × Z1 is measurable. In the same way, the subset B = {(ω, ω 0 , s) ∈ Ω × Ω × Z1 : ∆ρω − ∆ρω0 ∈ C1 (Xs )} of Ω × Ω × Z1 is measurable. By Lemma 4.9.1, for all ω, ω 0 ∈ Ω, the subset Bω,ω0 = {s ∈ Z1 : (ω, ω 0 , s) ∈ B} of Z1 has measure 0 or 1. Moreover, for every ω ∈ Ω the cocycle ρω is of type k + 1 by hypothesis and so by Lemma 4.9.1 the cocycle ∆ρ is of type k on Xs for µ1 -almost every s ∈ Z1 . Thus (P × µ1 )(A) = 1. Therefore, for µ1 -almost every s ∈ Z1 , using the inductive hypothesis applied to the system Xs and the map ω 7→ ∆ρω , we get (P × P ){(ω, ω 0 ) ∈ Ω × Ω : (ω, ω 0 , s) ∈ B} > 0. Therefore (P × P × µ1 )(B) > 0 and the subset C = {(ω, ω 0 ) ∈ Ω × Ω : µ1 (Bω,ω0 ) > 0} = {(ω, ω 0 ) ∈ Ω × Ω : µ1 (Bω,ω0 ) = 1} of Ω × Ω has positive measure under P × P. By Lemma 4.9.1 again, for (ω, ω 0 ) ∈ C, the cocycle ρω − ρω0 is a cocycle of type 2 on X. By the base step of the induction, C1 (X) has countable index in C2 (X), and so there exists ρ ∈ C2 (X) such that the set D = {(ω, ω 0 ) ∈ C : ρω − ρω0 − ρ ∈ C1 (X)} 141
satisfies (P × P )(D) > 0. Choose ω0 ∈ Ω so that the set Ω0 = {ω ∈ Ω : (ω0 , ω) ∈ D} has positive measure. Then for ω, ω 0 ∈ Ω0 , ρω − ρω0 ∈ C1 (X).
Corollary 4.9.7. (cf. [36], Corollary 9.7) Let (X, X, µ, T ) be an ergodic system and {Su : u ∈ U } a free action of a compact abelian group U on X by automorphisms. Let ρ : Zd ×X → T be a cocycle of type k for some integer k ≥ 2. Then there exist a closed subgroup U1 of U such that U/U1 is a toral group, and a cocycle ρ0 cohomologous to ρ with ρ0 ◦ Su = ρ0 for every u ∈ U1 . The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. Define U0 = {u ∈ U : ρ ◦ Su − ρ is a quasi-coboundary}. Clearly, U0 , is a measurable subgroup of U. The map u 7→ ρ ◦ Su − ρ is a measurable map from U to Ck (X) (and even to Ck−1 (X) by Corollary 4.7.5). By Theorem 4.9.6 there exists a subset U2 of U, with mU (U2 ) > 0, so that ρ ◦ Su − ρ ◦ Sv is a quasi-coboundary for every u, v ∈ U2 . We get immediately that U2 − U2 ⊆ U0 , and so mU (U0 ) > 0. Thus U0 is an open subgroup of U. By Lemma 4.15.10 applied to the action {Su : u ∈ U0 }, there exist a subgroup U1 of U0 and a cocycle ρ0 on X with the required properties. (Note that U/U1 is toral because U0 /U1 is toral and U/U0 is finite).
142
4.10
Systems of order k and nilmanifolds.
The main theorem of this section is the following. Theorem 4.10.1. (cf. [36], Theorem 10.1) Any Zd -system of order k ≥ 1 can be expressed as an inverse limit of a sequence of k-step nilsystems. We first reduce to toral systems, and then show that every toral system of oder k is a k-step nilsystem.
4.10.1
Reduction to toral systems.
Definition 4.10.2. An ergodic system (X, X, µ, T ) of order k ≥ 1 is toral if Z1 (X) is a compact abelian Lie group and for 1 ≤ j < k, Zj+1 (X) is an extension of Zj (X) by a torus. Theorem 4.10.3. (cf. [36], Theorem 10.3) Any Zd -system of order k ≥ 1 is an inverse limit of a sequence of toral systems of order k. We need a lemma analogous to Lemma 10.4 of [36]. Lemma 4.10.4. (cf. [36], Lemma 10.4) Let (X, X, µ, T ) be an ergodic system, U a torus and ρ : Zd × X → U a cocycle of type k + 1 for an integer k ≥ 0. Assume that X is an inverse limit of a sequence {Xi : i ∈ N} of systems. Then ρ is cohomologous to a cocycle ρ0 : Zd × X → U, which is measurable with respect to Xi for some i. The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. We show by induction on l that 143
(∗) For integers 0 ≤ l ≤ k, there exist il ∈ N and a cocycle ρl cohomologous to ρ that is measurable with respect to Zk−l (X) ∨ Xil . By Corollary 4.7.9, ρ is cohomologous to a cocycle which factors through Zk+1 (X). By Theorem 4.9.5, Zk+1 (X) is an extension of Zk (X) by a connected compact abelian group. By Corollary 4.7.10, there exists a cocycle ρ0 , cohomologous to ρ and measurable with respect to Zk (X), and therefore with respect to Zk (X) ∨ X1 . The claim (∗) holds for l = 0. Let 0 ≤ l < k and assume that (∗) holds for l. Let il and ρl be as in the statement of the claim. By Corollary 4.7.8, ρl is of type k + 1. As Zk−l (X) is an extension of Zk−l−1 (X) by a compact abelian group, by the first part of Lemma 4.15.2, Y is an extension of W by a compact abelian group V. We identify Y with W × V. As usual, for v ∈ V, we also let v : Y → Y denote the associated vertical rotation of Y above W. By Corollary 4.9.7, there exist a closed subgroup V1 of V, so that V /V1 is a compact abelian Lie group, and a cocycle ρ0 , cohomologous to ρl and thus to ρ, so that ρ0 (a, v · y) = ρ0 (a, y) for every v ∈ V1 , a ∈ Zd . We consider ρ0 as a cocycle defined on the factor W × V /V1 of Y. Since V /V1 is a compact abelian Lie group, its dual group V[ /V1 = V1⊥ is finitely generated. Choose a finite generating set {γ1 , . . . , γm } for V1⊥ . For 1 ≤ j ≤ m, consider γj as taking values in the circle group S and define the function fj on Y = W × V by fj (w, v) = γj (v). Since X is the inverse limit of the sequence {Xi }, there exists i ≥ il so that for 1 ≤ j ≤ m, E(fj |Xi ) 6= 0. Thus E(fj |W ∨ Xi ) 6= 0. By Lemma 4.15.2 the functions fj are measurable with respect to W ∨ Xi . But the 144
functions fj , 1 ≤ j ≤ m, together with the σ-algebra W, span the σ-algebra of the system W × V /V1 . As ρ0 is measurable with respect to this factor, it is measurable with respect to W ∨ Xi = Zk−l−1 ∨ Xi . Therefore, (∗) holds for l + 1 with il+1 = i. Property (∗) with l = k is the announced result.
Proof of Theorem 4.10.3. We proceed by induction. For k = 1 the result is proved in Section 4.8.1. Let k ≥ 1 be an integer and assume that the result holds for k. Let Y be a Zd -system of order k + 1. Write X = Zk (Y ). Then Y is an extension of X by a compact abelian group U and we let ρ : Zd × X → U be the cocycle defining this extension. By Theorem 4.9.5, U is connected and can be written as lim Uj , where each Uj is a torus. ←− d Let ρj : Z × X → Uj be the projection of ρ on the quotient Uj of U. By the inductive hypothesis, X can be written as an inverse limit lim Xi , where each ←− Xi is toral. By Lemma 4.10.4, for every j there exist ij and a Uj -valued cocycle ρ0j , measurable with respect to Xij , and cohomologous to ρj . We can clearly assume that the sequence {ij } is increasing. Each system Xij ×ρ0j Uj is toral and Y = X ×ρ U is clearly the inverse limit of these systems.
4.10.2
Building nilmanifolds.
Theorem 4.10.5. (cf. [36], Theorem 10.5) Let (X, X, µ, T ) be a toral Zd -system of order k ≥ 1. Then (1) G = G(X) is a Lie group and is k-step nilpotent. (2) Let G be the subgroup of G spanned by the connected component of the identity 145
and T. Then G admits a discrete co-cocompact subgroup Λ so that the system X is isomorphic to the nilmanifold G/Λ, endowed with Haar measure, and the action T is by left translations. The proof is similar to to the proof in [36], and we repeat the steps.
Conditions for lifting. We repeat the discussion in Section 10.2.1 of [36]. Throughout this section, k ≥ 1 is an integer and (Y, Y, ν, S) is a toral Zd -system of order k + 1. We write (X, X, µ, T ) for Zk (Y ), where Y is an extension of X by a torus U, given by a cocycle ρ : Zd × X → U of type k + 1. By the inductive hypothesis, G(X) is a Lie group. By Lemma 4.5.2, every element g of G(Y ) induces a transformation pk g of X, which belongs to G(X). We say that an element g of G(X) can be lifted to an element G(Y ) if there exists g ∈ G(Y ) with pk g = g. We now establish conditions for lifting. We use the maps F : X [k+1] → U and Φ : Y [k+1] → U introduced in Proposition 4.6.4: ∆k+1 ρ(a, x) = F ((T [k+1] )a x) − F (x), X Φ(x, u) = F (x) − s(ε)uε ,
(4.26)
ε∈Vk+1
under the identification of Y [k+1] with X [k+1] × U [k+1] . By Proposition 4.6.4, the σalgebra I[k+1] (Y ) is spanned by the σ-algebra I[k+1] (X) and the map Φ. Lemma 4.10.6. (cf. [36], Lemma 10.6) Let g : X → X. If g ∈ G(Y ) is a lift of g, then g is given by g · (x, u) = (g · x, u + ψ(x)) 146
(4.27)
where ψ : X → U is a map satisfying F ◦ g [k+1] − F = ∆k+1 ψ
(4.28)
The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. Let g ∈ G(X) and assume that g admits a lift g ∈ G(Y ). By Corollary 4.5.10, the vertical rotations of Y over X belong to the center of G(Y ) and thus commute with g. It follows that g has the form given by equation (4.27) for some ψ : X → U. As g ∈ G(Y ), the transformation g [k+1] of Y [k+1] acts trivially on I[k+1] (Y ) and thus leaves the map Φ invariant. This implies immediately that ψ satisfies Equation (4.28). Conversely, let g ∈ G(X), ψ : X → U be a map satisfying Equation (4.28) and let g be the measure preserving transformation given by Equation (4.27). Since ν [k+1] is conditionally independent over µ[k+1] and g [k+1] leaves the measure µ[k+1] invariant g [k+1] leaves the measure µ[k+1] invariant. Moreover, Equation (4.28) means exactly that the map Φ is invariant under g [k+1] . Since g ∈ G(X), g [k+1] acts trivially on I[k+1] (X). By Proposition 4.6.4, g [k+1] acts trivially on I[k+1] (Y ). By Corollary 4.6.6, g ∈ G(Y ).
Corollary 4.10.7. (cf. [36], Corollary 10.7) The kernel of the group homomorphism pk : G(Y ) → G(X) consists of the transformations of the form (x, u) 7→ (x, u + ψ(x)), where ψ ∈ Dk+1 (X, U ). (See Section 4.7 and Lemma 4.7.4.) (This is a direct result of Lemma 4.10.6.) In order to build lifts of elements of G(X), we progress from G(k−1) (X) to G(X) along the lower central series of G(X). For 1 ≤ j < k, we show that ‘many’ elements of 147
G(j) (X) satisfy a property stronger than the lifting condition of Lemma 10.6. We need some notation. [k+1]
Notation. Let β be an l-face of Vk+1 and ψ : X → U a map. We write ∆β
ψ :
X [k+1] → U for the map given by [k+1]
∆β
ψ(x) =
X
s(ε)ψ(xε ).
ε∈β
We have that [k+1]
∆β
[k+1]
ψ(x) = ±∆l ψ(ξβ
(x)),
where the sign depends on the face β. Lemma 4.10.8. (cf. [36], Lemma 10.8) Let j be an integer with 0 ≤ j < k. For g ∈ G(j) (X) and ψ : X → U, the following are equivalent: [k+1]
(1) For every (k + 1 − j)-face β of Vk+1 , F ◦ gβ [k+1]
(2) For every (k − j)-face α of Vk+1 , F ◦ gα
[k+1]
− F = ∆β [k+1]
− F − ∆α
ψ.
ψ is invariant on X [k+1] .
The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. (j)
Notation. We write G0 for the set of g ∈ G(j) (X) so that there exists ψ : X → U satisfying (1) and (2) of Lemma 4.10.8. Proof. The proof is similar to the proof of Lemma 4.10.6. Let g ∈ G(j) (X). Let ψ : X → U and let g be the measure preserving transformation of Y = X × U [k+1]
given by Equation (4.27). As g ∈ G(j) (X), the measure µ[k+1] is invariant under gα
whenever α is a (k − j)-face of Vk+1 . Also, ν [k+1] is invariant under g [k+1] because this α 148
measure is conditionally independent over µ[k+1] . So for a (k − 1 + 1)-face β, ν [k+1] is [k+1]
invariant under g β
.
The first property means that the function Φ (see Proposition 4.6.4) defined above is [k+1]
invariant under g β [k+1]
gβ
for every k + 1 − j-face β of Vk+1 . Moreover, by Corollary 4.5.8,
acts trivially on I[k+1] (X) because g ∈ G(j) (X). Therefore, the first property [k+1]
means that g β
acts trivially on I[k+1] (Y ) for any (k + 1 − j)-face β of Vk+1 .
maps Similarly, the second property means that for every (k − j)-face α of Vk+1 , g [k+1] α the σ-algebra I[k+1] (Y ) to itself. The equivalence of these properties follows from Lemma 4.5.3.
Note that for j = 0 the first property of Lemma 4.10.8 coincides with the condition (0)
given in Lemma 4.10.6. Therefore, G0 consists of the elements of G(X) which can (0)
be lifted to an element G(Y ) and G0 = pk (G(Y )) . (j)
More generally, let g ∈ G0 for some j and ψ satisfying the first property of Lemma 4.10.8. Then ψ obviously6 satisfies equation (4.28), and the transformation g of Y (j) (j) given by equation (4.27) is a lift of g in G(Y ). Therefore pk maps p−1 G onto G0 . 0 k Each element g of G(Y ) is given by equation (4.27) for g = pk g) and some ψ, and (j) p−1 G consists of those g for which the map ψ satisfies the conditions of Lemma 0 k (j) 4.10.8. Therefore, p−1 G is a closed subgroup of G(Y ). 0 k 6
Write Vk+1 as a disjoint union of (k + 1 − j)-faces Vk = [k+1]
(F ◦ gβ
S
β∈I
β. For disjoint α, β ∈ I, we have
− F ) ◦ gα[k+1] + (F ◦ gα − F ) = ∆k+1 ψ + ∆k+1 α ψ, β
[k+1]
[k+1] so that F ◦ gα∪β = ∆k+1 = ∆k+1 ψ. α∪β ψ. Continuing in this way, we find F ◦ g
149
Lifting results. Lemma 4.10.9. (cf. [36], Lemma 10.9) Each element of G(k−1) (X) can be lifted to an (k−1)
element of G(Y ). More precisely, G0
= G(k−1) (X).
Proof. Let g ∈ G(k−1) (X). Since G(X) is k-step nilpotent, g belongs to the center of G(X) and thus commutes with T and is an automorphism of X. Since G(Zk−1 ) is (k − 1)-step nilpotent, g induces the trivial transformation on Zk−1 . Thus g is a [k+1]
vertical rotation of X over Zk−1 . For every edge α of Vk+1 , the transformation gα
leaves the measure µ[k+1] invariant and commutes with each element of T [k+1] by Corollary 4.5.4. By equation (4.26), (F ◦ gα[k+1] − F ) ◦ (T [k+1] )a (x) − (F ◦ gα[k+1] − F )(x) = (∆k+1 ρ)(a, gα[k+1] x) − ∆k+1 ρ(x) X = (−1)|ε| (ρ(a, gxε ) − ρ(a, xε ))
(4.29)
ε∈α
= ±∆(ρ ◦ g − ρ) ◦ ξα[k+1] (x), for all a ∈ Zd , x ∈ X [k+1] . By Lemma 4.15.7, ∆(ρ ◦ g − ρ) : X 2 → U is a coboundary. As U is a torus, by Lemma 4.15.5, ρ ◦ g − ρ is a quasi-coboundary. Thus there exists ψ : X → U and φ ∈ hom(Zd , U ) with ρ(a, gx) − ρ(a, x) = ψ(T a x) − ψ(x) + φ(a) for all a ∈ Zd , x ∈ X.
(4.30)
Using this in equation (4.29), we get that for every edge α there exists a T -invariant map i : X [k+1] → U with F ◦ gα[k+1] − F = ∆[k+1] ψ + i. α 150
(k−1)
By Lemma 4.10.8, g ∈ G0
.
Remark 4.10.10. We need a technical lemma before we can proceed with the main lifting argument. This lemma is not needed in the case d = 1, because the cocycle rigidity theorem (Theorem 4.9.6) restricts the behavior of maps of the form Ω → B((X, U ), where Ω is a probability space, and U is a compact abelian group. When d > 1, Theorem 4.9.6 restricts the behavior of d-tuples of maps ρ1 , . . . , ρd : Ω → B((X, U ), where the ρi are related by the cocycle equation for Zd -actions. Since the key step (Proposition 10.10 in [36] for the case d = 1, Proposition 4.10.12 in our case) involves lifting individual maps, we need some restriction on individual maps, or at least on d-tuples of maps which do not necessarily satisfy the cocycle equations. The next lemma, which is an application of Corollary 4.15.13, satisfies this need. Lemma 4.10.11. Let (X, X, µ, T ) be a Zd -system of order k, let U be a torus, let (Ω, B, P ) be a probability space, and let A = {a1 , . . . , ad } be a generating set for Zd . For each a ∈ A, let fa,ω : X → U be a measurable family of maps. Suppose that for each ω ∈ Ω, there exists a map Fω : X [k] → U such that for each a ∈ A, ∆k fa,ω = Fω ◦ (T [k] )a − Fω . Then there is a measurable family of functions ga,ω : Zd × Zk−1 → U, Gω : X → U such that for each ω ∈ Ω, a ∈ A fa,ω (x) = ga,ω (πk−1 (x)) + Gω (T a x) − Gω (x). In particular, the functions fa,ω − ga,ω generate a cocycle for each ω, and ∆k (fa,ω − ga,ω ) = ∆k (fa,ω ) for each ω ∈ Ω, a ∈ A. 151
Proof. Let Fd denote the free group on the d generators a1 , . . . , ad . If we consider (X, X, µ, T ) as an Fd -system, then for each ω ∈ Ω, the maps (x, u) 7→ (T a x, fa,ω (x)+u) generate a free group action, which we denote by S. For each v ∈ Fd , the element Sv has the form Sv (x, u) = (T v x, σv,ω (x) + u), and σ must satisfy the cocycle equation (for Fd -actions). Hence we can apply Corollary 4.15.13 to conclude that there exist measurable families of functions {gv,ω : v ∈ Fd , ω ∈ Ω}, {Gω : ω ∈ Ω} such that σv,ω (x) = gv,ω (πk−1 (x)) + Gω (T v x) − Gω (x). Then ga,ω , a ∈ A and Gω satisfy the conclusion of the lemma.
Proposition 4.10.12. (cf. [36], Proposition 10.10) For an integer j with 0 ≤ j < (j)
k, G0 is open in G(j) (X). (j)
Proof. We proceed by induction downward on j. For j = k − 1, G0 = G(j) (X) by (j)
Lemma 4.10.9. Take j with 0 < j ≤ k − 1 and assume that G0 is open in G(j) (X). (j−1)
We prove now that G0 (j)
Since G0
is open in G(j−1) (X).
is an open subgroup of G(j) (X), it is also closed and it is locally com-
pact and Polish (actually it is a Lie group). We have noted that the continuous (j) (j) group homomorphism pk : p−1 G → G0 is onto. By Theorem A.1 of [36], this 0 k homomorphism admits a Borel cross-section. Let A = {a1 , . . . , ad } be a finite generating set for Zd , and let H :=
\
(j)
{g ∈ G(j−1) (X) : [g −1 ; T −a ] ∈ G0 }.
a∈A
By the inductive hypothesis, H is open in G(j−1) , and is locally compact. For each a ∈ A, let κa : H → G(Y ) denote the map obtained by composing the continuous 152
(j)
map g 7→ [g −1 ; T −a ] from H to G0 with the aforementioned Borel cross section. For g ∈ H, a ∈ A, κa (g) is given by equation (4.27) for some map ψg (a, ·) : X → U so that the properties of Lemma 4.10.8 are satisfied with [g −1 ; T −a ]. That is, for every (k + 1 − j)-face β of Vk+1 [k+1]
F ◦ [g −1 ; T −a ]β
− F = ∆k+1 β ψg (a, ·)
Define θg (a, x) = ψg (a, T a gx) + ρ(a, gx) − ρ(a, x). Let β be a (k + 1 − j)-face of Vk+1 . Then for a ∈ A, [k+1]
(F ◦ gβ
[k+1]
− F ) ◦ (T [k+1] )a (x) − (F ◦ gβ [k+1]
= (F ◦ [g −1 ; T −a ]β
− F )(x)
[k+1]
− F ◦ (T [k+1] )a gβ
[k+1]
)(x) − (F ◦ T [k+1] − F )(x)
(T [k+1] )a gβ
[k+1]
+ (F ◦ (T [k+1] )a gβ
− F ◦ gβ
[k+1]
[k+1] a = ∆k+1 ) gβ β ψg (a, (T
[k+1]
[k+1]
x) + ∆k+1 ρ(a, gβ
)(x)
x) − ∆k+1 ρ(a, x)
= ∆k+1 β θg (a, x) [k+1]
= ±∆k+1−j θg ◦ ξβ
(a, x).
[k+1] Thus the maps (a, x) 7→ ∆k+1 . As β θg (a, x) define a coboundary of the system X [k+1]
already noted, the cocycle (∆k+1−j θg ) ◦ ξβ
is equal to this coboundary or to its
opposite and thus is a coboundary. By Lemma 4.15.7, ∆k+1−j θg is a coboundary of the system X [k+1−j] and so by Lemma 4.10.11 there is a measurable family of maps Mg : Zd × Zk−1 → U such that θg − Mg ◦ πk−1 is a cocycle of type k + 1 − j ≤ k on X, and ∆k (θg − Mg ◦ πk−1 ) = ∆k θg . Write τg for θg − Mg ◦ πk−1 . Since the maps κa defined above are Borel, the maps g 7→ ψg (a, ·) from H to B(X, U ) (the space of functions from X to U ) are Borel, 153
and the map g 7→ τg is a Borel map from H to the group Ck+1−j (X, U ) of U -valued cocycles of type k + 1 − j on X. Choose a probability measure λ on H, equivalent to the Haar measure of G(j−1) (X) and apply Theorem 4.9.6. Then there exists a measurable subset B of H, with λ(B) > 0, so that τg − τh is a quasi-coboundary for every (g, h) ∈ B × B. Let g, h ∈ B. Let θ : X → U and φ ∈ hom(Zd , U ) be such that τg (a, x) − τh (a, x) = θ(T a x) − θ(x) + φ(a). For any (k + 1 − j)-face β of Vk+1 , by the last equation we get that [k+1]
(F ◦gβ
[k+1]
−F ◦hβ [k+1]
Thus, F ◦ gβ
[k+1]
)◦(T [k+1] )a −(F ◦gβ [k+1]
− F ◦ hβ
[k+1]
[k+1]
[k+1] a ) −∆k+1 ) = (∆k+1 β θ)◦(T β θ.
[k+1] − ∆k+1 . As h ∈ β θ is an invariant function on X
G(j−1) (X), the transformation hβ the function F ◦(gh−1 )β
[k+1]
−F ◦hβ
maps the σ-algebra I[k+1] (X) to itself. Therefore,
−1 [k+1] −F −∆k+1 -invariant. The second property β (θ◦h ) is T (j−1)
of Lemma 4.10.8 is satisfied, which implies gh−1 ∈ G0 (j−1)
Therefore B · B −1 ⊆ G0
.
. Since H is open in G(j−1) , B has positive Haar measure in (j−1)
G(j−1) and it follows that G0
(j−1)
also has positive Haar measure in G(j−1) . Since G0
is a Borel subgroup of G(j−1) (X), it is an open subgroup.
W now repeat the argument from the end of Section 10 of [36] to show that G is a Lie group and that Y is a quotient of G by a cocompact subgroup. 4.10.3
End of the proof.
Before we complete the proof of Theorem 4.10.5, we restate it for convenience. Theorem 4.10.5. Let (X, X, µ, T ) be a toral Zd -system of order k ≥ 1. Then 154
(1) G = G(X) is a Lie group and is k-step nilpotent. (2) Let G be the subgroup of G spanned by the connected component of the identity and T. Then G admits a discrete co-cocompact subgroup Λ so that the system X is isomorphic to the nilmanifold G/Λ, endowed with Haar measure, and the action T is by left translations. The end of the proof is essentially a verbatim reproduction of Section 10.2.3 of [36]. We give it for completeness. Recall that k ≥ 1 is an integer and we assume that the properties of Theorem 4.10.5 hold for every toral Zd -system of order k. Let (Y, Y, ν, S) be a toral system of oder k + 1. We write (X, X, µ, T ) = Zk (Y ). By the inductive hypothesis, the conclusions of Theorem 4.10.5 hold for this system. Let G and Λ be as in this Theorem and let (0)
G0 be as in the preceding subsection. (0)
(0)
(1) By Proposition 4.10.12 used with j = 0, the group G0 is open in G0 (X) = G(X) and is thus a Lie group. The restriction map pk : G(Y ) → G(X) is a continuous group (0)
homomorphism and maps G(Y ) onto G0 . Its kernel is Dk+1 (X, U ) by Corollary 4.10.7 (0)
and thus is a Lie group. Since G0 and Dk+1 (X, U ) are both Lie groups, G(Y ) is a Lie group by Corollary A.2 and Lemma A.3 of [36]. (2) Let H be the subgroup of G(Y ) spanned by the connected component of the identity and {S a , a ∈ Zd }. The image under pk of the connected component of the identity of G(Y ) is included in the connected component of the identity of G(X); (0)
moreover pk (S a ) = T a and thus pk (H) ⊂ G. Since pk maps G(Y ) onto G0 , it is an (0)
open map and pk (H) is an open subgroup of G0 and thus also of G(X). Therefore 155
pk (H) contains the connected component of the identity in G(X) and so it contains G. Now, pk (H) = G. On the other hand, for every u ∈ U, the corresponding vertical rotation belongs to G(Y ) and defines an embedding of U in G(Y ). H ∩ U is an open subgroup of U and since U is connected, U ⊂ H. By the inductive assumption, X = G/Λ. This means that G acts transitively on X and that Λ is the stabilizer of the point x1 of X, image of the identity element of G under the natural projection G 7→ G/Λ = X. Choose a lift y1 of X1 in Y and consider the map f : H → Y given by f (h) = h · x1 . Since U ⊂ H, the range of this map is invariant under all vertical rotations. The projection of this range on X is onto. Therefore f is onto. This defines a bijection of H/Γ onto Y, where Γ is the stabilizer of y1 in H. This bijection commutes with the actions of H on Y and H/Γ. The measure on H/Γ corresponding to ν through this bijection is invariant under the action of H and thus is the Haar measure of H/Γ. Thus we are left only to check that Γ is discrete and cocompact in H. Clearly, Γ · U = p−1 k (Λ). Since Γ ∩ U is trivial, Γ is discrete. This also implies that H/ΓU is homeomorphic to G/Λ and thus is compact. Since U is compact, Γ is cocompact in H.
4.11
Characteristic factors.
We continue to assume that (X, X, µ, T ) is an ergodic Zd system.
156
Here we imitate the proofs of Section 12 of [36] and show that the factors Zk are characteristic for expressions of the form k k Y 1 XY Mi a fi ◦ T − E(fi |Z) ◦ T Mi a = 0. |FN | a∈F i=1 i=1 N
where the Mi are endomorphisms of Zd such that the image of each Mi , Mi −Mj , i 6= j has finite index in Zd . If the image of the endomorphism M : Zd → Zd has finite index in Zd , then the action a = T M a has at most [M Zd : Zd ] = det M ergodic components. TM given by TM
Theorem 4.11.1. Let (X, X, µ, T ) be an ergodic system, and let k ≥ 1. Let M1 , . . . , Mk be endomorphisms of Zd such that Mi , Mi − Mj , i 6= j has finite index in Zd . Assume that f1 , . . . , fk are bounded functions on X with kfj k∞ ≤ 1 for j = 1, . . . , k. Assume further that each fi is supported on an atom of the algebra A generated by the ergodic components of the TMi , TMi −Mj , i 6= j. Then there exists a constant C depending only on the Mi such that k
1 X Y
lim sup fj ◦ T Mj a ≤ min C · |||fl |||k 1≤l≤k |F | L2 (µ) N →∞ N j=1 a∈F
(4.31)
N
for all Følner sequences FN ⊂ Zd . Proof. We proceed by induction. Let M = M1 . For k = 1, we consider the ergodic decomposition of TM . The ergodic components of TM are sets Ai of positive measure in X, and by ergodicity, T acts transitively on these sets, so they all have the same measure m. Since f is supported on an atom of A, we can assume that f is supported
157
on an ergodic component A of TM . Let FN be a Følner sequence. By the ergodic theorem, we have Z Z 1 X 1 X 1 Ma 1Ai lim f1 ◦ T = f1 dµ = 1A f dµ N →∞ |FN | m m Ai a∈F N
so that Z 1 X
1 1 M a lim f1 ◦ T = √ f1 dµ = √ |||f1 |||1 . 2 N →∞ |FN | m m L (µ) a∈F N
Let k ≥ 1 and assume that the inequality (4.31) holds for k. Choose 2 ≤ l ≤ k + 1 (the case l = 1 is similar.) Let ξa =
k+1 Y
fj ◦ T Mj a
j=1
Computing hξa , ξa+h i yields Z hξa , ξa+h i =
(f1 · f1 ◦ T
M1 h
)
k+1 Y
(fj · fj ◦ T Mj h ) ◦ T (Mj −M1 )a dµ
j=2
so that lim sup N →∞
1 X hξa , ξa+h i ≤ C|||fl · fl ◦ T Ml h |||k . |FN | a∈F
(4.32)
N
Now we average the expression on the right-hand side of (4.32) over h, and compare it to the average where T h replaces T Mj h . We note that, given a Følner sequence FN in Zd , the image Ml FN is a Følner sequence in Ml Zd , and if {ai }ni=1 is a fundamental S domain for Zd /Ml Zd , then GN := ni=1 ai + FN is a Følner sequence in Zd . Also, P P a∈GN ψ(a) ≤ n supi a∈ai +FN ψ(a). Fixing such a GN , we have X h∈FH
|||fl · fl ◦ T Ml h |||k =
X
|||fl · fl ◦ T h |||k ≤ n
h∈Ml FH
X h∈GH
158
|||fl · fl ◦ T h |||k ,
so that lim sup H→∞
1 X |||fl · fl ◦ T Ml h |||k |FH | h∈F H
≤ lim sup n H→∞
1 X |||fl · fl ◦ T h |||k |GH | h∈G
(4.33)
H
1 X k |||f1 · f1 ◦ T h |||2k |GH | h∈G
≤ lim sup n H→∞
!1/2k .
H
N k Writing f˜ = ε∈Vk fl (ε) and expanding |||f1 · f1 ◦ T h |||2k yields |||f1 · f1 ◦
k T h |||2k
=
Z O
fl (xε ) · fl (T h xε ) dµ[k] (x)
ε∈Vk
Z =
f˜ · f˜ ◦ (T [k] )h dµ[k] .
Averaging over GH we find lim sup H→∞
1 X k 2k+1 |||f1 · f1 ◦ T h |||2k = |||f |||k+1 , |GH | h∈G H
so that (4.33) implies that lim sup H→∞
1 X fl · fl ◦ T Ml h k ≤ n|||fl |||2k+1 . |FH | h∈F H
Combining this with (4.32) we see that lim sup H→∞
1 X 1 X lim sup hξa , ξa+h i ≤ Cn|||fl |||2k+1 |FH | h∈F N →∞ |FN | a∈F N
H
The van der Corput lemma now implies the result.
Theorem 4.11.1 now implies Theorem 4.1.2, since for k > 1, the σ-algebras Zk contain the algebra generated by the ergodic components of the TM . 159
Proof of Theorem 4.1.2. For k = 1 Theorem 4.1.2 follows from the mean ergodic theorem for Zd -actions. For k > 1, it suffices to prove that k 1 XY lim fj ◦ T Mj a = 0 N →∞ |FN | a∈F j=1 N
if E(fj |Zk−1 ) = 0 for some j, say j = 1. Assuming this, let A be the algebra of sets generated by the ergodic components of the TMi , TMi −Mi0 , i 6= i0 , and let {A1 , . . . , Av } P be the atoms of A. Write f1 = vr=1 gr , where gr = 1Ar f1 . Since Zk−1 includes the Kronecker factor Z1 , Zk−1 also includes A, so that E(gr,j |Zk−1 ) = 0 for each r. We Q P Q can write the summands kj=1 fj ◦ T Mj a as vr=1 gr ◦ T M1 a kj=2 fj ◦ T Mj a . By Lemma 4.4.3, we have |||gr |||k = 0 for each r, so by Theorem 4.11.1, we have k 1 XY lim fj ◦ T Mj a = 0. N →∞ |FN | a∈F j=1 N
4.12
The Zk are characteristic factors.
In this section, we generalize Theorem 4.1.2 to handle averages of expressions of the Q form ki=1 fi ◦ Tia , where the Ti are commuting actions of Zd . We say that actions S, T of Zd commute if S a T b = T b S a for all a, b ∈ Zd . If T is a Zd -action, we denote the action a 7→ T −a by T −1 . If T is an ergodic action, let Zk (T ) denote the factor of (X, X, µ, T ) defined in Definition 4.4.1. We will prove the following theorem.
160
Theorem 4.12.1. Let (X, X, µ) be a standard measure space, and let Ti , i = 1, . . . , k be commuting, measure preserving actions of Zd on (X, X, µ). Suppose that the actions Ti , Ti Tj−1 , i 6= j are ergodic. Let k > 0. Then Zk (Ti ) = Zk (Tj ) =: Zk , for i 6= j, and (Zk , µk , T1 , . . . , Tk ) can be given the structure of an inverse limit of nilsystems, where the Ti act by translations. For bounded fi : X → C and all Føler sequences FN in Zd , we have k k Y 1 XY a fi ◦ Ti − E(fi |Zk ) ◦ Tia = 0. lim N →∞ |FN | i=1 a∈F i=1 N
The case d = 1 is proved in [21]. The proof follows the proof in [21]. If S is an ergodic action of Zd on (X, X, µ), [k]
let µS be the measures associated with (X, X, µ, S), as in Section 4.3, let G(S) be the group associated to (X, X, µ, S) as in Definition 4.5.1, and let ||| · |||k,S denote the kth seminorm associated to (X, X, µ, S) as in Section 4.3.6. Let I[k] (S) denote the S [k] -invarant σ-algebra of µ[k] . We will show that if S and T are commuting ergodic actions on (X, X, µ), then ||| · |||k,S = ||| · |||k,T and G(S) = G(T ). Proposition 4.12.2. ([21], Proposition 3.1) Assume that T and S are commuting measure preserving actions of Zd on a probability space (X, X, µ) and that both T and S are ergodic. Then for all a ∈ Zd , S ∈ G(T ), G(T ) = G(S), and for all integers [k]
[k]
k ≥ 1 and all f ∈ L∞ (µ), µT = µS , I[k] (T ) = I[k] (S), and |||f |||k, T = |||f |||k, S, and Zk (S) = Zk (T ). Proof. By Lemma 4.5.5, S a ∈ G(X, T ) for all a ∈ Zd . We use induction on k to [k]
[k]
show that µT = µS and I[k] (T ) = I[k] (S). The statement is obvious for k = 0. [k+1]
Suppose that it holds for some integer k ≥ 1. By the definition of µT 161
[k+1]
and µS
,
[k+1]
we have µT [k+1]
µT
[k+1]
= µS
[k+1]
= µS
. Since S ∈ G(T ), we have that S [k+1] leaves the measure [k+1]
invariant and acts trivially on IT
. Hence, I[k+1] (T ) ⊂ I[k+1] (S). By
symmetry, I[k+1] (S) ⊂ I[k+1] (T ). This completes the induction. The equalities G(T ) = G(S), ||| · |||k,T = ||| · |||k,S and Zk (T ) = Zk (S) now follow from the definitions.
To show that (Zk , Zk , µk , T1 , . . . , Tl ) is an inverse limit of k-step nilsystems, we repeat the arguments in Section 4 of [21]. Let (Y, Y, ν, S1 , . . . , Sl ) be a measure preserving system with l commuting actions Si of Zd . Given cocycles ρ1 , . . . , ρl : Zd ×Y → U, where U is a compact abelian group, write ρ˜ = (ρ1 , . . . , ρl ), and can define the extension Y ×ρ˜ U = (Y × U, ν × mU , T1 , . . . , Tl ), where Tia (y, u) = (Sia y, ρi (a, y) + u). We say that ρ˜ is cohomologous to ρ˜0 if there is a function f : X → U such that ρi (a, x) = ρ0i (a, x)+f (Tia x)−f (x) for all a ∈ Zd , x ∈ X. We say that ρi is a coboundary if there exists f such that ρi (a, x) = f (Tia x) − f (x). It will be useful to consider the system (X, X, µ, T1 , . . . , Tl ) as an ergodic Zld system in the natural way: let R be the Zld -action given by R(a1 ,...,al ) = T1a1 · · · Tlal . If X is an extension of Y as above, then R is given by the cocycle whose values at the generators b ai = (0, . . . , 0, ai , 0, . . . , 0) of Zld are given by ρi (b ai , x) = ρi (ai , x). Note that ρ˜ is a coboundary if and only if σ is. We generalize Theorem 4.1 of [21] to the case where (X, X, µ, T ) is a Zd dynamical system. Theorem 4.12.3. ([21], Theorem 4.1) Any system (X, X, µ, T1 , . . . , Tl ) of order k is an inverse limit of a sequence (Xi , µi , T1 , . . . , Tl ) of toral systems of order k.
162
We need the following lemma. Lemma 4.12.4. ([21], Lemma 5.1) Let k, l ≥ 1 be integers and let T1 , . . . , Tl be commuting ergodic measure preserving Zd -actions on a measure space (X, X, µ). Then the system (Zk+1 , µk+1 , T1 , . . . , Tl ) is an extension of the system (Zk , µk , T1 , . . . , Tl ) by a connected compact abelian group. The proof is identical to the proof in [21]. We reproduce it for completeness. Applying Part (ii) of Theorem 4.9.5, we get that there exist a connected compact abelian group V, a cocycle ρ1 : Zd ×Zk → V, a measure preserving bijection φ : Zk+1 → Zk ×V that preserves Zk such that T1a (φ(x)) = φ((T10 )a (y, v)) for y ∈ Zk , v ∈ V, a ∈ Zd and a measure preserving action S1 : Zk → Zk such that (T10 )a (y, v) = (S1a (y), v + ρ1 (a, y)). For i = 2, . . . , l, define (Ti0 )a = φ−1 Tia φ. Since both φ and Ti preserve Zk , we have that Ti0a has the form (Ti0 )a (y, v) = (Sia (y), Qi,a (y, v)), for some measurable transformation Qi,a : Zk ×V → V for i = 2, . . . , l. By Corollary 4.5.10 all maps Ru : Zk ×V → Zk ×V defined by Ru (y, v) = (y, v + u) belong to the center of G(T10 ). By Proposition 4.12.2, (Ti0 )a ∈ G(T10 ) and so (Ti0 )a commutes with Ru for all u ∈ V. This can only happen if Qi,a has the form Qi,a (y, v) = v + ρi (a, y) for some measurable ρi : Y → V. This completes the proof.
Lemma 4.12.5. Let l ≥ be an integer and let T1 , . . . , Tl be commuting ergodic measure preserving actions of Zd on a probability space (X, X, µ). Let {Sv : v ∈ V } be a free action of a compact abelian group V on X that commutes with Ti for i = 1, . . . , l. 163
Let U be a finite dimensional torus and let ρ˜ = (ρ1 , . . . , ρl ) : X → U l be an l-cocycle of type k for some integer k ≥ 2. Then there exists a closed subgroup V 0 of V such that V /V 0 is a compact abelian Lie group, and there exists an l-cocycle ρ0 = (ρ01 , . . . , ρ0l ), cohomologous to ρ˜, such that ρ0i ◦ Sv = ρ0i for every v ∈ V 0 . Proof. Consider the Zld action R given by R(a1 ,...,al ) = T1a1 · · · Tlal . Let σ be the cocycle extension representing the extension of R generated by the cocycle extensions of the Ti by the ρi . That is σ((0, . . . , a . . . , 0), x) = ρi (a, x). Then σ is of type k. By Corollary 4.9.7 there exists a closed subgroup V 0 ⊆ V such that V /V 0 is a compact abelian Lie group and a cocycle σ 0 cohomologous to σ so that σ 0 (a, Sv x) = σ 0 (a, x) for all a ∈ Zld , x ∈ X. We can then take ρ0i (a, x) to be σ 0 ((0, . . . , ai , . . . , 0), x).
Lemma 4.12.6. Let l ≥ 1 be an integer, let T1 , . . . , Tl be commuting ergodic measure preserving actions of Zd on a probability space (X, X, µ), let U be a finite dimensional torus, and let ρ˜ : X → U l be an ergodic l-cocycle of type k and measurable with respect to Zk for some integer k ≥ 1. Assume the system (X, X, µ, T1 , . . . , Tl ) is an inverse limit of the systems {(Xi , µi , T1 , . . . , Tl )}i∈N . Then ρ˜ is cohomologous to a cocycle ρ˜0 : X → U l , which is measurable with respect to Xi for some i. The proof is identical to the proof of Theorem 10.4 in [36].
4.12.1
Proof that Zk is an inverse limit of nilsystems.
Now we can show, by induction, that the systems (Zk , µk , T1 , . . . , Tl ) are inverse limits of nilsystems. Assume this is so for a fixed k ≥ 1, and let (X, X, µ, T1 , . . . , Tl ) be a Zd system of order k +1. By Lemma 4.12.4, we get that the system (X, X, µ, T1 , . . . , Tl ) is 164
an extension of (Zk , µk , T1 , . . . , Tl ) by a connected compact abelian group V. We can consider Zk = (Zk , µk , T1 , . . . , Tl ) and x = (X, X, µ, T1 , . . . , Tl ) as Zld -systems, where X is an extension of Zk by a cocycle of type k + 1, and Zk is a Zd -system of order k. As a Z ld -system, X is a Zd -system of order at most k, and therefore is isomorphic to an inverse limit of nilsystems.
4.12.2
Proof that Zk is characteristic.
To prove Theorem 4.12.1, we need an inequality comparing the averages of the Qk a i=1 fi ◦ Ti to the seminorms |||fi |||l . Proposition 4.12.7. ([21], Proposition 3.2) Let l ≥ 1 be an integer. Assume that T1 , . . . , Tl are commuting ergodic measure preserving actions of Zd on a probability space (X, X, µ) such that Ti Tj−1 is ergodic for all i 6= j. Let f1 , . . . , fl ∈ L∞ , |fi | ≤ 1 for all i. l
1 XY
a fi ◦ Ti ≤ C min |||fi |||l . lim sup 1≤i≤l |FN | a∈F i=1 L2 (µ) n→∞ N
Proof. For l = 1, the statement is just the ergodic theorem for Zd -actions. Assume the statement holds for some l ≥ 1. Let 2 ≤ j ≤ l. xa =
l+1 Y
fi ◦ Tia .
i=1
We have Z hxa , xa+h i =
f1 · f1 ◦ T1h
l+1 Y (fi · fi ◦ Tih ) ◦ Tia T1−a dµ. j=2
Averaging over FN and applying Cauchy-Schwartz we have l+1 Y l+1
1 XY
1 X
h a −a lim sup hxa , xa+h i ≤ lim sup (fi · fi ◦ Ti ) ◦ Ti T1 |FN | a∈F i=2 j=2 L2 (µ) N →∞ |FN | N →∞ a∈F N
N
165
Let Si = Ti T1−1 , i = 2, . . . , l+1. The maps Si commute and are ergodic by assumption, and Si Sj−1 = Ti Tj−1 , so the Si Sj−1 are ergodic by assumption, for i 6= j, and the seminorms ||| · |||l+1,Si are all equal to the ||| · |||l+1 . Then by the induction hypothesis lim sup N →∞
1 X hxa , xa+h i ≤ |||fj · fj ◦ Tjh |||l . |FN | a∈F N
We then have 1 X 1 X lim sup hxa , xa+h i ≤ lim sup lim sup N →∞ |FN | H→∞ H→∞ |FH | a∈F h∈F H
N
1 X l kfj · fj ◦ Tjh k2l |FH | h∈F
!1/2l
H
= |||fj |||2l+1 . The van der Corput lemma now implies the result.
4.13
Necessity of the conditions on the endomorphisms.
Here we give an example that shows that the condition on the endomorphisms Mi , Mi − Mj is necessary in Theorem 4.1.2 to conclude that the factor Zk is an inverse limit of nilsystems. This is essentially the example given in [21] to show that the ergodicity hypotheses are necessary for the main theorem Let Mi : Zd → Zd , i = 1, . . . , k be endormorphisms, and suppose that the image of M1 does not have finite index in Zd . Consider Zd as a subgroup of Rd . Let V denote the image of M1 , and let r be the rank of V (as a Z-module). Let {v1 , . . . , vr } be a basis of the R-span of V, and let {vr+1 , . . . , vd } be a basis of V ⊥ . Define an action of Zd as follows: Let (Y1 , Y1 , µ1 , S1 ) be a mixing Rr -system (so in particular S1t is a mixing transformation for each t ∈ Rr \ {0}). Let (Y2 , Y2 , µ2 , S2 ) be a mixing Rr−d 166
action, and define an Rd -system (Y1 × Y2 , Y1 × Y2 , µ1 × µ2 , R) by R(c1 v1 +···+cd vd ) (x, y) = (c ,...,cr )
(S1 1
c
x, S2r+1
,...,cd
y). Let T be the Zd action defined by restricting R to Zd . Let
g : Y → R be any bounded function, and define f1 : X × Y :→ R by f1 (x, y) = g(y), so that f1 is T M1 a -invariant for all a ∈ Zd . Taking f2 , · · · , fk to be identically 1, we have k 1 XY fi ◦ T Mi a = f1 lim N →∞ |FN | a∈F i=1 N
d
for every finite FN ⊂ Z . This implies that every characteristic factor for the scheme (T1M1 a , · · · , TkMk a ) must at least contain the σ-algebra X × Y. Since the action of T is mixing, and the σ-algebra X × Y is nontrivial, the corresponding factor must be mixing, and cannot be an inverse limit of nilmanifolds. For an example showing the necessity of the condition that the image of Mi −Mj have finite index for i 6= j, suppose that (M2 − M1 ) has infinite index in Zd . Let f0 ∈ L2 , and consider the averages Z k Y 1 X f0 fi ◦ T Mi a dµ. |FN | a∈F i=1
(4.34)
N
Applying T −M1 a to the integrand, we can rewrite the integrand as Z f1 · f0 ◦ T
−M1 a
k Y
fi ◦ T (Mi −M1 )a dµ.
i=2
Writing Mi0 for Mi − M1 , i = 1, . . . , k, we see that we are considering weak limits of P Q 0 averages of the form |F1N | a∈FN ki=1 fi ◦ T Mi a , so the necessity that (M2 − M1 )Zd have finite index in Zd is shown by the above example.
167
4.14
Appendix, part 1.
The material in this section is mostly adapted from Appendices A and B of [36]. We first define the term Polish group, which in turn depends on the definition of Polish space. A Polish space is a separable completely metrizable topological space. A Polish group is a topological group whose topology is that of a Polish space. Theorem 4.14.1. (cf. [3], Theorem 1.2.4) Let G be a Polish group and H a closed subgroup. Then there is a Borel measurable function sH : G/H → G such that sH (xH) ∈ xH, i.e., sH is a Borel selector for the (left-) cosets of H. We refer to the function sH above as a Borel section. Corollary 4.14.2. ([36], Corollary A.2) Let H be a closed normal subgroup of the Polish group G. If H and G/H are locally compact, then G is locally compact. If H and G/H are compact, then G is compact.
4.14.1
Lie groups.
Lemma 4.14.3. Let G be a locally compact group and let H be a closed normal subgroup. If H and G/H are Lie groups then G is a Lie group. As remarked in [36], this follows from the characterization of Lie groups in Chapters 3 and 4 of [51].
168
4.15
Appendix, part 2.
The material in this section is mostly adapted from Appendix C of [36]. One exception is a variation of [36], Lemma C.5, which we have adapted from [68]. Let (X, X, µ, T ) be a Zd -system, and let U be a compact abelian group. If ρ : Zd ×X → U We say that ρ is ergodic if the corresponding action on X × U is ergodic. Write C(X, U ) for the cocycles of (X, X, µ, T ) taking values in U. Then C(X, U ) is a Polish group under the topology of convergence in probability. Recall that a coboundary is a cocycle of the form ρ(a, x) = f ◦ T a − f. Lemma 4.15.1. (cf. [36], Lemma C.1) Let Γ be a discrete group and let (X, X, µ, T ) be an ergodic Γ-system, U a compact abelian group and ρ : Γ × X → U a cocycle b , the cocycle of (X, X, µ, T ). Then ρ is a coboundary if and only if for every χ ∈ U χ ◦ ρ : Γ × X → T is a coboundary. For a proof, see Theorem 5.2 of [50]. Lemma 4.15.2. (cf. [36], Lemma C.2) Let (X, X, µ, T ) and (Y, Y, ν, S) be ergodic systems, U a compact abelian group, ρ : Zd × X → U an ergodic cocycle and W the extension of X be U associated to ρ. Assume that W and Y are factors of the same ergodic system K and let L and M be the factors of K associated to the invariant sub-σ algebras L = X ∨ Y and M = W ∨ Y, respectively. Then M is an extension of L by a closed subgroup V of U. b and consider γ as taking values in S. Define a function fγ on W by Let γ ∈ U fγ (x, u) = γ(u). If E(fγ |L) 6= 0, then fγ is measurable with respect to L and γ ∈ V ⊥ . The proof found in [36] suffices. 169
4.15.1
Measurability properties.
Let U be a compact abelian group, and define R : B((X, U ) → C(X, U ) by (Rf )(a, x) = f (T a x) − f (x). Then R is a continuous map from the Polish group U X to the closed subgroup of d ×X
UZ
consisting of cocycles, so the set of coboundaries is a Borel subgroup of the
set of cocycles. Lemma 4.15.3. (cf. [36], Lemma C.3) Let (X, X, µ, T ) be a (possibly nonergodic) system, (Y, Y, ν) a (standard) probability space, and y 7→ µy a weakly measurable map from Y to the space of probability measures on X. Assume that · For every y ∈ Y, the measure µy is invariant under T. · µ=
R Y
µy dν(y).
Let (Ω, P ) be a (standard) probability space and let ω 7→ ρω be a measurable map from ω to the set of cocycles from X to S. Then: (1) The set A = {(ω, y) ∈ Ω × Y : ρω is a coboundary of (X, X, µy , T )} is measurable. (2) For ω ∈ Ω, ρω is a coboundary of (X, X, µ, T ) if and only if the set Aω = {y ∈ Y : (ω, y) ∈ A} satisfies ν(Aω ) = 1. 170
Proof. For a bounded function (defined everywhere) on X, we write Bω,f for the set of points where the averages 1 Nd
N −1 X
ρω ((n1 , . . . , nd ), x)f (T (n1 ,...,nd ) x)
(4.35)
n1 ,...,nd =0
converge as N → ∞. Define the function ψω,f on Bω,f to be the limit of these averages. The set Bω,f is invariant under T a for all a ∈ Zd , and the function ψω,f satisfies ψω,f (T a x) = ψω,f (x)ρω (a, x)
(4.36)
for x ∈ Bω,f , due to the cocycle equation ρ(n, T a x) = ρ(n + a, x)ρ(a, x). Define Cω,f = {x ∈ Bω,f : ψω,f (x) 6= 0}. Then Cω,f is invariant under T. For every bounded function f on X, the subset Cf = {(ω, x) ∈ Ω × X : x ∈ Cω,f } is measurable in Ω × X. We show now that µ(Bω,f ) = 1. Let X × S1 be endowed with the transformation associated to the cocycle ρω and let ψ be the function defined on X × S1 by φ(x, u) = f (x)u. By applying the pointwise ergodic theorem on the system X × S1 and the function φ, we get that the averages (4.35) converge almost everywhere. That is µ(Bω,f ) = 1. Therefore, the function ψω,f is defined µ-almost everywhere, and satisfies (4.36) µ-almost everywhere. By the same argument, for every y ∈ Y , the same properties hold with µy substituted for µ.
171
Choose a countable family {fj : j ∈ J} of bounded functions on X that is dense in L2 (µ) and dense in L2 (µy ) for every y ∈ Y. Define Cω =
[
Cω,fj and C =
j∈J
[
Cfj .
j∈J
We claim that A = {(ω, y) ∈ Ω × Y : µy (Cω ) = 1}.
(4.37)
Let ω ∈ Ω and y ∈ Y so that (ω, y) ∈ A. There exists f : X → S1 so that ρω (a, x) = f (T a x)f (x) for µy -almost every x ∈ X and all a ∈ Zd . By construction, ψω,f = f µy -a.e. Choose a sequence {jk } in J so that fjk → f in L2 (µy ). The sequence of functions {ψω,fjk } converges in L2 (µy ) to ψω,f = f, which is of modulus 1. By the S definition of these sets, µy ( ∞ k=1 Cω,fjk ) = 1 and thus finally µy (Cω ) = 1. Conversely, assume that µy (Cω ) = 1. This set is the union for j ∈ J of the T -invariant sets Cω,Fj . Thus we can find a sequence {Dj } of measurable subsets of X, invariant and pairwise disjoint, with Dj ⊆ Cω,fj for every j and
[
Dj = Cω .
j∈J
Define a function f on Cω by f (x) = fj (x) for x ∈ Dj . As the sets Dj are invariant, it follows from the construction that for every j and everyx ∈ Dj we have ψω,f (x) = ψω,fj (x) 6= 0. Then ψω,f 6= 0 on Cω and so µω -almost everywhere. By dividing the two sides of equation (4.36) by |ψω,f |, we get that ρω is a coboundary of (X, X, µω , T ) and that (ω, y) ∈ A. Our claim (4.37) is proved and the first part of Lemma 4.15.3 follows. 172
(2) If ρω is a coboundary of (X, X, µ, T ), there exists f : Zd × X → S1 with ρω = R f ◦ T · f , µ-almost everywhere. As µ = µy dν(y), for ν-almost every y the same relation holds µy -almost everywhere and ρω is a coboundary of (X, X, µy , T ). Conversely, assume that for ν-almost every y the cocycle ρω is a coboundary of (X, X, µ, T ). Define the sets Cω,fj and Cω as above. For ν-almost every y we have (ω, y) ∈ A and thus µy (Cω ) = 1. It follows that µ(Cω ) = 1. Use the sets Dj and the function f defined above, with the measure µ substituted for µy . The function ψω,f is defined and nonzero µ-almost everywhere and satisfies Equation (4.36) µ-almost everywhere. Therefore, ρ is a coboundary of (X, X, µ, T ).
Corollary 4.15.4. (cf. [36], Corollary C.4) Let (X, X, µ, T ), (Y, Y, ν) and µω be as in Lemma 4.15.3. Let U be a compact abelian group and ρ : X → U a cocycle. Then the subset Aρ = {y ∈ Y : ρ if a coboundary of (X, X, µy , T )} of Y is measurable. The cocycle ρ is a coboundary of (X, X, µ, T ) if and only if ν(Aρ ) = 1. We say that ρ is a quasi-coboundary if it is the sum of a coboundary and a homomorphism from Zd into the range of ρ. Lemma 4.15.5. (cf. [36], Lemma C.5) Let (X, X, µ, T ) be an ergodic system, U a torus and ρ : X → U a cocycle. If the map (a, x, x0 ) 7→ ρ(a, x) − ρ(a, x0 ) is a coboundary of (X × X, µ × µ, T × T ), then ρ is a quasi-coboundary.
173
Lemma 4.15.6. (cf. [36], Lemma C.6) Let (X, X, µ, T ) be an ergodic Zd -system, U a compact abelian group and ρ a cocycle. Assume that the map (x, x0 ) 7→ ρ(x) : X × X → U is a coboundary on (X × X, µ × µ, T × T ). Then ρ is a coboundary. We present the proof from [36] with obvious modifications. Proof. By Lemma 4.15.1, we can reduce to the case where U = S. Write (Z, t1 , . . . , td ) for the Kronecker factor of X and π : X → Z for the natural projection. By hypothesis, there exists a function f : X × X → S with f (T a x, T a x0 )f (x, x0 ) = ρ(a, x) for all a ∈ Zd , x, x0 ∈ X. The function defined on X × X × X by (x, x0 , x00 ) 7→ f (x, x0 )f (x, x00 ) is invariant under T × T × T and thus is measurable with respect to Z × Z × Z. It follows that the function f is measurable with respect to X × Z. Taking the Fourier transform of f with respect to the second variable, we can write f (x, x0 ) =
X
gχ (x)χ(x0 ).
(4.38)
b χ∈Z
Then f (x, x0 )f (x, x00 ) =
X
gχ (x)gθ (x)χ(π(x0 ))θ(π(x00 )).
b χ,θ∈Z
As this function is invariant under T × T × T, by uniqueness of the Fourier transform b a = (n1 , . . . , nd ) ∈ Zd we get that for every χ, θ ∈ Z, gχ (T a x)gθ (T a x)gα (x)gθ (x) = χ(tn1 1 · · · tnd d )θ(tn1 1 · · · tnd d ). The function x 7→ gχ (x)gθ (x) is an eigenfunction of X for the eigenvalue (n1 , . . . , nd ) 7→ χ(t1n1 , . . . , tnd d )θ(tn1 1 , . . . , tnd d ) so there exists a constant cχ,θ with gχ (x)gθ (x) = cχ,θ χ(π(x))θ(π(x)). 174
b there By fixing θ, we see that there exists a function φ on X and for every χ ∈ U exists a constant cχ so that gχ (x) = cχ φ(x)χ(π(x)). Using the values of the functions gχ in equation (4.38), we see that there exists a function g on Z with f (x, x0 ) = φ(x)g(π(x) − π(x0 )). As f is of modulus 1, the functions g and φ have constant modulus and so we can assume that |φ| = 1. Now ρ(a, x) = φ(T a x)φ(x).
Lemma 4.15.7. (cf. [36], Lemma C.7) Let (X, X, µ, T ) be an ergodic system, 1 ≤ l ≤ k integers and let α be an l-face of Vk . Let U be a compact abelian group and let [k]
ρ : Zd × X [l] → U a cocycle. If the cocycle ρ ◦ ξα : Zd × X [k] → U is a coboundary of (X [k] , X[k] , µ[k] , T [k] ), then ρ is a coboundary of (X [l] , µ[l] , T [l] ). The proof is virtually identical to the proof for the case d = 1 in [36]. We reproduce it for completeness. Proof. We begin by the case l = 0. Here ρ : Zd × X → U. Assuming that for some vertex ε of Vk the cocycle (a, x) 7→ ρ(a, xε ) is a coboundary of X [k] , we have to show that ρ is coboundary on X. By permuting coordinates, we can restrict to the case that ε is the vertex 0. We proceed by induction on k. For k = 1, the result is exactly Lemma 4.15.6. Take k ≥ 1 and assume that the result holds for k. Assume that the cocycle x 7→ ρ(x0 ) is a coboundary of X [k+1] . We use the ergodic decomposition (4.3) of µ[k] and the formula (4.4) for µ[k+1] . By Corollary 4.15.4, for almost every ω the cocycle (a, x) 7→ ρ(a, x0 ) is 175
[k]
a coboundary on the Cartesian square of (X [k] , µω , T [k] ). This cocycle depends only on the first coordinate of this square and by Lemma 4.15.6 we get that the map (a, x0 ) 7→ [k]
ρ(a, x00 ) is a coboundary of the system (X [k] , µω T [k] ). As this holds for almost every ω, the map (a, x0 ) 7→ ρ(a, x00 ) is a coboundary of the system (X [k] , X[k] , µ[k] , T [k] ) by Corollary 4.15.4. By the induction hypothesis, ρ is a coboundary of X. This completes the proof when l = 0. Consider the case that l > 0. We use the ergodic decomposition given by formula (4.4) for µ[l] and by Lemma 4.3.2 we get µ
[k]
Z =
[k−l] (µ[k] dPl (ω). ω )
Ωl
We use Corollary 4.15.4 and the first part of the proof with k − l substituted for k [l]
and (X [l] , µω , T [l] ) substituted for (X, X, µ, T ). The result follows.
4.15.2
Cocycles and groups of automorphisms.
Let (X, X, µ) be a probability space, G a compact abelian group and (g, x) 7→ g · x an action of G on X by measure preserving transformations. This action is said to be free if there exists a probability space (Y, Y, ν) and a measurable bijection j : Y × G → X mapping ν × mG to µ, with j(y, gh) = g · j(y, h) for y ∈ Y and g, h ∈ G. Lemma 4.15.8. (cf. [36], Lemma C.8) Let {Sg : g ∈ G} be a free action of the compact abelian group G on the probability space (X, X, µ) and let g 7→ φg be a measurable map from G to the space of measurable functions from X to S so that φgh = φg · (φh ◦ g) for every g, h ∈ G. 176
(4.39)
¯ Then there exists φ so that φg = (φ ◦ Sg ) · φ. Lemma 4.15.8 is identical to Lemma C.8 in [36], so we omit the proof. Lemma 4.15.9. (cf. [36], Lemma C.9) Let (X, X, µ, T ) be an ergodic system, U a compact abelian group and let (u, x) 7→ u · x be a free action of U on X by automorphisms. Let ρ be a cocycle of (X, X, µ, T ) with the property that ρ ◦ Su − ρ is a coboundary for every u ∈ U. Then there exists an open subgroup U0 of U and a cocycle ρ0 , cohomologous to ρ, with ρ0 ◦ Su = ρ0 for every u ∈ U0 . Proof. By hypothesis, for every u ∈ U, there exists f : X → T with ρ(a, u · x) − ρ(a, x) = f (T a x) − f (x) for all a ∈ Zd , x ∈ X.
(4.40)
Write Su,f for the measure preserving transformation of X × T given by Su,f (x, t) = (u · x, t + f (x)). Let V denote the group of all such transformations, and let K be the subset of V consisting of the transformations Su,f , where u, f satisfy equation (4.40). Then K is a closed subgroup of V. Let p denote the natural projection Su,f 7→ u. By the argument in the proof of Lemma C.9 in [36], there is an open subgroup U0 of U such that p−1 (U0 ) is abelian, and the restriction of p to p−1 (U0 ) admits a cross section which is a continuous group homomorphism. This cross section has the form u 7→ Su,fu and u 7→ fu is a continuous map from U0 to TX , with ρ(a, u · x) − ρ(a, x) = fu (T a x) − fu (x) for all u ∈ U0 fuv (x) = fu (x) + fv (u · x) for all u, v ∈ U0 .
(4.41) (4.42)
Since the action of U on X is free, by Equation (4.42) and Lemma C.8 of [36], there exists f : X → T so that fu = f ◦ u − f for every u ∈ U0 . Write ρ0 (a, x) = 177
ρ(a, x) − f (T a x) + f (x). Then ρ0 is a cocycle cohomologous to ρ and by equation (4.41), ρ0 (a, u · x) = ρ0 (a, x) for u ∈ U0 .
Lemma 4.15.10. (cf. [36], Lemma C.10) Let (X, X, µ, T ) be an ergodic Zd -system, U a compact abelian group and (u, x) 7→ u·x a free action of U on X by automorphisms. Let ρ : Zd × X → U be a cocycle, so that ρ ◦ u − ρ is a quasi-coboundary for every u ∈ U. Then there exists a closed subgroup U1 of U so that U/U1 is toral and there exists a cocycle ρ0 , cohomologous to ρ, with ρ0 ◦ Su = ρ0 for every u ∈ U1 . Proof. We write Su x for u · x. By hypothesis, for every u ∈ U, there exists f : X → T, φ ∈ hom(Zd , T) with ρ(a, Su x) − ρ(a, x) = f (T a x) − f (x) + φ(a) for all a ∈ Zd .
(4.43)
We write Su,f for the transformation of X × T given by Su,f (x, t) = (Su x, t + f (x)), and let U n B(X) denote the group of all such Su,f . Let H be the subset of U n B(X) consisting of transformations Su,f so that u and f satisfy equation (4.43) for some φ ∈ hom(Zd , T). Clearly H is a closed subgroup of U n B(X). By hypothesis, the projection p : H → U is onto and its kernel is {S1,f : f is an eigenfunction of T }. Thus, ker(p) is homeomorphically isomorphic to the group A(Z) of affine functions on the Kronecker factor Z of X. This group can be identified with T ⊗ Zb and in particular, it is locally compact. By Corollary A.2 of [36], H is locally compact. A direct computation7 shows that the commutator subgroup H0 of H is included in 7
−1 −1 Su,f Sv,g Su,f (x, t) = (x, t + f (x) − f (v · x) + g(u · x) − g(x)). Using the identities from equation Sv,g (4.43) we see that f (x) − f (v · x) + g(u · x) − g(x) is T -invariant, and therefore constant.
178
the subgroup T of H. Thus K = H/T is a locally compact abelian group. We write q : K → U for the continuous group homomorphism induced by p. For Su,f ∈ H, the homomorphism φ appearing in equation (4.43) is well defined and the map ψ : Su,f 7→ φ induces a continuous group homomorphism from H to hom(Zd , T). This homomorphism is trivial on T and it induces a character χ of K = H/T. By the structure theorem for locally compact abelian groups, K admits an open subgroup L isomorphic to K ⊕ Rm , where K is a compact abelian group and d ≥ 0 is an integer. We identify L with K ⊕ Rm and write K0 = K ∩ ker(χ) and U0 for the closed subgroup q(K0 ) of U. For u ∈ U0 , there exists by definition f : X → T so that Su,f ∈ H and ψ(Su,f ) = 0. In other words, u and f satisfy equation (4.43) with φ = 0 for all a. By Lemma 4.15.9, there exist an open subgroup U1 of U0 and a cocycle ρ0 , cohomologous to ρ, with ρ0 ◦ u = ρ0 for every u ∈ U1 . It remains to show that U/U1 is a toral group. As L is open in K and q is an open map (being a homomorphism of locally compact groups), q(L) is an open subgroup of U and thus U/q(L) is finite. Now q(L)/q(K) is a quotient of L/K = Rm and is compact and thus is a torus. Also, K/K0 is isomorphic to χ(K), which is a closed subgroup of T and so is equal to T or finite, and q(K)/U0 is a quotient of K/K0 and so is either finite or isomorphic to T. Finally, U0 /U1 is open and the proof is complete.
179
4.15.3
Variations of Lemma 4.15.5.
We will need a variation of Lemma 4.15.5 due to Ziegler ([68]). Note that we consider actions of an arbitrary countable group Γ in the following theorem. If T is an action of Γ on a set Y, by ker T we mean {γ ∈ Γ : Tγ = id}, where id is the identity map on Y. Theorem 4.15.11. (cf. [68], Theorem 3.8.) Let Γ be a countable discrete group, and let Y = (Y, Y, µ, T ) be an ergodic Γ-system. Let H be a compact abelian group, and let W = Y ×ρ H be an ergodic extension of Y by a connected abelian group. Let ν be a measure corresponding to a k-fold self-joining of Y, and let µ be the relative product of µ × mH over ν.8 Let F : Y × H k → S be a measurable function. Let k ∈ N, and let σi : Γ × Y × H → S, i = 1, . . . , k be measurable functions. Suppose k Y
σi (γ, y, hi ) =
i=1
F (Tγ x, h1 + ρ(γ, y), . . . , hk + ρ(γ, y)) F (y, h1 , . . . , hk )
for all (γ, y, h1 , . . . , hk ). Then for i = 1, . . . , k there exist measurable functions fi : Γ × Y → S, Fi : Y × H → S such that σi (γ, y, h) = fi (γ, y)
Fi (Tγ (y, h)) . Fi (y, h)
In the proof we use the notion of the Mackey group of an isometric extension, and the fact that isometric extensions are spanned by finite-rank submodules. Notation: If V is a d-dimensional vector space over C, we write Ud for the group of unitary operators on V, and write C(Ud ) for the center of Ud . Let PV to denote 8
That is,
R Nk
i=1
fi dµ =
R Qk
i=1
R
fi (yi , hi ) dmH (hi ) dν(y1 , . . . , yk ).
180
the corresponding projective space, and write P : Ud → Ud /C(Ud ) for the natural quotient map. We will use the following Lemma from [68]. Lemma 4.15.12. ([68], Lemma 3.5) Let H be a compact abelian group, and A : H → Ud a measurable function. If P ◦ A is a homomorphism, then A(H) is a commuting set of matrices. Proof of Theorem 4.15.11. For each i, consider the system Xi = W ×σi S, and let X be the relatively independent product of the Xi over Y, with relative product measure µX . Define a function F˜ by F˜ (y, h1 , . . . , hk , ζ1 , . . . ζk ) = F (y, h1 , . . . , hk )
k Y
ζi−1
(4.44)
i=1
One can check by computation that F˜ is invariant under the action of T and therefore N ˆ ˆ is measurable with respect to ki=1 Y i , where Yi is the maximal isometric extension of Y in Xi for each i. Recall that isometric extensions are spanned by finite rank modules. By the description given in [11], the space of T -invariant functions in L2 (µX ) has an orthonormal basis spanned by mutually orthogonal modules of the form M1 ×· · ·×Mk , where each Mi is a finite rank module over L∞ (Y ). Each Mi has an orthonormal basis Bj = {ψi,j }dj=1 , with the property that ψi,1 ◦ Tγ (y, hi , ζi ) ψi,1 (y, hi , ζi ) .. .. = U (γ, y) , . . ψi,d ◦ Tγ (y, hi , ζi ) ψi,d (y, hi , ζi ) 181
where each U (γ, y) is a unitary d×d matrix. Then B1 ⊗B2 ⊗· · ·⊗Bk is an orthonormal N ˆ basis for L2 ( ki=1 Y i ), so we can write F˜ (y, h1 , . . . , hk , ζ1 , . . . ζk ) =
X
gj,n (y)
j,n (n)
k Y
(n)
ψi,j (y, hi , ζi ).
i=1 (n)
Write ψi,j,k (y, hi ) for the Fourier transform of ψi,j : Z (n) (n) ψi,j,k = ψi,j (y, h, ζ)ζ −k dζ,
(n) ψi,1 (y, hi , ζi )
~ (n) for the vector-valued function (y, hi , ζi ) 7→ and write ψ i,j
.. .
. By Equa (n) ψi,d (y, hi , ζi )
(n)
tion (4.44), for each i, there exists j, n so that ψi,j,−1 is not identically zero. Computing ψi,j,−1 ◦ Tγ , we find ~ i,j,−1 . ~ (n) (Tγ (y, hi )) = Ui,j,n (γ, y)ψ| σi−1 (γ, y, hi )ψ i,j,−1 Dropping the indices for now, we write ~ γ (y, h)) = U (γ, y)ψ(y, ~ h). σ −1 (g, y, h)ψ(T
(4.45)
~ h) : h ∈ H}. Then For each y, let Vy be the subspace of Cd spanned by {ψ(y, y 7→ dim(Vy ) is a measurable function, and invariant under the action of T. By the ergodicity of T, dim(Vy ) is constant almost everywhere, so write d for this constant. ~˜ be the projection of ψ ~ on PV, and let U˜ denote the projection of U on PUd . Let ψ Form the group extension W ×U˜ PUd . Then ~˜ γ (y, h)) = U˜ (γ, y)ψ(y, ~˜ h). ψ(T 182
Claim. The group extension W ×U˜ PUd has trivial Mackey group. To prove this claim, we assume that Y has the structure of a compact metric space, ~˜ is continuous, with and apply Lusin’s theorem to find a compact C ⊆ Y × H where ψ µY × µH (C) > 1 − ε for a given ε > 0. Let K be the Mackey group of the extension W ×U˜ PUd , and let k ∈ K. We will show by way of contradiction that k = id, so let B, B 0 be disjoint open neighborhoods of id ∈ PUd and k, respectively. Let η be an ergodic component of W ×U˜ PUd . There is a map Φ : W → PUd so that η is the image of Haar measure under the map Φ∗ given by (w, u) 7→ (w, Φ(w)−1 u). Let (y, h) ∈ C, and let R be a neighborhood of (y, h) such that |U (g, y)| < δ whenever Tγ (y, h) ∈ C ∩ R. By ergodicity, if η(Φ∗ [(C ∩ R) × B 0 ]) > 0, then there exists γ such that η(Φ∗ [(C ∩ R) × B] ∩ Tγ Φ∗ [(C ∩ R) × B 0 ]) > 0. But Tγ Φ∗ [(C ∩ R) × B 0 ] is disjoint from Φ∗ [(C ∩ R) × B] for sufficiently small δ. This is the desired contradiction, establishing the claim. ˜ : Y → PUd such By the claim, (γ, y) 7→ U˜ (γ, y) is a coboundary, so there exists M ˜ (Tγ (y, h)) = U (γ, y)M ˜ (y, h). We then have, for all h0 ∈ H, that M ˜ (Tγ (y, h + h0 )) = U (γ, y)M ˜ (y, h + h0 ) M ˜ (Tγ (y, h)) = U (γ, y)M ˜ (y, h) M so that ˜ (y, h), ˜ (Tγ (y, h + h0 ))−1 M ˜ (Tγ (y, h)) = M ˜ (y, h + h0 )−1 M M
183
(4.46)
˜ (y, h + h0 )−1 M ˜ (y, h) is a T -invariant function of (y, h). By erwhich means that M godicity, it is constant (depending on h0 ), which we denote by ˜ 0) = M ˜ (y, h + h0 )−1 M ˜ (y, h). A(h By Fubini’s theorem, and the change of variables (h, h0 ) 7→ (h, h0 − h), there exists h0 such that ˜ (y, h) = M ˜ (y, h0 )A˜−1 (h − h0 ), M
(4.47)
for a.e. (y, h). ˜ 0 ) is a homomorphism H → Ud . To see this, write The function A(h ˜ 0 + h00 ) = M ˜ (y, h + h0 + h00 )−1 M ˜ (y, h) A(h ˜ (y, h + h0 + h00 )−1 M ˜ (y, h + h0 )M ˜ (y, h + h0 )−1 M ˜ (y, h) =M ˜ 00 )A(h ˜ 0 ). = A(h Let P : Ud → PUd denote the natural projection. Then there is a measurable function A : H → Ud so that P ◦ A = A˜ and ˜ A(H) ⊆ P −1 A(H). By Lemma 4.15.12, A(H) is a commuting set of matrices. Substituting equation (4.47) into equation (4.46) we find ˜ (Tγ y, h0 )A˜−1 (h + ρ(γ, y) − h0 ) = M ˜ (Tγ y, h + ρ(g, y)) M ˜ (y, h) = U˜ (γ, y)M ˜ (y, h0 )A˜−1 (h − h0 ). = U˜ (γ, y)M 184
Thus ˜ (Tγ y, h0 )A˜−1 (ρ(γ, y))M ˜ −1 (y, h0 ), U˜ (γ, y) = M or U (γ, y) = M (Tγ y, h0 )A(−ρ(γ, y))M −1 (y, h0 )d(γ, y), where d(γ, y) is a scalar matrix. As A(H) is a commuting set, it is simultaneously diagonalizable: A(h) = N −1 D(h)N. Therefore U (γ, y) = M (Tγ y, h0 )N −1 D(−ρ(γ, y))N M −1 (y, h0 )d(γ, y). Let M 0 (y) = M (y, h0 ). Substituting into equation (4.45), we find −1 ~ y, h + ρ(γ, y)) = D(−ρ(γ, y))d(γ, y)N M 0−1 (y)ψ(y, ~ h). σ −1 (γ, y, h)N M 0 (Tγ y)ψ(T
~ gives the desired result. Considering coordinates of ψ
We actually want a version of Theorem 4.15.11 which is “measurable in parameters.” This is the content of the next corollary. If (Y, Y, ν) is a measure space then B(Y, S) denotes the collection of maps Y → S. With the topology of convergence in measure and the group operation of pointwise addition,B(Y, S) is a Polish group. Corollary 4.15.13. Let Γ be a countable discrete group, and let Y = (Y, µ, T ) be an ergodic Γ-system. Let H be a compact abelian group, and let W = Y ×ρ H be an ergodic extension of X by a connected abelian group. Let Ω be a probability space, let 185
k ∈ N, and let σi,ω : Γ × Y × H → S, i = 1, . . . , k be measurable functions, so that ω 7→ σi,ω is measurable. Suppose that for each ω ∈ Ω, there exists a function F such that k Y
σi,ω (γ, y, hi ) =
i=1
F (Tγ x, h1 + ρ(γ, y), . . . , hk + ρ(γ, y)) F (y, h1 , . . . , hk )
for all (γ, y, hi ). Then for i = 1, . . . , k there exist measurable functions ki,ω : Y → S, Giω : Y × H → S such that σi,ω (γ, y, h) = ki,ω (g, y)
Gi,ω (Tγ (y, h)) , Gi,ω (y, h)
and the maps ω 7→ ki,ω , ω 7→ Gi,ω are measurable maps from ω to B(Y, S). For the proof, we need the following adaptation of [68], Lemma 3.10. Lemma 4.15.14. Let Γ be a countable discrete group, let Y = X ×ρ H be measure preserving Γ-system which is an ergodic abelian extension of X, and F : X × H → S, g : Γ × X → S measurable functions such that g(γ, x) =
F ◦ Tγ (x, h) F (γ, x)
ˆ and k : Z → S such that for all γ ∈ Γ. Then there exists χ ∈ H F (x, h) = k(x)χ(h). ˆ Proof. Expand F as a Fourier series over G: F (x, h) =
X ˆ χ∈G
186
kχ (x)χ(x).
ˆ γ ∈ Γ, we have For all χ ∈ G, kχ (Tγ x)χ(h)χ(ρ(γ, x)) = g(γ, x)kχ (x)χ(h). Since T is ergodic, |kχ | is constant almost everywhere, for all χ. Since |F | = 1, there b such that kχ 6= 0. If there are distinct χ, χ0 ∈ G b such that kχ 6= 0, then exists χ ∈ G χ0 χ(ρ(γ, x)) is the coboundary kχ (Tg x)kχ (x)kχ0 (Tg x)kχ0 (x). Since ρ is an ergodic cocycle, (χ0 χ) ◦ ρ is a coboundary iff χ0 χ = 1. Hence F (x, h) = k(x)χ(h).
Corollary 4.15.13 now follows from Theorem 4.15.11 and the next lemma, which is essentially [68], Lemma 3.12. Lemma 4.15.15. Let (Y, Y, ν, S) be an ergodic abelian extension of the Γ-system (Z, Z, η, T ), and let (Ω, B, P ) be a probability space. If ω 7→ fω is a Borel measurable function from Ω to B(Γ × Y, S), and for all ω ∈ Ω, there are functions gω ∈ B(Γ × Z, S), Fω ∈ B(Γ × Y, S) such that fω (γ, y) = gω (z)
Fω (Sγ y) for all γ ∈ Γ. Fω (y)
(4.48)
Then there is a µ measurable choice of ω 7→ gω , ω 7→ Fω satisfying (4.48). Proof. Let B(Γ × Z, S) denote the subset of B(Γ × Y, S) consisting of those maps f : Γ × Z × H → S that do not depend on the H coordinate. Then B(Γ, Z, S) is a
187
¯ = B(Γ × Y, S)/B(Γ × Z, S). By Theorem 1.2.4 closed subgroup of B(Γ, Y, S). Let B ¯ → B. Equation (4.48) implies that of [3], there is a measurable section s : B F¯ω (Sγ y) f¯x (γ, y) = ¯ for all γ ∈ Γ. Fω (y) ¯→B ¯ by ϕ(f¯)/f¯. If ϕ(f¯) = ϕ(¯ Define ϕ : B g ), then for some function h ∈ B(Γ×Z, S1 ), f (Sγ y) g f (y) g
By Lemma 4.15.14, this implies that
f g
= h(γ, z).
belongs to a countable set, up to multiplication
¯ is a measurable set and there by a function depending only on z. By Lusin [46], ϕ(B) ¯ →B ¯ such that is a measurable function ψ : ϕ(B) ϕ ◦ ψ = id |ϕ(¯b) . If ψ(f¯ω ) = F¯ω , then F¯ω (Sγ y) f¯ω (γ, y) = φ ◦ ψ(f¯ω )(γ, y) = . F¯ω The composition ω 7→ fω 7→ f¯ω 7→ F¯ω 7→ s(F¯ω ) gives a measurable choice of Fω , and gω is a quotient of measurable functions.
188
CHAPTER 5 MULTIPARAMETER CORRELATION SEQUENCES
5.1
Introduction
In this chapter we analyze correlation sequences such as
R
f ·f ◦T n S m ·f ◦T n−m S n+2m dµ,
where T and S are commuting, measure preserving transformations of a probability space (X, X, µ), and f ∈ L∞ (µ). Our results generalize Theorem 1.9 from [8], which we repeat here. First we define the term “nilsequence.” Definition 5.1.1. Let k ≥ 1 be an integer and let X = G/Λ be a k-step nilmanifold. Let φ be a continuous real (or complex) valued function on X and let a ∈ G and e ∈ X. The sequence {φ(an · e)} is called a basic k-step nilsequence. A k-step nilsequence is a uniform limit of basic k-step nilsequences. Theorem 5.1.2. ([8], Theorem 1.9) Let (X, X, µ, T ) be an ergodic Z-system, let f ∈ L∞ (µ) and let k ≥ 1 be an integer. The sequence Z If (k, n) :=
f · f ◦ T n · · · · · f ◦ T kn dµ
is the sum of a sequence tending to 0 in uniform density and a k-step nilsequence.
189
By a sequence tending to 0 in uniform density we mean a sequence a(n) with N −1 X 1 |a(n)| = 0. lim N −M →∞ N − M n=M
R The case k = 1 is a consequence of Bochner’s Theorem: If (1, n) = f · f ◦ T n dµ = R exp(inθ) dν(θ) for some Borel measure ν on T. Writing ν = νc + ν0 , where νc is T R R purely atomic and ν0 is atomless, we have If (1, n) = exp(inθ) dνc (θ)+ exp(inθ) dν0 (θ). R One can easily show that r(n) = exp(inθ) dν0 (θ) tends to 0 in density, and that R a(n) := exp(inθ) dνc (θ) can be uniformly approximated by trigonometric polynomiP als λ∈F exp(inλ), where F is a finite subset of the atoms of νc . Since every trigonometric polynomial is of the form φ(Rαn x) for some group rotation (Z, Z, m, Rα ), this discussion demonstrates Theorem 5.1.2 in the case k = 1. Even for the case k = 2, the theory of characteristic factors is needed to establish Theorem 5.1.2, in the sense that no proof is known which does not use said theory. In this chapter, we generalize Theorem 5.1.2 to analyze correlation sequences of the form Z If (M1 , . . . , Mk , a) :=
f · f ◦ T M1 a · · · · f ◦ T Mk a , dµ,
(5.1)
where (X, X, µ, T ) is an ergodic Zd -system, and the Mi are endomorphisms of Zd satisfying the following restrictions: the image of each Mi has finite index in Zd , and the image of each Mi − Mj , i 6= j has finite index in Zd . Our proof will follow the broad outline of the proof of Theorem 5.1.2 in [8], but will be simplified for two reasons: (i) we do not attempt to characterize those sequences (5.1) k R which are frequently as large as f dµ , and (ii) we can apply Leibman’s recent results ([44]) on orbits of measures supported on subnilmanifolds of a nilmanifold. 190
Before we state our main result, we must make a distinction between “zero density” and “uniform zero density” in Zd , and we must define a notion of nilsequence compatible with actions of Zd . We say that a sequence u : Zd → C tends to 0 in density if 1 N →∞ (2N + 1)d lim
X
u(a1 , . . . , ad ),
−N ≤ai ≤N,i=1,...d
and we say that u tends to 0 in uniform density if 1 N −M →∞ (N − M )d lim
X
u(a1 , . . . , ad ) = 0.
M ≤ai ≤N −1 1≤i≤d
Definition 5.1.3. Let k ≥ 1 be an integer and let X = G/Λ be a k-step nilmanifold. Let φ be a continuous real (or complex) valued function on X and let t1 , . . . , td ∈ G be commuting elements of G, and e ∈ X. The sequence {φ(an · e)} is called a basic k-step d-nilsequence. A k-step d-nilsequence is a uniform limit of basic k-step nilsequences. We will usually refer to d-nilsequences as “nilsequences,” the prefix d being clear from the context. The main result of this chapter is the following theorem. Theorem 5.1.4. Let (X, X, µ, T ) be an ergodic Zd system, where (X, X, µ) is a probability space and T is an ergodic action of Zd . Let M1 , . . . , Mk be endormorphisms of Zd such that the image of each Mi , Mi − Mj , i 6= j has finite index. Let f0 , f1 , . . . , fk ∈ L∞ (µ). Then Z I(f1 , . . . , fk , M1 , . . . , Mk , a) :=
f0 ·
k Y
fi ◦ T Mi a dµ
i=1
is the sum of a k-step d-nilsequence and a sequence that goes to 0 in density. 191
The first steps of our proof will follow Section 4 of [8]. The main technical tool in the proof will be Theorem 4.1.2, which will play the role Theorem 10.1 of [36] plays in [8]. We say that that a Zd -system (X, X, µ, T ) is a k-step d-nilsystem if X = G/Λ, where G is a k-step nilpotent Lie group, Λ is a discrete, co-compact subgroup, and T is an action of Zd on X by T (a1 ,...,ad ) x = ta11 · · · tadd x, where each ti ∈ G. The next two sections contain the proof of Theorem 5.1.4. In section 5.2.1, we reduce to the case where (X, X, µ, T ) is an inverse limit of k-step d-nilsystems.
5.2
Reduction to nilsystems.
Here we apply Theorem 4.1.2 to prove that it is enough to consider only inverse limits of nilsystems in the proof of Theorem 5.1.4. To accomplish this, we need a description of the ergodic components of the systems (X, X, µ, TMi ), where TMi is the Zd -action given by a 7→ T Mi a . Let (X, X, µ, T ) be an ergodic Zd -system, and M : Zd → Zd an endormorphism with finite index image. Claim. Suppose that f : X → C is supported on an ergodic component A of TM . Let µs be an ergodic component of (X × X, µ × µ, T × T ). Then f ⊗ f is supported on an ergodic component of (X × X, µs , TM × TM ). Proof. Consider (X, X, µ, T ) as an extension of its Kronecker factor (Z1 , Z1 , µ1 , T ), and let π denote the factor map. Given an ergodic component A of (X, X, µ, TM ), we see that 1A has a finite orbit under T, and therefore A ∈ Z1 . It follows that 192
if A = {Ai }ni=1 are the ergodic components of (X, X, µ, TM ), and B = {Bi }ni=1 are the ergodic components of (Z, µ, TM ), then A = {π −1 (Bi )}ni=1 . Note that the ergodic components of (Z, µ, TM ) are cosets of some finite-index subgroup W of Z. Recall that ergodic components of (X × X, µ × µ, T × T ) are sets of the form {(x, y) : (π(x), π(y)) ∈ (0, s)+∆Z }, where ∆Z is the diagonal subgroup of Z ×Z. For µ1 -almost every s, the system (X ×X, µs , T ×T ) is ergodic. The system Zs = (Z1 ×Z1 , µs , T ×T ) is isomorphic to Z0 = (∆Z , µ0 , T × T ), with the isomorphism given by (x, y) 7→ (x, y) + (0, s). It follows that the ergodic components of Zs are sets of the form (0, s) + (z, z) + ∆W , and by the preceding paragraph the ergodic components of (X × X, µs , TM × TM ) are sets of the form {(x, y) : (π(x), π(y)) ∈ (0, s) + ∆B } where B is an ergodic component of (Z1 , µ, TM ). Finally, if A is an ergodic component of (X × X, µ × µ, TM × TM ), then for a fixed s, (π(a), π(a0 )) ∈ (0, s) + (z, z) + ∆W implies that π(a) ∈ z + W, so that a ∈ A implies π(a) ∈ z + W. Then (π(a), π(a0 )) ∈ (0, s) + (z 0 , z 0 ) + ∆W implies that π(a) ∈ z 0 + W, so z 0 + W = z + W, and (0, s) + (z, z) + ∆W = (0, s) + (z 0 , z 0 ) + ∆W . This shows that A × A meets at most one ergodic component of µs . So if f is supported on an ergodic component of (X, X, µ, TM ) then for each s, f is supported on an ergodic component of (X × X, µs , TM × TM ). This proves the claim.
Lemma 5.2.1. Let M1 , . . . , Mk be endomorphisms of Zd such that Mi Zd , (Mi −Mj )Zd has finite index in Zd for each i, j. Let fi be bounded functions supported on atoms of the algebra generated by the ergodic components of the TMi , TMi −Mj , i 6= j. Let Z Jk (a) =
f0 ·
k Y i=1
193
fi ◦ TiMi a dµ.
Then Z f0 ·
R(a) :=
k Y
fi ◦
TiMi a
Z dµ −
i=1
E(f0 |Zk )
k Y
E(fi |Zk ) ◦ TiMi a dµ
i=1
goes to 0 in uniform density. Proof. We will show that |Jk (a)|2 goes to 0 in density under the assumption that E(fj |Zk ) = 0 for some j. Assume, for simplicity, that the fi are real-valued. We have 2
Z
|Jk (a)| =
f0 ⊗ f0 ·
k Y
fi ⊗ fi (TiMi a x, TiMi a y) dµ × µ(x, y).
i=1
Write µ×µ =
R
position µ[k+1]
µs dµ1 (s) for the ergodic decomposition of µ1 (s). Recalling the decomR [k] RR R k [k] 2k+1 = µs dµ1 (s), we have |||f |||k+1 = f dµs dµ1 (s) = |||f |||2s,k dµ1 (s).
Averaging |Jk (a)|2 and applying Fatou’s Lemma yields lim sup N →∞
1 X |Jk (a)|2 ≤ |FN | a∈F n Z Z k Y 1 X f0 ⊗ f0 · fi ⊗ fi (TiMi a x, TiMi a y) dµs dµ1 (s). lim sup N →∞ |FN | i=1 a∈F
(5.2)
n
We apply Theorem 4.11.1 to the system (X × X, µs , T × T ) to conclude that k Y 1 X lim sup f0 ⊗ f0 · fi ⊗ fi (TiMi a x, TiMi a y) dµs ≤ C min|||fi |||s,k . i N →∞ |FN | i=1 a∈F N
The assumption that E(fj |Zk ) = 0 is equivalent to the assumption that |||fj |||k+1 = 0, R k which means that |||f |||2s,k dµ1 (s) = 0, and that |||fj |||s,k = 0 for µ1 -almost every s. P Then (5.2) implies that lim supN →∞ |F1N | a∈Fn |Jk (a)|2 = 0.
194
5.3
Nilmanifolds and orbits of diagonal measures.
By Lemma 5.2.1, a correlation sequence Z f0 ·
I(M1 , . . . , Mk , f1 , . . . , fk , a) =
k Y
fi ◦ T Mi a dµ
i=1
can be replaced by a correlation sequence I (M1 , . . . , Mk , f˜1 , . . . , f˜k , a) = 0
Z f0 ·
k Y
fi ◦ S Mi a dν,
i=1
where (Y, Y, ν, S) is an inverse limit of nilsystems, introducing an error which tends to zero in uniform density. So in the proof of Theorem 5.1.4, we will assume that (X, X, µ, T ) is an inverse limit of nilsystems. Let us fix M1 , . . . , Mk , f1 , . . . , fk ∈ L∞ (µ), and abbreviate I(M1 , . . . , Mk , f1 , . . . , fk , a) by I(a). Our immediate goal will be establish a decomposition I(a) = ψ(a) + φ(a) + b(a), where ψ(a) is a basic k-step d-nilsequence, φ(a) tends to zero in density, and |b(a)| < ε uniformly, for a given ε > 0. The main tool for this step is the following Theorem from [44]. We first state the definition of “polynomial sequence” from [44]. Definition 5.3.1. If G is a group, a map g : Zd → G is an (m-parameter) polynomial p (n)
sequence in G if g(n) = a11
p (n)
· · · ar r
, where a1 , . . . , ar ∈ G and p1 , . . . , pr are
polynomials Zd → Z. Theorem 5.3.2. ([44], Theorem 9) Let X = G/Γ be a connected nilmanifold, and let Y be a subnilmanifold of X. Let g : Zd → G be a polynomial sequence, let f ∈ 195
C(X), and let φ(n) =
R g(n)Y
f, n ∈ Zm . There exists a basic nilsequence ψ such that
φ(n) − ψ(n) → 0 in density. Here
R g(n)Y
f is an abbreviation for
R
f (g(n) · x) dνY , where νY is Haar measure of
the subnilmanifold Y. The sequences ta1d · · · · · takd which correspond to our Zd actions on nilmanifolds are polynomial sequences in the sense of [44]. See [44] for more details. Note that the conclusion is not that φ(n) − ψ(n) → 0 in uniform density. We now pass to a factor of X which is a nilsystem, at the cost of introducing a uniformly bounded error in our estimation of I(a). This can be accomplished by replacing each fi appearing in the definition of I(a) by a continuous function gi defined on a factor (Z, Z, ν, T ) of X which is a nilsystem, say Z = G/Γ, where G is a k-step nilpotent Lie group, Γ is discrete and co-compact, and T a z = ta11 · · · tadd · z for z ∈ Z. Let ε > 0, and write I 0 (a) =
R Z
g0
Qk
i=1
gi ◦ T Mi a dν, with the gi ∈ C(Z) chosen so
that |I 0 (a) − I(a)| < ε for all a. Consider the correlation sequence I 0 (a) as a sequence R of the form g(a)Y f dσ, where Y is the diagonal of Z k+1 , and σ is the Haar measure of Y. We can take Pd
g(a) = (t1
i=1
(1)
mi1 ai
Pd
· · · td
i=1
(1)
mid ai
Pd
, . . . , t1
i=1
(k)
mi1 ai
Pd
· · · t1
i=1
(1)
mid ai
)
where mlij is the (i, j)-th entry of the matrix representing Ml . Then g(a) is a polynomial sequence in the sense of [44], and Y is a subnilmanifold of Z, so by Theorem 5.3.2, the sequence I 0 (a) can be written in the form ψ(a) + φ(a), where ψ(a) is a basic
196
k-step d-nilsequence, and φ(a) tends to zero in density. We can therefore write the original correlation sequence I(a) in the form I(a) = ψ(a) + φ(a) + b(a), with |b(a)| < ε, as desired. We proceed as in Section 7 of [8]. For each r ∈ N, write I(a) = ψr (a) + φr (a) + br (a), where ψr (a) is a basic k-step d-nilsequence, φr (a) tends to zero in density, and |br (a)| < 1r . For any r, s we then have |ψr (a) − ψs (a)| = |φr (a) − φs (a) + br (a) − bs (a)|, so that |ψr (a) − ψs (a)|
ε⊂[d]
is both left- and right syndetic in Γd . From this we will deduce a generalization of Chu’s syndeticity result, Theorem 1.3 of [15]. Corollary 6.1.3. Let Γ be an amenable group, d ∈ N, E ⊂ Γd , and let (ΨN )N ∈N be a left Følner sequence for Γd . Let δ = dΦ (E). Then for all λ > 0, the set n \ o d g ∈ Γd : dΦ (g1−ε1 , . . . , gd−εd ) · E > δ 2 − λ ε∈{0,1}d
is both left- and right syndetic in Γd . In Section 6.3, we prove the special case of Theorem 6.1.1 where d = 2 and the Følner sequences in the averages are assumed to be two-sided. In this case, we give 199
a description of the minimal characteristic factors for the averages in (6.1). We also present an example, due to A. Leibman, showing that the minimal characteristic factors in this case are far from being inverse limits of nilsystems.
6.1.1
Remark.
Bergelson, McCutcheon, and Zhang have shown in [11] that whenever (X, X, µ) is a probability space, Γ is an amenable group, and T, S are commuting, measure preserving actions of Γ on (X, X, µ), the averages 1 X f1 ◦ Tg · f2 ◦ Tg Sg N →∞ |ΦN | g∈Φ lim
N
converge in L1 (µ), for all bounded f1 , f2 . The case d = 2 of Theorem 6.1.1 is closely analogous to this result.
6.2
Preliminary lemmas and notation.
Notation. For the rest of this chapter, we adopt the following notation: If X = (X, X, µ, T ) is a Γ system and Y = (Y, Y, ν, S) is a factor of X, we let WY denote the closed linear subspace of L2 (µ) spanned by functions of the form Z x 7→
H(x, z)φ(z) dµx (z),
where H ∈ L∞ (µ×Y µ) is T ×T -invariant, φ ∈ L∞ (µ), and x 7→ µx is the disintegration of µ over Y.
200
One can verify that WY is T -invariant by writing Z Z H(Tg x, z)φ(z) dµTg x (z) = H(Tg x, Tg z)φ(Tg z) dµx (z) Z = H(x, z)φ(Tg z) dµx (z). We will use the following version of Lemma 7.6 from [23]. Lemma 6.2.1. Let Γ be a countable amenable group, and let X = (X, X, µ, T ) be a Γ-system with factor Y. If f ∈ L∞ (µ), f ⊥ WY , then for all Følner sequences P (ΦN )N ∈N for Γ, limN →∞ |ΦN1 |2 g,h∈Φn kE(f ◦ Tg · f ◦ Th |Y)k = 0. Proof. It suffices to prove that limN →∞
1 |ΦN |2
P
g,h∈Φn
kE(f ◦ Tg · f ◦ Th |Y)k2 = 0.
Writing x 7→ µx for the disintegration of µ over Y, we have kE(f ◦ Tg · f ◦ Th |Y)k2 = RR R f (Tg z)f (Th z) dµx (z) f¯(Tg w)f¯(Th w) dµx (z) dµ(x). Using the Fubini theorem, we write 2
kE(f ◦ Tg · f ◦ Th |Y)k =
Z Z
f (Tg z)f (Th z)f¯(Tg w)f¯(Th w) dµx × µx (z, w) dµ(x).
Applying Th−1 to the integrand and averaging over ΦN × ΦN , we have
2 X 1
E(f ◦ T · f ◦ T |Y)
g h N →∞ |ΦN |2 g,h∈ΦN Z Z = f (z)H(z, w)f¯(w) dµx × µx (z, w) dµ(x), lim
where H is the orthogonal projection of f ⊗ f¯ onto the space of T × T -invariant functions in L2 (µ ×Y µ). Applying the Fubini theorem again, we can write the last RR R integral as f (z) H(z, w)f¯(w) dµx (w) dµx (z) dµ(x), which is 0, by hypothesis.
201
6.3
The case d = 2
In this section we establish a special case of Theorem 6.1.1, where d = 2 and we consider only two-sided Følner sequences. Every countable amenable group admits a two-sided Følner sequence, by Theorem 2 of [18]. We now fix a probability space (X, X, µ), an amenable group Γ, and commuting, µ-preserving actions T and S on X. We write IT , IS ⊆ X for the T -invariant and Sinvariant σ-algebras, respectively, and IT , IS for the corresponding factors. Note that IT and IS are both T - and S-invariant, since T and S are commuting actions. Write x 7→ µτ (x) for the disintegration of µ over IT , and x 7→ µσ(x) for the disintegration of µ over IS . Let WS/T denote the space WIT in (X, X, µ, S), and WT /S denote the space WIS in (X, X, µ, T ). Note that both WT /S and WS/T are T - and S-invariant. The main result in this section is the following theorem. Theorem 6.3.1. Let Γ be an amenable group, (X, X, µ) a probability space, and let T and S be commuting, µ-preserving actions of Γ on X. Then (1) for all f1 , f2 , f3 ∈ L∞ (µ) and all two-sided Følner sequences (ΦN )N ∈N , (ΨN )N ∈N , the limit 1 1 N →∞ |ΦN | |ΨN |
L := lim
X (g,h)∈ΦN ×ΨN
exists in L2 (µ).
202
f1 ◦ Tg · f2 ◦ Sh · f3 ◦ Tg Sh
(6.2)
(2) The limit is the function L(x) =
R
H(x, z)f2 (z) dµσ(x) (z), where H is the or-
thogonal projection of f1 ⊗ f3 onto the space of T × T -invariant functions in L2 (µ ×IS µ). Functions of the form L(x) =
R
H(x, z), f2 (z) dµσ(x) (z), where H is a T × T -invariant
function in L2 (µ ×IS µ), span a dense subspace of WT /S , so the limits in (6.2) span a dense subspace of WT /S . By reversing the roles of S and T in Part 2 of Theorem R 6.3.1, we see that the limit L is also given by x 7→ K(x, z)f1 (z) dµτ (x) (z), where K is the orthogonal projection of f2 ⊗ f3 on the space of S × S-invariant functions in L2 (µ ×IT µ). Hence the space WS/T also spanned by the limits in (6.2). Hence WS/T = WT /S . To prove Theorem 6.3.1 we employ the method of characteristic factors. Let us define the notion of characteristic factors for the present situation. We say that Y = (Y, A, ν, S, T ) is a characteristic factor for the scheme (Tg , Sh , Tg Sh ) if for all f1 , f2 , f3 ∈ L∞ (µ) and all two-sided Følner sequences (ΦN )N ∈N , (ΨN )N ∈N , writing ΘN = ΦN × ΨN , 1 N →∞ |ΘN |
X
lim
Tg f1 · Sh f2 · Tg Sh f3 − Tg E(f1 |A) · Sh E(f2 |A) · Tg Sh E(f3 |A) = 0.
(g,h)∈ΘN
We say that (Y1 , A1 , ν1 , S 0 , T 0 ) and (Y2 , A2 , ν2 , S 00 , T 00 ) are partial characteristic factors for the scheme (Tg , Sh , Tg Sh ) if for all f1 , f2 , f3 ∈ L∞ (µ) and all two-sided Følner sequences (ΦN )N ∈N , (ΨN )N ∈N , writing ΘN = ΦN × ΨN , 1 N →∞ |ΘN | lim
X
Tg f1 · Sh f2 · Tg Sh f3 − Tg E(f1 |A1 ) · Sh E(f2 |A2 ) · Tg Sh f3 = 0.
(g,h)∈ΘN
203
6.3.1
Reduction to partial characteristic factors.
Write P f for the orthogonal projection of f on WT /S , and write Qf for the orthogonal projection of f on WS/T . Let (ΦN )N ∈N , (ΨN )N ∈N be two-sided Følner sequences for Γ. Let f1 , f2 , f3 ∈ L∞ (µ), and define a sequence AN :=
1 1 |ΦN | |ΨN |
X
Tg f1 · Sh f2 · Tg Sh f3 ,
(g,h)∈Φn ×Ψn
and auxiliary sequences (1)
BN := (2)
BN := CN :=
1 1 |ΦN | |ΨN | 1 1 |ΦN | |ΨN | 1 1 |ΦN | |ΨN |
X
Tg (P f1 ) · Sh f2 · Tg Sh f3
(g,h)∈Φn ×Ψn
X
Tg f1 · Sh (Qf2 ) · Tg Sh f3
(g,h)∈Φn ×Ψn
X
Tg (P f1 ) · Sh (Qf2 ) · Tg Sh f3 .
(g,h)∈Φn ×Ψn (i)
Lemma 6.3.2. For i = 1, 2, we have limN →∞ AN −BN = 0, and limN →∞ AN −CN = 0, in the L2 (µ) norm. (1)
Proof. We prove that limN →∞ AN − BN = 0. The case i = 2 is similar, and the last assertion follows from the first two. Consider the case P f1 = 0. The general case follows from this case and the decomposition f1 = P f1 + f10 , where f10 ⊥ WT /S . We must now show that limN →∞ AN = 0. We apply Lemma 2.8.1 with H = Γ × Γ, and ug,h := Tg f1 · Sh f2 · Tg Sh f3 . Then for g, j, j 0 , k, k 0 ∈ Γ, hujg,kh , uj 0 g,k0 h i Z = Tg (Tj f1 · Tj 0 f¯1 ) · Sh (Sk f2 · Sk0 f¯2 ) · Tg Sh (Tj Sk f3 Tj 0 Sk0 f¯3 ) dµ. 204
(6.3)
Applying Tg−1 Sh−1 to the integrand we have hujg,kh , uj 0 g,k0 h i Z = Sh−1 (Tj f1 · Tj 0 f¯1 )Tg−1 (Sk f2 · Sk0 f¯2 ) · (Tj Sk f3 Tj 0 Sk0 f¯3 ) dµ.
(6.4)
Averaging over ΦN × ΨN and applying Theorem 2.5.1 yields X 1 hujg,kh , uj 0 g,k0 h i N →∞ |ΦN ||ΨN | g,h∈ΦN ×ΨN Z = E(Tj f1 · Tj 0 f¯1 |IS ) · E(Sk f1 · Sk0 f1 |IT ) · (Tj Sk f3 Tj 0 Sk0 f¯3 ) dµ. lim
(6.5)
Note that kE(Sk f1 · Sk0 f1 |IT ) · (Tj Sk f3 Tj 0 Sk0 f¯3 )k is bounded by some constant α > 0, so applying Cauchy-Schwarz to the right-hand side of (6.5), we have that X 1 huj 0 g,kh , uj 0 g,k0 h i ≤ αkE(Tj f1 · Tj 0 f¯1 |IS )k. lim N →∞ |ΦN ||ΨN |
(6.6)
g,h∈ΦN ×ΨN
averaging over (j, k), (j 0 , k 0 ) ∈ ΦM × ΨM and applying Lemma 6.2.1 we have X 1 1 1 lim M →∞ |ΦM |2 |ΨM |2 N →∞ |ΦN ||ΨN | 0 lim
X
hujg,kh , uj 0 g,k0 h i = 0.
(6.7)
j,j ∈ΦM (g,h)∈ΦN ×ΨN k,k0 ∈ΨM
By Lemma 2.8.1, limN →∞ AN = 0.
6.3.2
An invariant measure on X × X × X.
In this subsection we prove Part 1 of Theorem 6.3.1. By Lemma 6.3.2, the general case of Theorem 6.3.1 will follow from the case where f1 ∈ WT /S and f2 ∈ WS/T . Since each f ∈ WT /S can be approximated by a linear comR bination of functions of the form x 7→ H(x, z)ψ(z) dµσ(x) , where H ∈ L∞ (µ ×IS µ) 205
is T × T -invariant and ψ ∈ L∞ (µ), it suffices to prove Theorem 6.3.1 when f1 has this form. Similarly, the general case of Theorem 6.3.1 will follow from the case where f2 R has the form x 7→ K(x, z)φ(z) dµτ (x) (z), where K is S ×S-invariant and φ ∈ L∞ (µ). R R We now assume that f1 (x) = H(x, z)ψ(z) dµσ(x) , f2 (x) = K(x, z)φ(z) dµτ (x) (z), where H, ψ, K and φ are as described above. In this case, we have Z f1 (Tg x) =
H(Tg x, z)ψ(z) dµσ(Tg x) (z) Z
=
H(Tg x, Tg z)ψ(Tg z) dµσ(x) (z) Z
= and similarly f2 (Sh x) =
R
H(x, z)ψ(Tg z) dµσ(x) (z),
K(x, z)φ(Sh z) dµτ (x) (z). We then have
f1 (Tg x)f2 (Sh x)f3 (Tg Sh x) Z Z = H(x, z)ψ(Tg z) dµσ(x) (z) K(x, z)φ(Sh z) dµτ (x) (z)f3 (Tg Sh x), which can be rewritten as f1 (Tg x)f2 (Sh x)f3 (Tg Sh x) Z = H(x, z1 )K(x, z2 )ψ(Tg z1 )φ(Sh z2 )f3 (Tg Sh x) dµσ(x) × µτ (x) (z1 , z2 ).
(6.8)
We now consider the integrand in (6.8) as an element of L∞ (λ), where λ is the measure on X × X × X defined by Z λ :=
µσ(x) × µτ (x) × δx dµ(x),
(here δx is the probability measure defined by δx (A) = 1 if and only if x ∈ A). An R R equivalent definition of λ is given by φ1 ⊗ φ2 ⊗ φ3 dλ = E(φ1 |IS )E(φ2 |IT )φ3 dµ for 206
bounded φi . It is easy to see from this definition that λ is T × id ×T - and id ×S × S-invariant. Also, note that the map D : L2 (λ) → L2 (µ) given by D(M )(x) = R M (z1 , z2 , x) dµσ(x) (z1 ) dµτ (x) (z2 ) is linear and bounded: if kφ1 ⊗ φ2 ⊗ φ3 kL2 (λ) ≤ 1, then kD(φ1 ⊗ φ2 ⊗ φ3 )k2L2 (µ) 2 Z Z = φ1 (z1 )φ2 (z2 )φ3 (x) dµσ(x) (z1 ) dµ(x)
Z Z ≤
|φ1 (z1 )|2 |φ2 (z2 )|2 |φ3 (x)|2 dµσ(x) (z1 ) × µτ (x) (z2 ) dµ(x)
= kφ1 ⊗ φ2 ⊗ φ3 k2L2 (λ) ≤ 1. This shows that D is bounded on a dense subspace of L2 (λ), namely the space spanned by functions of the form φ1 ⊗ φ2 ⊗ φ3 for bounded φ1 , φ2 , φ3 . We see from (6.8) that Tg f1 · Sh f2 · Tg Sh f3 = D(H · K · Tg ψ ⊗ Sh φ ⊗ Tg Sh f3 ).
(6.9)
It follows from (6.9) and the boundedness of D that X 1 1 Tg f1 · Sh f2 · Tg Sh f3 N →∞ |ΦN | |ΨN | g∈Φ ,h∈Ψ lim
N
N
2
will exist in L (µ) if X 1 1 (T × id ×T )g (id ×S × S)h φ1 ⊗ φ2 ⊗ f3 N →∞ |ΦN | |ΨN | g∈Φ ,h∈Ψ lim
N
N
2
exists in L (λ). This last limit exists as a consequence of the mean ergodic theorem applied to the Γ × Γ-action (g, h) 7→ (T × id ×T )g (id ×S × S)h . 207
6.3.3
Evaluation of the limit and the minimal characteristic factor.
To prove Part 2 of Theorem 6.3.1, we show that limits of averages over products of Følner sequences can be evaluated as iterated limits. Claim. Let {ug,h }g,h∈Γ be a bounded sequence indexed by Γ × Γ such that for all two-sided Følner sequences (ΦN )N ∈N , (ΨN )N ∈N , the limits 1 N →∞ |ΦN ||ΨN |
X
L = lim
ug,h
(g,h)∈ΦN ×ΨN
1 X 1 X lim ug,h N →∞ |ΦN | M →∞ |ΨM | g∈Φ h∈Ψ
L0 = lim
N
M
exist. Then L = L0 . Proof. Let Φ, Ψ be Følner sequences for Γ. For each n, choose MN so that for all P P g ∈ ΦN , k |ΨM1 | h∈ΨM ug,h − limM →∞ |Ψ1M | h∈ΨM ug,h k < N1 . Then N
N
X 1 X 1 1 X lim ug,h ug,h − lim M →∞ |ΨM | N →∞ |ΦN ||ΨMN | N →∞ |ΦN | g∈Φ h∈Ψ (g,h)∈ΦN ×ΨMN N M X 1 X 1 1 X = lim ug,h − lim ug,h N →∞ |ΦN | M →∞ |ΨM | |Ψ MN | g∈Φ h∈Ψ h∈Ψ
L − L0 = lim
N
MN
M
= 0. This proves the claim.
We now evaluate the limit L in Theorem 6.3.1. Let f1 , f2 , f3 ∈ L∞ (µ), and let
208
(ΦN )N ∈N , (ΨN )N ∈N be two-sided Følner sequences for Γ. By the above remark, we have 1 X 1 X lim f1 (Tg x) · Sh (f2 (x) · f3 (Tg x)) N →∞ |ΦN | M →∞ |ΨM | g∈Φ g∈Ψ
L(x) = lim
N
M
1 f1 (Tg x) · Eµ (f2 · Tg f3 |IS )(x) N →∞ |ΦN | g∈ΦN Z 1 X = lim f1 (Tg x)f2 (z)f3 (Tg z) dµσ(x) (z) N →∞ |ΦN | g∈ΦN Z H(x, z) f2 (z) dµσ(x) (z), = lim
X
where H is the orthogonal projection of f1 ⊗ f3 on the space of T × T -invariant functions in L2 (µ ×IS µ). Since functions of the form x 7→
R
H(x, z)f (z) dµσ(x) (z) span a dense subset of
WS/T , Part 2 of Theorem 6.3.1 implies that every characteristic factor for the scheme (Tg , Sh , Tg Sh ) must include WS/T . 6.3.4
An example.
Here we show that the characteristic factor for the scheme (Tg , Sh , Tg Sh ) can be quite complicated. We specialize to the case Γ = Z. Let (X, X, µ, T 0 ) and (Y, A, ν, S 0 ) be two ergodic Z-systems, where T 0 and S 0 are the transformations T10 and S10 , and let K be a compact group. Let τ : X → K and σ : Y → K be measurable maps from X and Y to K, respectively. Define commuting actions T and S on X × Y × K by T (x, y, k) = (T 0 x, y, τ (x)k) and S(x, y, k) = (x, S 0 y, kσ(y)−1 ). Form the system (X × Y, X ⊗ A, µ × ν × mK , T, S), where mK is Haar measure on K. 209
If τ and σ are chosen so that the system Z = (X ×Y ×K, X⊗A⊗K, µ×ν ×mK , T, S) is ergodic as a Z2 -system, then L2 (µ × ν × mK ) is equal to WS/T , and therefore the system Z is its own minimal characteristic factor for the scheme (T n , S m , T n S m ). For most choices of K, τ, and σ, Z will not resemble a system of rotations on a compact quotient of a nilpotent lie group, contrasting with the results of [36] and Chapter 4.
6.4
The box joinings and seminorms.
In this section we introduce self-joinings of X analogous to the joinings introduced in Section 2 of [35]. We use the notation of [35]. k
Given a set X and an integer k, we consider the set X ∗ := X 2 as being indexed by the collection of strings of 0s and 1s of length k. We write {(xε ) : ε ∈ {0, 1}k } for a k
point of X 2 . If k ≥ 2 and η ∈ X 2
k−1
, we write η0 = η1 . . . ηk−1 0 and η1 = η1 . . . ηk−1 1.
k
We can also consider X 2 as being indexed by subsets of {0, 1}k , and identify a string ε ∈ {0, 1}k with the set {i ∈ {1, . . . , k} : εi = 1}. The string consisting of all 0s then corresponds to the empty set, and we write ∅ = 00 . . . 0 ∈ {0, 1}k . N k If fε , ε ∈ {0, 1} are functions on X, we define the function ε∈{0,1}k fε by O
fε (x) :=
ε∈{0,1}k
6.4.1
Y
fε (xε ).
ε∈{0,1}k
Box measures.
Let (X, X, µ) be a standard probability space, let Γ be an amenable group, and let T (1) , . . . , T (d) be d commuting, µ-preserving actions of Γ on X. Let Ij ⊂ X be the σ-algebra of T (j) -invariant sets. We define a measure µ∗ on X associated to 210
(T (1) , . . . , T (d) ). The measure µ∗ will be defined by induction on d. For k = 0, we let µ0 = µ. For k = 1, we let µk = µ ×I1 µ. If µk−1 is defined for k ≤ d, we define µk as µ ×Ik−1 µ, and we let µ∗ = µd . Equivalently, we may define µ∗ by Z O fε dµ∗ ε∈{0,1}d
:= lim
Nd →∞
1
X
(d)
|ΦNd |
(d)
gd ∈ΦN
1
· · · lim d
X
lim
N2 →∞ |Φ(2) | N1 →∞ N2 g ∈Φ(2) 2 N
1 (1)
|ΦN1 |
2
Z
Y
X (1) 1
g1 ∈ΦN
(Tg(1) )(1−ε1 ) · · · (Tgd )(1−εd ) fε dµ, 1
ε∈{0,1}d
for bounded fε . The equivalence of the two definitions of µ∗ can be seen by d applications of the mean ergodic theorem (Theorem 2.5.1). We now show that the measure µ∗ does not depend on the order of (T (1) , . . . , T (d) ). In the case d = 2, this can be seen from Theorem 6.3.1 and the second definition of µ∗ . Lemma 6.4.1. Let σ : [d] → [d] be a permutation, and let µ0 be the measure associated to (T (σ(1)) , . . . , T (σ(d)) ) in the way that µ∗ is associated to (T (1) , . . . , T (d) ). Then µ0 = µ∗ . Proof. We proceed by induction on d. If σ(d) = d, the lemma follows from the induction hypothesis. If σ exchanges d and d − 1, we apply Theorem 6.3.1 with T = (T (d−1) )⊗2
d−1
, S = (T (d) )⊗2
d−1
. Then µ∗ is the measure associated to (µd−2 , T, S)
in the way that µ∗ is associated to (µ, T (1) , . . . , T (d) ), and µ0 is the measure associated to (µd−2 , S, T ). By Theorem 6.3.1 and the first definition of µ∗ , we have µ0 = µ ∗ . k
Given a set η ⊆ d, |η| = k we can define a measure µ∗η on X 2 as the marginal of µ∗ 211
associated to the side η. By Lemma 6.4.1, the measure µ∗η is invariant under those coordinate permutations given by σ : [d] → [d] with σ(η) = η.
6.4.2
Box seminorms.
Given a bounded X-measurable function f, we define the seminorm |||f ||| by Z O 1/2d |||f ||| := f dµ∗ . ε∈{0,1}d
For each η ⊆ [d], |η| = k we define the seminorm |||f |||η by Z O 1/2k ∗ |||f |||η = gε dµ , ε∈{0,1}d
where gε = f if εi = 0 for all i ∈ / η, and gε = 1X if εi = 1 for some i ∈ / η. The seminorms ||| · |||η are associated to (T (i) : i ∈ η) in the way that ||| · ||| is associated to (T i : i ∈ [d]). Proposition 6.4.2. For fε ∈ L∞ (µ), ε ∈ {0, 1}d , Z O Y ∗ fε dµ ≤ |||fε |||. ε∈{0,1}d
ε∈{0,1}d
Furthermore, ||| · ||| is a seminorm on L∞ (µ). The proof of this Proposition is essentially identical to the proof of Proposition 2 in [35], and very similar to the proof of Lemma 4.3.10. The following is Lemma 2 from [35], adapted to the setting of actions of amenable groups. The proof relies on the reasoning behind the van der Corput lemma, which was stated as observation 2.8.2. 212
(1)
(d)
Lemma 6.4.3. Let f∅ ∈ L∞ (µ), and let (ΦN )N ∈N , . . . , (ΦN )N ∈N be left Følner sequences in Γ. Then for all δ > 0, there exists N0 ∈ N such that for all fε ∈ L∞ (µ), ∅ = 6 ε ∈ {0, 1}d with kfε kL∞ (µ) ≤ 1
X Z
1 (1)
Y
(Tg(1) )1−ε1 1
(d)
|ΦN1 | · · · |ΦNd |
gi ∈ΦiN i i=1,...,d
. . . (Tg(d) )1−εd fε d
dµ ≤ |||f∅ ||| + δ,
(6.10)
ε∈{0,1}d
whenever N1 , . . . , Nd > N0 . Proof. Let C > 0 be a positive constant. Assume that kf∅ kL∞ (µ) ≤ 1. Let J be the average on the left-hand side of (6.10). Write the integrand Y
(Tg(1) )1−ε1 . . . (Tg(d) )1−εd fε 1 d
ε∈{0,1}d
as Y
(Tg(1) )1−ε1 . . . (Tg(d) )1−εd fε · 1 d
εd =0
Y
(Tg(1) )1−ε1 . . . (Tg(d) )1−εd fε . 1 d
εd =1
Abbreviate the factor
Q
(1) 1−ε1 εd =0 (Tg1 )
(d)
(d)
. . . (Tgd )1−εd fε as (Tgd )Fg1 ,...,gd−1 , and the sec-
ond factor as Fg01 ,...,gd−1 . With these abbreviations, J can be rewritten as 1
X
(1) (d−1) |ΦN1 | · · · |ΦNd−1 |
(i)
gi ∈ΦN i i=1,...,d−1
Z
1
X
(d) |Φd | (d) gd ∈ΦN
Tg(d) Fg1 ,...,gd−1 · Fg01 ,...,gd−1 dµ. d
d
Since Fg01 ,...,gd−1 is uniformly bounded by 1, the Cauchy-Schwarz inequality implies that 1
2
|J| ≤
(1)
(d−1)
|ΦN1 | · · · |ΦNd−1 |
2 X X
1
(d) Tgd Fg1 ,...,gd−1 .
(d) L2 (µ) |ΦNd | (i) (d)
gi ∈ΦN i i=1 ...,d−1
213
gd ∈ΦN
d
By Observation 2.8.2, we can replace the average inside the norm above by
1
2 X (d)
Thd gd Fg1 ,...,gd−1
(d) |ΦNd | (d) hd ∈Φm
(d)
and introduce an error of at most C. Note that the norm kThd gd Fg1 ,...,gd−1 k is independent of gd , and in fact
2
X
(d) Thd gd Fg1 ,...,gd−1 =
(d) hd ∈Φm
X hd ,kd
∈Φ(d)
Z
(d) −1 Fg1 ,...,gd−1 d kd
Th
· Fg1 ,...,gd−1 dµ.
It follows that |J|2 is bounded by C+
1 (1)
X
X
(d−1)
|ΦN1 | · · · |ΦNd−1 |
gi ∈ΦN i hd ,kd ∈Φm i i=1,...,d−1 (g )
Z
Y
(d) 1−εd gε −1 ) d kd
(Tg(1) )1−ε1 . . . (Tg(d) )1−εd−1 (Th 1 d−1
ε∈{0,1}d
dµ
where gε is given by gη0 = gη1 = fη0 for η ∈ {0, 1}d−1 . Repeating the above argument with εd−1 instead of εd , and with εd−2 , . . . , ε1 , we have that X Z Y 1 (1) 2d (Th k−1 )1−ε1 . . . (Thd k−1 )1−εd f∅ dµ |J| ≤ dC + (1) (d) 2 d 1 1 2 |Φm1 | · · · |Φmd | (i) ε∈{0,1}d hi ,ki ∈Φmi i=1,...,d
By the definition of the seminorm ||| · |||, we can choose m1 , m2 , . . . , md ≤ M so that d
the average above has absolute value at most (|||f∅ ||| + δ/2)2 . Since C was arbitrary, the desired inequality follows.
6.5
The Magic Extension
We now repeat the construction of [35], Section 2.6. 214
Write X ∗ = X 2
d−1
× X2
d−1
and write points of X ∗ as (x0 , x00 ), where x0 , x00 ∈ X 2
d−1
,
where x0 = (xη0 : η ∈ {0, 1}d−1 ) and x00 = (xη1 : η ∈ {0, 1}d−1 ). (i)
d
For 1 ≤ i ≤ d, S (i) denotes the action of Γ on X 2 given by g 7→ Sg , where Tg(i) xε if i ∈ ε (i) . Sg (x)ε = xε if i ∈ /ε Note that (T (d) )⊗2
d−1
× id⊗2
d−1
= S (d) . For g ∈ Γd , and ε ⊂ [d], let ∗ Rε·g =
Y
Sg(i) . i
i∈ε
We can also define seminorms ||| · |||∗η for η ⊂ [d] in the way that the seminorms ||| · |||η were defined on L∞ (µ). Corresponding to each of these seminorms, we define as in [15] the σ-algebras Zε by Z∗ε :=
_
I(S (i) ),
i∈ε
where I(S (i) ) is the σ-algebra of S (i) -invariant in (X 2 , µ∗ ). d
The system (X, X, µ, T (1) , . . . , T (d) ) is a factor of (X ∗ , X, µ, S (1) , . . . , S (d) ) with factor map π given by π(xεη ) = xε . Hence, to prove Theorem 6.1.1, it suffices to prove that the averages 1
X
Y
fε (1) (d) |ΦN | · · · |ΦN | (1) (d) ε⊂[d] g∈ΦN ×···×ΦN
∗ ◦ Rε·g
(1)
(d)
converge for all fε ∈ L∞ (µ∗ ) and all left Følner sequences (ΦN )N ∈N , . . . (ΦN )N ∈N for Γ. The useful properties of (X ∗ , X∗ , µ, S (1) , . . . , S (d) ) are encapsulated in the following version of Theorem 3.2 from [15]. 215
Lemma 6.5.1. Let ε ⊂ [d]. If f ∈ L∞ (µ∗ ) and Eµ∗ (f |Z∗ε ) = 0 then |||f |||∗ε = 0. In the case ε = d, a proof of Lemma 6.5.1 would be the same as the proof of Theorem 3 in [35], with two minor differences. The first difference is that one must average over left Følner sequences instead of intervals, and the second is that one must appeal to Lemma 6.4.3 instead of Lemma 2 of [35]. For general ε ⊂ d, the proof of Lemma 6.5.1 is essentially the same as the proof of Theorem 3.2 in [15]. Lemma 6.5.1 is useful in conjunction with the next proposition, a version of Proposition 2.2 from [15]. Proposition 6.5.2. Let r ∈ N with r ≤ d, let fε , ε ⊂ [d] be bounded functions on X, (d)
(1)
and let (ΦN )N ∈N , . . . , (ΦN )N ∈N be Følner sequences for Γ. Then
X Y 1
lim sup (1) f ◦ R ≤ min |||fε |||ε ε ε·g (d) ε⊂d L2 (µ) N →∞ |ΦN | · · · |ΦN | (d) ε⊂[d] (1) 0
