CONVERGENCE OF AVERAGES IN ERGODIC THEORY
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Sergey Butkevich, M.S.
*****
The Ohio State University
2001
Dissertation Committee: Approved by
Vitaly Bergelson, Adviser Gerald Edgar Adviser Alexander Leibman Department of Mathematics UMI Number: 3023260
______
UMI Microform 3023260 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ______
Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 ABSTRACT
Von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theo-
rem lie in the foundation of Ergodic Theory. Over the years there have been many
generalizations of the two, most recently a version of pointwise ergodic theorem for
measure-preserving actions of amenable groups due to Elon Lindenstrauss. In the first
chapter, we extend some of Lindenstrauss’ results to measure-preserving actions of
countable left-cancellative amenable semigroups and to averaging along more general
types of Følner sequences.
In the next three chapters, we study convergence of Ces`aro averages of the form
s 1 X Y (i) Tg fi |Ln| g∈Ln i=1
for measure-preserving actions of countable amenable groups, where (Ln) is a Følner sequence. We extend some of the results obtained by D. Berend and V. Bergelson for joint properties of Z-actions to joint properties of actions of countable amenable groups. In particular, we obtain a criterion for joint ergodicity of actions of countable amenable groups by automorphisms of a not necessarily abelian compact group.
In the last chapter of this dissertation, we investigate convergence of Ces`aro av- erages for two non-commuting measure-preserving transformations along a regular sequence of intervals, i.e. we study averages of the form
b 1 Xn T kf · Skg, (1) bn − an k=an
ii where bn − an → ∞. Mean convergence of this expression depends both on the trans-
∞ formations T and S and the regular sequence of intervals [an, bn]n=1. We investigate the relationship between convergence of the expressions (2) and certain combinatorial
∞ properties of the regular sequences of intervals [an, bn]n=1.
iii ACKNOWLEDGMENTS
I wish to thank my advisor, Professor Bergelson, for suggesting the topics of this dissertation to me and leading me through this project. I thank Professor Leibman for the meticulous care with which he examined the dissertation. His numerous and most pertinent remarks allowed me to greatly improve the text. I consider all the remaining mistakes as my own.
I thank Marco Renedo for numerous valuable discussions and suggestions toward the improvement of this project. I thank Paul Larick, Steve Cook, Inna Korchagina, and Alexander Gorodnik for their encouragement and suggestions. I am also grateful to Itzhak Gelander for explaining Elon Lindenstrauss’ results to me.
Finally, I wish to thank my wife Irina for her love and understanding.
iv VITA
October 18, 1968 ...... Born in Moscow, Russia
2000 ...... M.S., Computer and Information Sci- ences, The Ohio State University 1999 ...... M.S., Mathematics, The Ohio State University 1991 ...... Diploma, Applied Mathematics, Moscow Aviation Institute 1992-present ...... Graduate Teaching Associate, The Ohio State University 1991-1992 ...... Software Engineer, Moscow Milan Sys- tems 1991 ...... Researcher, Moscow Aviation Institute
FIELDS OF STUDY
Major Field: Mathematics Specialization: Ergodic Theory
v CONVERGENCE OF AVERAGES IN ERGODIC THEORY
By
Sergey Butkevich, Ph.D.
The Ohio State University, 2001 Vitaly Bergelson, Adviser
Von Neumann’s Mean Ergodic Theorem and Birkhoff’s Pointwise Ergodic Theo- rem lie in the foundation of Ergodic Theory. Over the years there have been many generalizations of the two, most recently a version of pointwise ergodic theorem for measure-preserving actions of amenable groups due to Elon Lindenstrauss. In the first chapter, we extend some of Lindenstrauss’ results to measure-preserving actions of countable left-cancellative amenable semigroups and to averaging along more general types of Følner sequences.
In the next three chapters, we study convergence of Ces`aro averages of the form
s 1 X Y (i) Tg fi |Ln| g∈Ln i=1 for measure-preserving actions of countable amenable groups, where (Ln) is a Følner sequence. We extend some of the results obtained by D. Berend and V. Bergelson for joint properties of Z-actions to joint properties of actions of countable amenable
1 groups. In particular, we obtain a criterion for joint ergodicity of actions of countable
amenable groups by automorphisms of a not necessarily abelian compact group.
In the last chapter of this dissertation, we investigate convergence of Ces`aro av-
erages for two non-commuting measure-preserving transformations along a regular
sequence of intervals, i.e. we study averages of the form
b 1 Xn T kf · Skg, (2) bn − an k=an where bn − an → ∞. Mean convergence of this expression depends both on the trans-
∞ formations T and S and the regular sequence of intervals [an, bn]n=1. We investigate the relationship between convergence of the expressions (2) and certain combinatorial
∞ properties of the regular sequences of intervals [an, bn]n=1.
2 TABLE OF CONTENTS
Page
Abstract ...... ii
Acknowledgments ...... iv
Vita...... v
Chapters:
1. Pointwise Ergodic Theorems for Amenable Semigroups ...... 1
1.1 Definitions and Preliminary Results ...... 1 1.2 Reduction of Main Theorem ...... 9 1.3 Wiener’s Covering Lemma and Its Applications ...... 16 1.4 Rosenblatt-Wierdl’s Covering Lemma and Its Applications ...... 20 1.5 Lindenstrauss’ Covering Lemma and Its Applications ...... 24 1.6 A Version of Lindenstrauss’ Covering Lemma for Semigroups and Its Applications ...... 39 1.7 Open Problems ...... 47
2. Joint Ergodicity of Group Actions ...... 49
2.1 Definitions and Preliminary Results ...... 49 2.2 Some Spectral Properties of Measurable Actions ...... 53 2.3 Van der Corput’s Inequality for Amenable Groups ...... 55 2.4 Main Theorem on Joint Ergodicity ...... 57 2.5 Commutative Case ...... 60 2.6 Other Approaches to Joint Ergodicity ...... 61 2.7 Total Ergodicity ...... 66
vi 3. Joint Weak and Strong Mixing ...... 73
3.1 Definitions and Preliminary Results ...... 73 3.2 Abelian Case, Joint Eigenfunctions ...... 81 3.3 Joint Strong Mixing ...... 83
4. Joint Properties of Actions by Automorphisms of Compact Groups . . . 84
4.1 Preliminary Results ...... 84 4.2 Joint Ergodicity of Actions by Group Automorphisms ...... 88 4.3 Joint Weak Mixing of Actions by Group Automorphisms ...... 94 4.4 Joint Strong Mixing of Actions by Group Automorphisms ...... 96 4.5 Examples and Some Open Problems ...... 97
5. Joint Ergodicity along Sequences of Intervals ...... 101
5.1 Definitions and Preliminary Results ...... 101 5.2 A Universal Counterexample ...... 105 5.3 A Partial Ordering on L ...... 108 5.4 Examples ...... 110 5.5 Some Properties of (L, ≺) ...... 112 5.6 Other Types of Convergence ...... 115 5.7 Other Types of Sequences ...... 117
Appendices:
A. Some Facts from the Theory of Semigroups ...... 123
B. Some Facts about Amenable Groups and Semigroups ...... 131
C. Embedding Amenable Semigroups into Amenable Groups ...... 136
D. A Characterization of Følner Sequences in Z ...... 140
E. Basic Results of Character and Representation Theory ...... 142
E.1 Character Theory ...... 142 E.2 Representation Theory of Compact Groups ...... 143
vii F. Some Facts from Ergodic Theory and Functional Analysis ...... 149
Bibliography ...... 153
Index ...... 157
viii CHAPTER 1
POINTWISE ERGODIC THEOREMS FOR AMENABLE SEMIGROUPS
1.1 Definitions and Preliminary Results
In this chapter, a semigroup means a countable semigroup. A semigroup S is called left cancellative if for all x, a, b ∈ S one has
xa = xb =⇒ a = b.
Let A and B be subsets of a semigroup S. We define
A−1B def= {x | Ax ∩ B 6= ?}. (1.1)
Note that in a semigroup expressions involving A−1 are in general not associative, e.g.
(A−1B)c 6= A−1(Bc).
Example 1.1. Consider the additive semigroup N. We have:
(−[90, 100] + [1, 10]) + 100 = ?, but
−[90, 100] + ([1, 10] + 100) = [1, 20].
1 Since we will be using expressions of the form A−1(Bc) quite often, we define
A−1Bc def= A−1(Bc) = {x | Ax ∩ Bc 6= ?}. (1.2)
We study amenable semigroups. A semigroup S is called left amenable if there
exists a left invariant mean M on the space of all bounded complex-valued functions
on S. See Appendix B for the definition of invariant mean and a brief review of some
of the properties of amenable groups and semigroups.
Definition 1.2. By a left Følner sequence (Fn) in S, we mean a sequence of finite
subsets of S such that for every x ∈ S
|F 4 xF | lim n n = 0. n→∞ |Fn|
Since we only consider left amenable semigroups and left Følner sequences, the word “left” will sometimes be omitted. We shall denote the set of all left Følner sequences in S as F(S).
Proposition 1.3. A left cancellative semigroup is left amenable iff it contains a left
Følner sequence.
A proof can be found in Appendix B.
Remark 1.4. In this chapter, we only consider amenable semigroups that are left cancellative since we want to make use of Følner sequences as defined above. See
Remark 1.22 for more discussion on this issue.
Definition 1.5. An action T of a semigroup S on a probability measure space
(X, B, µ) is a homomorphism g 7→ Tg from S into the semigroup of measure-preserving transformations of X onto itself.
2 A dynamical system is the quadruple (X, B, µ, T ) such that T is an action of a
semigroup S on a probability measure space (X, B, µ).
Definition 1.6. A dynamical system (X, B, µ, T ) is ergodic if for every A ∈ B with
−1 0 < µ(A) < 1 there exists g ∈ S such that µ(Tg A 4 A) > 0.
For each action T of a semigroup S on (X, B, µ), we can define a right unitary
representation (that we shall also denote by T ) of S on the space H = L2(X) by
(Tgf)(x) = f(Tgx).
The following theorem was first proven for Z-actions in [vN32] and later general-
ized to more general groups and semigroups by other authors.
Let H be a Hilbert space and let T be a right unitary representation of a left
cancellative left amenable semigroup S in H. A vector v ∈ H is called T -invariant if
Tgv = v for all g ∈ S
Theorem 1.7 (Mean Ergodic Theorem). Let T be a right unitary representation
of a left cancellative left amenable semigroup S in a Hilbert space H, let (Fn) be a
Følner sequence in S, and let P : H → H be the orthogonal projection of H onto the
subspace of the vectors that are invariant with respect to the representation T . Then
for every f ∈ H 1 X Tgf → P f |Fn| g∈Fn in H-norm.
The first pointwise ergodic theorem was established in [Bir31]. We give a modern formulation of it.
Theorem 1.8 (Pointwise Ergodic Theorem for Z-actions). Let T be a measure- preserving transformation of a probability measure space (X, B, µ). Then for every
3 f ∈ L1(X) the sequence N 1 X f(T kx) N + 1 k=0 converges a.e.
Since Theorem 1.7 holds for every Følner sequence in Z and, in particular, for every sequence of intervals [an, bn] with bn −an → ∞, one naturally asks the question: along which sequences of intervals in Z does the pointwise ergodic theorem hold? This question turns out to be non-trivial. A satisfactory answer to it has been finally given in 1992 in [RW92] (see Theorem 1.13).
Another question that inspired a great deal of research is whether one can prove some pointwise ergodic theorems for more general amenable groups and semigroups.
Since even in Z there exist Følner sequences along which pointwise ergodic theorem does not hold (see Example 1.14 below), one can not possibly hope to prove a point- wise ergodic theorem for every Følner sequence in a general group (since it is not true even in Z.)
A positive answer to this question for a certain class of nested rectangles in Zd was given by Norbert Wiener [Wie39]. For more general groups and semigroups some positive results were obtained in [Cal53], [Tem67], [Eme74], [OW83]. The latter results made use of Følner sequences satisfying the so called Tempelman’s Condition:
Definition 1.9 (Tempelman’s Condition). A Følner sequence (Fn) in an amenable semigroup S is said to satisfy Tempelman’s Condition if
(1) F1 ⊂ F2 ⊂ F3 ⊂ ...
(2) There exists a constant C such that for all n and a ∈ S, one has
−1 |Fn Fna| ≤ C|Fn| (1.3)
4 Remark 1.10. If S is a group, then the condition (1.3) may be written as
−1 |Fn Fn| ≤ C|Fn| (1.4)
There exist several weaker versions of Tempelman’s Condition. The following condition is given in [Tem92] and [RW92].
Definition 1.11 (Weak Tempelman’s Condition). A Følner sequence in an amenable semigroup is said to satisfy Weak Tempelman’s Condition if there exists a constant C such that
n [ F −1F a < C|F | (1.5) k n n k=1 for all n > 0 and a ∈ S.
Remark 1.12. If S is a group, then the condition (1.5) may be written as
n [ F −1F < C|F | (1.6) k n n k=1
In [RW92] Rosenblatt and Wierdl proved the following
Theorem 1.13. Let (In) be a sequence of intervals in Z such that |In| is increasing.
Then pointwise ergodic theorem holds along the sequence (In) iff it satisfies Weak
Tempelman’s Condition.
It is interesting to consider a few examples of sequences of intervals in Z that satisfy and do not satisfy Weak Tempelman’s Condition.
Example 1.14.
• In = [0, n] is good. (C = 2)
5 • Every nested sequence of intervals is good. (C = 2)
2 2 • In = [n , n + n] is bad.
n n n • In = [4 , 4 + 2 ] is still bad. √ 2n 2n 2n • In = [2 , 2 + 2 ] is good.
It is easy to see that every sequence of intervals [an, bn] in Z, such that bn−an → ∞, contains a subsequence satisfying Weak Tempelman’s Condition. (This is a special case of Theorem 1.20 below.)
The following two questions still remain open.
Question 1.15. Is there a necessary and sufficient condition on a Følner sequence in Z that guarantees that the pointwise ergodic theorem holds along it?
Question 1.16. Is there an analogue of Theorem 1.13 for non-abelian groups or for abelian groups of infinite rank (such as the multiplicative group of positive rationals)?
Remark 1.17. Unfortunately, existence of Følner sequences satisfying Weak Tempel- man’s Condition has only been proven for amenable groups of polynomial growth.
An example of a finitely-generated amenable group in which there is no Følner se- quence satisfying Weak Tempelman’s condition is given in [Lin00]. The following
finer condition was proposed in [Shu88].
Definition 1.18 (Shulman’s Condition). A Følner sequence in a group is called tempered if there exists a constant C such that
n−1 [ F −1F < C|F | (1.7) k n n k=1 for all n.
6 Remark 1.19. It is not trivial to come up with a suitable form of Shulman’s Condition
for semigroups. This issue shall be discussed in Section 1.6.
Theorem 1.20 ([Lin99]). Every Følner sequence in an amenable group has a tem-
pered subsequence. In particular, every amenable group has a tempered Følner se-
quence.
In [Shu88] a pointwise ergodic theorem has been proven for tempered Følner se-
quences and for functions in L2. In [ST00] this result is extended to Lα for α > 1.
However, the decisive step in the quest for a pointwise ergodic theorem for amenable
groups has been made in [Lin99].
Theorem 1.21 ([Lin99], [Lin00]). Let G be an amenable group acting on a measure
1 space (X, B, µ) and let (Fn) be a tempered Følner sequence. Then for every f ∈ L (X)
the limit 1 X lim f(Tgx) n→∞ |Fn| g∈Fn exists a.e.
In Section 1.2 we develop machinery that can be used to reduce a certain class of pointwise ergodic theorems for semigroup actions to some covering lemmas for these semigroups.
In the Sections 1.3–1.6, we consider four different covering lemmas, in increasing difficulty.
In Section 1.3, we prove a version of Wiener’s Covering Lemma that is used to prove a pointwise ergodic theorem along Følner sequences satisfying Tempelman’s
Condition. Although this result is well-known, we present it here since it makes it much easier to understand the more difficult covering lemmas presented later. We also
7 make the following modest contribution. Namely, we treat the case of a nested Følner
sequence that does not satisfy Tempelman’s Condition and consider the question of
pointwise convergence for expressions of the form
1 X −1 f(Tgx) maxa∈S |Fn Fna| g∈Fn (Theorem 1.30, Corollary 1.31).
In Section 1.4, we prove a covering lemma that we call Rosenblatt-Wierdl Covering
Lemma. It is used to prove a pointwise ergodic theorem for Følner sequences satisfying
Weak Tempelman’s Condition. It also plays a key role in the study of pointwise convergence of the expressions of the form
1 X Sn −1 f(Tgx) maxa∈S F Fna i=1 i g∈Fn (Theorem 1.37, Corollary 1.38).
The next two sections are devoted to generalizations of Theorem 1.21.
In Section 1.5, we prove a version of Covering Lemma by Lindenstrauss that we use to prove a pointwise ergodic theorem for amenable groups along not necessarily tempered Følner sequences for the expressions of the form
1 X f(T x) Sn−1 −1 g maxa∈S F Fna i=1 i g∈Fn (Theorem 1.44, Corollary 1.45)
In Section 1.6, we generalize Theorem 1.21 to left cancellative left amenable semi- groups. We introduce a notion of tempered Følner sequences for left cancellative semigroups (Definition 1.52). We prove a pointwise ergodic theorem along tempered
Følner sequences in such semigroups. However, we have not been able to prove a pointwise ergodic theorem along non-tempered Følner sequences for the expressions
8 of the form 1 X f(T x). Sn−1 −1 g maxa∈S F Fna i=1 i g∈Fn Also, we have not been able to obtain an analogue of Theorem 1.20 for semigroups.
See Section 1.6 for more discussion on these issues.
In Section 1.7, we discuss some open problems.
Remark 1.22. We only consider left cancellative amenable semigroups. The reason is that a non-left-cancellative amenable semigroup may not posses a Følner sequence
(in the sense of Definition 1.2). It always satisfies Følner Condition (Definition B.10), but this is not enough to prove Mean Ergodic Theorem (Theorem 1.7) or Transfer
Principle (Proposition 1.27).
1.2 Reduction of Main Theorem
In this section, we reduce a pointwise ergodic theorem for an action of an amenable semigroup to a purely combinatorial statement pertaining to the semigroup itself.
This procedure has been used by many authors, for instance, in [Pet83], [RW95],
[Tem92], to prove certain statements about pointwise convergence of averages. For now, let us fix a countable left cancellative left amenable semigroup S. Let a Følner sequence (Fn) in S, and probability measure space (X, B, µ), on which S acts be
1 given. Let (qn) be a sequence of positive numbers. Let f ∈ L (X). We consider convergence of the following ergodic averages:
1 X Tgf. qn g∈Fn
We shall be primarily interested in the case when qn = |Fn|, but we shall consider the general case as well.
9 The following notation shall be used:
1 X An(Fn, qn)[f] = Tgf. qn g∈Fn
Mt(Fn, qn)[f](x) = max An[|f|](x) n≤t
M(Fn, qn)[f](x) = sup An[|f|](x). n
In most cases, the sequences (Fn) and (qn) are fixed and we shall use the abbrevi- ated notation An[f], Mt[f], and M[f] instead of An(Fn, qn)[f], Mt(Fn, qn)[f](x), and
M(Fn, qn)[f](x) respectively.
1 We consider the left action L of S on itself given by Lgx = gx. This action induces a right action on the set of the functions ϕ : S → C with finite support. We define
˜ 1 X An[ϕ] = Lgϕ. qn g∈S ˜ ˜ Mt[ϕ](x) = max An[|ϕ|](x) n≤t ˜ ˜ M[ϕ](x) = sup An[|ϕ|](x). n
Let a Følner sequence (Fn) and a sequence of factors (qn) be fixed.
We formulate the following three types of statements (that may or may not be true):
• Pointwise Convergence (PWC). For every f ∈ L1(X) the limit
lim An[f](x) n→∞
1Here, by action we mean a group action in purely algebraic sense, not an action in the sense of Definition 1.5. There is no measure involved.
10 exists a.e.
• Maximal Inequality (MAX IE). There exists c > 0, depending on the
1 sequence (Fn), but not on X, such that for every f ∈ L (X), one has
c µ{x | M[f](x) > λ} ≤ ||f|| . (1.8) λ 1
for all λ > 0.
• Maximal Inequality on S (MAX IE on S). There exists a constant c
(depending on (Fn)) such that for every function ϕ : S → C with finite support
and every λ > 0,
c {a ∈ S | M˜ [ϕ](a) > λ} ≤ kϕk . (1.9) λ 1
Note that it is enough to prove MAX IE and MAX IE on S for the case λ = 1.
We shall prove that
MAX IE on S =⇒ MAX IE
We shall also prove that, if limn→∞ |Fn|/qn exists and is finite, then
MAX IE =⇒ PWC.
Lemma 1.23. Let a left cancellative left amenable semigroup S act on a probability
2 measure space X. Then L (X) = Hinv ⊕ Herg, where
2 Hinv = {f ∈ L (X) | Tgf = f ∀g ∈ S}
∞ Herg = Span{h − Tgh | h ∈ L (X), g ∈ S}.
11 ∞ Proof. Assume that f ⊥ Herg. Then for all g ∈ S and all h ∈ L (X) we have:
(f, h − Tgh) = 0
(f, h) = (f, Tgh)
∗ ∗ (f, h) = (Tg f, h), where Tg is the adjoint operator.
In particular, for all A ∈ B, we have:
Z Z ∗ f dµ = Tg f dµ A A
∗ Hence, f = Tg f a.e. for all g ∈ S. In view of Lemma F.9, it follows that f = Tgf a.e. for all g ∈ S and, therefore, f ∈ Hinv.
Reversing the steps, we see that if f ∈ Hinv, then f ⊥ Herg. Thus, Hinv is the orthogonal complement of the closed subspace Herg. Hence,
2 L (X) = Hinv ⊕ Herg.
Proposition 1.24. Assume that limn→∞ |Fn|/qn exists and is finite. Then
MAX IE =⇒ PWC.
Remark 1.25. This fact is well-known for the case when qn = |Fn|. The proof can be found, fir example, in [Pet83]. The proof of the more general fact that we give here is very similar to that classical proof. Of course. the case when qn 6= |Fn|, but
0 < limn→∞ |Fn|/qn < ∞ is not important. The case when limn→∞ |Fn|/qn = 0 is important. As we shall see below, it is possible that for a given Følner sequence (Fn),
PWC does not hold for qn = |Fn|, but holds for a sequence of denominators (qn) such that limn→∞ |Fn|/qn = 0.
12 Proof. Let
1 C = {f ∈ L (X) | An[f] converges a.e. }
1 D = {f + (h1 − Tg1 h1) + ... (hm − Tgm hm) | f ∈ L (X),Tgf = f ∀g ∈ S,
∞ h1, . . . , hm ∈ L (X), g1, . . . .gm ∈ S}.
We need to prove that C = L1(X). We claim that D ⊂ C.
Namely, let us first consider the case when qn = |Fn|. If f = Tgf for all g ∈ S,
∞ then, clearly, f ∈ C. If f = h − Tah for some h ∈ L (X) and a ∈ S, then for all
x ∈ X,
1 X 1 X Tg(h − Tah)(x) = (Tgh − Tagh)(x) |Fn| |Fn| g∈Fn g∈Fn
1 X |Fn 4 aFn| ≤ |Tgh(x)| ≤ khk∞ −−−→ 0, |Fn| |Fn| n→∞ g∈Fn4aFn
since (Fn) is a left Følner sequence. So, if qn = |Fn|, then D ⊂ C.
The fact the D ⊂ C in the case, when qn 6= |Fn|, but limn→∞ |Fn|/qn exists and is finite, follows from the following simple observation. Let (an) be a sequence of complex numbers such that the limit
a lim n n→∞ |Fn|
exists. Assume that |F | lim n = α < ∞. n→∞ qn Then a a |F | a lim n = lim n · lim n = α lim n . n→∞ qn n→∞ |Fn| n→∞ qn n→∞ |Fn| Thus, D ⊂ C and it is enough to prove that:
13 (1) D is dense in L1
(2) C is closed.
Indeed:
(1) By Lemma 1.23, L2 ∩ D is dense in L2 and, therefore, in L1. Hence, D is dense
in L1.
1 (2) Let fk ∈ C, k = 1, 2,... , and fk → f in L as k → ∞. We will show that the
sequence An[f] is fundamental a.e. Indeed,
|Am[f] − An[f]| ≤ |Am[fk] − An[fk]| + |Am[f − fk]| + |An[f − fk]| ∀k.
The first term on the right approaches 0 a.e. as m, n → ∞. Hence, for every
λ > 0,
µ{lim sup |Am[f] − An[f]| > λ} ≤ µ{2 sup |An[f − fk]| > λ} m,n→∞ n
≤ µ{2 sup An[|f − fk|] > λ} n 2c ≤ kf − f k (by MAX IE). λ k 1
Since kf − fkk → 0, the sequence An[f] is fundamental a.e and, therefore,
limn→∞ An[f](x) exists a.e.
Lemma 1.26. Let S be a left cancellative left amenable semigroup and g1, . . . , gn ∈ S.
Then for every ε > 0 there exists a finite set K ⊂ S such that
|g1K ∩ · · · ∩ gnK| > (1 − ε)|K|.
14 Proof. Let (Fn) be a Følner sequence. Let ε > 0 be given. There exists an index r
such that |Fr 4 gkFr| < (ε/n)|Fr| for 1 ≤ k ≤ n. Let K = Fr. Then
|g1K ∩ · · · ∩ gnK| ≥ |K ∩ g1K ∩ · · · ∩ gnK|
≥ |K| − |K 4 g1K| − · · · − |K 4 gnK)|
> |K| − n(ε/n)|K|
= (1 − ε)|K|.
Proposition 1.27. MAX IE on S =⇒ MAX IE.
Proof. Let f ∈ L1(X), and λ > 0 be given. It is enough to prove that for all t
c µ{x ∈ X | M [f](x) > λ} ≤ kfk . t λ 1
Let t > 0 and x ∈ X be given. Let K ⊂ S be a “big” finite set to be chosen later.
We define ( f(T x), if a ∈ K, ϕ(a) = a 0, otherwise Note that 1 X ˜ Mt[f](Tax) = max f(Tgax) = Mt[ϕ](a), n≤t qn g∈Fn S 0 provided that ga ∈ K for all g ∈ n≤t Fn. In other words, a must be in K , where \ K0 = g−1K. S g∈ n≤t Fn By Lemma 1.26, for a fixed t and ε, we can choose K so that |K0| > (1 − ε)|K|. Now, let
P = {y ∈ X | Mt[f](y) > λ}
˜ Π = {a ∈ S | Mt[ϕ](a) > λ}
15 Then
X X 1P (Tax) = 1Π(a) ≤ |Π| a∈K0 a∈K0 c ≤ kϕk (by MAX IE on S) λ 1 c X = |f(T x)| λ g g∈K Taking integral from both sides of the above inequality, we obtain
c |K0| · µ(P ) ≤ |K|kfk . λ 1
When ε → 0, |K|/|K0| → 1. Thus, we have
c µ{x ∈ X | M [f](x) > λ} = µ(P ) ≤ kfk . t λ 1
Making t → ∞, we conclude that
c µ{x ∈ X | M[f](x) > λ} ≤ kfk . λ 1
Remark 1.28. Proposition 1.27 is often referred to as “Transfer Principle”. While it appeares implicitly in [Wie39] and works by other authors, it was Calder´onwho first noticed its importance and stated it explicitly in [Cal68].
1.3 Wiener’s Covering Lemma and Its Applications
The Maximal Inequality on S may be proven using a covering lemma of this or an- other kind. For a Følner sequence satisfying Tempelman’s Condition (Definition 1.9) one uses the following covering lemma due to Wiener [Wie39] generalized by Tempel- man in [Tem67]. Here we present an even more general version of it that allows us
−1 to prove a maximal inequality for qn = maxa∈S |Fn Fna| even if the Følner sequence does not satisfy Tempelman’s Condition.
16 Lemma 1.29 (Wiener’s Covering Lemma). Let S be a discrete semigroup. Let
(Fn) be a nested Følner sequence. Let E ⊂ S be a finite set and let for each a ∈ E a positive integer n(a) be given. Then there exists a subset E0 ⊂ E such that
0 (1) Fn(a)a ∩ Fn(b)b = ? ∀a, b ∈ E , a 6= b,
[ −1 (2) E ⊂ Fn(a)Fn(a)a. a∈E0
Proof. Let a1 ∈ E be an element such that n(a1) is maximal and let
C = F −1 F a . 1 n(a1) n(a1) 1
Note that a1 ∈ C1. Let E1 = E \ C1.
If E1 is non-empty, choose an a2 ∈ E1 so that n(a2) is maximal and let
C = F −1 F a . 2 n(a2) n(a2) 2
Note that a2 ∈ C2. Let E2 = E1 \ C2.
Keep doing it, i.e. set
C = F −1 F a (1.10) i n(ai) n(ai) i and Ei = Ei−1 \ Ci. Since E is finite, this procedure will terminate and we set
0 E = {a1, a2,... }. Condition (2) is satisfied by construction. We claim that condition
(1) is satisfied as well.
Indeed, let a, b ∈ E0 and a 6= b. We may assume that b has been chosen before a. Then n(a) ≤ n(b). Suppose that Fn(a)a ∩ Fn(b)b 6= ?, This means that there exist
−1 elements x ∈ Fn(a) ⊂ Fn(b) and y ∈ Fn(b) such that xa = yb. Then a = x yb, i.e.
−1 −1 a ∈ Fn(b)Fn(b)b. But according to our construction, the set Fn(b)Fn(b)b is deleted from E before a is chosen. Contradiction.
So, both conditions (1) and (2) hold and we are done.
17 Theorem 1.30. Let S be a left cancellative left amenable semigroup acting on a probability measure space (X, B, µ). Let (Fn) be a nested left Følner sequence in S and f ∈ L1(X). Let
−1 qn = max |Fn Fna|. a∈S
Then
1 µ M(F , q )[f](x) > λ ≤ kfk . (1.11) n n λ 1
Proof. In view of Proposition 1.27, it is enough to prove that the MAX IE on S holds for every ϕ ∈ l1(S) with constant c = 1. We may assume that ϕ is real-valued and non-negative. Also, may assume that λ = 1. Let
E = {a ∈ S | M˜ [ϕ](a) > 1}.
Since ϕ has finite support, E is a finite set. Notice that a ∈ E iff there exists n > 0 such that
X X qn < ϕ(ga) = ϕ(g). (1.12)
g∈Fn g∈Fna
Let n(a) be the smallest n such that the above holds. By virtue of Wiener’s Covering
0 0 Lemma, we can choose E ⊂ E so that the sets Fn(a) ·a for a ∈ E are pairwise disjoint and
[ X X |E| ≤ F −1 F a ≤ F −1 F a ≤ q . (1.13) n(a) n(a) n(a) n(a) n(a) a∈E0 a∈E0 a∈E0
Taking sum of (1.12) over E0, we obtain
X X X qn(a) ≤ ϕ(g) (1.14) 0 0 a∈E a∈E g∈Fn(a)a
18 0 Since the sets Fn(a)a are pairwise disjoint for a ∈ E , it follows that
X X X ϕ(g) ≤ ϕ(g) = kϕk1. (1.15) 0 a∈E g∈Fn(a)a g∈S Combining (1.13), (1.14), and (1.15), we see that
|E| < kϕk1. (1.16)
Thus, MAX IE on S holds and we are done.
Corollary 1.31. If, in addition to the assumptions of Theorem 1.30, the limit
|F | lim n n→∞ qn exists and is finite, then 1 X f(Tgx) qn g∈Fn converges a.e.
Corollary 1.32. Let G be an amenable group acting on a probability measure space
(X, B, µ). Let (Fn) be a nested left Følner sequence in S such that
|F | lim n n→∞ −1 |Fn Fn| exists and is finite and let f ∈ L1(X). Then
1 X −1 f(Tgx) |Fn Fn| g∈Fn converges a.e.
−1 −1 Proof. In a group, |Fn Fna| = |Fn Fn|.
−1 −1 Remark 1.33. In order to have |Fn Fna| = |Fn Fn|, it is not enough to assume that S is cancellative or even embeddable into a group, as Example 1.1 in the beginning of this chapter shows.
19 Corollary 1.34. Let S be a left cancellative left amenable semigroup acting on a
probability measure space (X, B, µ). Let (Fn) be a left Følner sequence satisfying
Tempelman’s condition and let f ∈ L1(X). Then
1 X f(Tgx) |Fn| g∈Fn converges a.e.
−1 Proof. According to Theorem 1.30, the MAX IE holds for qn = maxa∈S |Fn Fna|.
0 Since qn ≤ C|Fn|, when we replace qn by qn = |Fn|, MAX IE still holds. By Proposi- tion 1.24, it follows that 1 X f(Tgx) |Fn| g∈Fn converges a.e.
Remark 1.35. Corollary 1.34 has been proven in [Tem67]. As far as I know,
Lemma 1.29 and Theorem 1.30 has only been proven for Følner sequences satisfy- ing Tempelman’s Condition.
1.4 Rosenblatt-Wierdl’s Covering Lemma and Its Applications
It is possible to generalize the results obtained in the previous section by dropping the assumption that the Følner sequence (Fn) is nested. Then we have to make use
Sn −1 −1 of expressions of the form i=1 Fi Fn instead of Fn Fn. The results that we present below have been obtained by Rosenblatt and Wierdl in [RW92] for the case S = Z.
The book [Tem92] contains some of these results for the case of general semigroup actions as well.
20 The proofs are very similar to the proofs given in the previous section. The main new ingredient is that in order to make the sets Ci disjoint, we have to set
n(ai) [ −1 Ei = Fi Fn(ai)ai i=1 instead of
E = F −1 F a . i n(ai) n(ai) i
Lemma 1.36 (Rosenblatt-Wierdl’s Covering Lemma). Let S be a countable semigroup and let (Fn) be a Følner sequence. Let E ⊂ S be a finite set and let for each a ∈ E a positive integer n(a) be given. Then there exists a subset E0 ⊂ E such that
0 (1) Fn(a)a ∩ Fn(b)b = ? ∀a, b ∈ E , a 6= b,
n(a) [ [ −1 (2) E ⊂ Fi Fn(a)a. a∈E0 i=1
Proof. Let a1 ∈ E be an element such that n(a1) is maximal and let
n(a1) [ −1 C1 = Fi Fn(a1)a1. i=1
Note that a1 ∈ C1. Let E1 = E \ C1.
If E1 is non-empty, choose an a2 ∈ E1 so that n(a2) is maximal and let
n(a2) [ −1 C2 = Fi Fn(a2)a2. i=1
Note that a2 ∈ C2. Let E2 = E1 \ C2.
Keep doing it, i.e. set
n(ak) [ −1 Ck = Fi Fn(ak)ak (1.17) i=1
21 and Ek = Ek−1 \Ck. Since E is finite and at each step we remove at least one element
0 from it, this procedure will terminate and we set E = {a1, a2,... }. Condition (2) is satisfied by construction. We claim that condition (1) is satisfied as well.
Indeed, let a, b ∈ E0 and a 6= b. We may assume that b has been chosen before a. Then n(a) ≤ n(b). Suppose that Fn(a)a ∩ Fn(b)b 6= ?. This means that there exist
−1 elements x ∈ Fn(a) and y ∈ Fn(b) such that xa = yb. Then a = x yb, i.e. n(b) −1 [ −1 a ∈ Fn(a)Fn(b)b ⊂ Fi Fn(b)b i=1 that is supposed to be deleted from E before a is chosen. Contradiction.
Thus, both conditions (1) and (2) hold and we are done.
Theorem 1.37. Let S be a left cancellative left amenable semigroup acting on a probability measure space (X, B, µ). Let (Fn) be a left Følner sequence in S and f ∈ L1(X). Let
n [ −1 qn = max Fi Fna . a∈S i=1 Then 1 µ M(F , q )[f](x) > λ ≤ kfk . (1.18) n n λ 1 Proof. In view of Proposition 1.27, it is enough to prove that the MAX IE on S holds
for every ϕ ∈ l1(S) with constant c = 1. We may assume that ϕ is real-valued and
non-negative. Also, may assume that λ = 1. Let
E = {a ∈ S | M˜ [ϕ](a) > 1}.
Since ϕ has finite support, E is a finite set. Notice that a ∈ E iff there exists n > 0
such that
X X qn < ϕ(ga) = ϕ(g). (1.19)
g∈Fn g∈Fna
22 Let n(a) be the smallest n such that the above holds. By virtue of Tempelman’s
0 0 Covering Lemma, we can choose E ⊂ E so that the sets Fn(a) · a for a ∈ E are
pairwise disjoint and
n(a) n(a) [ [ −1 X [ −1 X |E| ≤ F Fn(a)a ≤ F Fn(a)a ≤ qn(a). (1.20) i i a∈E0 i=1 a∈E0 i=1 a∈E0 Taking sum of (1.19) over E0, we obtain
X X X qn(a) ≤ ϕ(g) (1.21) 0 0 a∈E a∈E g∈Fn(a)a
0 Since the sets Fn(a)a are pairwise disjoint for a ∈ E , it follows that
X X X ϕ(g) ≤ ϕ(g) = kϕk1. (1.22) 0 a∈E g∈Fn(a)a g∈S Combining (1.20), (1.21), and (1.22), we see that
|E| < kϕk1.
Thus, MAX IE on S holds and we are done.
Corollary 1.38. If, in addition to the assumptions of Theorem 1.37, the limit
|F | lim n n→∞ qn exists and is finite, then 1 X f(Tgx) qn g∈Fn converges a.e.
Corollary 1.39. Let S be an amenable group acting on a probability measure space
(X, B, µ). Let (Fn) be a left Følner sequence in S such that
|Fn| lim n n→∞ S −1 i=1 Fi Fn 23 exists and is finite and let f ∈ L1(X). Then
1 X Sn −1 f(Tgx) F Fn i=1 i g∈Fn
converges a.e.
Corollary 1.40. Let S be a left cancellative left amenable semigroup acting on a
probability measure space (X, B, µ). Let (Fn) be a left Følner sequence satisfying
Weak Tempelman’s condition and let f ∈ L1(X). Then
1 X f(Tgx) |Fn| g∈Fn converges a.e.
Remark 1.41. Corollary 1.40 has been proven in [Tem92]. For the case S = Z,
Lemma 1.36 can be found in [RW92].
1.5 Lindenstrauss’ Covering Lemma and Its Applications
As we have already noted in Section 1.1, Weak Tempelman’s Condition is too
strong. We would like to obtain analogues of results from the previous section for
Følner sequences satisfying the weaker Shulman’s Condition (Definition 1.18). The
only difference between Weak Tempelman’s and Shulman’s Conditions is the range
Sn−1 −1 of summation. Namely, we use the expression i=1 Fi Fn in Shulman’s Condition Sn −1 as opposed to i=1 Fi Fn in Weak Tempelman’s Condition. This difference is signif- icant. Indeed, as we noted above (Remark 1.17), there may be no Følner sequences satisfying Weak Tempelman’s Condition in a general amenable group. However, according to Theorem 1.20, in every amenable group, a Følner sequence satisfying
Shulman’s Condition always exists.
24 This difference also leads to an obstacle in proving a Covering Lemma. To prove
Lemma 1.36, we need to choose disjoint sets of the form Fn(a)a. Since it might well
happen that n(a) = n(b) for a 6= b, to be sure that Fn(a)a ∩ Fn(b)b = ?, we have to
Sn(a) −1 Sn(a)−1 −1 subtract i=1 Fi Fn(a)a, not just i=1 Fi Fn(a)a, from E. How can we achieve Sn(a)−1 −1 the desired result by subtracting i=1 Fi Fn(a)a? One possibility is to arrange for all n(a)’s to be distinct. It is not clear at all how to achieve that. Another possibility is to group all a’s with same value of n(a) into sets, call them A’s, and then at each step work with the whole set A instead of just one element a. But how to do it so that the sets Fn(a)a with a from same set A are disjoint? Having all this in mind, we can better appreciate the following covering lemma due to Lindenstrauss [Lin00].
Here, we present a slightly generalized version of this lemma suitable for proving the
Sn−1 −1 maximal inequality with denominators qn = i=1 Fi Fn even if the Følner sequence does not satisfy Shulman’s Condition.
Let us make some notational remarks before we formulate the covering lemma. We consider a countable amenable group G. We introduce set-valued random variables
B(a). These are functions from some measure space, which we denote as Ω, into the class of all finite subsets of G. Let P be the probability measure on Ω. If X is a random variable and P is an event, i.e. a measurable subset of Ω, then E(X) denotes the expectation and E(X | P ) denotes the conditional expectation with respect to
the event P .
Lemma 1.42 (Lindenstrauss’ Covering Lemma). Let G be a countable group,
(Fn) a Følner sequence, E ⊂ G a finite set, and let for each a ∈ E a positive integer
n(a) be chosen. Let δ > 0 be given. Then it is possible to find set valued random
variables B(a) for a ∈ E such that
25 (1) B(a) is either Fn(a)a or ?.
P (2) If we set Λ: G → N to be the random function Λ(g) = a∈E 1B(a)(g), then
E(Λ(g) | Λ(g) > 0) < 1 + δ.
(3) For some γ > 0 that depends on δ, n(a)−1 E X [ −1 Fi B(a) > γ|E|.
a∈E i=1
The proof makes use of the following simple lemma.
Pn 2 Lemma 1.43. Let Y = i=1 Xi, where the Xi’s are i.i.d. random variables which are 1 with probability p and 0 with probability 1 − p. Then
E(Y | Y ≥ 1) ≤ 1 + (n − 1)p.
Proof. We use induction to prove this lemma. For n = 1 the above inequality holds trivially. To prove the general case, we note that
E(Y | Y ≥ 1) = P(X1 = 1)E(Y | X1 = 1 and Y ≥ 1))
+ P(X1 = 0)E(Y | X1 = 0 and Y ≥ 1))
(1.23)
Observe that
E(Y | X1 = 1 and Y ≥ 1)) = 1 + (n − 1)p (1.24) and
E(Y | X1 = 0 and Y ≥ 1)) ≤ 1 + (n − 2)p (1.25)
2“independent and identically distributed”
26 by induction hypothesis.
Combining (1.23), (1.24), and (1.25), we obtain
E(Y | Y ≥ 1) ≤ P(X1 = 1)(1 + (n − 1)p) + P(X1 = 0)(1 + (n − 2)p)
= p + (n − 1)p2 + (1 − p) + (1 − p)(n − 2)p = 1 + (n − 2)p + p2 ≤ 1 + (n − 1)p.
Proof of Lindenstaruss’ Covering Lemma. Let N = maxa∈E n(a). For 1 ≤ j ≤ N we let
Aj = {a ∈ E | n(a) = j}.
We start from j = N. For all a ∈ AN we let (independently from all other b ∈ AN )
( δ FN a with probability pN = , B(a) = |FN | (1.26) ? with probability 1 − pN
Assume that we have already defined B(a) for all a ∈ Ak, j < k ≤ N. Then for each
a ∈ Aj we do the following (independently from all other b ∈ Aj)
0 0 • If Fja is disjoint from all B(a ) for all a ∈ Ak, j < k ≤ N, then
( δ Fja with probability pj = , B(a) = |Fj | (1.27) ? with probability 1 − pj
• Otherwise B(a) = ?.
It is enough to consider the case when δ is a rational number. In this case, the random variables B(a) defined above may be implemented by means of a finite set Ω with normalized counting measure. Therefore, every subset of Ω is measurable.
We first show that
E(Λ(g) | Λ(g) > 0) < 1 + δ.
27 Let Λj be the random function
X Λj(g) = 1B(a)(g).
a∈Aj
0 We remark that for j 6= j the events Λj(g) > 0 and Λj0 (g) > 0 are mutually exclusive.
We now define for every g ∈ G and j
N [ [ −1 0 −1 A(j, g) = Aj \ Fj B(a ) ∩ Fj g. 0 0 j =j+1 a ∈Aj0
This represents the set of all a ∈ Aj such that g ∈ Fja and such that Fja is disjoint from B(a0) for all a0. By our construction, |A(j, g)| is the number of sets B(b), with b ∈ Aj, such that the probability to contain the element g is positive. Clearly,
|A(j, g)| ≤ |Fj|. Using Lemma 1.43, we see that
E(Λj(g) | Λj(g) > 0) = E(Λj(g) | Λj(g) ≥ 1) ≤ 1 + pj|A(j, g)| ≤ 1 + pj|Fj| = 1 + δ.
Hence,
E(Λ(g) | Λ(g) > 0) ≤ 1 + δ.
We use induction to show that n(a)−1 E X [ −1 Fi B(a) > γ|E| (1.28)
a∈E i=1 where
δ γ = . (1.29) δ + 1
If N = 1, n(a)−1 ! ! E X [ −1 E X X E X Fi B(a) = |B(a)| = 1B(a)(g)
a∈A1 i=1 a∈A1 a∈A1 g∈F δ = |A1||F1| = δ|A1| = δ|E| > γ|E|. |F1|
28 and thus (1.28) is satisfied.
Now, assume that (1.28) holds for N˜ = N − 1. Note that the distribution on B(a) ˜ for a ∈ Aj, j < N is the same as one would obtain for N = N − 1 and
˜ [ −1 Aj = Aj \ Fj B(a). a∈AN By the induction hypothesis, for all ω ∈ Ω,
j−1 N−1 N−1 E X X [ −1 [ ˜ Fi B(a) ≥ γ Aj . j=1 a∈Aj i=1 j=1
Note that
N−1 N−1 N−1 N−1 N−1 [ ˜ [ [ [ −1 [ [ [ −1 Aj = Aj \ Fj B(a) ≥ Aj − Fj B(a) . (1.30) j=1 j=1 a∈AN j=1 j=1 a∈AN j=1 It follows that
j−1 N−1 N−1 N−1 E X X [ −1 [ [ [ −1 Fi B(a) ≥ γ Aj − γ Fj B(a) . (1.31) j=1 a∈Aj i=1 j=1 a∈AN j=1
Adding P SN−1 F −1B(a) to both sides of (1.31) and taking expectation, we a∈AN i=1 i have:
n(a)−1 N−1 N−1 !
E X [ −1 [ E X [ −1 Fi B(a) ≥ γ Aj + (1 − γ) Fi B(a) . (1.32)
a∈E i=1 j=1 a∈AN i=1 Clearly,
N−1 ! δ E X [ −1 X E Fi B(a) ≥ (|B(a)|) = |AN | |FN | = δ|AN |. (1.33) |FN | a∈AN i=1 a∈AN Because of (1.29), δ(1 − γ) = γ. Hence, combining (1.32) and (1.33), we have
n(a)−1 N−1 ! N
E X [ −1 [ [ Fi B(a) ≥ γ Aj + |AN | = γ Aj = γ|E|.
a∈E i=1 j=1 j=1
29 Theorem 1.44. Let G be an amenable group acting on a probability measure space
(X, B, µ). Let (Fn) be a left Følner sequence in G and let
n−1 [ −1 qn = Fi Fn . i=1
There exists a constant c > 0 such that for all f ∈ L1(X).
c µ M(F , q )[f](x) > λ ≤ kfk . (1.34) n n λ 1
Proof. The proof is quite similar to the proof of Proposition 1.37.
In view of Proposition 1.27, it is enough to prove that the MAX IE on G holds for every ϕ ∈ l1(G). We may assume that ϕ is real-valued and non-negative. Also, may assume that λ = 1. Let
E = {a ∈ G | M˜ [ϕ](a) > 1}.
Since ϕ has finite support, E is a finite set. Notice that a ∈ E iff there exists n > 0 such that
X X qn < ϕ(ga) = ϕ(g). (1.35)
g∈Fn g∈Fna
Let n(a) be the smallest n such that the above holds. We apply Lindenstrauss’
Covering Lemma with δ = 1. There exist a positive number γ and set-valued random variables B(a) for a ∈ E such that:
(1) B(a) is either Fn(a)a or ?,
(2) E(1B(a)(g)) ≤ 2, n(a)−1 E X [ −1 (3) Fi B(a) > γ|E|.
a∈E i=1
30 Then ! ! X X X X X E ϕ(g)1B(a)(g) = ϕ(g)E 1B(a)(g) ≤ 2 ϕ(g) = 2kϕk1. (1.36) a∈E g∈G g∈G a∈E g∈G
On the other hand, ! E X X E X X ϕ(g)1B(a)(g) = ϕ(g) . (1.37) a∈E g∈G a∈E g∈B(a)
If B(a) = ?, then
n(a)−1 X [ −1 ϕ(g) = 0 = F B(a) = 0. i g∈B(a) i=1
If B(a) = Fn(a)a, then by virtue of (1.35),
n(a)−1 n(a)−1 X [ −1 [ −1 ϕ(g) > qn(a) = F Fn(a) = F B(a) . i i g∈B(a) i=1 i=1 So, in either case,
n(a)−1 X [ −1 ϕ(g) ≥ F B(a) . (1.38) i g∈B(a) i=1 Thus, n(a)−1 E X X E X [ −1 ϕ(g) ≥ Fi B(a) > γ|E|. (1.39)
a∈E g∈B(a) a∈E i=1 Combining (1.36), (1.37), and (1.39), we see that
2 |E| < kϕk . γ 1
Corollary 1.45. If, in addition to the assumptions of Theorem 1.44, the limit
|F | lim n n→∞ qn 31 exists and is finite, then 1 X f(Tgx) qn g∈Fn converges a.e.
Corollary 1.46. Let G be an amenable group acting on a probability measure space
(X, B, µ). Let (Fn) be a left Følner sequence satisfying Shulman’s condition (Defini- tion 1.18) and let f ∈ L1(X). Then
1 X f(Tgx) |Fn| g∈Fn converges a.e.
Remark 1.47. As we can see, for the ergodic averages of the form
1 X f(Tgx) (1.40) qn g∈Fn there exist several alternatives:
Sn−1 −1 (1) If qn = | i=1 Fi Fn| and limn→∞ |Fn|/qn = α > 0, then, according to Corol- lary 1.45, (1.40) converges a.e. to a function f:
1 X f(x) = lim (Tgf)(x). n→∞ qn g∈Fn Taking integral of both sides, we obtain
Z |F | Z Z f dµ = lim n · f dµ = α f du. X n→∞ qn X X
Sn−1 −1 (2) If qn = | i=1 Fi Fn| ≤ C|Fn|, but limn→∞ |Fn|/qn does not exist, then (1.40) might not converge. For example, consider a constant function f and the se-
quence (qn) given by ( |Fn|, if n is odd qn = 2|Fn|, if n is even Then clearly (1.40) does not converge.
32 Sn−1 −1 (3) If qn = | i=1 Fi Fn| and |F | lim n = 0, n→∞ qn then (1.40) converges a.e. to 0. Indeed, (1.40) converges to a function f a.e.
by Corollary 1.46. Let g denote the operation of taking the limit as n → ∞ of
An(Fn, qn)[g]. It is clear that if h is a bounded function, then h(x) = 0 for all
1 x. Let f ∈ L (X) be given and let (hm) be a sequence of bounded functions
that converges to f in L1. Then
f = (f − hm) + hm.
and
f ≤ (f − hm) + hn . (1.41) 1 1 1
Note that
Z 1 1 Z X X Tg(hm − f) dµ ≤ |Tg(hm − f)| dµ. (1.42) X qn qn X g∈Fn g∈Fn
We know that qn > |Fn| and
Z Z m→∞ |Tg(hm − f)| dµ = |hm − f| dµ −−−→ 0. X X
Taking (1.41) and (1.42) into account, we conclude that kfk1 = 0.
(4) What happens if
1 n−1 [ −1 lim Fi Fn = ∞, (1.43) n→∞ q n i=1
we do not know in general. It is shown, in [RW92], that when G = Z and (Fn)
consists of intervals in Z one has the following result: If (1.43) holds, then for
33 every Z-action on a non-atomic measure space (X, B, µ), there exists a function
f ∈ L1(X) such that
lim sup |An(Fn, qn)f(x)| = ∞ for a.e. x ∈ X. n→∞
It is possible to obtain a similar result for Zd-actions. However, we do not have
yet a machinery to produce such negative results for actions by more general
amenable groups.
(5) Even more complicated is the case when
1 n−1 [ −1 lim sup Fi Fn = ∞, (1.44) n→∞ q n i=1 but
1 n−1 [ −1 lim inf Fi Fn < ∞, (1.45) n→∞ q n i=1 In this case our ergodic averages may or may not converge depending on the
thickness of the subsequence of N along which convergence in (1.45) occurs.
Example 1.48. The following example is given in [RW92]. Let a non-atomic proba- bility measure space (X, B, µ) be fixed. Consider the additive group Z and the Følner
sequence (Fn) given by
n n Fn = [2 , 2 + n].
Then n−1 [ n2 (F − F ) ≥ . n i 2 i=1 2 Since |Fn|/n → 0, in view of Remark 1.47,
2n+n 1 X lim f(T kx) = 0 for a.e. x n→∞ n2 k=2n 34 for all f ∈ L1(X) and every aperiodic measure-preserving transformation T of X.
On the other hand (by the result of Rosenblatt and Wierdl mentioned in part (4)
of Remark 1.47), for every aperiodic measure-preserving transformation T of X and
every δ > 0, there exists f ∈ L1(X) such that
n 1 2 +n X k lim sup 2−δ f(T x) = ∞ a.e. n→∞ n k=2n
However, it is possible to find a subsequence of (Fn), so that the ergodic averages
An(Fn, |Fn|) converge. Consider
an an Gn = [2 , 2 + an],
where the sequence of natural numbers (an) is defined recursively as follows: a1 = 1
an and an+1 = 2 , n = 1, 2,... . Then
n−1 [ an an−1 an (Gn − Gi) ⊂ [2 − 2 − an−1, 2 + an] i=1
and
n−1 [ an−1 an−1 an−1 an−1 (Gn − Gi) ≤ an + 2 + an−1 = 2 + 2 + an−1 ≤ 3 · 2 ≤ 3an = 3|Gn|. i=1
Hence, according to Theorem 1.44, for every non-atomic dynamical system
(X, B, µ, Z), 2an +a 1 X n lim f(T kx) = 0 for a.e. x. n→∞ an k=2an This also follows from [RW92, Theorem 1.7].
× Example 1.49. Consider the multiplicative group of positive rational numbers Q+.
It can be also thought of as L Z. This is an abelian (and, therefore, amenable), infinitely generated group.
35 × We consider the following Følner sequence in Q+:
k1 k2 kn Fn = p1 p2 . . . pn | −n ≤ k1, k2, . . . , kn ≤ n , (1.46)
where p1, p2,... are the prime numbers ordered by their magnitude. It is easy to see
× n that (Fn) is a Følner sequence in Q . Note that |Fn| = (2n + 1) .
Let us evaluate
n−1 [ −1 qn = Fi Fn . i=1 Since our Følner sequence is nested,
−1 qn = |Fn−1Fn|
k1 k2 kn = p1 p2 . . . pn | −2n + 1 ≤ k1, k2, . . . , kn−1 ≤ 2n − 1, 0 ≤ kn ≤ n
n−1 n−1 = (4n − 1) (2n + 1) < 2 |Fn|.
Hence, according to Theorem 1.44, for every dynamical system (X, B, µ, Q×) and every f ∈ L1(X), the sequence
1 X n−1 f(Tgx) 2 |Fn| g∈Q× converges a.e. In view of Theorem 1.20, we should be able to find a subsequence of
(Fn) along which the ergodic averages A(Fn, |Fn|) would converge a.e. We can take the Følner sequence (Gk) defined as Gk = F22k . Then
k 2k 22 |Gk| = (2 · 2 + 1) and
k−1 k k−1 −1 2k 2k−1 22 2k 22 −22 |Gk−1Gk| = (2 · 2 + 2 · 2 + 1) (2 · 2 + 1) .
36 Let n = 22k−1 . Then 22k = n2 and
k−1 k k−1 |G−1 G | (2 · 22k + 2 · 22k−1 + 1)22 (2 · 22k + 1)22 −22 k−1 k = k 2k |Gk| (2 · 22 + 1)2 (2n2 + 2n + 1)n(2n2 + 1)n2−n = (2n2 + 1)n2 (2n2 + 2n + 1)n = (2n2 + 1)n 2n n = 1 + → e. 2n2 + 1
Thus, according to Corollary 1.46, one has a regular pointwise convergence theorem along the Følner sequence (Gk).
Example 1.50. Another interesting example is the so called Lamplighter Group. Let
M G = Z n Z2. Z
with the group operation defined as
(i, a) · (j, b) = (i + j, σja + b),
where σ is the left shift:
(σx)k = xk+1.
˜ L ˜ Let G = Z Z2 and let ek ∈ G be defined as
ek = (... 0 ... 1 ... 0 ... ). k
Let ( 2n )
X Fn = (j, b) ∈ G |j| ≤ n, b = βkek, where βk = 0 or 1 . k=−2n
We claim that (Fn) is a left Følner sequence in G. Let g ∈ G be an arbitrary element. Pm Then g = (i, a) for some i ∈ Z and a = k=−m αkek. Let f ∈ Fn. Then f = (j, b),
37 P2n where b = k=−2n βkek. So, we have:
m−j 2n ! j X X gf = (i, a)(j, b) = i + j, σ a + b = i + j, αkek + βkek . k=−m−j k=−2n
If n > m, then −2n < −m − j < m − j < 2n. Hence,
m−j 2n 2n X X X αkek + βkek = γkek, k=−m−j k=−2n k=−2n
where γk is either 0 or 1. It follows that if n > m, then gf ∈ Fn unless |i + j| > n.
For a fixed g and Fn, the number of elements gf with f ∈ Fn such that |i + j| > n is
no greater than |i| · 24n+1. Hence,
4n+1 |gFn 4 Fn| ≤ |i| · 2 , whereas
4n+1 |Fn| = (2n + 1)2 .
Therefore, |gF 4 F | |i| n n ≤ −→ 0. |Fn| 2n + 1 n→∞
So, (Fn) is a left Følner sequence in G. One can easily see that (Fn) does not satisfy
Tempelman’s Condition. Let us see whether it satisfies Shulman’s Condition. Let us
−1 write a formula for Fm . P2m If g = (i, a), where a = k=−2m αkek, then
2m+i ! −1 −1 X g = (i, a) = −i, αkek . k=−2m+i
Thus, ( 2m+i )
−1 X Fm = (i, a) |i| ≤ m, a = αkek, where αk = 0 or 1 k=−2m+i
38 and
( 2m+i−j 2n )
−1 X X Fm Fn = (i + j, a + b) |i| ≤ m, |j| ≤ n, a = αkek, b = βkek . k=−2m+i−j k=−2n (1.47) Note that in (1.47), 2m + i − j < 3m + n and −2m + i − j > −3m − n. It follows that, if n ≥ 3m, then
2m+i−j 2n 2n X X X αkek + βkek = γkek. (1.48) k=−2m+i−j k=−2n k=−2n and
( 2n )
−1 X Fm Fn = (l, c) |l| ≤ m + n, c = γkek . k=−2n Consequently,
−1 4n+1 |Fm Fn| = (2n + 2m + 1)2 and
|F −1F | 2m m n = 1 + (1.49) |Fn| 2n + 1
k −1 Let Gk = Fnk , where nk = 3 . Then nk = 3nk−1. Hence, to estimate Gm Gn with n > m, we can make use of (1.49). Then
k−1 S −1 −1 −1 i=1 Gi Gk |F F | k−1 Gk−1Gk nk−1 nk 2 · 3 = = = 1 + k < 2. |Gk| |Gk| |Fn| 2 · 3 + 1
Thus, the Følner sequence (Gk) satisfies Shulman’s Condition and we have the regular
(qk = |Gk|) pointwise ergodic theorem along (Gk).
1.6 A Version of Lindenstrauss’ Covering Lemma for Semigroups and Its Applications
Since Tempelman’s covering lemma (Lemma 1.36) holds not only for a group
but for an arbitrary semigroup as well, it is natural to try to prove Lindenstrauss’
39 Covering Lemma for semigroups. However, here we face some serious difficulties. Let
us consider the following example.
Example 1.51. Assume that we have a badly non-cancellative semigroup. For in-
stance, let the semigroup S consist of all finite subsets of Z together with the empty
set with the semigroup operation being the set intersection. Further, assume that for
all a ∈ E, n(a) = 1. Let F1 contain only sets from the compliment of the union of the
sets belonging to E. Then for all a ∈ E, F1a = {?}. If an analogue of Lindenstrauss’
Covering Lemma held for this situation, it would have been possible to define such
probability distribution on the sets B(a) that expectation of their intersections would
be less that 1 + δ and union is bigger than some fixed constant times |E|. To make
it so that
E(Λ(g) | Λ(g) > 0) < 1 + δ,
we have to let ( {?}, with probabilty δ/|E|, B(a) = ?, otherwise Then ! X δ E |B(a)| = |E| = δ. |E| a∈E But we want X E( |B(a)|) > γ|E|. a∈E for some fixed constant γ. Impossible.
So, to be able to use Lindenstrauss’ approach, we have to assume some degree of cancellativity. We were able to prove the Covering Lemma below under assumption that S is left-cancellative. Note that without left cancellativity assumption some other
40 results needed for a pointwise convergence theorem such as the Transfer Principle
(Proposition 1.27) would not hold as well.
We were not able to obtain the analogues of the results from the previous section
for the general sequence of factors qn. We only consider the case when qn = |Fn| and
impose an analogue of Shulman’s Condition (Definition 1.18) on (Fm) right from the
beginning.
Definition 1.52 (Shulman’s Condition for Semigroups). A Følner sequence in
a left-cancellative semigroup S is called tempered if there exists a constant C such that for all n n−1 [ F −1F a < C|F | for all a ∈ S k n n k=1 Remark 1.53. It is easy to see that for groups the above definition is equivalent to
Definition 1.18.
Lemma 1.54 (Lindenstrauss’ Covering Lemma for Semigroups). Let S be a left-amenable left-cancellative semigroup, (Fn) a tempered left Følner sequence, E ⊂ S a finite set, and let for each a ∈ E a positive integer n(a) be chosen. Let δ > 0 be given. Then it is possible to find set valued random variables Φ(a) for a ∈ E such that
(1) Φ(a) is either Fn(a) or ?.
(2) Let B(a) = Φ(a)a and Λ: G → N be the random function
X Λ(g) = 1B(a)(g). a∈E Then
E(Λ(g) | Λ(g) > 0) < 1 + δ.
41 (3) For some γ > 0 that depends on δ, ! X E |Φ(a)| > γ|E|. (1.50) a∈E
Proof of the Lemma. Let N = maxa∈E n(a). For 1 ≤ j ≤ N we let
Aj = {a ∈ E | n(a) = j}.
We start from j = N. For all a ∈ AN we let (independently from all other b ∈ AN )
( δ FN a with probability pN = , Φ(a) = |FN | (1.51) ? with probability 1 − pN
Assume that we have already defined Φ(a) for all a ∈ Ak, j < k ≤ N. Then for each
a ∈ Aj we do the following (independently from all other b ∈ Aj)
0 0 • If Fja is disjoint from all Φ(a ) for all a ∈ Ak, j < k ≤ N, then
( δ Fja with probability pj = , Φ(a) = |Fj | (1.52) ? with probability 1 − pj
• Otherwise Φ(a) = ?.
We first show that
E(Λ(g) | Λ(g) > 0) < 1 + δ.
Let Λj be the random function
X Λj(g) = 1B(a)(g).
a∈Aj
0 We remark that for j 6= j the events Λj(g) > 0 and Λj0 (g) > 0 are mutually exclusive.
We now define for every g ∈ S and j
N [ [ −1 0 −1 A(j, g) = Aj \ Fj B(a ) ∩ Fj g. 0 0 j =j+1 a ∈Aj0
42 −1 (where Fj g := {a ∈ S | g ∈ Fja}). This represents the set of all a ∈ Aj such that
0 0 g ∈ Fja and such that Fja is disjoint from B(a ) for all a . By our construction, the quantity |A(j, g)| is the number of sets B(b), with b ∈ Aj, such that the probability to
−1 contain the element g is positive. Clearly, |A(j, g)| ≤ |Fj g|. By left-cancellativity,
−1 |Fj g| ≤ |Fj|. Hence, |A(j, g)| ≤ |Fj|. Using Lemma 1.43, we see that
E(Λj(g) | Λj(g) > 0) = E(Λj(g) | Λj(g) ≥ 1) ≤ 1 + pj|A(j, g)| ≤ 1 + pj|Fj| = 1 + δ.
Hence,
E(Λ(g) | Λ(g) > 0) ≤ 1 + δ.
We use induction to show that ! X E |Φ(a)| > γ|E| (1.53) a∈E for
δ γ = , (1.54) 1 + Cδ where C is the constant from Definition 1.52.
If N = 1, ! ! X X X δ E |Φ(a)| = E 1Φ(a)(g) = |A1||F1| = δ|A1| = δ|E| > γ|E|. |F1| a∈A1 a∈A1 g∈F and thus (1.53) is satisfied.
Now, assume that (1.28) holds for N˜ = N − 1. Note that the distribution on Φ(a) ˜ for a ∈ Aj, j < N is the same as one would obtain for N = N − 1 and
˜ [ −1 Aj = Aj \ Fj B(a). a∈AN
43 By the induction hypothesis, N−1 N−1 E X X [ ˜ |Φ(a)| ≥ γ Aj . j=1 a∈Aj j=1
Since (Fn) is tempered,
N−1 N−1 N−1 N−1 N−1 [ ˜ [ [ [ −1 [ [ [ −1 Aj = Aj \ Fj B(a) ≥ Aj − Fj B(a) j=1 j=1 a∈AN j=1 j=1 a∈AN j=1
N−1 [ X ≥ Aj − C |Φ(a)|. (1.55) j=1 a∈AN It follows that N−1 N−1 E X X [ X |Φ(a)| ≥ γ Aj − γC |Φ(a)| . (1.56) j=1 a∈Aj j=1 a∈AN
Adding P |Φ(a)| to both sides of (1.56) and taking expectation, we have: a∈AN
! N−1 !
E X [ E X |Φ(a)| ≥ γ Aj + (1 − γC) |Φ(a)| . (1.57) a∈E j=1 a∈AN Clearly, ! X X δ E |Φ(a)| = E (|Φ(a)|) = |AN | |FN | = δ|AN |. (1.58) |FN | a∈AN a∈AN Hence,
! N−1
X [ E |Φ(a)| ≥ γ Aj + (1 − γC)δ|AN |. (1.59) a∈E j=1
Because of (1.54), δ(1 − γC) = γ. Hence, combining (1.57) and (1.58), we have
! N−1 ! N
E X [ [ |Φ(a)| ≥ γ Aj + |AN | = γ Aj = γ|E|. a∈E j=1 j=1
44 Theorem 1.55. Let S be a left-amenable left-cancellative semigroup acting on a
probability measure space (X, B, µ). Let (Fn) be a tempered left Følner sequence in S.
There exists a constant c such that for all f ∈ L1(X).
c µ M(F )[f](x) > λ ≤ kfk . (1.60) n λ 1
Proof. In view of Proposition 1.24 and Proposition 1.27, it is enough to prove that
the MAX IE on S holds for every ϕ ∈ l1(S). We may assume that ϕ is real-valued
and non-negative. Also, may assume that λ = 1. Let
E = {a ∈ S | M˜ [ϕ](a) > 1}.
Since ϕ has finite support, E is a finite set. Notice that a ∈ E iff there exists n > 0
such that
X X |Fn| < ϕ(ga) = ϕ(g). (1.61)
g∈Fn g∈Fna Let n(a) be the smallest n such that the above holds. We apply Lemma 1.54 with
δ = 1. There exist a positive number γ and set-valued random variables Φ(a) for
a ∈ E such that:
(1) Φ(a) is either Fn(a) or ?,
(2) E(1Φ(a)a(g)) ≤ 2, ! X (3) E |Φ(a)| > γ|E|. a∈E Then ! ! X X X X X E ϕ(g)1Φ(a)a(g) = ϕ(g)E 1Φ(a)a(g) ≤ 2 ϕ(g) = 2kϕk1. a∈E g∈S g∈S a∈E g∈S (1.62)
45 On the other hand, ! E X X E X X ϕ(g)1Φ(a)a(g) = ϕ(g) . (1.63) a∈E g∈S a∈E g∈Φ(a)a
If Φ(a) = ?, then X ϕ(g) = 0 = |Φ(a)| = 0. g∈Φ(a)a
If Φ(a) = Fn(a), then by virtue of (1.61),
X ϕ(g) > |Fn(a)| = |Φ(a)| . g∈Φ(a)a
So, in either case,
X ϕ(g) ≥ |Φ(a)| . (1.64) g∈B(a)
Thus, ! X X X E ϕ(g) ≥ E |Φ(a)| > γ|E|. (1.65) a∈E g∈Φ(a) a∈E
Combining (1.62), (1.63), and (1.65), we see that
2 |E| < kϕk . γ 1
Corollary 1.56. Let S be an left amenable left cancellative semigroup acting on a probability measure space (X, B, µ). Let (Fn) be a left Følner sequence in S satisfying
Shulman’s condition and let f ∈ L1(X). Then
1 X f(Tgx) |Fn| g∈Fn converges a.e.
46 Example 1.57. Consider the multiplicative semigroup N×. A left Følner sequence
(Fn) is given by
k1 k2 kn Fn = p1 p2 . . . pn | −n ≤ k1, k2, . . . , kn ≤ n , (1.66)
Similarly to Example 1.49, we can see that (Fn) is not tempered Følner sequence, but a subsequence of it is.
This example can be generalized. Let S be a cancellative (from both sides) left amenable semigroup. Then, according to Theorem C.3, it is embeddable into an amenable group G that is generated by S. Moreover, according to Proposition C.5, every Følner sequence in S is a Følner sequence in G as well. Hence, for cancellative left amenable semigroups we can use all the results that we obtained for groups, in particular, Theorem 1.20. Thus, we obtain
Theorem 1.58. Every Følner sequence in a cancellative left amenable semigroup has a tempered subsequence. In particular, every cancellative left amenable semigroup has a tempered Følner sequence.
1.7 Open Problems
Question 1.59. Does an analogue of Theorem 1.58 hold for left cancellative, but not right cancellative semigroups? One example of such semigroup is “Lamplighter
Semigroup” described in Example B.9. As one can check, a natural Følner sequence in this semigroup satisfies Tempelman’s Condition, so the pointwise ergodic theorem trivially holds. What happens for other left cancellative semigroups?
Rosenblatt and Wierdl [RW92] proved a pointwise ergodic theorem along se- quences of intervals in Z under a condition that is identical to the condition used
47 by Lindenstrauss. Their result only holds for sequences of intervals in Z (not for general Følner sequences). According to [RW92], this results also holds for sequences of cubes in Zd. So, it is a quite limited result compared to Theorem 1.44. However,
Rosenblatt and Wierdl’s result is stronger in the following sense.
Question 1.60. Rosenblatt and Wierdl showed that for a sequence of intervals the condition of being tempered is, in certain sense, necessary and sufficient for hav- ing a pointwise ergodic theorem along this sequence. Can this result be extended?
Unfortunately, as a simple example shows, even in Z one may find a non-tempered
Følner sequence along which the pointwise ergodic theorem holds. Thus, we must refine Shulman’s condition (1.7) to make it necessary and sufficient in general case.
However, this seems to be a very hard problem.
Question 1.61. Another possibility is to consider some analogues of intervals in more general groups and try to see whether Shulman’s condition is a necessary condition for pointwise convergence in this class of sequences. Of course, this class has to be natural enough for this investigation to make sense. One possibility is to consider the class R of Følner sequences such that one can tile the group by shifts of an element of this Følner sequence. This is exactly the class for which Rokhlin’s Lemma [Pet83, cf. p. 48] holds. Is Shulman’s Condition necessary and sufficient for Følner sequences from the class R?
48 CHAPTER 2
JOINT ERGODICITY OF GROUP ACTIONS
2.1 Definitions and Preliminary Results
Let G be a countable amenable group and (X, B, µ) a probability measure
space. We are interested in studying joint ergodic properties of several G-actions
T (1),...,T (s) on X. By this we mean studying the convergence of the ergodic aver-
ages of the form s 1 X Y (i) fi Tg x |Ln| g∈Ln i=1 Qs (i) for a Følner sequence (Ln) and x ∈ X. For brevity, we shall often just write i=1 Tg fi meaning that s ! s Y (i) Y (i) Tg fi (x) = fi Tg x . i=1 i=1 The symbol T (0) shall denote the identity G-action,
(0) def Tg = Id for all g ∈ G.
Definition 2.1. Let G be a countable amenable group and T (1),...,T (s) be measure-
preserving G-actions on X. The G-actions T (1),...T (s) are called
49 (1) weakly jointly ergodic (w.j.e.) if for all sets A0,...,As ∈ B and for every Følner
sequence (Ln) in G
s ! s 1 X \ (i) Y lim µ Tg (Ai) = µ(Ai) (2.1) n→∞ |Ln| g∈LN i=0 i=0
(2) jointly ergodic (j.e.) if for all sets Ai,Bj ∈ B, 1 ≤ i, j ≤ s and for every Følner
sequence (Ln) in G
s ! s 1 X \ (i) (i) Y lim 2 µ Tg (Ai) ∩ Th (Bi) = µ(Ai)µ(Bi) (2.2) n→∞ |Ln| g,h∈Ln i=1 i=1 The following theorem gives more justification to the word “weak” in Defini- tion 2.1.
Theorem 2.2. T (1),...,T (s) are
(1) w.j.e. iff
s s 2 Z 1 X Y (i) w.−L Y Tg fi −−−→ fi dµ (2.3) |Ln| N→∞ g∈Ln i=1 i=1 X (2) j.e. iff
s s 2 Z 1 X Y (i) L Y Tg fi −−−→ fi dµ (2.4) |Ln| N→∞ g∈Ln i=1 i=1 X
∞ for all f1, . . . , fs ∈ L (X, B, µ) and any Følner sequence of sets (Ln).
Proof. For simplicity, assume that all functions are real.
(1) “If” part. Let the condition (2.3) hold and let A0,A1,...,As ∈ B be given. Then one has:
s ! Z s ! 1 X Y (i) 1 X Y (i) lim Tg fi, 1A0 = lim Tg 1Ai · 1A0 dµ n→∞ |Ln| n→∞ |Ln| g∈Ln i=1 g∈Ln X i=1 s Y Z Z = 1Ai dµ · 1A0 dµ i=1 X X
50 Therefore, (2.1) holds.
“Only if” part. Let T (1),...T (s) be w.j.e. Clearly, (2.3) holds for characteristic
functions of measurable sets. These functions form a dense subset in L∞(X). A rou-
tine check shows that for every j with 1 ≤ j ≤ s and for fixed f1, . . . , fj−1, fj+1, . . . , fs,
∞ the set of functions fj for which (2.3) holds is closed in L . Therefore, (2.3) holds
∞ for all f1, . . . , fs ∈ L .
(2) “If” part. Let the condition (2.4) hold. Let Ai,Bi ∈ B, 1 ≤ i ≤ s, be given and ˜ let fi = 1Ai , fi = 1Bi . Then one has: s s 2 Z 1 X Y (i) L Y Tg fi −−−→ fi dµ |LN | N→∞ g∈LN i=1 i=1 X s s 2 Z 1 X Y (i) ˜ L Y ˜ Tg fi −−−→ fi dµ |LN | N→∞ g∈LN i=1 i=1 X Therefore, s s ! s Z s Z 1 X Y (i) X Y (i) ˜ Y Y ˜ 2 Tg fi , Th fi −−−→ fi dµ · fi dµ |LN | N→∞ g∈LN i=1 h∈LN i=1 i=1 X i=1 X and (2.2) follows.
“Only if” part. One has:
2 s s Z Z s 1 X Y (i) Y 1 X Y (i) (i) Tg fi − fi dµ = 2 Tg fi · Th fi dµ |LN | |LN | g∈L i=1 i=1 g,h∈L i=1 N X 2 N X 2 Z s ! s Z s Z 2 X Y (i) Y Y − Tg fi dµ · fi dµ + fi dµ |LN | g∈LN X i=1 i=1 X i=1 X 2 Z s s Z 1 X Y (i) (i) Y = 2 Tg fi · Th fi dµ − fi dµ |LN | g,h∈LN X i=1 i=1 X s Z Z s ! s Z Y 1 X Y (i) Y − 2 fi dµ · Tg fi dµ − fi dµ . (2.5) |LN | i=1 X g∈LN X i=1 i=1 X
51 This proves the condition (2.4) for characteristic functions if we set fi = 1Ai . The
general case follows by using the density argument.
Remark 2.3. Theorem 2.2 also holds if we replace L2-convergence by Lp-convergence
p ∞ and consider functions f1, . . . , fs from L (X) instead of L (X) for p ≥ 2(s + 1).
Corollary 2.4. If G-actions T (i), 1 ≤ i ≤ s, are j.e., then they are w.j.e.
Remark 2.5. It is clear that if several actions are w.j.e., then each of them is ergodic.
In [BB84], [Ber85b], and [BB86], D. Berend and V. Bergelson studied joint er-
godic properties of several Z-actions. They have obtained several criteria for weak
and strong joint ergodicity of such actions. We prove analogues of their results in
Sections 2.4 and 2.5. In [BB86], it has been shown that the notions of j.e. and w.j.e.
are equivalent for Z-actions. In Section 2.4, we prove this result for actions of count-
able abelian groups and discuss whether this result holds for non-abelian groups. In
Section 2.6 we extend notions of joint functions and sets introduced by D. Berend
in [Ber85b] for Z-actions to general amenable groups and discuss their properties.
In Section 2.7, we investigate connections between notions of joint ergodicity, total
ergodicity, and total joint ergodicity.
The notions of joint ergodicity in context of commuting actions of general groups
were first introduced by V.Bergelson and J.Rosenblatt in [BR88a]. However, their
notion of mutual ergodicity is slightly different from the notion of joint ergodicity that
we study here. We shall make use of some of the results obtained in [BR88a] below.
In Sections 2.2 and 2.3 we present some technical results that will be needed later.
52 2.2 Some Spectral Properties of Measurable Actions
Here we present some spectral properties of measurable actions. If G is an abelian group, then Gˆ denotes the dual group (the group of characters) of G. The reader may consult Appendix E for a review of basic results from character g theory that are used below.
Definition 2.6. A function f ∈ L2(X) is called an eigenfunction of G-action T if there exists a character λ of G such that Tgf = λ(g)f for all g ∈ G. The character λ is called an eigenvalue of T .
The set of all eigenvalues of an action is called its spectrum. If the action is ergodic, then its spectrum is a subgroup of the unit circle.
Definition 2.7. Let G be a countable abelian group. We say that G-actions
T (1),...,T (s) have mutually disjoint spectra if the following condition is satisfied.
(i) Qs If eigenvalues λi of T , 1 ≤ i ≤ s, are such that i=1 λi = 1, then λi = 1 for all 1 ≤ i ≤ s.
In case of two G-actions T and S, this just means that T and S have no common eigenvalues except 1. If actions T (i), 1 ≤ i ≤ s, have mutually disjoint spectra, then, in particular, every pair of transformations T (i), T (j) with 1 ≤ i < j ≤ s have no common eigenvalues except 1.
We shall be using direct products of several measurable actions. On the unitary level, a direct product corresponds to the tensor product of the actions’ unitary representations. For that reason, we shall normally be using the symbol ⊗ as opposed to × to denote the direct product of actions.
53 Proposition 2.8. Let G be a countable abelian group. Let T (i), 1 ≤ i ≤ s, be G-
B Ns (i) (i) actions on (X, , µ). The action i=1 T is ergodic iff each T is ergodic and they have mutually disjoint spectra.
Proof. “Only if” direction is easy. If T (1) ⊗ · · · ⊗ T (s) is ergodic then, clearly, each
T (i) is ergodic. If they do not have mutually disjoint spectra, i.e. if for 1 ≤ i ≤ s ˆ there exist λi ∈ G, not all equal to 1, and square-integrable functions fi such that (i) Qs Tg fi = λi(g)fi for all g ∈ G and i=1 λi = 1, then the function f = f1 ⊗ f2 ⊗ · · · ⊗fs is a non-constant invariant function of T (1) ⊗ · · · ⊗ T (s).
To prove the “if” part, we first consider the case s = 2. Assume that our hy- pothesis is satisfied. Let f ∈ L2(X × X) be a T (1) ⊗ T (2)-invariant function. By
Lemma F.10, f can be expanded in L2 as
X f = un ⊗ vn n
(1) (2) 0 0 so that Tg un = λn(g)un, Tg vn = λn(g)vn for all g ∈ G, and λnλn = 1 for all n.
(1) (2) 0 Since T and T have mutually disjoint spectra, λn = 1 and λn = 1 for all n. So, all functions un and vn are constants. Therefore, f = const as well and, consequently,
T (1) ⊗ T (2) is ergodic.
The general case follows by using induction on s. Namely, let us have s ergodic
G-actions T (1),...,T (s) with mutually disjoint spectra, s > 2. If the actions T (i),
Qs−1 1 ≤ i ≤ s − 1, have eigenfunctions fi and eigenvalues λi, such that i=1 λi = 1, then all λi = 1 since otherwise we set fs to be a constant and λs = 1 and get a contradiction with T (1),...,T (s) having mutually disjoint spectra. So, by the induction hypothesis,
T (1) ⊗ · · · ⊗ T (s−1) is ergodic. T (s) is also ergodic. It remains to show that they have mutually disjoint spectra. Assume that there exists λ ∈ Gˆ that is an eigenvalue
54 (1) (s−1) (s) Qs−1 for both Tg ⊗ · · · ⊗ Tg and T . In view of Remark F.11, λ = i=1 λi, where
(1) (s−1) (1) (s) λ1, . . . , λs−1 are eigenvalues of the actions T ,...,T . Since T ,...,T have
mutually disjoint spectra, it follows that λ1 = ··· = λs−1 = 1. Thus, the actions
T (1) ⊗ · · · ⊗ T (s−1) and T (s) have mutually disjoint spectra. Applying the case s = 2,
we conclude that T (1),...,T (s) is ergodic.
Lemma 2.9. If T (1),...,T (s) are weakly jointly ergodic actions of a countable abelian group G, then they have mutually disjoint spectra.
Proof. Assume they do not. Then they have eigenfunctions fi and corresponding Qs to them eigenvalues λi, 1 ≤ i ≤ s, such that i=1 λi = 1 and at least one of the R eigenvalues, say, λj 6= 1. Then fj 6= const and X fj dµ = 0. So, the right side of (2.3) is zero. On the other hand, the left side of (2.3) does not converge to zero.
Contradiction.
Corollary 2.10. If actions T (1),...,T (s) of a countable abelian group are weakly jointly ergodic, then their direct product is ergodic.
Is there a generalization of the Corollary above for the case when G is not abelian?
The answer is positive if the actions T (1),...,T (s) commute.
Lemma 2.11. If T (1),...,T (s) are weakly jointly ergodic commuting actions of a
Ns (i) countable group G, then the action i=1 T is ergodic.
Proof. See [BR88a, Theorem 2.6].
2.3 Van der Corput’s Inequality for Amenable Groups
The following proposition can be found in [BR88a].
55 Lemma 2.12. Let G be a countable amenable group and let (Ln) be a left Følner
sequence in G. Let A : G → H be a function where H is a Hilbert space. Assume that
A(G) is bounded. Let 1 X S1(N) = A(g) |LN | g∈LN and
1 X 1 X S2(N,H) = A(hg) |LH | |LN | h∈LH g∈LN for N,H ∈ N. Then for all H
lim kS1(N) − S2(N,H)k = 0 N→∞
Proof. Assume that A : G → H is a bounded function, α = maxg∈G A(g). Let a
Følner sequence (Ln), a number H ∈ N, and ε > 0 be given. By definition of Følner sequence, there exists N0 ∈ N such that |LN 4 hLN | < (ε/α)|LN | for all N > N0 and
h ∈ LH . Therefore,
1 X 1 X 1 X kS1(N)−S2(N,H)k = A(g) − A(hg) |LN | |LH | |LN | g∈LN h∈LH g∈LN
1 X 1 X 1 X 1 X ≤ A(g) − A(hg) |LN | |LH | |LH | |LN | g∈LN h∈LH h∈LH g∈LN
1 1 X X = (A(g) − A(hg)) |LN | |LH | h∈LH g∈LN 1 1 X X 1 1 ε ≤ kA(g)k < |LH | |LN | α = ε |LN | |LH | |LN | |LH | α h∈LH g∈LN 4hLN
for all N > N0.
The following Lemma is an abstract analogue of the well-known Van der Corput’s inequality.
56 Lemma 2.13 (Van der Corput’s Inequality). Let G be a countable amenable
group and let (Ln) be a left Følner sequence in G. Let A : G → H be a bounded
function where H is a Hilbert space. For every H ∈ N and ε > 0 there exists N0 ∈ N
such that for all N > N0 2 1 1 1 X X X A(g) ≤ 2 (A(ug),A(vg)) + ε (2.6) |LN | |LN | |LH | g∈LN g∈LN u,v∈LH Proof. First, observe that in view of Lemma 2.12,
1 X 1 X 1 X lim A(g) − A(hg) = 0. N→∞ |LN | |LN | |LH | g∈LN g∈LN h∈LH By Cauchy-Schwartz inequality, 2 2 1 X 1 X 1 X 1 X A(hg) ≤ A(hg) |LN | |LH | |LN | |LH | g∈LN h∈LH g∈LN h∈LH 1 X 1 X X = 2 (A(ug),A(vg)) . |LN | |LH | g∈LN u∈LH v∈LH
2.4 Main Theorem on Joint Ergodicity
Theorem 2.14. Let T (1),...,T (s) be actions of a countable abelian group G on a
probability space (X, B, µ). The following conditions are equivalent:
(1) T (1),...,T (s) are jointly ergodic.
(2) T (1),...,T (s) are weakly-jointly ergodic.
(3) (a) T (1) ⊗ · · · ⊗ T (s) is ergodic. Z s s Z 1 X Y (i) Y (b) Tg fi dµ −−−→ fi dµ |LN | N→∞ g∈LN X i=1 i=1 X ∞ for every f1, f2, . . . , fs ∈ L (X) and every Følner sequence of sets (LN )
in G.
57 Proof. The implications (1) ⇒ (2) and (2) ⇒ (3b) are trivial. The implication (2) ⇒
(3a) follows from Corollary 2.10.
∞ (3) ⇒ (1): We need to show that given f1, f2, . . . , fs ∈ L (X) and a Følner sequence (LN ) one has
s s 2 Z 1 X Y (i) L Y Tg fi −−−→ fi dµ |LN | N→∞ g∈LN i=1 i=1 X
It may be assumed that each fi is a real function bounded by 1 in absolute value and
R Qs (i) 2 that X fi dµ = 0. Let A(g) = i=1 Tg fi, g ∈ G. Then A : G → L (X) is a bounded function because s s Y Y kA(g)k ≤ T (i)f = kf k ≤ 1. 2 g i ∞ i ∞ i=1 i=1
Let us fix a positive integer H and ε > 0. By Lemma 2.13, there exists N0 ∈ N
such that for all N > N0 one has
s 2 Z s 1 X Y 1 X 1 X Y (i) (i) T (i)f ≤ T f · T f dµ + ε g i 2 g+u i g+v i |LN | |LN | |LH | g∈LN i=1 2 g∈LN u,v∈LH X i=1 1 X = 2 S(N, u, v) + ε, |LH | u,v∈LH (2.7) where Z s 1 X Y (i) (i) S(N, u, v) = Tg+ufi · Tg+vfi dµ. |LN | g∈LN X i=1 From the condition (3b), we see that
d Z Y (i) (i) S(N, u, v) −−−→ Tu fi · Tv fi dµ (2.8) N→∞ i=1 X
s Bs s Qd B (i) ∞ s Let (X , , µ ,T ) = i=1(X, , µ, T ). Let F ∈ L (X ) be defined by
F = f1 ⊗ · · · ⊗ fs.
58 Holding H fixed, it now follows from (2.7) and (2.8) that
s 2 1 X Y 1 X Z lim T (i)f ≤ T F · T F dµs + ε. (2.9) g i 2 u v N→∞ |LN | |LH | g∈LN i=1 2 u,v∈LHXs Since T is ergodic,
Z Z !2 1 X s 1 X s 2 TuF · TvF dµ = Thf dµ −−−→ 0. (2.10) |LH | |LH | H→∞ u,v∈LHXs Xs h∈LH From (2.9) and (2.10), we conclude that
s s 2 Z 1 X Y (i) L Y Tg fi −−−→ fi dµ. |LN | N→∞ g∈DN i=1 i=1 X
Remark 2.15. For the case of Z-actions, Theorem 2.14 has been proven by V. Bergel- son and D. Berend in [BB86].
Remark 2.16. In the proof of Theorem 2.14 we used the fact that G is abelian only once — when we quoted Corollary 2.10. If the group G is non-abelian, but
T (1),...,T (s) commute, the 3 conditions of Theorem 2.2 are still equivalent since we can use Lemma 2.11 instead of Corollary 2.10. Another case when our 3 conditions are equivalent is when ergodicity of the tensor product of T (1),...,T (s) can be taken for granted. If X is a compact group and T (1),...,T (s) are actions of continuous
group-automorphisms, then weak joint ergodicity of T (1),...,T (s) implies ergodicity
Ns (i) (1) (s) (1) (s) of i=1 T . Namely, if T ,...,T are w.j.e, then each of T ,...,T is ergodic. As we shall see later (Proposition 4.10), each action T (1),...,T (s) is weakly mixing
Ns (i) and, in view of Theorem 3.3, it follows that i=1 T is ergodic. So, for this case (actions by continuous automorphisms of a compact group) the three types of conver-
gence — strong, weak, and even weaker type of convergence (3b) — are equivalent.
59 2.5 Commutative Case
Now, let us consider the case when the G-actions T (1),...,T (s) commute. By that
(i) (j) (j) (i) we mean that Tg Tg = Tg Tg for all 1 ≤ i < j ≤ s and g ∈ G. Then one can prove
the following generalization of the results obtained in [BB86]
Lemma 2.17. If T (1),...,T (s) are commuting actions on (X, B, µ) of a countable
Ns (i) (i) (1)−1 abelian group G such that i=1 T is ergodic and each T T , 1 < i ≤ s, is
Ns (i) (1)−1 ergodic, then i=2(T T ) is ergodic.
Ns (i) (1)−1 Proof. Assume, to the contrary, that i=2(T T ) is non-ergodic. Then, accord- ˆ ing to Proposition 2.8, there exist λi ∈ G, 2 ≤ i ≤ s, not all of which are 1, with
Qs 2 i=2 λi = 1, and functions fi ∈ L (X) such that
(i) (1)−1 Tg Tg fi = λi(g)fi, i = 2, 3, . . . , s. g ∈ G
(1) (i) (1)−1 Since T and T T commute and latter is ergodic, by Lemma F.7, fi forms
an eigenfunction of T (1), say
(1) Tg fi = ηi(g)fi ∀g ∈ G, i = 2, 3, . . . , s.
Consequently,
(i) T fi = λiηifi, i = 2, 3, . . . , s.
It follows that the function F ∈ L2(Xs) defined by
f2(x2)f3(x3) . . . fs(xs) s F (x1, x2, . . . , xs) = , (x1, x2, . . . , xs) ∈ X f2(x1)f3(x1) . . . fs(x1)
Ns (i) is a non-constant i=1 T -invariant function.
Theorem 2.18. Let T (1),...,T (s) be commuting actions of a countable abelian group
G on a probability space (X, B, µ). The following conditions are equivalent:
60 (1) T (1),...,T (s) are w.-j.e.
(2) T (1),...,T (s) are j.e.
(3) T (1) ⊗ · · · ⊗ T (s) is ergodic, and T (i), T (j) are j.e. for every i 6= j.
(4) T (1) ⊗ · · · ⊗ T (s) is ergodic, and all T (i)T (j)−1, i 6= j, are ergodic.
Proof. The implications (1) ⇒ (2), (2) ⇒ (3), and (3) ⇒ (4) are trivial in view of
Theorem 2.14.
(4) ⇒ (1): We shall use induction on s. The case s = 1 is trivial. If commuting
G-actions T (1),...,T (s), s > 1, are given, such that T (1) ⊗ · · · ⊗ T (s) is ergodic, and
(i) (j)−1 Ns (i) (1)−1 all T T , i 6= j, are ergodic, then, according to Lemma 2.17, i=2(T T ) is ergodic. Then, by the induction hypothesis, T (2)T (1)−1,T (3)T (1)−1,...,T (s)T (1)−1 are
jointly ergodic. But the latter condition, since T (1),...,T (s) commute, is equivalent
to the condition (3b) of Theorem 2.14. Hence, T (1),...,T (s) are j.e.
2.6 Other Approaches to Joint Ergodicity
Following D. Berend [Ber85b, p. 261–262], we introduce the following notions.
Definition 2.19. Given a a set of G-actions T (1),...,T (s) on a probability measure- space (X, B, µ), a joint function is an s-tuple of measurable functions from X to C∗.
A joint invariant function is a joint function such that
s s Y (i) Y Tg fi = fi for all g ∈ G. i=1 i=1
A constant joint function is a joint function (f1, . . . , fs) such that fi = const, 1 ≤ i ≤ s.
61 Definition 2.20. Let a set of G-actions T (1),...,T (s) on (X, B, µ) be given. We say
that this set has property:
(J0) if T (1),...,T (s) are j.e.
B Qs (J1) if for every A0,A1,...,As ∈ with i=0 µ(Ai) > 0 there exists g ∈ G such
(1) (s) that µ(A0 ∩ Tg A1 ∩ · · · ∩ Tg As) > 0 (this may be called “joint transitivity”)
(J2) if the only joint invariant functions for T (1),...,T (s) are constants.
In the case of a single G-action (s = 1), it is well known that the three conditions
above are equivalent (compare with Theorem F.3). When s > 1, the situation is more
complicated.
Proposition 2.21. =⇒ (J1) =⇒ (J0) T1 × · · · × Ts is ergodic =⇒ (J2) =⇒ Proof. The implication (J0) =⇒ (J1) is trivial.
(J0) =⇒ (J2): Let (f1, . . . , fs) be a non-constant joint invariant function for
T (1),...,T (s). Let 1 f0 = Qs . i=1 fi 1 Assume first that each fi takes only positive values. We claim that fi ∈ L (X),
∞ 0 ≤ i ≤ s. For, otherwise we can take functions hi ∈ L (X), 0 ≤ i ≤ s, with hi ≤ fi
and s Y Z hi dµ > 1. i=1 X
By Theorem 2.2, for every left Følner sequence (Fn), Z s Z s s Z 1 X Y (i) 1 X Y (i) Y Tg fi dµ ≥ Tg hi dµ −−−→ hi dµ > 1, |Fn| X |Fn| X n→∞ X g∈Fn i=0 g∈Fn i=0 i=1
62 δ δ δ which is a contradiction. Since (f0 , f1 , . . . , fs ) is also a joint invariant function for
(1) (s) p T ,...,T for every δ > 0, this implies that fi ∈ L (X) for every p > 0 and
each i. Therefore, in view of Remark 2.3, we may apply Theorem 2.2 to functions
f0, f1, . . . , fs and obtain
Z s s Z 1 X Y (i) Y 1 = Tg fi dµ −−−→ fi dµ. (2.11) |Fn| X n→∞ X g∈Fn i=0 i=1
2 Substituting fi for fi, 0 ≤ i ≤ s, we similarly get
s Z Y 2 1 = fi dµ. (2.12) i=1 X
Combining (2.11) and (2.12), we have
s Z s Z 2 Y 2 Y 2 1 = fi dµ = fi dµ . (2.13) i=1 X i=1 X
Since Z Z 2 2 2 fi dµ ≥ fi dµ , 0 ≤ i ≤ s, X X it follows that Z Z 2 2 2 fi dµ = fi dµ , 0 ≤ i ≤ s, X X
so that fi = const, 0 ≤ i ≤ s.
Now, let us consider the general case. If (f1, . . . , fs) is a joint invariant function
(1) (s) for T ,...,T , then so is (|f1|,..., |fs|), and by the preceding part each |fi| is a
constant. Hence, we may assume that |fi| = 1, 1 ≤ i ≤ s. The same reasoning as
before, shows that
Z Z
1 = |fi| dµ = fi dµ , 0 ≤ i ≤ s, X X
and so fi = const, 1 ≤ i ≤ s.
63 (J1) =⇒ T1 × · · · × Ts is ergodic: It is clear that if (J1) holds, then all actions
Ti, 1 ≤ i ≤ s, are ergodic. Assume that T1 × · · · × Ts is not ergodic. Then, according
to Proposition 2.8, there exist non-constant functions fi, 1 ≤ i ≤ s, such that
s s Y (i) Y Tg fi = fi, g ∈ G. (2.14) i=1 i=1
(1) (s) The s-tuple f1, . . . , fs forms a joint invariant function for (T ,...,T ).
(J2) =⇒ T1 × · · · × Ts is ergodic: It is clear that if (J2) holds, then all actions Ti,
1 ≤ i ≤ s, are ergodic. Assume that T1 × · · · × Ts is not ergodic. Then, according to
(i) Proposition 2.8, there exist non-constant eigenfunctions fi of actions T , 1 ≤ i ≤ s
such that
s s Y (i) Y Tg fi = fi, g ∈ G. (2.15) i=1 i=1
(i) We may assume that |fi(x)| = 1 for x ∈ X and 1 ≤ i ≤ s. (Since T is ergodic,
|fi| = const and so can not be 0. Thus, we may divide by |fi|.) Qs Set f0 = i=1 fi. Define measures νi on the unit circle in the complex plane by
−1 νi = µ ◦ fi , i = 0, 1, . . . , s.
For each 0 ≤ i ≤ s pick a point αi in the support of νi. If fi is non-constant, then Qs νi is not a point mass, and hence we may assume that α0 6= i=1 αi. We can find Qs neighborhoods Ii of αi, 1 ≤ i ≤ s, such that if βi ∈ Ii, 0 ≤ i ≤ s, then β0 6= i=1 βi. −1 Qs Qs Setting Ai = fi (Ii) we have i=0 µ(Ai) = i=0 νi(Ii) > 0. However, for all g ∈ G,
s ( s ) \ (i)−1 Y (1) (s) ? Tg Ai = x : fi(x) ∈ I0,Tg f1(x) ∈ I1,...,Tg fs(x) ∈ Is = . i=0 i=1
More can be said in the case when T (1),...,T (s) commute.
64 Theorem 2.22. If T (1),...,T (s) are commuting actions of a countable abelian group
G, then the conditions (J0), (J1), and (J2) are equivalent.
Proof. In view of Theorem 2.18, it is enough to show that any of the conditions (J1) and (J2) implies that T (i)T (j)−1 are ergodic for all i 6= j.
If (J1) holds, then for every A, B ∈ B with µ(A)µ(B) > 0 we have
(i) (j)−1 (i) (j) µ(Tg Tg A ∩ B) = µ(Tg A ∩ Tg B)
(i) (j) = µ(X ∩ · · · ∩ X ∩ Tg A ∩ X ∩ · · · ∩ X ∩ Tg B ∩ X ∩ · · · ∩ X) > 0
for some g ∈ G. Hence, T (i)T (j)−1 is ergodic.
Now, let (J2) holds. Assume that for some i 6= j, T (i)T (j)−1 is not ergodic. Then
(i) (j)−1 there exists a non-trivial invariant set A for T T . Let f(x) = 1A(x) + 1 and h(x) = 1/f(x). Then the s-tuple
i j (1,..., 1, f, 1,..., 1, h, . . . , 1) is a non-constant joint invariant function for (T (1),...T (s)).
Remark 2.23. In the non-commutative case, (J1) 6⇒ (J0) in general. In Chapter 5,
we shall give an example (Example 5.6) of two transformations, S and TE, for which
the Ces`aro averages converge to the right limit along a Følner sequence in Z and,
therefore, satisfy the condition (J1), but do not converge to the right limit along
another Følner sequence and, therefore, are not j.e.
It remains an open question whether any of the implications (J1) =⇒ (J2), (J2)
=⇒ (J1), or (J2) =⇒ (J0) holds in non-commutative case.
65 Remark 2.24. One can define joint invariant sets in the following way: An s-tuple of
(1) (s) measurable sets (A1,...,As) is called joint invariant set for T ,...,T if
s s \ (i) \ Tg Ai = Ai mod µ for all g ∈ G. (2.16) i=1 i=1
Now, we say that a set of G-actions T (1),...,T (s) satisfies the condition
(1) (s) (J3) if T ,...,T have no joint invariant sets A1,...,As with
s Y 0 < µ(Ai) < 1. i=1
It is clear that (J1) =⇒ (J3) and (J2) =⇒ (J3). However, nothing else is known about the relation of the condition (J3) to the other conditions even in the commutative case.
Remark 2.25. For the case of Z-actions, most of the results of this section can be found in [Ber85b].
2.7 Total Ergodicity
Throughout this section G will be a (not necessarily abelian) countable group. By cosets of a subgroup of G we shall always understand its left cosets.
Definition 2.26. Let T be an action of a group G. Let Γ be a subgroup of G. The
subaction TΓ is the restriction of the map T to the subgroup Γ.
Definition 2.27. A G-action T is called totally ergodic if for every normal subgroup
Γ ⊂ G of a finite index the subaction TΓ is ergodic. T is called strongly totally ergodic
if for every infinite subgroup Γ ⊂ G the subaction TΓ is ergodic.
66 Since Z does not contain any subgroups of infinite index, in the case of Z-actions, the notions of total ergodicity and strong total ergodicity are equivalent (and coincide with the well-known definition of total ergodicity for a single transformation). This, however, is not the case for the actions of more general groups.
Example 2.28. Let A be any invertible totally ergodic transformation of a measure space (X, B, µ). Consider the Z2-action T on (X, B, µ) generated by A and I, the
m identity transformation. In other words, T(m,n) = A . Then T is ergodic since there are no non-trivial sets invariant with respect to A and, therefore, there are no non- trivial sets invariant with respect to the action T . Let Γ ⊂ Z2 be a subgroup of finite
n0 index. It contains an element g = (m0, n0) with n0 6= 0. Then Tg = A . Since A is
n0 totally ergodic, the transformation A is ergodic. Hence, TΓ is ergodic and so, T is totally ergodic. On the other hand, the infinite subgroup of Z2 generated by (1, 0) acts on X by identity transformations and, therefore, is not ergodic. Hence, T is not strongly totally ergodic.
Of course, one can ask whether there exist examples of ergodic but not totally ergodic actions. The simplest example is a Z-action given by the transformation A on a space consisting of two points interchanging the two points. Then A is ergodic, but A2 is not. A more interesting example is given below.
Q∞ Example 2.29. Let X = n=2 Zn and let the transformation A be given by
(A(x))n = xn + 1 ( mod n ).
Then T is ergodic. However, not only T is not totally ergodic, but TΓ is not ergodic for every proper subgroup of Z.
67 We would like to obtain some equivalent definitions of total ergodicity. If G is abelian, the following theorem gives two such definitions.
Theorem 2.30. Let T be a measurable action by a countable abelian group G. The following conditions are equivalent:
(1) T is totally ergodic.
2 (2) For every non-constant function f ∈ L (X) the set {Tgf}g∈G is infinite.
ˆ (3) If λ ∈ G is an eigenvalue of the unitary action T , λ 6= 1, then {λ(g)}g∈G is an infinite subset of C.
Proof. The implication (2) ⇒ (3) is trivial. To show that (1) ⇒ (2) assume that for a non-constant function f the set {Tgf}g∈G is finite. Then the set
Γ = {g ∈ G | Tgf = f} is a subgroup of G of a finite index, so T is not totally ergodic.
(3) ⇒ (1): Let T be non-totally ergodic. Then there exists a subgroup Γ ⊂
G with [G : Γ] = n < ∞ and a function f 6= const such that Tγf = f for all
γ ∈ Γ. Let g1Γ, g2Γ, . . . , gnΓ be the cosets of Γ, where g1 = e. Look at the finite- H dimensional Hilbert space generated by the functions Tgk f, 2 ≤ k ≤ n. The space H is an invariant subspace for the unitary operators Tg2 ,...,Tgn . The operators H Tg2 |H ,...,Tgn |H have a common basis in made out of their eigenfunctions. Let
F be one of these eigenfunctions, F 6= const. Then Tgk F = λkF for some λk ∈ C,
2 ≤ k ≤ n. Let the function λ : G → C be defined as λ(g) = λk if g ∈ gkΓ where we set λ1 = 1. It is easy to see that so defined λ in an eigenvalue of T . Clearly, the set
{λ(g)}g∈G contains no more than n elements.
68 There exists a generalization of the above theorem for the case when G is not
2 necessarily abelian. Let L0(X) denote the orthogonal complement of the subspace of constant-functions in L2.
Theorem 2.31. Let T be a measurable action by a countable group G. The following conditions are equivalent:
(1) T is totally ergodic.
2 (2) For every function f ∈ L0(X) the set {Tgf}g∈G is infinite.
H 2 (3) For every finite-dimensional T -invariant subspace in L0(X) the set of the unitary operators Tg|H g∈G is infinite.
Proof. The implications (1) ⇒ (2) and (2) ⇒ (3) are clear.
(3) ⇒ (1): Let T be non-totally ergodic. Then there exists a normal subgroup
Γ ⊂ G with [G : Γ] = n < ∞ and a function f 6= const such that Tγf = f for all
γ ∈ Γ. Let g1Γ, g1Γ, . . . , gnΓ be the cosets of Γ, where g1 = e. Look at the finite- H dimensional Hilbert space generated by the functions Tgk f, 0 ≤ k ≤ n − 1. The H space is an invariant subspace for the unitary operators Tg2 ,...,Tgn .
If g ∈ G, then g = giγ for some γ ∈ Γ and some i, 1 ≤ i ≤ n. Since Γ is a normal subgroup of G, it follows that
0 TgTgk f = Tgiγgk f = Tgigkγ = Tgigk f = Tgi Tgk f.
(where γ0 ∈ H).
Therefore, for every g ∈ G, one has Tg|H = Tgi |H for some i, 1 ≤ i ≤ n. Thus, the set Tg|H g∈G contains no more than n distinct elements.
69 A simple sufficient condition for total ergodicity of arbitrary group actions is given
by the following theorem.
Theorem 2.32. If the subaction TH for an infinite subgroup H ⊂ G is totally ergodic,
then T is totally ergodic.
Proof. Assume T is not totally ergodic, i.e. there exists a normal subgroup Γ ⊂ G of a finite index such that TΓ is non-ergodic. Look at H ∩ Γ. This is a normal subgroup of H. H ∩ Γ has a finite index as a subgroup of H because
H /H ∩ Γ ∼ HΓ/ Γ ⊂ G/Γ.
However, TH∩Γ is non-ergodic as a subaction of TΓ. This is a contradiction with the assumption that TH is totally ergodic.
Corollary 2.33. If the transformation Th for an element h ∈ G of infinite order is totally ergodic (as a Z-action), then T is totally ergodic.
Proof. Take H = {hn|n ∈ Z} and apply Theorem 2.32.
Remark 2.34. The condition of Theorem 2.32 is not only sufficient, but also necessary: take the group G itself as H. On the other hand, the condition of the Corollary is not necessary as the example below shows.
Example 2.35. This is an example of a totally ergodic Z2-action T such that no
2 transformation T(m,n) is ergodic. Let T be the 2-dimensional torus and let τ1, τ2 be √ √ 2 the measurable transformations of T given by τ1(x, y) = x + 2, y + 2 2 mod 1 √ √ and τ2(x, y) = x + 2 2, y + 2 mod 1.
70 2 Let (m, n) ∈ Z with (m, n) 6= 0. We claim that T(m,n) is non-ergodic. Namely, √ √ √ √ T(m,n)(x) = x + (m + 2n) 2, y + (2m + n) 2 . Since (m + 2n) 2 and (2m + n) 2 are rationally dependent, T(m,n) is non-ergodic.
On the other hand, assume that the Z2-action T is non-ergodic. Then there exists
2 2 2 a non-constant function f ∈ L (T ) such that T(m,n)f = f for all (m, n) ∈ Z . In particular, τ1f = T(1,0)f = f and τ2f = T(0,1)f = f. Note that f can be written as
X 2πi(ax+by) f(x, y) = Ca,be . (a,b)∈Z2
The condition that τ1f = f and τ2f = f implies that a + 2b = 0 and 2a + b = 0 which means that a = b = 0, so f = const. The contradiction proves that T is ergodic.
Analogously, one can show that T is totally ergodic.
Definition 2.36. Let T (1),...,T (s) be actions of a countable group G on a measurable space (X, B, µ). The actions T (1),...,T (s) are called totally jointly ergodic if for every
(1) (s) subgroup of a finite index Γ ⊂ G the actions TΓ ,...,TΓ are j.e.
In [Ber85b] it has been proved that if commuting Z-actions T (1),...,T (s) are j.e., then they are totally jointly ergodic. However, for more general group actions, total joint ergodicity does not follow from joint ergodicity.
Example 2.37. Let X and A be as in Example 2.29. Let us consider two Z2-actions
S and S0 defined as follows. The action S is generated by the identity transformation
m 0 I and the transformation A, i.e. S(n,m) = A . The action S is generated by A and
0 n 0 I, i.e. S (n,m) = A . Then, obviously, S and S are ergodic. We claim, moreover, that S and S0 are j.e. Namely, by Theorem 2.18, it is enough to show that S × S0
0−1 0−1 0−1 m−n and SS are ergodic. The action SS given by SS (n,m)x = A x is ergodic.
71 0 0 m n Moreover, the action S × S given by S × S(n,m)(x, y) = (A x, A y) is ergodic by Proposition 2.8. However, neither S nor S0 is totally ergodic (for S take the subgroup
Γ = {(m, 2n) | n, m ∈ Z} ⊂ Z2).
It might still be possible to prove total joint ergodicity of totally jointly ergodic
commuting actions for general abelian groups if one assumes, in addition, that one of
the actions is totally ergodic. The proof of this statement would be more complicated
than the one given in [Ber85b] for Z-actions.
72 CHAPTER 3
JOINT WEAK AND STRONG MIXING
3.1 Definitions and Preliminary Results
Let G be a countable amenable group and ∆ ⊂ G. If a Følner sequence L = (Ln)
is given, then the upper density of the set ∆ with respect to the sequence L is
|∆ ∩ Ln| d¯L(∆) = lim . (3.1) n→∞ |Ln|
The density of the set ∆ with respect to the sequence L is
|∆ ∩ Ln| dL(∆) = lim (3.2) n→∞ |Ln|
(if the limit exists). The upper density of the set ∆ is
d¯(∆) = sup d¯L(∆). (3.3) L∈F(G)
If dL(∆) exists for all L ∈ F(G) and does not depend on L, we call that number the density of the set ∆, and write d(∆).
Let G be a countable amenable group and let (ag) be a sequence in a Banach space B indexed by elements of G.
We say that limg→∞ ag = g if for all ε > 0 the set S = {g ∈ G : kag − ak < ε} is
finite.
73 We say that D-limg→∞ ag = a if for all ε > 0 the set S = {g ∈ G : kag − ak < ε}
is of density 1.
The following lemma is a generalization of a fact that is well-known for numerical
sequences indexed by natural numbers (see, for example, [Pet83, Lemma 6.2]).
Lemma 3.1. Let G be a countable amenable group and let (ag) be a bounded se-
quence in a Banach space B indexed by elements of G. The following conditions are
equivalent:
(1) D-lim ag = a. g→∞
(2) For every Følner sequence (Ln) one has
1 X lim kag − ak = 0. (3.4) n→∞ |Ln| g∈Ln
(3) For every L ∈ F(G) there exists a subset EL ⊂ G with dL(EL) = 0 and such
that lim a = a. g→∞ g g∈ /EL
Proof. (1) ⇒ (2): Let ε > 0 be given and let (Ln) be a Følner sequence in G. If
S = {g ∈ G : kag − ak < ε} , then, by the definition of D-lim, one has
|S ∩ L | lim n = 1. n→∞ |Ln|
Let α be an upper bound for kag − ak. One has:
1 X |S ∩ Ln| |(G \ S) ∩ Ln| kag − ak ≤ · ε + · α |Ln| |Ln| |Ln| g∈Ln |S ∩ L | ≤ ε + 1 − n α ≤ 2 ε |Ln|
74 for big enough n. Hence,
1 X lim kag − ak = 0. n→∞ |Ln| g∈Ln
(2) ⇒ (3): Let a Følner sequence L = (Ln) be given and let
1 E = g ∈ G | ka − ak > . m g m
Then E1 ⊂ E2 ⊂ ... , and each Em has density 0 because of the assumption. There-
fore, one can choose a sequence of natural numbers i1 < i2 < ··· < im−1 < im < . . . such that
|Em ∩ Ln| 1 < for all n > im−1. |Ln| m Let ∞ i im−1 ∞ [ [m [ [ EL = E0 ∪ Em ∩ Lk \ Lk , where E0 = G \ Ln m=2 k=im−1+1 k=1 n=1 Clearly,
lim a = 0. g→∞ g g∈ /EL
On the other hand, if im−1 < n ≤ im, then
|EL ∩ L | |E ∩ L | 1 n ≤ m n < . |Ln| |Ln| m
Hence, dL(EL) = 0.
(3) ⇒ (1): Let ε > 0 and a Følner sequence L = (Ln) be given and let EL ⊂ G
be a set satisfying
dL(EL) = 0 and lim a = a. g→∞ g g∈ /EL
Then there exists a finite set Φ ⊂ G such that |ag − a| < ε for all g ∈ G \ (EL ∪ Φ).
It is clear that S = {g ∈ G : |ag − a| < ε} ⊃ G \ (EL ∪ Φ) and, therefore,
dL(S) ≥ dL(G \ (EL ∪ Φ)) = 1 − dL(EL ∪ Φ) = 1.
75 Since this is true for every L ∈ F(G), we can conclude that d(S) = 1.
Definition 3.2. Let T be a measurable action of a countable amenable group G on a measure space (X, B, µ). The action T is called weakly mixing if for all A, B ∈ B one has:
D-lim µ(TgA ∩ B) = µ(A)µ(B) g→∞
Theorem 3.3 ([BR88b]). Let T be a measure-preserving action of a countable amenable group G on a probability measure space (X, B, µ). The following statements
about the action T are equivalent:
(1) T is weakly mixing.
2 (2) For all f1, f2 ∈ L0(X) one has D-lim (Tgf1, f2) = 0. g→∞
2 (3) For all f1, f2 ∈ L0(X) and for every (Ln) ∈ F(G) one has:
1 X lim |(Tgf1, f2)| = 0. n→∞ |Ln| g∈Ln
(4) T ⊗ T is weakly mixing.
(5) T ⊗ T is ergodic.
(6) T ⊗ S is ergodic for every ergodic G-action S.
(7) When considered as a representation of the group G in L2(X), T has no nontriv-
ial finite-dimensional subrepresentations (or, in the case when G is abelian,—
T has no non-constant eigenfunctions).
Remark 3.4. In [BR88b], weakly mixing actions of groups that are not necessarily amenable are considered. (The case of amenable groups has already been investigated
76 in [Dye65].) We are not interested in the non-amenable case. This is why we have
formulated Theorem 3.3 only for amenable groups.
Proposition 3.5 ([BR88a]). Let S be a weakly mixing action of an amenable group
G and T be an ergodic G-action on a non-atomic separable probability space (X, B, µ).
If T and S commute, then T is also weakly mixing.
Definition 3.6. Let T (1),...,T (s) be actions of a countable amenable group G on a measure space (X, B, µ). The actions T (1),...,T (s) are called
(1) weak-jointly weakly mixing (w-j.w.m.) – if
s ! s \ (i) Y B D-lim µ Tg Ai = µ(Ai),A0,A1,...,As ∈ ; g→∞ i=0 i=0
(2) jointly weakly mixing (j.w.m.) – if
s ! s \ (i) (i) Y B D-lim µ Tg Ai ∩ Th Ai = (µ(Ai)µ(Bi)) ,A1,B1,...,As,Bs ∈ . g,h→∞ i=1 i=1
Proposition 3.7. If T (1),...,T (s) are (weak)-jointly weakly mixing, then they are
(weakly) jointly ergodic.
Proof. This follows from the definition of joint ergodicity and Lemma 3.1.
Theorem 3.8. Let T (1),...,T (s) be actions of a countable amenable group G on a probability measure space (X, B, µ). The following conditions are equivalent:
(1) The G-actions T (1),...,T (s) on (X, B, µ) are w-j.w.m. (resp. j.w.m.)
(2) The G-actions
T (1) ⊗ T (1),T (2) ⊗ T (2),...,T (s) ⊗ T (s)
on (X × X, B × B, µ × µ) are w-j.w.m. (resp. j.w.m.)
77 (3) For every measure space (Y, B, ν) and all w.j.e. (resp. j.e.) G-actions
S(1),...,S(s) on (Y, B, ν), the G-actions
T (1) ⊗ S(1),T (2) ⊗ S(2),...,T (s) ⊗ S(s)
on (X × Y, B × B, µ × ν) are w.j.e. (resp. j.e.)
(4) The G-actions
T (1) ⊗ T (1),T (2) ⊗ T (2),...,T (s) ⊗ T (s)
on (X × X, B × B, µ × µ) are w.j.e. (resp. j.e.)
Proof. We shall prove only the “weak-joint” part of the statements. The proof of the
“joint” part is analogous.
(1) ⇒ (2): Let A0,B0,A1,B1,...,As,Bs ∈ B be given. Then
s ! s s ! s \ (i) Y \ (i) Y D-lim µ Tg Ai = µ(Ai) and D-lim µ Tg Bi = µ(Bi). g→∞ g→∞ i=0 i=0 i=0 i=0
By Lemma 3.1, for every Følner sequence L = (Ln) there exist E1,E2 ⊂ G with dL(E1) = 0, dL(E2) = 0 and such that
s ! s \ Y lim µ T (i)A = µ(A ) g→∞ g i i g∈ /E1 i=0 i=0 and s ! s \ Y lim µ T (i)B = µ(B ). g→∞ g i i g∈ /E2 i=0 i=0
Let E = E1 ∪ E2. Then dL(E) = 0 and
s ! \ lim (µ × µ) T (i) ⊗ T (i) (A × B ) g→∞ g i i g∈ /E i=0 s ! s ! s s \ \ Y Y = lim µ T (i)A · lim µ T (i)B = µ(A ) · µ(B ) g→∞ g i g→∞ g i i i g∈ /E i=0 g∈ /E i=0 i=0 i=0
78 Hence, T (1) ⊗ T (1),T (2) ⊗ T (2),...,T (s) ⊗ T (s) are w-j.w.m.
(2) ⇒ (1): Trivial.
(1) ⇒ (3): Let weakly jointly ergodic actions S(1),...,S(s) on (Y, B, ν) be given. In
order to show that T (1) ⊗ S(1),T (2) ⊗ S(2),...,T (s) ⊗ S(s) are weakly jointly ergodic,
it is enough to show that for all A0,A1,...,As ∈ B and B0,B1,...,Bs ∈ B and every
Følner sequence (Ln) one has
s ! s 1 X \ (i) (i) Y lim (µ × ν) T ⊗ S g (Ai × Bi) = µ(Ai)ν(Bi). (3.5) n→∞ |Ln| g∈Ln i=0 i=1
But the left-hand side of (3.5) is
s ! s ! 1 X \ (i) \ (i) lim µ Tg Ai ν Sg Bi n→∞ |Ln| g∈Ln i=0 i=0 ( s s ! 1 X Y \ (i) = lim µ(Ai) · ν Sg Bi n→∞ |Ln| g∈Ln i=1 i=0 " s ! s # s !) \ (i) Y \ (i) + µ Tg Ai − µ(Ai) · ν Sg Bi . (3.6) i=0 i=1 i=0
(1) (s) Qs Since S ,...,S are w.j.e., the first term of (3.6) tends to i=1 µ(Ai)ν(Bi). The second term is dominated by
s ! s 1 \ Y µ T (i)A − µ(A ) , |L | g i i n i=0 i=1
which tends to 0 as n → ∞ because T (1),...,T (s) are w-j.w.m.
(3) ⇒ (4): Trivial.
79 (4) ⇒ (1): If A0,A1,...,As ∈ B and (Ln) is a Følner sequence, then
" s ! s #2 1 X \ (i) Y µ Tg Ai − µ(Ai) |Ln| g∈Ln i=0 i=1 " s !#2 1 X \ (i) = µ Tg Ai |Ln| g∈Ln i=0 s s ! s !2 Y 1 X \ (i) Y − 2 µ(Ai) · µ Tg Ai + µ(Ai) |Ln| i=1 g∈Ln i=0 i=1 s ! 1 X \ (i) (i) = (µ × µ) T ⊗ T g (Ai × Ai) |Ln| g∈Ln i=0 s s ! s !2 Y 1 X \ (i) (i) Y − 2 µ(Ai) · (µ × µ) T ⊗ T g (Ai × X) + µ(Ai) . |Ln| i=1 g∈Ln i=0 i=1
Since T (1) ⊗ T (1),T (2) ⊗ T (2),...,T (s) ⊗ T (s) are w.j.e., as n → ∞ this expression
tends to
s s s s !2 Y Y Y Y (µ × µ)(Ai × Ai) − 2 µ(Ai) · (µ × µ)(Ai × X) + µ(Ai) i=1 i=1 i=1 i=1 s !2 s !2 s !2 Y Y Y = µ(Ai) − 2 µ(Ai) + µ(Ai) = 0. (3.7) i=1 i=1 i=1
In view of Lemma 3.1, it follows that T (1),...,T (s) are w-j.w.m.
Corollary 3.9. Let T (1),...,T (s) be actions of a countable abelian group G. The actions T (1),...,T (s) are weak-jointly weakly mixing iff they are jointly weakly mixing.
Proof. This follows from Theorem 3.8 and Theorem 2.14.
Corollary 3.10. Let T (1),...,T (s) be commuting actions of a countable amenable
group G. The actions T (1),...,T (s) are weak-jointly weakly mixing iff they are jointly
weakly mixing.
80 Corollary 3.11. Let T (1),...,T (s) be actions of a countable amenable group G by continuous automorphisms of a compact group X. The actions T (1),...,T (s) are weak-jointly weakly mixing iff they are jointly weakly mixing.
Corollaries 3.10 and 3.11 follow from Theorem 3.8 in view of Remark 2.16.
3.2 Abelian Case, Joint Eigenfunctions
Theorem 3.12. Let T (1),...,T (s) be commuting actions of a countable abelian group
G on a non-atomic separable measure space (X, B, µ). The following conditions are
equivalent:
(1) T (1),...,T (s) are w-j.w.m.
(2) T (1),...,T (s) are j.w.m.
(3) T (i) is weakly mixing for every i, and every pair of actions T (i),T (j) is j.w.m.
for every i 6= j.
(4) T (i) is weakly mixing for every i, and T (i)T (j)−1 is weakly mixing for every i 6= j.
(5) T (1),...,T (s) are j.e., and there exists a weakly-mixing G-action T commuting
with T (i) for some i.
Proof. The equivalence of (1) and (2) is established in Corollary 3.9. The fact that (2),
(3), and (4) are equivalent follows from Theorem 2.18, and part (4) of Theorem 3.8
(Joint weak mixing of T (i)’s is equivalent to joint ergodicity of T (i) ⊗T (i)’s. Therefore,
we can make use of the criteria of joint ergodicity.). The implication (4) ⇒ (5) is
trivial. The implication (5) ⇒ (4) follows from Proposition 3.5, Theorem 2.18, and
the fact that our actions are commuting.
81 Definition 3.13. Let T (1),...,T (s) be an action of a countable group G on a proba-
bility space (X, B, µ). Joint function (f1, f2, . . . , fs) is called a joint eigenfunction of
T (1),...,T (s), if there exists λ ∈ Gˆ (which is called a joint eigenvalue of T (1),...,T (s)), such that s s Y (i) Y Tg fi = λ(g) fi, g ∈ G. i=1 i=1 Proposition 3.14. Let T (1),...,T (s) be actions of a countable amenable group G. If
T (1),...,T (s) are j.w.m., then they have no joint eigenfunctions except constants.
(i)s Proof. Suppose (f1, f2, . . . , fs) is a T i=1-joint eigenfunction. For 1 ≤ i ≤ s de-
fine Fi : X × X by Fi(x, y) = fi(x)/fi(y). It is clear that (F1,F2,...,Fs) is a
(i) (i)s T ⊗ T i=1-joint invariant function, so by Proposition 2.21 it is a constant. Hence,
(f1, f2, . . . , fs) is constant as well.
Theorem 3.15. Let T (1),...,T (s) be commuting actions of a countable abelian group
G. The actions T (1),...,T (s) are j.w.m. iff they have no joint eigenvalues except
constants.
Proof. In view of Proposition 3.14, we need to prove the “if” part only. If the ac-
tions T (1),...,T (s) have no non-constant joint eigenfunctions, then by Theorem 2.22,
T (1),...,T (s) are j.e. Assume that T (1),...,T (s) are not j.w.m. Then by Theorem 3.12
(1) T is not weakly mixing. By Theorem 3.3, T1 has a non-constant eigenfunction, say,
(i)s f1. Let fi = 1 for 2 ≤ i ≤ s. Then (f1, f2, . . . , fs) is a non-constant T i=1-joint eigenfunction.
Remark 3.16. In [Ber85b], D. Berend introduced the notions of weak joint weak mixing, joint weak mixing, and joint eigenfunctions for Z-actions. He proved Theo- rem 3.12 and Theorem 3.15 for that case.
82 3.3 Joint Strong Mixing
An action T of a countable group G is called (strongly) mixing if limg→∞ µ(TgA ∪
B) = µ(A)µ(B) for all A, B ∈ B.
Definition 3.17. The actions T (1),...,T (s) of a countable group G are called weak- jointly strongly mixing (w-j.s.m.) if for all A0,A1,...,As ∈ B one has s ! s \ (i) Y lim µ Tg Ai = µ(Ai). g→∞ i=0 i=1 A set ∆ ⊆ G2 is of eventually bounded fibers if there exists C > 0 such that
|{h ∈ G | (g, h) ∈ ∆}| ≤ C for all g ∈ G outside a finite set. Let (ag,h) be a sequence
2 in a Banach space B indexed by the elements of G . We shall write EC - limg,hag,h = a if for every ε > 0 there exists a set ∆ of eventually bounded fibers such that
2 kag,h − ak < ε for all (g, h) ∈ G \ ∆.
Definition 3.18. The actions T (1),...,T (s) of a countable group G are called jointly strongly mixing (j.s.m.) if for all A1,B1,...,As,Bs ∈ B one has s ! s \ (i) (i) Y EC - lim µ (Tg Ai ∩ Th Bi) = (µ(Ai)µ(Bi)). g,h i=1 i=1 Proposition 3.19. T (1),...,T (s) are
(1) weak jointly strongly mixing iff Z s s Z Y (i) Y ∞ lim Tg fi dµ = fi dµ, fo, f1, . . . , fs ∈ L (X); g→∞ X i=0 i=0 X (2) jointly strongly mixing iff Z s s Z Z Y (i) (i) Y EC - lim (Tg fi · Th gi) dµ = fi dµ · gi dµ , g,h X i=1 i=0 X X ∞ f1, g1, . . . , fs, gs ∈ L (X).
Proof. The proof is routine.
83 CHAPTER 4
JOINT PROPERTIES OF ACTIONS BY AUTOMORPHISMS OF COMPACT GROUPS
4.1 Preliminary Results
In this chapter, we shall be interested in the case when a countable (possibly non- abelian) group G acts on a compact group by its automorphisms. In this case, the action T is a homomorphism T : G → Aut(X). A number of important facts about ergodicity and mixing of such actions has been established (see, for example [Sch95]).
We give a brief summary of some of those results below.
We consider finite-dimensional continuous unitary representations of the com- pact group X. If finite-dimensional continuous unitary representations τ1 and τ2 are unitarily equivalent, we write τ1 ∼ τ2. A review of theory of continuous unitary representations of compact group can be found in Appendix E.
Theorem 4.1. Let T be an action of a countable group G by automorphisms of a compact group X. The action T is ergodic iff for every non-trivial continuous irreducible unitary representation τ of X on a Hilbert space H the group
Gτ = {g ∈ G | Tgτ ∼ τ} has infinite index in G.
84 Here Tgτ denotes the unitary representaion of X defined as (Tgτ)(x) = τ(Tgx).
Remark 4.2. This theorem has been proven in [Ber85a] under weaker assumptions
than in our version. In the case of Z-actions, this result is due to Halmos [Hal43] in
the case of abelian X and Kaplansky [Kap49] in the general case.
Theorem 4.3 ([Sch95]). Let T be an action of a countable group G by automor-
phisms of a compact group X. The action T is mixing iff for every non-trivial, continuous, irreducible, unitary representation τ of X on a Hilbert space H the group
Gτ = {g ∈ G | Tgτ ∼ τ}
is finite.
The following fact is curious.
Proposition 4.4. Let T be an action of a countable group G by group-automorphisms
of a compact group X. The action T is ergodic iff it is totally ergodic. T is mixing
iff it is strongly totally ergodic.
This follows directly from the two above theorems and Definition 2.27.
Theorem 4.1 and Theorem 4.3 admit simpler formulations in the case when X is
abelian. We shall denote the dual group of X as Xˆ and the dual right action of T
(defined in Appendix E) on Xˆ as Tˆ. Since a function γ can be thought of as element
of both L2(X) and Xˆ we shall use notation T γ when we think about γ as an element
of L2(X) and Tˆ γ when we think about γ as an element of Xˆ. Since X is compact,
Xˆ is a discrete countable group.
Corollary 4.5. Let T be an action of a countable group G by automorphisms of a
compact abelian group X. The action T is ergodic iff for every non-trivial character
85 γ ∈ Xˆ the group n o ˆ Gγ = g ∈ G Tgγ = γ has infinite index in G.
Corollary 4.6. Let T be an action of a countable group G by automorphisms of a compact abelian group X. The action T is mixing iff for every non-trivial character
γ ∈ Xˆ the group n o ˆ Gγ = g ∈ G Tgγ = γ is finite.
Remark 4.7. Since every infinite subgroup of Z has finite index, for Z-actions by automorphisms of compact groups ergodicity is equivalent to mixing. Since already the group Z2 has infinite subgroups of infinite index, this fact does not hold for general group actions.
Example 4.8. Let X = T2 be the 2-dimensional torus. Let us consider the Z2-action
T on X generated by two automorphisms of X, I and A, where I is the identity
n transformation and A is an ergodic (as a Z-action) automorphism: T(m,n) = A . One can easily see that so defined action T is ergodic but not mixing.
A natural question to ask is how weak mixing of the actions under consideration is related to their ergodicity and (strong) mixing. The following result is quite surprising
(considering Example 4.8).
Proposition 4.9. Let T be an action of a countable group G by automorphisms of a compact abelian group X. The action T is ergodic iff it is weakly mixing.
86 Proof. Assume that T is not weakly mixing. Then, by Theorem 3.3, T × T is not
ergodic. By Corollary 4.5 and because the dual group of X ×X is Xˆ ×Xˆ, there exists
γ ∈ Xˆ × Xˆ such that the group ˆ ˆ Gγ = g ∈ G T × T γ = γ g ˆ is of finite index in G. But γ = (γ1, γ2) for some γ1, γ2 ∈ X. Therefore, Gγ1 = n ˆ o g ∈ G | Tgγ1 = γ1 ⊃ Gγ. Hence, the action T is not ergodic.
The same fact holds for the general case when X is a compact group, not neces- sarily abelian.
Proposition 4.10. Let T be an action of a countable group G by automorphisms of a compact group X. The action T is ergodic iff it is weakly mixing.
Proof. Assume that T is not weakly mixing. Then by Theorem 3.3, T ⊗ T is not
ergodic. By Theorem 4.3, there exists a non-trivial, continuous, irreducible, unitary
representation τ of X × X on L2(X × X) = L2(X) ⊗ L2(X) such that the group
Gτ = {g ∈ G | (Tg ⊗ Tg)τ ∼ τ}
has a finite index in G. By Theorem E.5, there exist continuous irreducible unitary
2 representations of X on L (X), τ1 and τ2, such that τ is unitarily equivalent to the
X × X-representation given by τ1 ⊗ τ2. If (Tg ⊗ Tg)τ is unitarily equivalent to τ for
2 some g ∈ G, then there exists a unitary operator Ug from L (X × X) to itself such
that
(Tg ⊗ Tg)(τ1(x1) ⊗ τ2(x2)) Ug = Ug (τ1(x1) ⊗ τ2(x2)) , for all x1, x2 ∈ X.
Hence, (Tgτ1)Ug = Ugτ1 and (Tgτ2)Ug = Ugτ2. It follows that
Gτ1 = {g ∈ G | Tgτ1 ∼ τ1} ⊃ Gτ .
87 Therefore, the action T is not ergodic.
Remark 4.11. Proposition 4.9 is a special case of Proposition 4.10. Since the proof of
Proposition 4.9 is somewhat simpler than the proof of Proposition 4.10, we presented both proofs.
4.2 Joint Ergodicity of Actions by Group Automorphisms
Proposition 4.12. Let T (1),...,T (s) be actions of a countable amenable group G by
automorphisms of a compact abelian group X. The actions T (1),...,T (s) are j.e. iff
they are w.j.e.
Proof. See Remark 2.16
Definition 4.13. Let (ag)g∈G be a sequence in a set A indexed by elements of an amenable countable group G. Denote Sa = {g | ag = a} for a ∈ A. The sequence
(ag) is of density 0 multiplicity if d(Sa) = 0, i.e.
|Sa ∩ Ln| lim = 0 for every (Ln) ∈ F(G) and all a ∈ G. n→∞ |Ln|
Theorem 4.14. Let T (1),...,T (s) be actions of a countable amenable group G by automorphisms of a compact abelian group X. The actions T (1),...,T (s) are jointly ˆ Ps ˆ(i) ergodic iff for all γ1, γ2, . . . , γs ∈ X, not all of them 0, the set i=1 Tg γi is of g∈G density 0 multiplicity.
Proof. By Proposition 4.12, T (1),...,T (s) are j.e. iff they are w.j.e. Since linear com- binations of characters are dense in L2(X) and in view of Theorem 2.2, T (1),...,T (s) ˆ ˆ are weakly jointly ergodic iff for every γ1, γ2, . . . , γs ∈ X, not all 0, every γ0 ∈ X, and
88 every (Ln) ∈ F(G) we have
s ! ( s )
1 X Y (i) 1 X ˆ(i) Tg γi, γ0 = g ∈ Ln Tg γi = γ0 −−−→ 0, |Ln| |Ln| n→∞ g∈Ln i=1 L2(X) i=1
Qs (i) Ps ˆ(i) ˆ because i=1 Tg γi is i=1 Tg γi when thought of as an element of X. This means Ps ˆ(i) that i=1 Tg γi is of density 0 multiplicity. g∈G
Remark 4.15. In the statement of Theorem 4.14 it is enough to consider only such ˆ Ps Ps ˆ(i) Ps ˆ(i) γ1, . . . γs ∈ X that i=1 γi = 0. Namely, if i=1 Tg γi = i=1 Tg0 γi for some g0 ∈ G
0 ˆ(i) Ps 0 and all g ∈ ∆ ⊆ G, then for γi = γi − Tg0 γi, 1 ≤ i ≤ s, one has i=1 γi = 0 and
Ps ˆ(i) 0 i=1 Tg γi = 0 for all g ∈ ∆. Hence, we can formulate an equivalent condition for joint ergodicity:
Corollary 4.16. Let T (1),...,T (s) be actions of a countable amenable group G by au- tomorphisms of a compact abelian group X. The actions T (1),...,T (s) are jointly ˆ ergodic iff for all γ1, γ2, . . . , γs ∈ X, not all of them 0, the set
( s )
X ˆ(i) g ∈ G Tg γi = 0 i=1
has density 0.
For the case when s = 2 the condition becomes even simpler.
Corollary 4.17. Let T and S be actions of a countable amenable group G by automor-
phisms of a compact abelian group X. The actions T and S are jointly ergodic iff for
all γ ∈ Xˆ, γ 6= 0, the set n o ˆ ˆ g ∈ G T γ = Sγ
has density 0.
89 Remark 4.18. For Z-actions, Theorem 4.14 has been established in [Ber85b] and, in
more convenient form, in [BB86].
Now, we would like to extend this theorem to the case when X is a compact,
not necessarily abelian group. Let H0 be a fixed infinite-dimensional Hilbert space.
We shall consider continuous unitary representations τ of X in a finite-dimensional
subspace H of H0. (The reader may consult Appendix E for a brief review of repre-
sentation theory of compact groups.) If we fix a basis in H, then τ is given by a n×n
unitary matrix (tk,l), whose elements tk,l are continuous complex-valued functions on
X. We shall denote by U the set of all continuous unitary finite-dimensional repre- sentations of X in H0. Let Γ be the dual object of X, i.e. the set of all equivalence
classes of representations in U. If τ ∈ γ ∈ Γ, then the dimension of the representation
space of τ is denoted by dτ or dγ. If τ ∈ U and T is a G-action by automorphisms
of the group X, then Tgτ is another irreducible continuous unitary representation of
X in H given by (Tgτ)(x) = τ(Tgx). Thus, the action T gives rise to a dual right
action on U which induces, in turn, an action on Γ which shall also be denoted by
1 2 T . If τ1 ∼ (tk,l)k,l=1,n1 and τ2 ∼ (tk,l)k,l=1,n2 are two unitary representations of X in
finite-dimensional Hilbert spaces H1 and H2 respectively, then we can define their H H tensor product τ1 ⊗τ2 as the representation of X in 1 ⊗ 2 with the matrix (tk1k2l1l2 )
where t = t1 t2 , 1 ≤ k , l ≤ n ; 1 ≤ k , l ≤ n . Similarly, one can define k1k2l1l2 k1,l1 k2,l2 1 1 1 2 2 2 Ns the tensor product i=1 τi of s representations τ1, . . . , τs of X in finite-dimensional H H Ns H Hilbert spaces 1,..., s as a representation of X in the Hilbert space i=1 i. Let τ ∈ γ ∈ Γ and τ 0 ∈ γ0 ∈ Γ. The unitary representation τ ⊗τ 0 is not irreducible
in general. However, in view of the Theorem E.9, it can be decomposed as
0 τ ⊗ τ = m1τ1 ⊕ · · · ⊕ mrτr.
90 for some irreducible representations τi with multiplicities mi, 1 ≤ i ≤ r. We define a binary operation × on Γ as follows:
0 def γ × γ = {τ1, . . . , τr}.
(See Appendix E for more details.)
Let us consider the regular left representation τ of X in L2(X). By Peter-Weyl
Theorem (Theorem E.12) L2(X) can be represented as a direct sum of τ-invariant
finite-dimensional Hilbert spaces Hr, r = 1, 2,... , so that τ is decomposed onto a direct sum of finite-dimensional irreducible representations τr, r = 1, 2,... . Let us
2 r fix a basis in L (X) and let representations τr have matrices (tk,l)k,l=1,dr with respect
r to that basis, dr = dτr . According to Theorem E.12, the set of functions {tk,l} spans a dense linear subspace in L2(X).
We shall need the following definition.
Definition 4.19. Let {Ag | g ∈ G} be a family of non-empty subsets of a set A indexed by elements of a countable amenable group G. We say that the family of sets
(Ag)g∈G is weakly disjoint if for every a ∈ A the set {g ∈ G | a ∈ Ag} has density 0 in
G.
In other words, the sets Ag are weakly disjoint if for every subset P of G of positive T density, the set g∈P Ag is empty. Now, let T (1),...,T (s) be actions of a countable amenable group G by automor- phisms of X. Let γ1, . . . , γs ∈ Γ and let τi ∈ γi, 1 ≤ i ≤ s. Fix a g ∈ G and consider Ns (i) the continuous unitary representation τ = i=1 Tg τi. The representation τ does not have to be irreducible, but, in view of Theorem E.9, it can be represented as a direct
91 sum of irreducible representations:
τ = k1ψ1 ⊕ k2ψ2 ⊕ · · · ⊕ kmg ψmg (4.1)
0 for some ψi ∈ γi ∈ Γ with multiplicities ki > 0, 1 ≤ i ≤ mg, where mg is an integer number depending on g. On the level of the dual object Γ, we have:
(1) (s) 0 0 Tg γ1 × · · · × Tg γs = {γ1, . . . , γmg }.
Theorem 4.20. Let T (1),...,T (s) be actions of a countable amenable group G by automorphisms of a compact group X. The actions T (1),...,T (s) are jointly ergodic iff for all γ1, . . . , γs ∈ Γ, not all of them trivial, the family of sets
T (1)γ × · · · × T (s)γ g 1 g s g∈G is weakly disjoint.
Proof. “If” part. In view of Proposition 4.12, it is enough to prove that the actions
T (1),...,T (s) are weakly jointly ergodic. In view of Theorem 2.2 and because the
r 2 (1) (s) functions tk,l, 1 ≤ k, l ≤ dr, r = 1, 2,... , span a dense set in L (X), T ,...,T are weakly jointly ergodic if for all natural numbers r0, k0, l0, r1, k1, l1, . . . , rs, ks, ls such that 1 ≤ k , l ≤ d and not all tri are constants, and every Følner sequence i i ri ki,li
(Ln) ∈ F(G) one has
s ! 1 X Y T (i)tri , tr0 −−−→ 0. (4.2) g ki,li k0,l0 |Ln| n→∞ g∈Ln i=1 L2(X)
Let us fix g ∈ G and let t = Qs T (i)tri . Note that t is a coordinate function g i=1 g ki,li g Ns (i) of the representation τ = i=1 Tg τri . Since τ = k1ψ1 ⊕ k2ψ2 ⊕ · · · ⊕ kmg ψm(g) for some irreducible unitary representations ψ1, . . . , ψm(g), it follows that tg is a linear
92 combination of coordinate functions of ψ1, ψ2, . . . , ψmg . Because of the orthogonality
conditions (Theorem E.12, equation (E.5)), if none of the representations ψi, 1 ≤ i ≤
m , is unitarily equivalent to τ , then (t , tr0 ) = 0. If τ is unitarily equivalent to g r0 g k0,l0 r0 (1) (s) some ψi, 1 ≤ i ≤ m(g), then γr0 ∈ Tg γ1 × · · · × Tg γs, where γr0 3 τr0 . But the sets
T (1)γ × · · · × T (s)γ , g ∈ G, are weakly disjoint. Therefore, (t , tr0 ) = 0 only for a g 1 g s g k0,l0 set of g of density 0. Hence, (4.2) holds and T (1),...,T (s) are w.j.e.
“Only If” part. Assume that there exist γ1, γ2, . . . , γs, not all of them trivial, such
(1) (s) that the sets Tg γ1×· · ·×Tg γs, g ∈ G, are not weakly disjoint. This means that there
(1) (s) exists γ0 ∈ Γ and a subset S of G of positive density such that γ0 ∈ Tg γ1 ×· · ·×Tg γs Ns (i) for all g ∈ S. Let us consider the character of the representation τ = i=1 Tg τi:
s s Y (i) Y (i) χ(τ) = χ Tg τi = Tg χ(τi). (4.3) i=1 i=1
Let fi = χ(τi), 1 ≤ i ≤ s. Then the functions fi are continuous. Note that at least
one of τi’s, 1 ≤ i ≤ s, is non-trivial. Let it be τ1. Then f1 is not a constant. Moreover, R since τ1 6∼ 1, X f1 dµ = 0 (by virtue of Theorem E.9). On the other hand, in view of (4.1), one has:
mg X χ(τ) = ki χ(ψi). (4.4) i=1
If g ∈ S, then at least one of ψi’s, say, ψlg ∈ γ0, so if we let f0 = χ(γ0), then R ¯ def X χ(τ)f0 dµ = kg ≥ 1, where kg = klg is the multiplicity of ψlg in the direct sum decomposition (4.1) of τ. Combining (4.3) and (4.4), we obtain:
s ! 1 X Y (i) lim Tg fi, f0 ≥ dL(S). n→∞ |Ln| g∈Ln i=1 L2(X)
Choosing a Følner sequence L such that dL(S) = d(S) > 0 we get a contradiction.
93 Remark 4.21. We can modify the condition “the family of sets
T (1)γ × · · · × T (s)γ g 1 g s g∈G is weakly disjoint” by observing that, in view of Corollary E.11,
(1) (s) (1) (s) γ0 ∈ Tg γ1 × · · · × Tg γs iff 1 ∈ γ¯0 × Tg γ1 × · · · × Tg γs, where 1 denotes the equivalence class of the trivial representation. Hence, an equiv- alent criterion is: “for all γ0, γ1, . . . , γs ∈ Γ, not all of them trivial, the set
( s )
(i) g ∈ G 1 ∈ Tg γi i=0 has density 0”.
Remark 4.22. D. Berend in [Ber88] proved a spacial case of Theorem 4.20 for com- muting Z-actions. For that case he was able to obtain a simpler criterion of ergodicity of the actions T (1),...,T (s). Namely, he proved that T (1),...,T (s) are j.e. iff the only solution of the equation
(i) (j) Tn γ = Tn γ with 0 ≤ i < j ≤ s, γ ∈ Γ, and n ∈ Z, is γ = 1.
4.3 Joint Weak Mixing of Actions by Group Automorphisms
Theorem 4.23. Let T (1),...,T (s) be actions of a countable amenable group G by automorphisms of a compact abelian group. The actions T (1),...,T (s) are jointly weakly mixing iff they are weakly jointly ergodic.
Proof. By Proposition 3.7, if T (1),...,T (s) are j.w.m., then they are j.e. and, therefore, w.j.e. It remains to prove validity of the converse statement.
94 Assume that T (1),...,T (s) are not jointly weakly mixing. In view of Theorem 3.8
0 00 0 00 0 00 ˆ and Theorem 4.14, there exist γ1, γ1 , γ2, γ2 , . . . , γs, γs ∈ X, not all of them 0, a
0 00 ˆ subset S of G of positive density, and γ0, γ0 ∈ X such that s 0 00 X ˆ(i) ˆ(i) 0 00 (γ0, γ0 ) = Tg × Tg (γi, γi ) i=1
0 for all g ∈ S. WLOG one of γi, 1 ≤ i ≤ s, is non-zero. Since
s 0 X ˆ(i) 0 γ0 = Tg γi i=1 for all g ∈ S, it follows that T (1),...,T (s) are not j.e. Contradiction.
A generalization of this theorem exists for the case when X is a non-abelian
compact group. The proof is somewhat more complicated.
Theorem 4.24. Let T (1),...,T (s) be actions of a countable amenable group G by automorphisms of a compact group X. The actions T (1),...,T (s) are jointly weakly
mixing iff they are weakly jointly ergodic.
Proof. If T (1),...,T (s) are j.w.m., then they are w.j.e. It remains to prove the converse statement. Let T (1),...,T (s) be not jointly weakly mixing. In view of Theorem 3.8
and Theorem 4.20, there exist γ1, . . . , γs ∈ Γ(X × X), a subset S of G of positive
¢s (i) (i) density, and γ0 ∈ Γ(X × X) such that γ0 ∈ i=1(Tg ⊗ Tg )γi for all g ∈ S. Note
0 00 0 00 0 00 0 00 that there exist γ1, γ1 , γ2, γ2 , . . . , γs, γs ∈ Γ(X) such that γi = γi ⊗ γi , 1 ≤ i ≤ s. Let
s s ! s ! (i) (i) (i) 0 (i) 00 γ = T ⊗ T gγi = Tg γi ⊗ Tg γi . (4.5) i=1 i=1 i=1 Note that we deal with two quite different types of tensor product in (4.5). The
opertaion on the dual object denoted by the symbol ¢ corresponds to the tensor
95 product of two or more representations of X considered as a representation of X. The
operation ⊗ corresponds to the tensor product of two representations of X considered
as a representation of X × X.
¢s (i) 0 0 0 ¢s (i) 00 00 00 0 0 Let Tg γ = {δ , . . . , δ 0 } and Tg γ = {δ , . . . , δ 00 }, where ψ ∈ γ ∈ i=1 i 1 mg i=1 i 1 mg j j
00 00 0 00 Γ(X) and ψp ∈ γq ∈ Γ(X), 1 ≤ p ≤ mg, 1 ≤ q ≤ mg . Then
0 00 0 00 γ = δp ⊗ δq | p = 1, . . . , mg, q = 1, . . . , mg (4.6)
0 00 Ns (i) (i) Note that δp ⊗δq is an element of Γ(X×X). We know that γ0 ∈ i=1(Tg ⊗Tg )γi
0 00 for all g ∈ S. This means that for given g there exist p, 1 ≤ p ≤ mg, and q, 1 ≤ q ≤ mg
0 00 0 such that γp ⊗γq = γ0. But this means that there exists γ0 ∈ Γ(X) such that for each
0 0 (1) (s) g ∈ S one has γp ∈ γ0 (where p depends on g). Hence, by Theorem 4.20, T ,...,T are not j.e.
Remark 4.25. Equivalence of joint ergodicity and joint weak mixing for Z-actions by
automorphisms of abelian groups has been established in [Ber85b].
4.4 Joint Strong Mixing of Actions by Group Automorphisms
Definition 4.26. In the setup of Definition 4.13 the sequence (ag) is of
(1) finite multiplicity — if |Sa| < ∞, a ∈ A,
(2) bounded multiplicity — if supa∈A |Sa| < ∞.
Theorem 4.27. Let T (1),...,T (s) be actions of a countable amenable group G by automorphisms of a compact abelian group X. The actions T (1),...,T (s) are
Ps ˆ(i) (1) w-j.s.m. iff the sequence i=1 Tg γi is of finite multiplicity; g∈G 96 Ps ˆ(i) (2) j.s.m. iff the sequence i=1 Tg γi is of bounded multiplicity. g∈G
The proof is routine.
It is possible to obtain a similar condition for actions by automorphisms of non- abelian groups along the lines of Theorem 4.20.
Question 4.28. Is it true that if the G-actions T (1),...,T (s) by automorphisms of
an abelian compact group X are w-j.s.m., then they are j.s.m.? It is not even known
for Z-actions.
4.5 Examples and Some Open Problems
Z2 2 2 Example 4.29. Let X = (Z2) and let T,S ∈ Aut(Z ,X) be Z -actions on X
defined as follows: (T(p,q)(x))m,n = xm+p,n+q and (S(p,q)(x))m,n = xm+p+q,n+q. We
claim that so defined Z2-actions T and S are mixing, jointly ergodic, but not w-j.s.m. ˆ L ˆ ˆ Indeed, the group X is isomorphic to Z2 Z2. The dual actions T and S act on Xˆ by shifts. Let γ ∈ Xˆ, γ 6= 1. It is clear that γ can not be a periodic point for
either Tˆ or Sˆ. Hence, both T and S are mixing according to Theorem 4.3.
On the other hand,
ˆ ˆ 2 Tgγ = Sgγ iff g ∈ H = {(p, q) ∈ Z | q = 0}.
Since H is infinite, but of 0 density, T and S are j.e., but not w.-j.s.m.
Example 4.30. It is quite hard to come up with a not-trivial example of an ergodic
automorphism (or an ergodic action of a countable group by automorphisms) of a
non-abelian compact group.
The first example that comes to mind is to consider the group X = Y G where G is
Z2 a countable group and Y is a non-abelian compact group. For example, X = (S3) .
97 We can consider shift automorphisms of X. However, they are not different from
Z2 shifts on, say, (Z6) . Another family of automorphisms of X is given by inner
−1 automorphisms of the base group Y , namely Ah : Y → Y with Ah(y) = h yh, where h is a fixed element of Y . Then we can define the automorphism B on X given by
−1 (B(x))g = (hgh )g for all g ∈ G and use it, together with a shift automorphism, to define a Z2-action on X. However, on the level of the dual object of Y , B acts exactly the same as the identity operator. Thus, using shifts and inner automorphisms only, one can not define a non-trivial example of the kind that we need.
More interesting possibility stems from the so called Ledrappier’s example. In
Z2 [Led78], F. Ledrappier considered the shift invariant subgroup X of (Z2) given by
n Z2 2o X = x = (xm,n) ∈ (Z2) | xm,n + xm+1,n + xm,n+1 = 0 ∀ (m, n) ∈ Z . (4.7) and the Z2-actions by shifts on X. This action has very interesting dynamical prop- erties. See [Sch95] where this and similar dynamical systems studied in great detail.
One can consider a non-abelian analogue of (4.7)
n Z2 2o Z = x = (xm,n) ∈ (S3) | xm,n · xm+1,n · xm,n+1 = 1 ∀ (m, n) ∈ Z . and study shift actions on it. (Of course, one can consider another compact non- abelian group instead of S3.) In order to apply to Z the methods developed earlier in this chapter, it would be necessary to determine the structure of the dual object of Z.
Nevertheless, this example shows that there exist natural examples of group ac- tions by automorphisms of non-abelian groups with non-trivial dynamical properties.
In this chapter, we studied joint properties of group actions by a special class of transformations — group automorphisms. We were able to get some convenient
98 criteria for joint dynamic properties of such actions. There exist other classes of actions for which it might be possible to obtain similar results. Let us mention a few of them. Unfortunately, it was not possible for the author to study these cases in this work. All of them are good directions for further research.
(1) Actions by group rotations. We consider a compact group X and the action
by a countable group G given by Tgx = x · bg, where b is a homomorphism from
X to G and we write bg = b(g). The abelian case is not hard. Let we have s
(i) (i) rotation actions of an abelian group X given by Tg x = x + bg , 1 ≤ i ≤ s. One
(1) (s) ˆ can show that T ,...,T are jointly ergodic iff for all γ1, . . . , γs ∈ X, not all
of them 0, there exists g ∈ G such that
(1) (s) γ1 bg + ··· + γs bg 6= 0.
The non-abelian case is more complex and worth studying.
(2) Actions by affine transformations. An action by affine transformations of
a compact group X is a transformation T given by T x = Agx · bg, where Ag
is an automorphism of X and bg ∈ X for all g ∈ G. In [Hah63] and [Tho72]
the single Z-actions by affine transformations of abelian and non-abelian groups
respectively have been studied. As far as we know, affine actions by more general
groups have never been studied as neither been studied joint ergodic properties
of such actions. Unfortunately, the criteria for joint ergodicity of affine actions
turn out to be quite complicated to be practical, even in abelian case.
(3) Automorphisms on nil-manifolds have been studied extensively (see, for
example, [Aus70]). The problem of characterizing ergodic and joint ergodic
99 properties of group actions by such automorphisms is a natural one to study.
Here one has readily available interesting non-abelian examples.
100 CHAPTER 5
JOINT ERGODICITY ALONG SEQUENCES OF INTERVALS
5.1 Definitions and Preliminary Results
B 2 Suppose that we have a Lebesgue probability space (X, , µ). Let L0(X) be the orthogonal complement in L2(X) of the one-dimensional subspace of the constant
∞ def ∞ 2 functions. Also, L0 (X) = L (X) ∩ L0(X).
We consider sequences of intervals [an, bn] = {an, an+1, . . . , bn}, n = 1, 2,... , in N, where (an) and (bn) are sequences of natural numbers, bn ≥ an. By a regular sequence
∞ of intervals we mean a sequence of intervals [an, bn]n=1 in N such that bn − an → ∞ as n → ∞.
One way to characterize an ergodic measure-preserving transformation is as fol- lows:
2 Fact 5.1. A measure-preserving transformation T is ergodic iff for every f ∈ L0(X) one has
n 1 X L2 T kf −−−→ 0. (5.1) n n→∞ k=0
101 ∞ But one can also fix an arbitrary regular sequence of intervals [an, bn]n=1 instead
∞ of the sequence [0, n]n=1 and obtain the following:
Fact 5.2. A measure-preserving transformation T is ergodic iff for a fixed regular
∞ 2 sequence of intervals [an, bn]n=1 and every f ∈ L0(X) one has
bn 1 X L2 T kf −−−→ 0. (5.2) bn − an n→∞ k=an
To see that Fact 5.1 and Fact 5.2 indeed characterize ergodicity one uses von
Neumann Mean Ergodic Theorem (Theorem 1.7). One can use either of them as a
definition of an ergodic transformation.
In Chapter 2, we defined joint ergodicity of several group actions. In this chapter,
we would like to study some aspects of joint ergodicity of two N-actions, i.e. two
not necessarily invertible, measure-preserving transformations. We shall define joint
2 ergodicity for pairs of measure-preserving actions in terms of functions from L0(X). This is consistent with Definition 2.1, in view of Theorem 2.2 and Remark 2.3. We are interested in convergence of joint ergodic averages along regular sequences of intervals.
In Remark 5.27 we shall discuss how our results change if one considers convergence along Følner sequences of general type.
Definition 5.3. A pair of measure-preserving transformations (T,S) is called jointly
∞ ∞ ergodic along a sequence of intervals [an, bn]n=1 if for all functions f, g ∈ L0 (X) one has:
bn 1 X L2 T kf · Skg −−−→ 0. (5.3) bn − an n→∞ k=an
102 Definition 5.4. A pair of measure-preserving transformations (T,S) is called uni-
∞ formly jointly ergodic if for all functions f, g ∈ L0 (X) and every regular sequence of
∞ intervals [an, bn]n=1 one has:
bn 1 X L2 T kf · Skg −−−→ 0. (5.4) bn − an n→∞ k=an
Remark 5.5. In Chapter 2, we used the term jointly ergodic as opposed to uniformly jointly ergodic to refer to a notion of uniform joint ergodicity along all Følner sequence.
The reason was that we were not interested in the more subtle issue of convergence along individual Følner sequences. Since, in this chapter, we are going to investigate, how convergence of joint ergodic averages (5.4) depends on the choice of a regular sequence of intervals, we shall be using the terms uniformly jointly ergodic and jointly ergodic along a sequence of intervals as defined above.
Are Definition 5.4 and Definition 5.3 equivalent? In other words, is Definition 5.3
∞ invariant with respect to the choice of the regular sequence of intervals [an, bn]n=1? The answer is positive if T and S commute and are invertible. Repeating the argu-
ments that have been used in the proof of Theorem 2.18, one can show that a pair
of commuting invertible transformations T and S are jointly ergodic along a regular
sequence of intervals iff the transformations T × S and TS−1 are both ergodic. Since
this criterion is independent of the choice of the regular sequence of intervals, the
notion is independent of this choice as well. Invertibility of T and S is not important
since one can consider the natural extensions T 0 and S0 of the transformations T and
S and prove that T and S are jointly ergodic along a sequence of intervals iff T × S
and T 0S0−1 are ergodic as it is done in [BB84].
103 As we shall see below, when T and S do not commute, Definition 5.3 is not
∞ invariant with respect to the choice of the sequence of intervals [an, bn]n=1. Before we begin to discuss what happens in the non-commutative case, let us N ∞ introduce some more definitions. By we denote the sequence of intervals [0, n]n=1. Let L be the set of all regular sequences of intervals in N. Given E ⊂ N the upper L ∞ density of E along a regular sequence of intervals = [an, bn]n=1 is
|E ∩ [an, bn]| d¯L(E) = lim . n→∞ bn − an
Of course, d¯N(E) is just the familiar notion of upper density of E which is customarily
denoted as d¯(E).
The upper Banach density
∗ def d (E) = sup d¯L(E). L∈L
L ∞ Finally, if (cn) is a sequence of complex numbers and = [an, bn]n=1 ∈ L, then we
define upper Ces`aro limit of (cn) along L as
1 X CL - lim cn = lim cn. n→∞ n→∞ bn − an n∈Ln
Clearly, if E ⊂ N, then ¯ CL - lim 1E(n) = dL(E). n→∞
The Ces`aro limit of (cn) along L is defined as
1 X CL - lim cn = lim cn. n→∞ n→∞ bn − an n∈Ln
if the limit exits.
104 5.2 A Universal Counterexample
Example 5.6. Let E ⊂ N be given. Let D = E ∪ (−E). So defined set D is a symmetric subset of Z. We define two automorphisms, S and TE, of the abelian
Z compact group X = (Z2) as follows. The transformation S is the left shift on X, that is
(Sx)k = xk+1, x ∈ X, k ∈ Z.
Define a permutation π on Z by ( k, k ∈ D π(k) = −k, k 6∈ D.
Let TE be the automorphism of X given by
(TEx)k = xπ(π(k)+1).
Then
n (TE x)k = xπ(π(k)+n), n ∈ Z.
Therefore, xk−n, if k 6∈ D and −k + n 6∈ D, n x−k+n, if k 6∈ D and −k + n ∈ D, (T x)k = E x , if k ∈ D and k + n 6∈ D, −k−n xk+n, if k ∈ D and k + n ∈ D
So, the transformation TE is “asymptotically” the right shift if E is “small”. It is easy to see that so defined measure-preserving transformations S and TE are both L ∞ ergodic. Let = [an, bn]n=1 be a regular sequence of intervals. In view of (the proof ˆ of) Corollary 4.17, the Z-actions TE and S are j.e. along L iff for every 0 6= γ ∈ X the set
ˆn ˆn Pγ = { n ∈ Z | S γ = TE γ }
105 ˆ satisfies the condition dL(Pγ) = 0. The dual space X is the set of all sequences with elements in Z2 with only finitely many non-zero entries. The dual actions of S and
TE are given by
ˆ (Sγ)k = γk−1,
ˆ (TEγ)k = γπ(π(k)−1)), for γ ∈ Xˆ and k ∈ Z.
ˆn Hence, (TE γ)k = γπ(π(k)−n)). For all γ ∈ Xˆ let
Fγ = {k ∈ Z | γk = 1}, bγ = max{|k| | k ∈ Fγ}.
It is easy to check that
FSˆnγ ⊆ [−bγ + n, bγ + n], n ∈ N and that if n is such that ±Fγ − n ∩ D 6= ?, then
F ˆn ⊆ [−bγ − n, bγ − n], n ∈ N. TE γ
If dL(E) = 0, then the set
Hγ = {n ∈ N | ±Fγ − n ∩ D 6= ?} has density 0 along the sequence L. If γ 6= 0, then
ˆn ˆn {n > bγ | S γ = T γ} ⊆ {n > bγ | F ˆn ∩ F ˆn 6= ?} ⊆ Hγ. E S γ TE γ
So, if dL(E) = 0, then S and TE are j.e. along L.
Assume now that d¯L(E) > 0. Let
−2 −1 0 1 2 γ = (..., 0 , 0 , 1, 0, 0,... ).
ˆn ˆn L Then S γ = TE γ for all n ∈ E. Therefore, S and TE are not j.e. along .
106 Let us summarize our results as a proposition.
Proposition 5.7. Let E ⊂ N and let (X, B, µ) be a Lebesgue probability space. There
exist two measure-preserving transformations T and S of X such that given a regular
sequence of intervals L, T and S are j.e. along L iff d¯L(E) = 0.
Z Proof. Every Lebesgue space is metrically isomorphic to the space (Z2) with Haar
measure. The transformations S and TE described in Example 5.6 are jointly ergodic
along a regular sequence of intervals iff their images under this isomorphism are.
Remark 5.8. Our universal example is based on the example given by D. Berend and
V. Bergelson in [BB86]. They considered sets E ⊂ N such that d¯N(E) = 0, but
∗ S∞ 3 3 d (E) > 0 (for instance, one can take E = n=1[n , n + n]). For such a set, the
transformations S and TE are jointly ergodic along N (“jointly ergodic” according to
the terminology of [BB86]), but not uniformly jointly ergodic.
Our example is more general. For instance, we can construct two transformations
that are j.e. along a regular sequence of intervals, but not j.e. along N. Namely, it ¯ ¯ is enough to find a set E2 ⊂ N such that dN(E2) > 0 but dL(E2) = 0 for another
regular sequence of intervals L. Let
∞ [ 2 2 E2 = n − n + 1, n , n=1
If we represent the set E2 by a sequence of 1’s and 0’s, where 1’s correspond to the
elements of N that belong to E2, then we have
E2 ∼ 10110011100011110000 ...
¯ 1 Clearly, dN(E2) = 2 . Now, let
L 2 2 ∞ 2 = n + 1, n + n n=1. 107 Since E2 is disjoint from all intervals forming the sequence L2, it follows that
¯ dL2 (E2) = 0.
5.3 A Partial Ordering on L
∞ Let us fix a Lebesgue space X. Let JE ([an, bn]n=1) be the set of all pairs of
∞ measure-preserving transformations of X that are jointly ergodic along [an, bn]n=1. As
∞ ∞ we have seen in the previous section, the set JE ([an, bn]n=1) depends on [an, bn]n=1.
∞ ∞ ∞ However, it is clear that JE ([0, n]n=1) = JE ([0, 2n]n=1). What about JE ([0, n]n=1)
2 ∞ n ∞ and JE [0, n ]n=1 ? JE ([0, 2 ]n=1)? To address this question, we would like to find a partial ordering on the set of
∞ sequences of intervals corresponding to the partial ordering among JE ([an, bn]n=1) given by inclusion. In other words, we would like to find a relation “ ” on the set of
∞ 0 0 ∞ ∞ all sequences of intervals such that JE ([an, bn]n=1) ⊆ JE ([an, bn]n=1) iff [an, bn]n=1
0 0 ∞ [an, bn]n=1. The following natural relation happens to work.
Definition 5.9. We say that a regular sequence of intervals L is stronger than another
regular sequence of intervals L0 and write L L0, if for every set E ⊂ N one has:
d¯L(E) = 0 =⇒ d¯L0 (E) = 0, (5.5)
We say that two sequences of intervals, L and L0, are equivalent, denoted as
L ∼ L0, if we have both L L0 and L0 L.
0 0 Lemma 5.10. Let L, L ∈ L, L L . Then for every bounded sequence (cn) of
non-negative real numbers
CL - lim cn = 0 =⇒ CL0 - lim cn = 0 n→∞ n→∞
108 Proof. Let (cn) be a bounded sequence of positive real numbers. It can be approx-
imated in l∞-norm from below by finite linear combinations of indicator functions
of subsets of N, i.e for every ε > 0 there exist r ∈ N, Ek ⊂ N, and positive real Pr numbers αk, 1 ≤ k ≤ r, such that cn = k=1 αk1Ek (n) + εn, where 0 ≤ εn < ε. Since
cn ≥ αk1Ek (n) for 1 ≤ k ≤ r, it follows that dL(Ek) = CL -limn→∞ 1Ek (n) = 0 for
1 ≤ k ≤ r. Therefore, because of our assumption, dL0 (Ek) = 0 for all 1 ≤ k < r.
Now,
r r X X ¯ CL0 - lim cn = CL0 - lim αk1E (n) + ε ≤ αk dL0 (Ek) + ε = ε. n→∞ n→∞ k k=1 k=1 Since ε is an arbitrary positive number, we are done.
Proposition 5.11. JE (L) ⊆ JE (L0) iff L L0.
Proof. Let L L0 and let (T,S) ∈ JE (L). To prove that (T,S) ∈ JE (L0), it is
∞ enough to show that for all f, g ∈ L0 (X) the equation (5.3) holds. According to [BB86, Theorem 2.1], it is enough to prove weak convergence for (5.3) to hold (for
invertible transformations it follows form Theorem 2.2). Thus, it remains to show
2 2 that for all positove real-valued f, g ∈ L0(X) and h ∈ L (X), one has Z n n CL0 - lim T fS g h dµ = 0. n→∞ X
We set Z n n cn = T fS g h dµ X and apply Lemma 5.10.
The other direction follows from Proposition 5.7.
Remark 5.12. In particular, the notions of joint ergodicity along L and L0 are equiv- alent iff L ∼ L0.
109 5.4 Examples
L ∞ L N Example 5.13. Let = [0, 2n]n=1. Let us show that ∼ . Namely, let E ⊂ N
with d¯N(E) = 0. Then d¯L(E) = 0 because L is a subsequence of N. Conversely, let
E ⊂ N with d¯L(E) = 0. Note that for all n one has
|E ∩ [0, n]| |E ∩ [0, 2n]| ≤ 2 n 2n
which implies that d¯N(E) ≤ 2d¯L(E). Hence, d¯N(E) = 0 and L ∼ N.
L 2 ∞ L N L N Example 5.14. Let = [0, n ]n=1. Then ≺ since is a subsequence of . Let
us show that L N as well. Namely, let E ⊂ N with d¯L(E) = 0. Let ε > 0 be given.
Then there exists n0 ∈ N such that for all n > n0 one has
|E ∩ [0, n2]| < ε. n2
2 2 Let n > max(n0, 2). If n ≤ m ≤ (n + 1) , then
|E ∩ [0, m]| |E ∩ [0, (n + 1)2]| |E ∩ [0, (n + 1)2]| ≤ < 2 < 2ε. m m (n + 1)2
So, L ∼ N.
k ∞ N n ∞ N Using same arguments, one can show that [0, n ]n=1 ∼ and [0, k ]n=1 ∼ for every natural number k ≥ 2. However, it does not work if L is a very fast growing
subsequence of N as the next example shows.
L ∞ L N L N Example 5.15. Let = [0, n!]n=1. It is clear that ≺ . We claim that 6∼ . S∞ ¯ Namely, let E = n=1[n! , 2 · n! ]. To show that dN(E) > 0 it is enough to find a M N ¯ M ∞ ¯ 1 subsequence of such that dM(E) > 0. Let = [0, 2 · n!]n=1. Then dM(E) = 2 ¯ 1 and, therefore, dN(E) ≥ 2 . On the other hand, |E ∩ [0, (n + 1)! ]| |[0, 2 · n!] ∩ [0, (n + 1)! ]| 2 n! ≤ = −−−→ 0. (n + 1)! (n + 1)! (n + 1)! n→∞
110 Hence, d¯L(E) = 0.
√ L ∞ N L Example 5.16. Let = [n, n + d ne]n=1. We claim that . Indeed, let
∞ [ √ E = [n!, n! + n!]. n=1
Then dL(E) = 1, but dN(E) = 0. Hence, N 6∼ L.
To show that N ≺ L, let E ⊂ N with d¯N(E) > 0. We need to show that
d¯L(E) > 0. Indeed, since dN(E) > 0, there exists c > 0 such that
|[0, m] ∩ E| > c m
for an unbounded increasing sequence of numbers m. For every m one can represent √ the interval [0, m] as a union of disjoint intervals of the form [nk, nk + d nke]:
√ √ [1, m] = [n1, n1 + d n1e] ∪ · · · ∪ [nr, nr(m) + d nr(m)e].
√ A simple estimate shows that the number of the intervals on the right r(m) ≤ 3 m.
Since √ √ |[0, m] ∩ E| [n1, n1 + d n1e] ∩ E [nr(m), nr(m) + d nr(m)e] ∩ E = + ··· + > c, m m m there exists an index nk(m) such that √ [nk(m), nk(m) + d nk(m)e] ∩ E c > √ m 3 m
and, therefore, √ [nk(m), nk(m) + d nk(m)e] ∩ E c √ > . m 3
It is clear that limm→∞ k(m) = ∞. Thus, we obtained a subsequence
√ ∞ nk(m), nk(m) + d nk(m)e m=1 111 of L such that √ √ [nk(m), nk(m) + d nk(m)e] ∩ E [nk(m), nk(m) + d nk(m)e] ∩ E c √ ≥ √ > , nk(m) m 3
which means that d¯L(E) > 0.
2 2 ∞ √ ∞ Example 5.17. Let us consider the subsequence [n , n + n]n=1 of [n, n + d ne]n=1.
2 2 ∞ ∞ 2 2 ∞ ∞ It is easy to see that [n , n + n]n=1 6≺ [0, n]n=1 and [n , n + n]n=1 6 [0, n]n=1.
Let us summarize our examples below:
∞ 2 ∞ n ∞ ∞ [0, n]n=1 ∼ [0, n ]n=1 ∼ [0, 2 ]n=1 [0, n!]n=1. √ ∞ ∞ [0, n]n=1 n, n + d ne n=1.
∞ 2 2 ∞ [0, n]n=1 ⊥ [n , n + n]n=1.
√ ∞ Remark 5.18. The example of [n, n + d ne]n=1 is especially interesting since this is a regular sequence of intervals that is universally bad for the pointwise ergodic theorem
[AdJ75, RW92]. We shall discuss connections between our results and pointwise convergence of Ces`aroaverages for a single transformation later in this chapter.
5.5 Some Properties of (L, ≺)
Proposition 5.19. Let L ∈ L. Then there exist L0, L00 ∈ L such that
L0 L L00.
0 Proof. Let L = (Ln) ∈ L be given. Let us first construct L L. Namely, for
0 0 1 each n let Ln be any subinterval of Ln such that |Ln| = 2 |Ln| . Then the resulting L0 0 L S∞ S∞ 0 sequence = (Ln) is not equivalent to . Namely, for E = n=1 Ln \ n=1 Ln one has d¯L(E) = 1/2 whereas d¯L0 (E) = 0. On the other hand, for every E ⊂ N and ε > 0
0 |Ln ∩ E| |Ln ∩ E| |Ln ∩ E| 0 ≤ 0 ≤ (2 + ε) for n big enough |Ln| |Ln| |Ln| 112 0 and, therefore, d¯L0 (E) ≤ 2d¯L(E). So, L L .
To construct L00, let us first construct an auxiliary regular sequence of intervals M = (Mn). Let n1 = 1 and M1 = Ln1 . Let n2 be the smallest index such that
|Ln2 | ≥ 8 and such that Ln2 \ Ln1 contains an interval of length 2. Let M2 to be such
a subinterval. If M1,...,Mk−1 are already chosen, let nk be the smallest index such
3 Snk−1 that |Lnk | ≥ k and Lnk \ i=1 Li contains a subinterval of length k. Let Mk be such ¯ S∞ a subinterval. Then, as one can easily check, dL( k=1 Mk) = 0, whereas, certainly, ¯ S∞ M L L00 dM( k=1 Mk) = 1 and so 6≺ . Now, we define by setting Lk, if n = 2k, 00 Ln = Mk, if n = 2k + 1
Then L00 L and L00 M. Because M 6≺ L, it follows that L00 6≺ L and, therefore,
L00 6∼ L.
The method used to build L00 from L and M can be generalized. Let L0, L00 ∈ L.
We define L0 + L00 to be the regular sequence L with elements ( L0 , if n = 2k, L = k n 00 L k, if n = 2k + 1
So defined binary operation “+” is commutative and associative up to equivalence ∼.
It is clear that L0 + L00 L0, L00. Moreover, it is the weakest sequence that has this property.
Proposition 5.20. Let L0, L00 ∈ L. Then
(1) L0 + L00 L0 and L0 + L00 L00,
(2) if L L0 and L L00, then L L0 + L00.
113 ¯ Proof. (1) Let E ⊂ N be such that dL0+L00 (E) = 0. Then E has 0 density along every subsequence of L0 + L00, in particular, along L0 and L00. ¯ ¯ ¯ (2) Let dL(E) = 0. Then dL0 (E) = 0 and dL00 (E) = 0, i.e. for every ε > 0 there
exists n0 ∈ N such that for all n > n0 one has
0 00 |E ∩ Ln| |E ∩ Ln| 0 < ε and 00 < ε. |Ln| |Ln|
L0 L00 0 00 Let + = (Mn). Since for all n either Mn = Ln or Mn = Ln, it follows that for all n > n0 |E ∩ M | n < ε. |Mn|
Corollary 5.21. For all L0, L00 ∈ L, one has
JE (L0 + L) = JE (L0) ∩ JE (L00).
P∞ L(k) One can similarly define the sum k=1 of a countable set of regular sequences L(k), k ∈ N. Namely, if L(1) (1) (1) (1) = ( L1 ,L2 ,L3 ,... ) L(2) (2) (2) (2) = ( L1 ,L2 ,L3 ,... ) L(3) (3) (3) (3) = ( L1 ,L2 ,L3 ,... ) ··· P∞ L(k) (1) (1) (2) (3) (2) (1) we set k=1 = ( L1 ,L2 ,L1 ,L1 ,L2 ,L3 ,... ). It is easy to see that
∞ X L(k) L(k) k=1 for all k = 1, 2,... .
We have established some important facts about the set L.
Theorem 5.22. (1) The set L has neither minimal nor maximal element with re-
spect to the partial ordering ≺.
114 (2) The factor set L/∼ is uncountable.
Proof. (1) follows from Proposition 5.19. (2) follows from the following observation. If
L(k) P∞ L(k) , k ∈ N contains representatives of all elements of L/∼, then k=1 is another regular sequence of intervals that is not equivalent to any of L(k).
We have seen that for every finite or countable set A ⊂ L there exists L ∈ L that is stronger than all elements of A. Interestingly, one can not always find an element of L that is weaker than two given elements of L.
Example 5.23. Let
L0 0 2 2 ∞ L00 00 2 2 ∞ = (Ln) = [n , n + n]n=1 and = (Ln) = [n + n + 1, n + 2n]n=1.
Let ∞ ∞ 0 [ 0 00 [ 00 E = Ln and E = Ln. n=1 n=1 0 00 Then d¯L00 (E) = 0 for every E ⊂ E and d¯L0 (E) = 0 for every E ⊂ E . So, if
0 00 0 00 L ≺ L , L , then one would have d¯L(E) ≤ d¯L(E ∩ E ) + d¯L(E ∩ E ) = 0 for every
E ⊂ N, which is impossible. Therefore, there is no L such that L ≺ L0, L00.
Remark 5.24. Example 5.23 is an example of two sequences of intervals L0 and L00 for
which there is no regular sequence of intervals L with JE (L0) ∪ JE (L00) ⊂ JE (L).
5.6 Other Types of Convergence
The partial ordering ≺ and the equivalence relation ∼ on the set of all sequences
of intervals that we defined above were induced by the inclusion relation between
classes of systems that jointly ergodic along the sequences of intervals.
115 If we consider types of convergence different from (5.3), we might obtain different
types of partial ordering and equivalence relations on L. In view of Lemma 5.10,
one would expect that our equivalence relation “∼” is the finest among this type
of equivalence relation, i.e. the equivalence classes defines by a different type of
convergence of ergodic averages would be unions of the equivalence classes defined by
the relation “∼”. Of special interest is pointwise convergence of Ces`aro averages for
a single transformation.
Definition 5.25. An invertible measure-preserving transformation T of a Lebesgue
space X is called pointwise ergodic along a regular sequence of intervals L (in short,
L 1 T ∈ PWE ( )) if for all functions f ∈ L0(X) one has:
b 1 Xn T kf(x) −−−→ 0 for a.e. x. (5.6) bn − an n→∞ k=an
The relation “PWE (L) ⊆ PWE (L0)” also defines a partial ordering and an equiv- alence relation on the set of all sequences of intervals which we will denote as p and
∼p respectively.
In view of Birkhoff’s Pointwise Ergodic Theorem (Theorem 1.8), among these equivalence classes there exists a global minimal element — the equivalence class corresponding to the sequence of intervals N. Interestingly enough, there is also
a global maximal element — the equivalence class corresponding to the sequence
√ ∞ [n, n + d ne]n=1. This is because for every invertible transformation there exists f ∈ L1(X) such that (5.6) does not converges a.e. along this sequence ([AdJ75]).
116 The natural question to ask: Is there anything between these two? The short
answer is “almost nothing”. Roughly speaking, PWE (L) is non-empty iff the regular
sequence of intervals L satisfies Shulman’s Condition.
∞ More precisely, according to [RW92], if [an, bn]n=1 satisfies Shulman’s condition,
∞ then PWE ([an, bn]n=1) contains all ergodic transformations. If a regular sequence of
∞ intervals [an, bn]n=1 does not satisfy Shulman’s Condition and the sequence of numbers
∞ (bn − an) is increasing, then PWE ([an, bn]n=1) is empty. If the sequence bn − an is not
∞ increasing, but [an, bn]n=1 satisfies a stronger condition:
Sn−1 [an, bn] − [ai, bi] lim i=1 = ∞, n→∞ bn − an
then PWE ([an, bn]) is also empty.
∞ The question of how big the set PWE ([an, bn]n=1) can be for those sequences of intervals that fall into neither of the three above categories remains open, as far as
we know.
Let us note, however, that if we consider only sequences of intervals with increasing
length, then the set L/∼p contains only two elements — the maximal and the minimal
elements.
5.7 Other Types of Sequences
We studied convergence along sequences of regular intervals of non-negative inte-
gers. However, one may ask: why not to consider convergence along different (not
necessarily wider) classes of sequences of finite sets. Let us mention a few of the
possibilities:
(1) all Følner sequences in N
117 (2) sequences of intervals or Følner sequences in Z.
(3) nested sequences of finite sets of the form {p1, p2, . . . pk} where p1 < p2 < ··· <
pk.
(4) all sequences of finite subsets of N
(5) all Følner sequences in a countable amenable group G
In this section, we shall discuss how replacing “all sequences of intervals” by “all
Følner sequences in N” changes our results. We shall also mention, what happens if we consider sequences of intervals in Z instead of N. Extending our results to classes of sequences (3) – (6) is very interesting, but beyond the scope of this work.
Let F be the set of all Følner sequences in N. Clearly, L ⊂ F. As one can easily see, all of the facts that we have proved about L (Lemma 5.10 – Theorem 5.22) hold true for F without any changes. However, there remains two important questions:
(1) Are L/∼ and F/∼ identical? In other words, is it true that for every Følner
sequence F there exits a regular sequence of intervals L such that L ∼ F?
(2) Do the classes of uniformly convergent objects change if we move from sequences
of intervals to all Følner sequences? For instance: Is the set of all pairs of
transformations that are jointly ergodic along every regular sequence of intervals
identical to the set of all pairs of transformations that are jointly ergodic along
all Følner sequences?
It turns out that the answer to the first question is negative, but the answer to the second question is positive, which means that for most practical purposes sequences of intervals are as good as Følner sequences in Z.
118 To address these two questions, we need a characterization of all Følner sequences in N.
Let H ⊂ N be a finite set. Then H consists of a finite number of non-adjacent intervals, so that one can write
H = [a1, b1] ∪ · · · ∪ [an, bn] and this decomposition is unique. Let
`(H) = min (bk − ak). (5.7) k=1,...,n
We call the number `(H), the footprint of the set H.
A sequence of finite sets (Fn) in N is a Følner sequence iff it can be represented as Fn = Gn ∪ Hn so that
(1) `(Gn) → ∞ as n → ∞,
(2) |Hn|/|Fn| → 0 as n → ∞.
(For a proof, see Appendix D, Lemma D.1.)
Now we are in a position to prove the following theorem.
Theorem 5.26. For every F ∈ F there exists L ∈ L such that L F.
Proof. Let F = (Fn) ∈ F. In view of Lemma D.1, there exist sequences G = (Gn) and
H = (Hn) of finite subsets of N such that (Fn) = (Gn) ∪ (Hn) and
(1) `(Gn) → ∞ as n → ∞,
(2) |Hn|/|Fn| → 0 as n → ∞.
119 Each set Gn consists of a finite number, say r(n), of non-adjacent intervals. Let ¯ us denote them as In,1,In,2,...,In,r(n). For every E ∈ N with dF(E) > 0 we are
going to give a method of how to choose one of these intervals, we shall call this
interval In,kE (n), out of each set Gn, so that the resulting regular sequence of intervals ∞ L ¯L ¯F E = In,kE (n) n=1 is such that d E (E) > 0. Namely, let E ∈ N with d (E) > 0
be given. WLOG we may assume that dF(E) > 0 (just go to a subsequence). Then
there exist n0 ∈ N and α > 0 such that |E ∩ Gn|/|Gn| > α for all n > n0. We claim
that for each n > n0 there exists an index kE(n) such that |E ∩In,kE (n)|/|In,kE (n)| > α.
Namely, if not, then |E ∩ In,k|/|In,k| < α for some n > n0 and 1 ≤ k ≤ r(n). But
then |E ∩ G | |E ∩ I | + ··· + |E ∩ I | n = n,1 n,r(n) < α. |Gn| |In,1| + ··· + |In,r(n)|
Contradiction. This finishes the construction of the sequence LE. It is a regular
sequence of intervals with bn − an → ∞ because of the property (1) of the sequence
G.
Now, we are ready to construct the sequence L we are looking for. Take
L = I1,1,I1,2,...,I1,r(1),...,In,1,In,2,...,In,r(n),...
From what we have just proved it follows that for every E ∈ N with d¯F(E) > 0
there exists a subsequence nk such that limk→∞ |E ∩ Lnk |/|Lnk | > 0. Therefore,
limn→∞ |E ∩ Ln|/|Ln| > 0.
Remark 5.27. Theorem 5.26 gives a positive answer to our second question on the connection between F and L. Namely, it shows that for every Følner sequence F,
there exists a regular sequence of intervals along which convergence is at least as
hard as along F. Therefore, a sequence convergent along all Følner sequences is no
120 better than a sequence convergent along all sequences of intervals. In particular, if
one defines a new “uniform” density as
¯ dF(E) = sup dL(E), L∈F
∗ then for E ⊂ Z one has dF(E) = 0 iff d (E) = 0. (The question of whether always
∗ dF(E) = d (E) is more subtle.)
Using the method employed in the proof of Theorem 5.26, one can similarly show
0 that for all F ∈ F and every E ⊂ N there exists L ∈ L such that d¯L0 (E) ≤ d¯F(E). E E But one can not use it to construct a uniform sequence L0 ∈ L such that L0 ≺ F.
Does there exist such a sequence? It turns out that in general the answer is negative.
Sn 5 3 5 3 2 ∞ Example 5.28. Let Fn = m=1 In,m, where In,m = [n + mn , n + mn + n ]n=1.
Then F = (Fn) is a Følner sequence. We claim that there is no L ∈ L such that L ≺ F. L S∞ L F Namely, assume = (Ln) is such a sequence. Let A = n=1 Fn. Since ≺ , it
follows that for k big enough Lk ∩A 6= ? and, moreover, |Lk ∩A|/|Lk| → 1 as k → ∞.
For big k the distance between consecutive intervals forming the sets Fk is bigger than
the length of a single interval and, therefore, for big k each Lk intersects one and only
one of the sets In,m, say, In(k),m(k), and one has limk→∞ |Lk ∩In(k),m(k)|/|Lk| = 1. Now, L let us choose a subsequence (Lsk ) of such that (Lsk ) intersects each set Fn no more ¯ S∞ than once. Then |Lsk |/|Fn(k)| → 0 as k → ∞ and, consequently, dF ( k=1 Lsk ) = 0. S∞ On the other hand, obviously, dL ( k=1 Lsk ) = 1. Contradiction.
Remark 5.29. This examples gives a negative answer to our first question — there
exists F ∈ F that is not equivalent to any regular sequence of intervals.
121 Finally, let us discuss how our results may be generalized for sequences of intervals
in Z. One can suggest two definitions for partial ordering of regular sequences of
intervals in Z.
0 0 (1) For two regular sequence of intervals L and L in Z we shall write L z L , if
for every set E ⊂ Z one has:
d¯L(E) = 0 =⇒ d¯L0 (E) = 0,
(2) We say that a regular sequence of intervals L in Z is stronger than another
0 0 regular sequence of intervals L and write L s L , if for every symmetric set
E ⊂ Z one has:
d¯L(E) = 0 =⇒ d¯L0 (E) = 0,
Which of the two definitions is the “right” one?
If we use the first definition, then we can prove an analogue of Lemma 5.10 and, therefore,
∞ 0 0 ∞ ∞ 0 0 ∞ [an, bn]n=1 z [an, bn]n=1 =⇒ JE ([an, bn]n=1) ⊆ JE [an, bn]n=1 . (5.8)
On the other hand, if we use the second definition, then we can make use of our universal counterexample and, therefore,
∞ 0 0 ∞ ∞ 0 0 ∞ [an, bn]n=1 s [an, bn]n=1 ⇐= JE ([an, bn]n=1) ⊆ JE [an, bn]n=1 . (5.9)
It remains an open question whether the inverse implications for (5.8) or (5.9) hold.
122 APPENDIX A
SOME FACTS FROM THE THEORY OF SEMIGROUPS
In this appendix, we give some definitions and results from semigroups theory and about embedding semigroups into groups. Most of these results can be found in either [Lja63] or [CP61].
Definition A.1. A semigroup is a non-empty set S, in which for every ordered pair of elements x, y ∈ S there is another element called their product u = xy ∈ S, so that for all x, y, z ∈ S we have
(xy)z = x(yz).
Definition A.2. Let S1 and S2 be semigroups. A mapping ψ : S1 → S2 is called a homomorphism if ψ(xy) = ψ(x)ψ(y) for all x, y ∈ S1. A 1-1 homomorphism is called
an isomorphism.
Definition A.3. A semigroup S is called a group if for all a, b ∈ S there exist x and
y in S, such that
xa = b, ay = b.
Definition A.4. An equivalence relation ∼ on S is called a congruence if for all x1,x2,
y1, y2 ∈ S, one has:
x1 ∼ x2, y1 ∼ y2 =⇒ x1y1 ∼ x2y2.
123 Proposition A.5. An equivalence relation ∼ in a semigroup S is a congruence iff
for all a, b, s ∈ S one has:
a ∼ b =⇒ as ∼ bs and sa ∼ sb. (A.1)
Proof. Assume that ∼ is a congruence in S. Let a ∼ b and s ∈ S. Since s ∼ s, then
by definition, as ∼ bs and sa ∼ sb.
Conversely, assume that for all a, b.s ∈ S, (A.1) holds. Let a ∼ b and c ∼ d. Then
ac ∼ bc and bc ∼ bd. By transitivity of ∼, it follows that ac ∼ bd.
Proposition A.6. Let ∼ be a congruence in S. Then the factor-set S = S/∼ is a semigroup homomorphic to S.
Proof. Let X,Y,Z ∈ S. We claim that
XY ∩ Z 6= ? =⇒ XY ⊂ Z, (A.2) where multiplication of two subsets A,B of S is defined as
AB def= {ab | a ∈ A, b ∈ B}.
Indeed, assume that x1 ∈ X, y1 ∈ Y , and x1y1 ∈ Z. Let x2 ∈ X and y2 ∈ Y be some other elements. We know that x1 ∼ x2 and y1 ∼ y2. By the definition of congruence, it follows that x1y1 ∼ x2y2 and, therefore, x2y2 ∈ Z.
Since the equivalence classes are pairwise disjoint, it follows from (A.2) that for every X,Y ∈ S there exists a unique Z ∈ S such that XY ⊂ Z.
Let us define an operation ◦ in S as follows:
X ◦ Y = Z if XY ⊂ Z.
124 As the above discussion shows, this operation is well-defined. It is trivial to see that
the operation ◦ is associative.
The natural projection h : S → S is a homomorphism with respect to so defined semigroup operation in S. Indeed, if Z = h(xy), then xy ∈ Z and so h(x) ◦ h(y) =
Z.
Definition A.7. A nonempty set I is a right ideal of a semigroup S if IS ⊂ I.A
nonempty set I is a left ideal of a semigroup S if SI ⊂ I.
Definition A.8. A semigroup S is said to be left (right) reversible if every pair of
right (left) ideals have a non-empty intersection.
In other words, S is left reversible if for all a, b ∈ S there exist u, v ∈ S such that
au = bv.
Definition A.9. A semigroup S is called left cancellative if for all x, a, b ∈ S one has
xa = xb =⇒ a = b.
The semigroup is called right cancellative if for all x, a, b ∈ S one has
ax = bx =⇒ a = b.
Proposition A.10. Let S be a left reversible semigroup and let ≈ be the relation on
S defined by setting x ≈ y if there exists s ∈ S for which xs = ys. Then ≈ is a
congruence on S and the semigroup S = S/ ≈ is right cancellative.
Proof. It is clear that the relation ≈ is reflexive and symmetric. To show that it is transitive, assume that a ≈ b and b ≈ c. Then there exist s, r ∈ S such that as = bs
125 and br = cr. Since S is left reversible, there exist u, v ∈ S such that su = rv. Then asu = bsu = brv = crv = csu and thus a ≈ c.
To show that ≈ is a congruence, according to Proposition A.5, it is enough to show that for all s ∈ S
a ≈ b =⇒ as ≈ bs and sa ≈ sb.
Let a ≈ b. This means that ar = br for some r ∈ S. It follows that sar = sbr for all s ∈ S and so sa ≈ sb. Now, by left reversibility, there exist u, v ∈ S such that ru = sv. Then asv = aru = bru = bsv. Therefore, as ≈ bs. So, ≈ is a congruence in
S and, by Proposition A.6, the factor set S = S/≈ is a semigroup.
Assume that for some X1,X2, and Y in S, one has X1Y = X2Y . Then for every
0 0 0 x1 ∈ X and y ∈ Y there exists x2 ∈ X and y ∈ Y such that x1y = x2y . Since y ≈ y ,
0 0 there exists s ∈ S such that ys = y s and so x1ys = x2y s = x1ys. Hence, x1 ≈ x2. It
follows that X1 = X2. Consequently, the semigroup S is right cancellative.
Now, we define so called partial transformations that will be used in our subse- quent constructions. Let Ω be a set and let A, B ⊂ Ω. Then a mapping h from A onto B is called a partial transformation or partial mapping of the set Ω. We use the following notation: Dom h = A and Im h = B. If h1 and h2 are two partial transfor- mations of Ω, we may define their product to be the partial transformation h = h1h2
with Dom h = {x ∈ Dom h2 | h1(x) ∈ Dom h1} and such that h(x) = h1(h2(x)). This
operation defines a semigroup structure among partial transformations.
Now, let H be a semigroup of one-to-one partial mapping of some set Ω such that
for every h ∈ H, the inverse transformation h−1 is in H as well. Note that h−1 is an
inverse transformation and does not represent an inverse element to h with respect
126 to the semigroup multiplication operation. It is easy to see that H does not have to be a group. In what follows we construct a factor-group of the semigroup H.
First, we define a partial ordering in H. We say that h1 ≺ h2 if Dom h1 ⊂ Dom h2 and h1(x) = h2(x) for all x ∈ Dom h1.
Now, we define another relation in H: we write that h1 ∼ h2 if there exists t ∈ H such that t ≺ h1 and t ≺ h2. It is clear that so defined relation ∼ is reflexive and symmetric. It is also transitive. Namely, let h1 ∼ h2 and h2 ∼ h3. Then there exist
−1 t, s ∈ H such that t ≺ h1, t ≺ h2, s ≺ h2, and s ≺ h3. Let r = ts s. Then Dom r ⊂
Dom t and Dom r ⊂ Dom s. Also, ∀x ∈ Dom(r), r(x) = t(x) = h2(x) = s(x). Hence, r ≺ t ≺ h1 and r ≺ s ≺ h3. Therefore, h1 ∼ h3.
Thus, the relation ∼ is an equivalence relation. Moreover, as one can easily check, it is a congruence. Hence, by Proposition A.6, H˜ = H/∼ is a semigroup.
We claim that H˜ is actually a group. Indeed, let a, b ∈ H. We need to show that there exist x, y ∈ H such that xb ∼ a and yb ∼ a, which would imply thatx ¯¯b =a ¯ and y¯¯b =a ¯. Indeed, let x = ab−1 and y = b−1a. Then xb = ab−1b ≺ a and by = bb−1a ≺ a.
Thus, according to Definition A.3, H˜ is a group.
Let us summarize our results in a lemma.
Lemma A.11. Let H be a semigroup of one-to-one partial mappings of a set Ω such that for every h ∈ H, the inverse transformation h−1 is also in H. Let the equivalence relation ∼ in H be defined as described above. Then H/∼ is a group.
We use this lemma to prove the following theorem due to Ore [Ore31].
Theorem A.12. Let S be a left reversible semigroup with two-sided cancellation.
Then S is embeddable in a group.
127 Proof. Let S be the semigroup of all partial one-to-one transformations of S. For every a ∈ S we define sa ∈ S to be the partial transformation such that
Dom sa = S, sa(x) = ax (∀x ∈ S).
The transformation sa is one-to-one since the semigroup S is cancellative.
H S −1 Let us consider the subsemigroup of , generated by all sa and sa for all a ∈ S. A general element of H has the form
h = h h . . . h , where h = s or h = s−1, i = 1, . . . , m. 1 2 m i ai i ai
−1 −1 −1 H Since the partial transformation h = hm . . . h1 also belongs to , we may apply Lemma A.11 and conclude that the factor semigroup H˜ = H/∼ is a group.
We need one more fact about the semigroup H to finish the proof. Namely, we need to know that for every h ∈ H, Dom h 6= ?. We present the proof of this fact as
Lemma A.15 below. Note that this is the only place where we use left reversibility.
Let η : H → H˜ be the natural homomorphism. Let ψ : S → H be the homo- morphism defined as ψ(a) = sa. Consider the homomorphism ξ = ηψ. This is a homomorphism from the semigroup S into the group H˜ . We claim that ξ is one-to- one. Indeed, suppose that for some a, b ∈ S, a 6= b we have ξ(a) = ξ(b). This means that sa ∼ sb, i.e. there exists a partial transformation t ∈ H such that t ≺ sa and t ≺ sb. Take some z ∈ Dom t (such an element exists by Lemma A.15). Then
saz = tz, sbz = tz.
But
saz = az, sbz = bz.
Thus az = bz, which is a contradiction since S is cancellative. Thus, ψ is one-to-one, i.e. it is an isomorphism from S into the group H˜ .
128 Remark A.13. It can be shown that the condition of left-reversibility is not only necessary, but also a sufficient condition for a cancellative semigroup to be embeddable into a group (see [CP61, Theorem1.24]). For an example of a cancellative semigroup that is not embeddable into a group see Mal0cev’s Example in AppendixC.
Corollary A.14. A cancellative commutative semigroup is embeddable into a group.
Lemma A.15. Let H be as in the proof of Theorem A.12. Then for all h ∈ H,
Dom h and Im h are right ideals of the semigroup S.
Proof. Let h ∈ H. Then
h = h . . . h , where h = s or h = s−1, i = 1, . . . , m. 1 m i ai i ai
We shall use induction by m.
First, consider the case m = 1. Then either h = sa, in which case Dom h = S and
Im h = aS, or h = s−1a and then Im h = S and Dom h = aS. The sets S and aS are
clearly right ideals.
Now, let us consider the case m > 1. By symmetry, it is enough to prove that
Dom h is a right ideal of S. First, let us show that Dom h 6= ?. Indeed, by induction
hypothesis, there exist some a ∈ Im h2 . . . hm and b ∈ Dom h1. By left irreversibility,
there exist u, v ∈ S such that
au = bv = w (for some w ∈ S).
Since Im(h2 . . . hm) and Dom h1 are right ideals, w ∈ Im(h2 . . . hm) and w ∈ Dom h1.
Since w ∈ Im(h2 . . . hm), it follows that there exists c ∈ S such that h2 . . . hm(c) = w.
On the other hand, since w ∈ Dom h1, it follows that c ∈ Dom(h1h2 . . . hm) = Dom h,
which shows that Dom h is non-empty.
129 It remains to show that Dom h is a right ideal. Let x ∈ Dom h and let y ∈ S. We need to show that xy ∈ Dom h. Clearly, x ∈ Dom(h2 . . . hm). Since Dom(h2 . . . hm) is a right ideal, xy ∈ Dom(h2 . . . hm). Note that
(h2 . . . hm)(xy) = ((h2 . . . hm)(x)) · y
(this is not obvious, but can be easily checked by induction). Since (h2 . . . hm)(x) ∈
Dom h1, it follows that
(h2 . . . hm)(xy) = ((h2 . . . hm)(x)) · y ∈ Dom h1, i.e. xy ∈ Dom h. This finishes the proof.
130 APPENDIX B
SOME FACTS ABOUT AMENABLE GROUPS AND SEMIGROUPS
In this appendix, we present definitions and some of major results from the theory of amenable groups and semigroups. Main references on the subject are [Pat88],
[Gre69], and [Pie84]. We only consider discrete countable semigroups and groups.
The more general case of σ-compact locally-compact semigroups is quite similar.
Let S be a countable semigroup. Let B(S) be the Banach space of all bounded complex-valued functions on S equipped with sup norm kfk∞.
Definition B.1. A linear functional M : B(S) → C is a mean if
(1) M(f) = M(f) ∀f ∈ B(S)
(2) infx∈S f(x) ≤ M(f) ≤ supx∈S f(x) for all real-valued f ∈ B(S)
Definition B.2. A mean M is called left invariant if
∀f ∈ B(S) ∀a ∈ SM(af) = M(f),
where af(x) = f(ax). M is called right invariant if
∀f ∈ B(S) ∀a ∈ SM(fa) = M(f)
where fa(x) = f(xa).
131 Definition B.3. A semigroup S is called left (right) amenable if there exists a left
(right) invariant mean M on B(S). By an amenable semigroup we mean a semigroup
that is either left or right amenable.
Remark B.4. If S is a group, then it is left amenable iff it is right amenable. Indeed, if
M is a left invariant mean on B(S) we can define another mean M˜ (f) = M(f ∗), where
f ∗(x) = f(x−1). It is easy to see that M˜ is a right invariant mean. Thus, for groups
we shall use the term amenable. For semigroups, the situation is more complicated.
There exist examples of semigroups that are left, but not right amenable and vice
versa.
Theorem B.5. An abelian group is amenable.
A proof can be found in [Pat88, p. 14].
Theorem B.6. If S is a left (right) amenable semigroup and π is a homomorphism
from S onto another semigroup H, then H is left (right) amenable.
Proof. Let M be the left (right) invariant mean on B(G). Let M˜ be the functional on
B(H) defined as M˜ (f) = M(f ◦ π). It is clear that if f ∈ B(H), then f ◦ π ∈ B(G).
It is easy to check that so defined functional M˜ is a left (right) invariant mean on
H.
Lemma B.7. A left (right) amenable semigroup is left (right) reversible (see Defini- tion A.8).
Proof. We consider the “left amenable” case. The other case is similar. Let I and J be two disjoint right ideals in S and let a ∈ I and b ∈ J. If S is left amenable, then
132 there exists a left invariant mean M on B(S). Since 1S(x) ≥ 1I (x) + 1J (x) for all
x ∈ S, we have
M(1S) ≥ M(1I ) + M(1J ) ≥ M(1aS) + M(1bS) = M(1S) + M(1S),
which is impossible as M(1S) = 1.
Thus, in an amenable semigroup, we may define a relation a ≈ b if there exists s ∈ S such that as = bs which, according to Proposition A.10, is a congruence and the factor-semigroup S = S/≈ is right cancellative.
Proposition B.8. A left reversible semigroup S is left amenable iff S/ ≈ is left
amenable.
A proof can be found in [Pat88, p. 35].
Example B.9. Let us consider the Lamplighter Semigroup. It is defined as follows.
Let M S = N n Z2. N with the semigroup operation defined as
(i, a) · (j, b) = (i + j, σja + b),
where σ is the left shift. It is a left-cancellative left-amenable semigroup.
Because of loss of information due to left shift, this semigroup is left cancellative,
but not right cancellative. It is easy to see that it is left reversible. In fact, if we have
two elements of S, s1 = (i, a) and s2 = (j, b), there always exist big enough natural
numbers k and l so that
(i, a) · (k, 0) = (j, b) · (l, 0) = (i + k, 0).
133 Now, consider again two arbitrary elements of S,(i, a) and (j, b). It is clear that
(i, a) ≈ (j, b) iff i = j. Indeed, we can always choose a big enough natural number k
such that
(i, a) · (k, 0) = (i, a) · (k, 0) = (i + k, 0).
Thus, the factor semigroup S/≈ is isomorphic to the additive semigroup N. Since N is left amenable, it follows that S is left amenable as well.
A very important feature of amenable semigroups is the presence of a summing sequence of sets, a so called Følner sequence.
Definition B.10. We say that a semigroup S satisfies Følner Condition if for every
ε > 0 and every finite set K ⊂ S there exists a finite set F such that
|F \ xF | < ε ∀x ∈ K (B.1) |F |
Definition B.11. We say that a semigroup S satisfies Strong Følner Condition if for every ε > 0 and every finite set K ⊂ S there exists a finite set F such that
|F 4 xF | < ε ∀x ∈ K (B.2) |F |
Theorem B.12 (cf. [Pat88, p. 131]). If a semigroup S is left amenable, then it
satisfies Følner Condition.
Theorem B.13 (cf. [Pat88, p. 145]). If a semigroup S satisfies Strong Følner
Condition, then it is left amenable.
Theorem B.14 (cf. [Pat88, p. 145]). A left cancellative semigroup S is left
amenable iff it satisfies Strong Følner Condition.
134 Remark B.15. Because we want to be able to make use of Strong Følner Condition,
we normally consider left cancellative semigroups only.
By a left Følner sequence (Fn) in S, we mean a sequence of finite subsets of S such that for every x ∈ S |F 4 xF | lim n n = 0. n→∞ |Fn| Remark B.16. The word “left” will often be omitted and we would say just “Følner sequence” meaning “left Følner sequence”.
Proposition B.17. A left cancellative semigroup is left amenable iff it contains a left Følner sequence.
Proof. This is an easy consequence of Theorem B.14.
135 APPENDIX C
EMBEDDING AMENABLE SEMIGROUPS INTO AMENABLE GROUPS
The main result presented in this appendix is Theorem C.3 due to Wilde and Witz
[WW67] stating that every amenable cancellative semigroup is embeddable into an amenable group generated ny this semigroup.
Definition C.1. Let G be a group and let S be a subsemigroup of G. We say that
S generates G if G is generated by the set S ∪ S−1, where S−1 is the set of inverse elements (with respect to the group operation in G) of elements of S.
Lemma C.2. If a discrete countable group G is generated by an amenable semigroup
S, then G is amenable.
Proof. We consider the case when S is left amenable. Let M be a left invariant mean ˜ ˜ on B(S). We define the left invariant mean M on B(G) as follows: M(f) = M(f|S).
Clearly, M˜ is a mean. Let us check that M˜ is left invariant. Indeed, for all a ∈ S,
˜ ˜ M(f(ax)) = M(f(ax)|S) = M(f|S) = M(f) and
˜ −1 −1 −1 M(f(a x)) = M(f(a x)|S) = M(f(aa x)|S) = M(f|S) = M(f).
136 Since S generates G, it follows that for all b ∈ G, M˜ (f(bx)) = M˜ (f), i.e. that M˜ is
left invariant.
Theorem C.3. Every countable cancellative amenable semigroup S is embeddable into a countable amenable group that is generated by S.
Proof. We consider left amenable case. By Lemma B.7, for all a, b ∈ S, there exist x, y ∈ S such that ax = by. Then by Theorem A.12, the cancellative semigroup S is embeddable into a group G. We need to show that G is amenable. In the proof of Theorem A.12 we constructed the group H˜ and the isomorphism ξ from S into
H˜ . The semigroup ξ(S) is an amenable subsemigroup of H˜ . From the way H˜ was constructed, it is clear that ξ(S) generates H˜ . Thus, by Lemma C.2, H˜ is amenable.
The right amenable case is completely symmetric. We would need to use a “left” analogue of Theorem A.12, left ideals instead of right ideals, etc.
Remark C.4. To prove that the group G is amenable, we could argue a bit differently.
Namely, since S,S−1 ⊂ G, then even if G if not generated by S (which is the case, but not clear from formulation of Theorem A.12), still there is a subgroup in G that is generated by S that is amenable by Lemma B.7.
Now, assume that (Fn) is a Følner sequence in a left amenable semigroup S and
−1 we constructed an amenable group G such that S ∪ S generate G. Is (Fn) a left
Følner sequence in G? The answer is positive.
Proposition C.5. Let a semigroup S generate a group G and let (Fn) be a left Følner sequence in S. Then (Fn) is a left Følner sequence in G.
137 Proof. Since (Fn) is left Følner sequence in S, then for all x ∈ S,
|xF 4 F | n n → 0. (C.1) |Fn|
Then
|x−1F 4 F | |F 4 xF | n n = n n → 0. (C.2) |Fn| |Fn|
−1 Every element g ∈ G can be represented as g = x1x2 . . . xm, where xi ∈ S ∪ S . To
prove that for all g ∈ G,
|gF 4 F | n n → 0. (C.3) |Fn| we use induction by m. The case m = 1 is clear by virtue of (C.1) and (C.2). Assume that we have proven (C.3) for some m. Then
|x x . . . x F 4 F | |x . . . x F 4 x−1F | 1 2 m n n = 2 m n 1 n (C.4) |Fn| |Fn| |x . . . x F 4 F | |x−1F 4 F | ≤ 2 m n n + 1 n n → 0. (C.5) |Fn| |Fn| by induction hypothesis.
Mal0cev’s Example. The condition that the semigroup is amenable is essential in
Theorem C.3. There exist examples of cancellative semigroups not embeddable into groups. First such example belongs to Mal0cev [Mal37]. Examples of this kind are constructed in the following way. First, one comes up with a certain property that always holds in a group, but might not hold in a cancellative semigroup. Mal0cev used the following property:
Condition M. ax = by, cx = dy, au = bv =⇒ cu = dv.
Then one constructs a cancellative semigroup in which this property does not hold. Mal’cev considered the free semigroup S on 8 generators, denoted as a, b, c,
138 d, x, y, u, and v, with relations ax = by, cx = dy, and au = bv. S is a cancellative semigroup. However, cu 6= dv, and so Condition M is not satisfied. Thus, S is not embeddable into a group.
139 APPENDIX D
A CHARACTERIZATION OF FØLNER SEQUENCES IN Z
In this appendix, we give a convenient characterization of Følner sequences in Z.
The proposed characterization in this form seems to be new.
We use the following notation. Let H ⊂ Z. Then H consists of a finite number of non-adjacent intervals, so that one can write
H = [a1, b1] ∪ · · · ∪ [an, bn] and this decomposition is unique. Let
`(H) = min (bk − ak). (D.1) k=1,...,n
We call the number `(H), the footprint of the set H.
Lemma D.1. A sequence of finite sets (Fn) in Z is a Følner sequence iff it can be represented as Fn = Gn ∪ Hn so that
(1) `(Gn) → ∞ as n → ∞,
(2) |Hn|/|Fn| → 0 as n → ∞.
140 Proof. “If” part. Let Fn = Gn ∪ Hn that satisfy the conditions (i) and (ii). Let r ∈ Z
be given. Because of our assumptions, for every ε > 0 there exists n0 ∈ N such that
for all n > n0 one has `(Gn) > 1/ε and |Hn|/|Fn| < ε. Note that
|Fn 4 (Fn + r)| ≤ |Gn 4 (Gn + r)| + 2|Hn|.
Since `(Gn) > 1/ε, the set Gn contains no more than ε|Gn| gaps between its intervals.
Therefore, for all n > n0
|F 4 (F + r)| |G 4 (G + r)| 2|H | 2εr|G | n n ≤ n n + n ≤ n | + 2ε ≤ (2 + 2r)ε. |Fn| |Fn| |Fn| |Fn
Since ε is arbitrary, the first part of lemma is proved.
”Only if” part. Assume the statement is not true, i.e. there exists a Følner sequence (Fn) for which the decomposition above is impossible. This means that
∃ε > 0 ∀n0 > 0 ∃n > n0 such that Fn can not be written as Fn = Hn ∪ Gn with
`(Gn) > 1/ε and |Hn|/Fn| < ε. Hence, Fn contains a lot of small intervals. Namely,
0 0 let Fn be the union of all intervals of length less than 1/2ε. Then |Fn|/|Fn| > ε
0 0 (otherwise, we could let Hn = Fn, Gn = Fn \Fn, and this would be the decomposition
0 that we assumed does not exist). It follows that the set Fn (and, therefore, the set
ε2 Fn as well) contains at least 2 |Fn| intervals. Hence,
|F 4 F + 1| ε2 n n ≥ . |Fn| 2
Contradiction.
The result contained in Lemma D.1 can be extended to more general amenable
groups.
141 APPENDIX E
BASIC RESULTS OF CHARACTER AND REPRESENTATION THEORY
E.1 Character Theory
Here we present some of the basic results of Character Theory. Proofs can be found, for example, in [HR60a].
Let X be a locally compact abelian group. Let Xˆ be the collection of all continuous
homomorphisms from X into the unit circle T. The members of Xˆ are characters of X. Under the operation of pointwise multiplication of functions Xˆ is an abelian
group. With the compact open topology Xˆ becomes a locally compact abelian group.
The group Xˆ is called the dual group of X.
One can show that Tˆn = Zn and each γ ∈ Tˆn is of the form:
2πi(x1p1+···+xnpn) n γ(x1, . . . , xn) = e for some (p1, . . . pn) ∈ Z .
More generally, for a compact group X, the dual group Xˆ is discrete.
The following facts hold.
\ ˆ ˆ Theorem E.1. X1 × X2 = X1 × X2.
142 Theorem E.2. Let Xi, i ∈ I, be a non-void family of locally compact abelian groups.
Then \ Y M ˆ Xi = Xi. i∈I i∈I Z L∞ In particular, the dual group for Zp is isomorphic to i=−∞ Zp.
Theorem E.3. If X is a compact abelian group, then the elements of Gˆ form an orthonormal basis for L2(X, µ), where µ is the Haar measure.
If A : X → X is an automorphism, we can define the dual automorphism Aˆ :
Xˆ → Xˆ by Aγˆ (x) = γ(Ax), x ∈ X, where γ ∈ Xˆ. Then Aˆ is an automorphism of Xˆ.
E.2 Representation Theory of Compact Groups
Now we present some of the basic facts of the theory of unitary representations of
compact groups. Proofs can be found in either [HR60b] or [NS82].ˇ
Let X be a compact group. A (continuous) unitary representation τ of X in a complex Hilbert space H is a homomorphism from X to the group of all continuous unitary operators of H so that the resulting map X ×H → H is continuous. For each
g ∈ X, τ(g) is a unitary operator on H. If H is finite-dimensional, then, with respect to a fixed basis, a unitary representation can be defined by a unitary matrix (tk,l,
1 ≤ k, l ≤ dτ , where dτ = dim H, and tk,l are continuous complex-valued functions
on X. The functions tk,l are called coordinate functions of the representation τ. The
number dτ is called the dimension of the representation τ. A representation τ is called
finite-dimensional if dτ < ∞.
0 0 0 A subspace H of H is called τ-invariant if for all g ∈ X, τgH ⊂ H . A repre-
sentation τ is called irreducible if it has no closed invariant subspaces other than {0}
and all H.
143 Theorem E.4. Every irreducible continuous unitary representation of a compact group X is finite-dimensional.
Two unitary representations of X, τ on H and τ 0 on H0, are called (unitarily) equivalent if there exists a bounded unitary operator U : H → H0 such that
τ 0(g)U = Uτ(g) for all g ∈ X.
We write τ ∼ τ 0.
Let U(X) be the set of all continuous irreducible unitary representations of X
(in some fixed infinite-dimensional Hilbert space) and let Γ(X) be the set of all equivalence classes of U(X). The set Γ(X) is called the dual object of X. The symbol
1 will sometimes denote the trivial irreducible representation given by the identity operator on 1-dimensional Hilbert space.
A tensor product τ1 ⊗ τ2 of two finite-dimensional unitary representations τ1 and
τ2 on the Hilbert spaces H1 and H2 respectively is the unitary representation on the space H1 ⊗ H2 defined as
(τ1 ⊗ τ2(g)) (v1 ⊗ v2) = τ1(v1) ⊗ τ2(v2), for all v1 ∈ H1, v2 ∈ H2.
If τ ∼ (t1 ) and τ ∼ (t2 ), then τ ⊗ τ is given by the matrix (t ), 1 k,l k,l=1,n1 2 k,l=1,n2 1 2 k1,k2,l1,l2 where t = t1 t2 for 1 ≤ k , l ≤ n , 1 ≤ k , l ≤ n . One can show that k1,k2,l1,l2 k1,l1 k2,l2 1 1 1 2 2 2 τ ⊗τ 0 ∼ τ 0⊗τ. Similarly, one defines a tensor product of finitely many representations.
Q Theorem E.5. Let Xi be a non-void family of compact groups and let X = i∈I Xi.
Let {i1, . . . , im} be a finite subset of I and let τik be irreducible unitary representations
of Xik , 1 ≤ k ≤ m. Then the mapping
(gi) 7→ τi1 (gi1 ) ⊗ · · · ⊗ τim (gim ) (E.1)
144 is an irreducible unitary representation of X. Moreover, every irreducible unitary
representation of X is equivalent to a representation of the form (E.1).
Let H be a Hilbert space and let Hi, i ∈ I, be a family of pairwise orthogonal
non-zero closed linear subspaces of H whose union spans a dense linear subspace of H H L H . Then is isometric to i∈I i. Let τ be a representation of a compact group
X on H such that each subspace Hi is invariant under τ. Let τi(g) be the linear
operator τ(g) with its domain restricted to Hi. Then τi is a unitary representation
of X in Hi for each i. We say that τ is the direct sum of the representations τi:
M τ = τi. (E.2) i∈I
If {σj}j∈J is the family of mutually inequivalent representations τi and if mj is the multiplicity of σi in the decomposition (E.2), then we also write:
M τ = mjσj. j∈J
Let τ be a finite-dimensional representation of X with a matrix elements (tk,l),
1 ≤ k, l ≤ n, with respect to a fixed basis. The conjugate representationτ ¯ is a
unitary representation of X with matrix elements (t¯k,l). If γ ∈ Γ(X) and τ ∈ γ, then
γ¯ denotes the equivalence class containing the representationτ ¯.
Let τ be a finite-dimensional representation of a compact group X. The character
χτ of the representation τ is the function on X defined as
χτ (g) = tr τ(g).
The notion of “character of a representation” differs from the one of “character of
a group”. Every character of a group is the character of the 1-dimensional represen-
tation defined by it. However, the character of a representation of a compact group
is a character of the group iff the representation is 1-dimensional.
145 Theorem E.6. Let τ and τ 0 be representations of a compact group X. One has:
(1) χτ¯ = χτ ,
(2) χτ⊕τ 0 = χτ + χτ 0 ,
(3) χτ⊗τ 0 = χτ χτ 0 .
Theorem E.7. Two continuous unitary finite-dimensional representations τ and τ 0 of a compact group X are equivalent iff
χτ = χτ 0 .
Theorem E.8. Every continuous unitary representation of a compact group X can be represented as a direct sum of irreducible finite-dimensional unitary representations.
Theorem E.9. Every finite-dimensional continuous unitary representation τ of a compact group X can be represented as a direct sum of irreducible unitary represen- tations of X:
τ = m1τ1 ⊕ · · · ⊕ mrτr. (E.3)
The decomposition (E.3) is unique up to a permutation of indexes. Let σ be an irreducible unitary representation of X. Then one has:
Z ( mj, if σ ∼ σj (j = 1, 2, . . . , r) χτ χ¯σ dµ = (E.4) X 0 otherwise
Let γ, γ0 ∈ Γ(X) and let τ ∈ γ and τ 0 ∈ γ0. Consider the unitary representation
τ ⊗ τ 0 of X. This representation is not irreducible in general. However, in view of the previous theorem, it can be decomposed as
0 τ ⊗ τ = m1τ1 ⊕ · · · ⊕ mrτr.
146 0 def We define γ × γ = {τ1, . . . , τr}.
If B ⊂ Γ, then
def [ γ × B = γ ⊗ γ0. γ0∈B
Theorem E.10. Let γ1, γ2 ∈ Γ(X). Then 1 ∈ γ1 × γ2 iff γ1 =γ ¯2.
Corollary E.11. Let γ ∈ Γ(X) and B ⊂ Γ(X). Then 1 ∈ γ × B iff γ¯ ∈ B.
The left regular representation of a compact group X is the unitary representation
τ of X in the Hilbert space L2(X, µ), given by τ(g)f(x) = f(g−1x). (The measure µ is the left-invariant Haar measure.)
Theorem E.12 (Peter-Weyl). Let τ be the left regular representation of a compact group X. Let {τi, i ∈ I} be the set of all mutually non-equivalent irreducible represen-
i tations of X with matrix elements tk,l, 1 ≤ k, l ≤ di = dτi , i ∈ I, with respect to some H 2 i basis. Let i be the subspace of L (X) spanned by the functions tk,l, 1 ≤ k, l ≤ di. 2 L H H Then L (X) = i∈I i, each space i is invariant with respect to the representation
τ, and the restriction of τ onto Hi is equivalent to diτi (the direct sum of di copies of the representation τi). In other words,
M τ = diτi. i∈I
Also,
Z ( 0 0 0 i i0 1/di, if i = i , k = k , and l = l tk,ltk0,l0 dµ = (E.5) X 0, otherwise √ i 2 The functions ( ditk,l), 1 ≤ k, l ≤ di, i ∈ I, form an orthonormal basis for L (X).
The equation (E.5) is often referred to as orthogonality conditions .
147 Let T be an automorphism of a compact group X and let τ be an irreducible
finite-dimensional unitary representation of X. Then we have another irreducible unitary representation that we denote as τT defined as (τT )(g) = τ(T g). Thus, T defines an operator on U(X) and Γ(X).
148 APPENDIX F
SOME FACTS FROM ERGODIC THEORY AND FUNCTIONAL ANALYSIS
Here we present some basic facts from ergodic theory. For some of these results, we give exact references. Others (at lest for Z-actions) may be found in textbooks such as [Pet83] or [CFS82].
Let S be a countable semigroup.
Definition F.1. An action T of S on a probability measure space (X, B, µ) is a homomorphism g 7→ Tg from S into the semigroup M(X) of measure-preserving transformations of X into itself.
A dynamical system is the quadruple (X, B, µ, T ) such that T is an action of a semigroup S on a probability measure space (X, B, µ).
The simplest case of an action is a N- or Z- action given by a single measure- preserving transformation T (that has to be invertible in the case of Z-action).
Definition F.2. A dynamical system (X, B, µ, T ) is ergodic if for every A ∈ B with
0 < µ(A) < 1 there exists g ∈ S such that µ(TgA 4 A) > 0.
A set A ∈ B is called an invariant set for the action T if TgA = A mod µ for all g ∈ G. A measurable function f is called an invariant function for the action T if f(T x) = f(x) a.e. for all g ∈ G.
149 Theorem F.3. The following statements are equivalent.
(1) T is ergodic.
(2) T has no non-constant measurable invariant functions.
(3) T has no invariant set A ∈ B with 0 < µ(A) < 1.
(4) For every A ∈ B with µ(A) > 0 one has ! [ −1 µ (Tg) A = 1. g∈S
(5) For every A, B ∈ B with µ(A) > 0, µ(B) > 0 there exists g ∈ S with
−1 µ((Tg) A ∩ B) > 0.
For each action T of a semigroup S on (X, B, µ), we can define a right unitary
representation (that we shall also denote by T ) of S on the space H = L2(X) by
(Tgf)(x) = f(Tgx).
Let H be a Hilbert space and let T be a right unitary representation of a left
cancellative left amenable semigroup S in H. A vector v ∈ H is called T -invariant if
Tgv = v for all g ∈ S
Theorem F.4 (Mean Ergodic Theorem). Let T be a right unitary representation
of a left cancellative left amenable semigroup S in a Hilbert space H, let (Fn) be a
Følner sequence in S, and let P : H → H be the orthogonal projection of H onto the
subspace of the vectors that are invariant with respect to the representation T . Then
for every f ∈ H 1 X Tgf → P f |Fn| g∈Fn in H-norm.
150 Lemma F.5. Let a left cancellative left amenable semigroup S act on a probability
2 measure space X. Then L (X) = Hinv ⊕ Herg, where
2 Hinv = {f ∈ L (X) | Tgf = f ∀g ∈ S}
∞ Herg = Span{h − Tgh | h ∈ L (X), g ∈ S}.
Remark F.6. Usually, one uses a somewhat more obvious splitting with
2 Hinv = {f ∈ L (X) | Tgf = f ∀g ∈ S}
2 Herg = Span{h − Tgh | h ∈ L (X), g ∈ S}.
Let T be a unitary operator on a Hilbert space H. A non-zero vector v ∈ H is
called an eigenvector of T if T v = λv for some λ ∈ C.
Lemma F.7. Let T be an ergodic transformation on a probability measure space
(X, B, µ) and let f ∈ L2(X) be its eigenfunction. Let T 0 be another transformation
on same space commuting with T . Then f is an eigenfunction of T 0 as well.
Proof. Let T f = λf, f 6= 0. Since T is ergodic, |f| is a non-zero constant a.e. Since
T and T 0 commute, T (T 0f) = λT 0f. Let
T 0f(x) g(x) = . f(x)
Then T g = g. Since T is ergodic, g is a constant a.e., say, g(x) = c. Then T 0f = cf a.e.
Definition F.8. Let T be a unitary operator in a Hilbert space a H. The adjoint operator is the unitary operator T ∗ in H having the property that for all u, v ∈ H
one has
(T u, v) = (u, T ∗v).
151 Lemma F.9 (cf. [Kre85, Lemma 1.2, p. 3]). Let T be a unitary operator in a
Hilbert space H and u ∈ H. Then u = T u holds iff u = T ∗u.
Lemma F.10 (cf. [Fur81, Lemma 4.18]). Let T and T 0 be unitary operators on
Hilbert spaces H and H0 respectively and let w ∈ H ⊗ H0 be an eigenvector of U ⊗ U 0:
(U ⊗ U 0)w = λw.
Then
X 0 w = cnun ⊗ un, n 0 0 0 0 0 where Uun = λnun, Unun = λnun and the sequences {un}, {un} are orthonormal.
Remark F.11. It follows that the eigenvalues of U ⊗ U 0 are given by λλ0, where λ
and λ0 are eigenvectors of U and U 0 respectively. Similarly, one can show that the
Qs eigenvalues of U1 ⊗ · · · ⊗ Us, where Ui are unitary operators, are given by i=1 λi,
where λi is an eigenvalue of the operator Ui, 1 ≤ i ≤ s.
If we have two dynamical systems (X, B, µ, S, T ) and (Y, B, ν, T 0), we can define
their direct product (X × Y, B × B, µ × ν, S, T × T 0) in a natural way. On unitary
level, this defines tensor product T ⊗ T 0 of the unitary representations T and T 0.
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156 Index
action, 2, 149 direct product, 152 ergodic, 3, 149 jointly ergodic, 49, 50 along a sequence of intervals, 102 uniformly, 103 weakly, 49, 50 jointly strongly mixing, 83 mixing, 83 pointwise ergodic, 116 strongly totally ergodic, 66 tensor product, 152 totally ergodic, 66 totally jointly ergodic, 71 weak-jointly strongly mixing, 83
Berend, D., 52, 59, 61, 82, 94 Bergelson, V., 52, 59 Birkhoff, G., 3
Ces`arolimit along a sequence of intervals, 104 Calder´on,A.-P., 16 character, 142 of a representation, 145 congruence, 123 covering lemma Lindenstrauss’, 26 for semigroups, 41 Tempelman’s, 21 Wiener’s, 17 density, 73 density 0 multiplicity, 88 dual group, 142
157 dual object, 144 dynamical system, 3, 149 ergodic, see action, ergodic
eigenfunction, 53 eigenvalue, 53
F(S), 2 footprint, 119, 140 Følner condition, 134 strong, 134 Følner sequence, 2, 135 in Z characterization, 140 tempered, 7, 41
group, 123 amenable, 132 generated by a semigroup, 136
homomorphism, 123
ideal, 125 invariant function, 149 invariant set, 149
∞ JE ([an, bn]n=1), 108 joint eigenfunction, 82 joint eigenvalue, 82 joint function, 61 joint invariant function, 61 joint invariant set, 66
L, 104 `(H), see footprint, see footprint lamplighter group, 37 lamplighter semigroup, 47, 133 Ledrappier’s example, 98 Ledrappier, F., 98 Lindenstrauss, E., 7, 25, 40 maximal inequality, 11, 12, 15 on S, 11, 15 Mal0cev’s example, 138
158 mean, 131 invariant, 131 mean ergodic theorem, 3, 150
N, 104
operator adjoint, 151
∞ PWE ([an, bn]n=1), 116 Peter-Weyl theorem, 147 pointwise ergodic theorem, 3, 11, 12, 18, 22, 30, 45
Rosenblatt, J., 5, 21, 47, 52 semigroup, 123 amenable, 132 embedding into group, 137 cancellative, 125 not embeddable into group, see Mal0cev’s example embedding into group, 127 left cancellative, 9, 125 reversible, 125 Shulman’s condition, 7, 32, 46 for semigroups, 41 Shulman, A., 32, 46 spectrum, 53 mutually disjoint, 53 subaction, 66
T (0), 49 Tempelman’s condition, 4, 20 weak, 5, 24 Tempelman, A., 4, 21 transfer principle, 16
unitary representation, 143 conjugate, 145 coordinate function, 143 direct sum, 145 equivalent, 144 irreducible, 143 orthogonality conditions, 147 regular, 147
159 tensor product, 144 upper Banach density, 104 von Neumann, J., 3, 150 weakly disjoint family of sets, 91 Wiener, N., 4, 17 Wierdl, M., 5, 21, 47
160