Closed Ideals in the Stone-ˇcech Compactification of a Countable Semigroup and Some Applications to Ergodic Theory and Topologi

Home , Pushforward measure

CLOSED IDEALS IN THE STONE-CECHˇ COMPACTIFICATION OF A COUNTABLE SEMIGROUP AND SOME APPLICATIONS TO ERGODIC THEORY AND TOPOLOGICAL DYNAMICS

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of

Philosophy in the Graduate School of the Ohio State University

Cory Christopherson, B.S.

Graduate Program in Mathematics

The Ohio State University

2014

Dissertation Committee:

Vitaly Bergelson, Advisor

Timothy Carlson

Alexander Leibman ⃝c Copyright by

Cory Christopherson

2014 ABSTRACT

We study the relationship between algebra in the Stone-Cechˇ compactiﬁcation

βS of a countable semigroup and dynamics. In particular, we establish a correspondence between the closed subsets of βS and certain types of recurrence in compact topological dynamical systems.

The notion of recurrence associated to a closed subset A of βS is most satisfying in the event that A is an ideal (or at least a semigroup) in βS. Those notions of recurrence which arise naturally in the study of topological dynamics all correspond to ideals.

This general theory is then applied to obtain results pertaining to recurrence in dynamical systems as well as combinatorial results about large sets in countable amenable groups.

ii ACKNOWLEDGMENTS

I would like to thank Timothy Carlson, John H. Johnson, Alexander Leibman,

Joel Moreira, and Warren Sinnott for reading earlier drafts and providing insightful, helpful comments. John H. Johnson, in particular, generously donated many hours of his time, for which I am extremely grateful.

I would also like to thank Fabrizio Polo for the many intriguing conversations we had about his work on weak twisting dynamical systems, which served as the primary motivation for Chapter 7 of this dissertation.

I owe special thanks to my advisor, Vitaly Bergelson. Dr. Bergelson’s mathe- matical influence on this dissertation cannot be overstated, and without his infinite patience and unwavering support I would never have finished it.

iii VITA

1980 ...... Born

2005 ...... B.S. in Mathematics, OSU

2005-Present ...... Graduate Teaching Associate, The Ohio State University

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Topological Dynamics, Topological Algebra in Stone-Cechˇ Compactiﬁcations, Ergodic Ramsey Theory

iv TABLE OF CONTENTS

Abstract...... ii Acknowledgments...... iii

Vita...... iv

CHAPTER PAGE

1 Introduction...... 1

1.1 D-sets in Abelian Groups: Deﬁnitions and Some Historical Context 2 1.1.1 Minimal Systems and Central Sets ...... 7 1.1.2 Invariant Measures, Positive Density, and D-sets ...... 9 1.2 NoncentralD-sets ...... 12 1.3 SomeExamples ...... 20 1.4 FurtherInvestigations ...... 23

2 DeﬁnitionsandBackground ...... 25

2.1 Algebra in the Stone-CechCompactiﬁcation.ˇ ...... 25 2.2 Topological Dynamics ...... 31 2.3 Amenability and Density ...... 33 2.4 Measure Preserving Dynamics and Ergodic Theory ...... 35

3 βS andTransitiveSubshifts ...... 42

3.1 ShiftSpace...... 42 3.2 A Metrizable Rendering of Beiglb¨ock’s Proof of Jin’s Theorem . . . 46

4 StrausExamplesinAmenableGroups...... 50

4.1 Introduction ...... 50 4.2 MainResults...... 55 4.3 ConcludingRemarks...... 65

5 Ideals in βS andForcingRecurrence ...... 67

5.1 Closed Ideals and Semigroups of βS ...... 67 v 5.2 An Algebraic Characterization of Sets that Force F-Recurrence...... 75 5.3 Anti-F-recurrence ...... 80

6 SetsthatForceEssentialRecurrence ...... 86

6.1 TheMainResult...... 86 6.2 Consequences...... 87 6.3 NoncentralD-sets ...... 89

7 AﬃneActionsofAmenableGroups ...... 91 7.1 Proximality and Strong Proximality ...... 92 7.2 Strong Proximality and Anti-essential-recurrence ...... 97 7.3 Mixing Properties of Measure Preserving Dynamical Systems . . . 101 7.4 Twisting and Weak Twisting ...... 112 8 Conclusion...... 125

Bibliography ...... 127

vi CHAPTER 1

INTRODUCTION

The following thesis is, broadly, a study of the class of all sets E ⊆ G having positive upper Banach density, where G is a countable amenable group. More specifically, we will concern ourselves with the so-called “D-sets” E ⊆ G. The question whether, for a fixed countable amenable group G, there exists a D-set E ⊆ G which is not a central set is fundamental to the study of D-sets. Furthermore, this question serves as a common motivation for the various (and, at first glance, seemingly unrelated) topics considered in later chapters.

In the work that follows we strive for generality, and this generality introduces certain technical diﬃculties. If, however, we restrict our attention to countable abelian groups G, the question becomes much more manageable. Therefore, in the remainder of this introductory chapter, we provide a short and relatively self-contained discussion of D-sets in a countable abelian group G1 .We will be able to prove that in any such group G there exists a D-set E ⊆ G which is not a central set. We then discuss some examples which illuminate the sometimes subtle relationship between the class of D-sets and the classes of IP sets and of sets having positive upper Banach density. We conclude the introduction by noting that certain

1The deﬁnitions of “D-set”, “central set”, and “IP set” will be provided shortly.

1 topics taken up in later chapters arise naturally when attempting to generalize the techniques of this chapter to more general amenable groups.

1.1 D-sets in Abelian Groups: Deﬁnitions and Some Histor-

ical Context

The notion of a central set E ⊆ N was introduced by H. Furstenberg (see [29, Chap- ter 8] ). Such sets arise naturally when using the theory of recurrence in topological dynamics to prove theorems in partition Ramsey theory. Furstenberg’s Central Sets

Theorem, stated below, shows that central sets have very strong combinatorial properties. We now present Furstenberg’s definition of a central set. Since we plan to generalize to abelian groups, we will work with the integers Z rather than N. This will have only superficial effects on the development, and the reader may recover the notion of acentralsubsetofN by removing the hypothesis “(X, T) is invertible” and replacing

Z by N throughout.

By a topological dynamical system (X, T) we mean a compact metric space X and a continuous map T : X → X.Thesystem(X, T) is invertible if T is a homeomorphism. A set Y ⊆ X is invariant if T (Y ) ⊆ Y (if (X, T) is invertible, then we require that T −1(Y )=Y ).

Let (X, T) be an invertible topological dynamical system. Given x ∈ X,theorbit closure O¯(x) is deﬁned by O¯(x)={T nx : n ∈ Z}. A closed invariant set Y ⊆ X is minimal if there are no proper closed invariant subsets of Y . It is well-known that, given a point x ∈ X, the orbit closure O¯(x) is minimal if and only if x is uniformly recurrent2. Finally, a pair of points (x, y) ∈ X × X is a proximal pair if for every

2For E ⊆ Z and t ∈ Z, let −t + E denote the set {n ∈ Z : t + n ∈ E}.AsetE ⊆ Z is syndetic

2 ϵ>0 there is n ∈ Z so that d(T nx, T ny) <ϵ, and a topological dynamical system is proximal if every pair (x, y) ∈ X × X is a proximal pair.

Deﬁnition 1.1.1 ([29, Deﬁnition 8.3]). AsetE ⊆ Z is central if there is some dynamical system (X, T),someproximalpair(x, y) of points with y uniformly recurrent, and some neighborhood U of y so that E = {n ∈ N : T nx ∈ U}.

Theorem 1.1.2 (Central Sets Theorem, [29, Proposition 8.21] ). Let E ⊆ Z be

i ∞ central. Let m ∈ N,andfori ≤ i ≤ m let (xn)n=1 be sequences in Z.Then,there

∞ ∞ is a sequence (an)n=1 in N and a sequence (Hn)n=1 of nonempty ﬁnite subsets of N such that:

1. For all n ∈ N one has max(Hn) < min(Hn+1). 2. For all ﬁnite F ⊆ N and every i ∈{1, 2,...,m} one has

i (an + xn) ∈ E. !n∈G n∈H # " "n Among the corollaries of the Central Sets Theorem is Rado’s famous theorem

([51]) characterizing the partition regular systems of linear equations over the natural numbers (See also [29, Chapter 8, Section 7]).

Many generalizations and improvements of the Central Sets Theorem have been obtained. In particular, it can be formulated and has been proven in any semigroup.

The reader is directed to [8], [25], [37], and [36].

In [14], the notion of a “D-set” E ⊆ Z is introduced, and it is shown in [4] that every D-set E ⊆ Z also satisfies the conclusion of the Central Sets Theorem. Such sets are defined in a way analogous to Definition 1.1.1, but the conditionthaty be

n if there exist ﬁnitely many natural numbers t1,...,tn so that Z = −ti + E. A point x ∈ X i=1 $ n is uniformly recurrent if for every neighborhood U of x,thesetRU (x)={n ∈ N : T x ∈ U} is syndetic.

3 uniformly recurrent is weakened. As will become clear, D-sets bear the same relation to the class of sets E ⊆ Z having positive upper Banach density as central sets bear to the class of piecewise syndetic sets. We will elaborate (and supply the formal deﬁnition of “D-set”) soon, but for the present discussion, it suﬃces to note that every central E ⊆ Z is a D-set.

The question now arises: Is the class D of D-sets, which is apriorilarger than the class Cen of central sets, actually larger? That is, is there a D-set E ⊆ Z which is not central? The ﬁrst example of such a set was constructed by Adams in

[1]. A dynamical construction of a noncentral D-set E ⊆ Z was given by Bergelson and Downarowicz ([14, Theorem 2.11]). Although we will be considering dynamical systems of a very diﬀerent nature than the one considered in [14], we will similarly approach the problem from a dynamical point of view.

We now supply the deﬁnitions necessary for the discussion of D-sets in abelian groups. These deﬁnitions extend naturally to the class of all amenable groups, see

Chapter 2.

Let G be a countable abelian group. We write the group operation as + and use

0G or, if the group G is clear from context, 0 to denote the identity in G. A topological dynamical system, written (X, G), is a compact metric space X together with a collection {Tg : g ∈ G} of continuous maps Tg : X → X such that for all g,h ∈ G and all x ∈ X one has Tg(Th(x)) = Tg+h(x) and for all x ∈ X one has

T0g (x)=x.Notethat,asaconsequence,eachmapTg : X → X is a homeomorphism. The deﬁnitions of proximality, minimality, and orbit closure given for G = Z above readily generalize. A probability measure µ on X is invariant if for all g ∈ G and

−1 all Borel sets A ⊆ X one has µ(Tg (A)) = µ(A). An invariant measure is ergodic

−1 if any Borel set A ⊆ X such that Tg (A)=A for all g ∈ G has the property that µ(A) ∈{0, 1}.Anysystem(X, G) has at least one invariant Borel probability

4 measure (see Theorem 2.4.5 below); the system (X, G) is uniquely ergodic if there is only one such measure.

For A ⊆ X and x ∈ X, we will ﬁnd the following notation useful:

RA(x)={g ∈ G : Tgx ∈ A}.

A point x ∈ X is recurrent if for every neighborhood U of x,thesetRU (x) contains a nonzero element g ∈ G.

It was discovered by V. Bergelson and N. Hindman that central setscanbechar- acterized by algebra in the Stone-Cechˇ compactiﬁcation βN of N ([17]). This characterization will be central to our discussion (no pun intended), so we give here a brief review of the relevant theory. For a more complete discussion, see [38]. See also

Chapter 2.

We realize the Stone-Cechˇ compactification βG of G as the set of all ultrafilters3 on G, with the elements of G identified with the principal ultrafilters, and with the topology determined by the base B = {A¯ : A ⊆ G},whereforA ⊆ G the set A¯ is given by A¯ = {p ∈ βG : A ∈ p}.NotethatforeveryA ⊆ G the set A¯ ⊆ βG is both closed and open. For p, q ∈ βG,definep + q by:

p + q = {E ⊆ G : {x ∈ G : −s + E ∈ q}∈p}.

Then p + q is an ultraﬁlter, and this deﬁnition of + extends the group operation in

G to make βG a compact right-topological semigroup. That is, for any p ∈ βG,the function ρp : βG → βG given by ρp(q)=q + p is continuous. Furthermore, this extension is unique.

For any semigroup T ,anonemptysetI ⊆ T is a left (right) ideal if for every t ∈ T one has t+I ⊆ I (I +t ⊆ I). A nonempty set I ⊆ T is an ideal if I is both a left ideal

3 recall that an ultrafilter on a set S is a collection q ⊆P(S) such that i) ∅ ∈/ q, ii) for any finite H ⊆ q one has H ∈ q, and iii) for all E ⊆ S either E ∈ q or (S \ E) ∈ q.Aprincipal ultrafilter on S is an ultrafilter% of the form q = {E ⊆ S : s ∈ E} for some s ∈ S. 5 and a right ideal. A (left, right) ideal I ⊆ T is minimal if it properly contains no (left, right) ideals. Any compact right-topological semigroup T has a unique smallest ideal, denoted by K(T ), which is the union of all minimal left ideals, and also is the union of all minimal right ideals (See, for instance, [38, Theorem 2.8]).

We are now prepared to state the promised characterization of central sets. Again, we state the result for central subsets of Z, despite the fact that it was stated in [17] for central subsets of N.

Theorem 1.1.3 ([17, Corollary 6.12] ). AsetE ⊆ Z is central if and only if there is some idempotent p ∈ K(βZ) so that E ∈ p.

The surprising connection that is established by Theorem 1.1.3 between topological dynamics and ultrafilters is quite fruitful. For example, it now follows immediately that the collection Cen of central sets satisfies the Ramsey property : given any partition E = E1 ∪ E2 of a central set, there is i ∈{1, 2} so that Ei is also central. Of course, this result can be proven directly from Furstenberg’s definition (see [29,

Theorem 8.8]). Note, however, that in light of Theorem 1.1.3 it is obvious. Another fact which may be hard to see from Furstenberg’s deﬁnition but which is obvious in light of Theorem 1.1.3 is that any superset of a central set is central. More generally, the idempotents in βG have both combinatorial and dynamical

∞ significance. Let (xn)n=1 be a sequence in G.Definethefinite sums set generated by

∞ (xn)n=1 by

∞ FS((xn)n=1)={ xn : H ⊆ N is ﬁnite and nonempty} n H "∈ ∞ and say that E ⊆ G is an IP set if there is some sequence (xn)n=1 such that

∞ FS((xn)n=1) ⊆ E.AsetE ⊆ G is an IP set if and only if there is some nonprincipal idempotent p = p + p ∈ βG with E ∈ p ([38, Theorem 5.12]). A point

(X, G) is recurrent if and only if for every neighborhood U of x the set RU (x) ⊆ G 6 is an IP set. The reader is encouraged to prove this last assertion by induction as an exercise. See also Chapter 5, Lemma 5.2.2.

For the remainder of this section, we investigate two important ideals in βG.In particular, we establish a correspondence between these ideals and certain types of recurrence in topological dynamical systems.

1.1.1 Minimal Systems and Central Sets

AsetE ⊆ G is syndetic if there is some ﬁnite set F ⊆ G such that G = −f + E. f∈F AsetE ⊆ G is thick if for every ﬁnite set F ⊆ G there is some g ∈$G so that

F + g ⊆ E. Given any syndetic A ⊆ G and any thick B ⊆ G, the intersection A ∩ B is inﬁnite. A set E ⊆ G is piecewise syndetic if there is a syndetic A ⊆ G and a thick

B ⊆ G so that E = A ∩ B. Recall that K(βG) is the unique smallest two-sided ideal in βG.AsetE ⊆ G is piecewise syndetic if and only if E¯ ∩ K(βG) ≠ ∅ ([38, Theorem 4.40]).

Let (X, G) be any topological dynamical system. As in the introduction, a point x ∈ X is uniformly recurrent if for every neighborhood U of x the set RU (x) ⊆ G is syndetic. A point x ∈ X is uniformly recurrent if and only if O¯(x) is a minimal subsystem of X. For this reason, uniformly recurrent points are also called minimal points. The following two theorems summarize the connection between minimal systems and K(βG). An idempotent p = p+p ∈ K(βG) is called a minimal idempotent.

Theorem 1.1.4 ([38, Theorem 19.23] or [12, Proposition 3.3 ]). For any topological dynamical system (X, G) and any x ∈ X, x is a minimal point if and only if there is some minimal idempotent such that p- lim Tgx = x. g

For G = Z, the following theorem was proven in [14]. The proof given there readily generalizes to countable groups. In essence, it states thatthecentralsetscan be characterized in the product system (X ×X, T ×T )ratherthanthesystem(X, T). 7 It is this characterization in terms of the product system which will be generalized in Chapter 5.

Theorem 1.1.5 ([14, Theorem 2.3]). Let E ⊆ G.Thefollowingtwoconditionson E are equivalent.

i) There is some minimal idempotent p ∈ K(βG) with E ∈ p.

ii) There is some compact topological dynamical system (X, G),somepair(x, y) of proximal points in X with y minimal, and some neighborhood U of (y,y) ∈ X × X such that E = RU (x, y).

Any set E ⊆ G satisfying the conditions of Theorem 1.1.5 is called a central set.

The following characterization of piecewise syndetic sets is well-known in G = Z.

It holds in any semigroup, see Lemma 2.1.7.

Lemma 1.1.6. Let E ⊆ G.ThenE is piecewise syndetic if and only if there is some syndetic set B ⊆ G so that for all ﬁnite H ⊆ B there is g ∈ G so that H + g ⊆ E.

Proof. Let E ⊆ G be piecewise syndetic. By [38, Thoerem 4.40], there is q ∈ K(βG) with E ∈ q.LetB = {g ∈ G : −g + E ∈ q}.ThenB is syndetic by [38, Theorem 4.39]. Let H ⊆ B be ﬁnite. Then {−h + E : h ∈ H}⊆q,so −h + E ∈ q.Let h H %∈ g ∈ −h + E.ThenH + g ⊆ E.NotethatH was an arbitrary ﬁnite subset of the h∈H syndetic% set B.

Now assume that there is some syndetic B ⊆ G so that for all ﬁnite H ⊆ B there is g ∈ G with H + g ⊆ E.ThenB is, in particular, piecewise syndetic, so there is some q ∈ K(βG) with B ∈ q, by [38, Theorem 4.40].

Also, the collection F = {− + Eb : b ∈ B} has the ﬁnite intersection property. So there is some ultraﬁlter p ∈ βG so that F⊆p.Foreachb ∈ B, −b + E ∈ p,so

B ⊆{x ∈ G : −x+ E ∈ p}.Thus{x ∈ G : −x+ E ∈ p}∈q and E ∈ p + q ∈ K(βG).

So E is piecewise syndetic by [38, Theorem 4.40] again.

8 1.1.2 Invariant Measures, Positive Density, and D-sets

Recall that, for a set E ⊆ N,thedensity of E is deﬁned as:

|{E ∩{1, 2,...,n}| d(E) = lim n→∞ n

(if the limit exists)4. This notion is generalized as follows.

A Følner sequence in G is a sequence F =(Fn)n∈N of ﬁnite subsets of G with the property that for any g ∈ G,onehas

|(F ∩ (g + F )| lim n n =1. n→∞ |Fn|

There exists some Følner sequence in every abelian countable group (see for example

[28, Main Theorem]). Given a Følner sequence F =(Fn)n∈N,asetE ⊆ G,andn ∈ N, |E ∩ Fn| deﬁne An(E,F)= .Now,theupper density of E with respect to F is given |Fn| by

d¯F (E) = lim sup An(E,F), n→∞ the lower density of E with respect to F is

dF (E) = lim sup An(E,F), n→∞ and if the limit exists then the density of E with respect to F is

lim An(E,F). n→∞

For E ⊆ G,theupper Banach density is given by:

∗ d(E) =sup{d¯F (E):F =(Fn)n∈N is a Følner sequence in G},

and we say that E ⊆ G has positive upper Banach density if d∗(E) > 0. Since a subsequence of a Følner sequence is again a Følner sequence, E ⊆ G has positive

4For finite set E, we denote the cardinality of E by |E|. 9 upper Banach density if and only if there is some Følner sequence F =(Fn)n∈N such that dF (E) exists and satisfies dF (E) > 0. One readily verifies that the collection L of all sets E ⊆ G having positive upper

Banach density is upward hereditary: If A ⊆ B ⊆ G and A ∈Lthen B ∈L.

Furthermore, L has the Ramsey property. It follows that L is a union of ultraﬁlters

(See Theorem 5.1.1 for a proof of this last assertion).

Deﬁne β(L)={p ∈ βG : p ⊆L}.Thenβ(L) is a closed two-sided ideal of βG ([37, Theorem 1.10]). Thus K(βG) ⊆ β(L). Any idempotent p = p + p ∈ β(L) is called an essential idempotent. Note that all minimal idempotents are essential idempotents.

Let (X, G) be any topological dynamical system. A point x ∈ X is called essentially recurrent if for every neighborhood U of x,onehasRU (x) ∈L. Just as uniform recurrence is characterized by minimality of the orbit closure, we have the following theorem characterizing essential recurrence.

Theorem 1.1.7. Let (X, G) be any topological dynamical system. Let x ∈ X.Then x is essentially recurrent if and only if for every open U ⊆ O¯(x) there is an invariant

Borel probability measure µ with supp(µ) ⊆ O¯(x) and such that µ(U) > 0.

Proof. For G = Z, this is [14, Theorem 2.6]. The proof given there easily generalizes to countable abelian groups G. The general idea is as follows.

Let U ⊆ O¯(x) be open. Then there is some h ∈ G such that Th(x) ∈ U.By continuity of Th, there is some neighborhood V of x with Th(V ) ⊆ U.Soforany

−1 invariant µ, µ(U)=µ(Th (U)) ≥ µ(V ). Thus it suﬃces to show that there is an invariant Borel probability measure µ supported on O¯(x) with µ(V ) > 0.

Let (Fn)n∈N be a Følner sequence such that

a = lim An(RV (x),F) > 0. n→∞ 10 For n ∈ N and f ∈ C(X), let

1 mn(f)= f(Tgx). |Fn| g∈F "n ∞ Then (mn)n=1 is a sequence in the set M(X) of all regular Borel probability measures on X, which we identify with a convex subset of C(X)∗ by the Reisz representation theorem. Endowed with the weak∗ topology, M(X) is a compact set, hence

∞ ∗ ∗ ∞ the sequence (mn)n=1 has (weak )-limit points, and any weak limit µ of (mn)n=1 will be invariant and satisfy a = µ(V ).

The following two theorems serve as analogues to Theorems 1.1.4 and 1.1.5. For

G = Z, Theorem 1.1.9 is proved in [14]. Theorem 1.1.8, in the case that G = Z, is implicit in [14]. We provide proofs (in a more general setting) in Chapter 5(Lemma

5.2.2 and Theorem 5.2.3).

Theorem 1.1.8. Let (X, G) be any compact topological dynamical system. Let x ∈ X.Thenx is essentially recurrent if and only if there is some essential idempotent p ∈ β(L) so that p- lim Tgx = x. g

Theorem 1.1.9 ([14, Theorem 2.8]). Let E ⊆ G.Thefollowingtwoconditionson

E are equivalent.

i) There is some essential idempotent p ∈ β(L) with E ∈ p.

ii) There is some compact topological dynamical system (X, G),somepair(x, y) of proximal points in X with y essentially recurrent, and some neighborhood U of

(y,y) ∈ X × X such that E = RU (x, y).

Now, finally, we define a set E ⊆ G to be a D-set if E satisfies the conditions of

Theorem 1.1.9. Since any uniformly recurrent point is essentially recurrent (because any syndetic set has positive upper Banach density), it is clear that every central set is a D-set. The following two theorems are the main results of this chapter. 11 Theorem 1.1.10. Let G be a countable abelian group. The the following are equivalent: i) There is some D-set E ⊆ G such that E is not piecewise syndetic.

ii) There is some set E ⊆ G having positive upper Banach density which is not piecewise syndetic.

iii) There is some compact metric topological dynamical system (X, G) which is proximal but which is not uniquely ergodic.

Theorem 1.1.10 will be proven in Section 1.2. In speciﬁc groups G, it is usually much easier to construct examples of sets E ⊆ G satisfying condition ii) than to directly construct sets E ⊆ G as in conditon i) of the above theorem. Hence it is useful to note that the easier construction will suﬃce.

Theorem 1.1.11. If G is any countable abelian group, then there is some E ⊆ G which has positive upper Banach density but which is not piecewise syndetic.

Proof. This is Theorem 1.2.5.

Corollary 1.1.12. If G is a countable abelian group, then there is some D-set E ⊆ G which is not piecewise syndetic. Consequently, E is not a central set.

1.2 Noncentral D-sets

Throughout this section, we ﬁx some countable abelian group G. Our ﬁrst goal is to show that there is some set E ⊆ G which has positive upper Banach density but which is not piecewise syndetic.

The following lemma is given as exercise 16 after Chapter 1 in [3]. A much stronger statement is proven in [31, Theorem 3.4]. The reader is also referred to [26, Theorem

1.2]. We provide a simple proof in Chapter 5 (see the discussion following Corollary

5.3.6).

12 Lemma 1.2.1. Let (X, G) be a compact topological dynamical system, with G an abelian group. Then (X, G) is proximal if and only if there is a ﬁxed point x0 ∈ X so that {x0} is the only minimal invariant subset of X.

It should be noted that Lemma 1.2.1 fails when the acting group G is not assumed to be abelian. Examples are given in [31], Section II.5. See also the discussion following Theorem 7.1.2.

Lemma 1.2.2. There is a compact abelian metric group K and an injective homomorphism j : G → K so that j(G) ⊆ K is dense.

Remark 1.2.3. For G = Z,Lemma1.2.2iseasy:LetK = {z ∈ C : |z| =1} and

in ∞ let j : Z → K be given by j(n)=e .IfG = ⊕n=1Hn,whereeachHn is ﬁnite, then

∞ again Lemma 1.2.2 is trivial: give each Hn the discrete topology, then K =Πn=1Hn is compact by Tychonoﬀ’s Theorem and contains G as a dense subset.

ˆ 1 ∞ Proof. Let G denote the group of all characters χ : G → S .Let(xn)n=1 be a sequence of all nonzero elements of G. Since Gˆ separates point, for every n ∈ N there ∞ 1 is χn ∈ Gˆ so that χn(gn) =1.Let̸ Y = S , and let j : G → Y be given by: n=1 (j(g))n = χn(g). Then Y is a compact abelian& metric group and j is an injective homomorphism. Letting K = j(G) ⊆ Y completes the proof.

Lemma 1.2.4. Let (X, d) be any compact metric space, and assume G acts on X in such a way that (X, G) is uniquely ergodic with unique invariant measure µ.Let

U ⊆ X be open with µ(∂U)=0.Letx ∈ X and let (Fn)n∈N be any Følner sequence in G.ThendF (RU (x)) = µ(U).

Proof. Let ϵ>0. Choose compact K ⊆ U and C ⊆ (X \ U¯)suchthatµ(U \ K) <ϵ and µ((X \ C) \ U¯) <ϵ.Letf ∈ C(X)besuchthatf(C)=0,f(U¯)=1and

13 f(X) = [0, 1], let h ∈ C(X)besuchthath(K)=1,h(X \ U)) = 1 and h(X) = [0, 1]. Then, by unique ergodicity5, 1 d¯F (RU (x)) = lim sup χu(Tgx) n→∞ |Fn| g F "∈ n 1 ≤ lim sup f(Tgx)= fdµ n→∞ |Fn| g∈F "n ' ≤ µ(U)+ϵ

and 1 dF (RU (x)) = lim inf χu(Tgx) n→∞ |Fn| g∈F "n 1 ≥ lim inf h(Tgx)= hdµ n→∞ |Fn| g∈F "n ' ≥ µ(U) − ϵ.

¯ Since ϵ>0 is arbitrary, we see that dF (RU (x)) = dF (RU (x)) = µ(U).

Theorem 1.2.5. Let F =(Fn)n∈N be any Følner sequence in G.Thereissomeset

A ⊆ G which is not piecewise syndetic and which satisﬁes d¯F (A) > 0.

Proof. By Lemma 1.2.2 there is some compact abelian metric group K and an injective homomorphism j : G → K such that j(G) is dense in K.Letd be any invariant metric on K.Foranyx ∈ K and any ϵ>0, let Bϵ(x) denote the open ball with

5 It is well-known that a compact topological dynamical system (X, T) is uniquely ergodic if and N−1 1 only if for all f ∈ C(X) and all x ∈ X one has lim f(T nx)= fdµ (see, for example, N→∞ N n=0 ' [29] Theorem 3.5). The usual proof readily generalizes to" show that given any countable abelian group G and any Følner sequence F =(Fn)n∈N the system (X, G) is uniquely ergodic if and only 1 if for all f ∈ C(X) and all x ∈ X one has lim f(Tgx)= fdµ. See also Proposition n→∞ |Fn| g∈Fn ' 2.4.6 and the discussion preceeding it. "

14 radius ϵ and center x. Setting Tg(k)=k + j(g) for all k ∈ K and g ∈ G,onemay consider (K, G) as a topological dynamical system. Since J(G) ⊆ K is dense, (K, G) is minimal and uniquely ergodic, with unique invariant probability measure given by the normalized Haar measure µ on K.

Since K is inﬁnite, µ is nonatomic. Thus, by regularity of µ, there is some sequence

∞ n+2 (Un)n=1 of open neighborhoods of 0K so that µ(Un) < 1/2 .Letrn > 0besuch that Brn (0K) ⊆ Un.

Fix n ∈ N.Then{∂(Bs(0K)) : 0

disjoint Borel subsets of K, so there is some sn ∈ (0,rn) with µ(∂(Bsn (0K))) = 0. Set

−n−2 Vn = Bsn (0K). Then 0 <µ(Vn) ≤ µ(Un) < 2 and µ(∂Vn)=0.

Let Bn = {g ∈ G : j(g) ∈ Vn}.LetF =(Fn)n∈N be any Følner sequence

−n−2 in G. By Lemma 1.2.4, dF (Bn)=dF (RVn (0K)) = µ(Vn) < 2 .Wenowshow

∞ that Bn intersects every IP set. Let (xk)k=1 be any sequence of elements of G.For k k ∈ N, let tk = xi.ThenbycompactnessofK,therearea>b∈ N so that i=1 " d(j(ta),j(tb))

∞ ∞ j(ta − tb) ∈ Vn,henceta − tb ∈ Bn ∩ FS((xk)k=1). Since (xn)n=1 ⊆ G is arbitrary, Bn intersects every IP set as asserted.

Now, for each n ∈ N, there is some Nn ∈ N such that if m>Nn then A(−gn + Nn −n−1 Bn,m) < 2 .LetEn =(−gn + Bn) \ ( Fi). Since Bn \ (gn + En) is ﬁnite, En i=1 intersects every IP set also. $ ∞ ∞

Let E = En +gn. Then for all m ∈ N,onehasA(E,m) ≤ A(En +gn,m) ≤ n=1 n=1 ∞ $ " 2−n−1 =1/2. So d(E) ≤ 1/2. n=1 "Let A = G \ E.Thend(¯ A) ≥ 1 − d(E) ≥ 1/2, so A has positive upper Banach density. Furthermore, since every shift of E intersects every IP set, no shift of A

15 contains an IP set. In particular, no shift of A is central, so A is not piecewise syndetic by [38, Theorem 4.43].

We now turn our attention to proving Theorem 1.1.10. The following lemma was ﬁrst proved for G = Z in [22], and will be extended to arbitrary semigroups in

Chapter 5.

Lemma 1.2.6. Let (X, G) be a compact topological system and let K ⊆ X be closed.

Assume that there is some x ∈ X so that E = RK (x) is piecewise syndetic. Then there is some uniformly recurrent point y ∈ K.

Proof. Let (X, G), K ⊆ X, x ∈ X,andE = RK (x) be as in the statement of the

Lemma. Then by [38, Theorem 4.43], there is some g0 ∈ G so that −g0 + E is a central set. Let p ∈ −g0 + E ∩ K(βG) be a minimal idempotent. Let z = p- lim Tgx. g

Then p- lim Tgz = p- lim Ts(p- lim Ttx)=(p + p)- lim Tgx = p- lim Tgx = z. Since g s t g g p is a minimal idempotent, z is uniformly recurrent. Hence, y = Tg0 z is uniformly recurrent. It only remains to show that y ∈ K.

Letg ¯0 denote the principal ultraﬁlter containing the set {g0}.ThenE ∈ (¯g0 + p), and (g ¯0 + p)- lim Tgx = Tg (p- lim Tgx)=Tg z = y. So for every neighborhood U of y, g 0 g 0

RU (x) ∈ (¯g0 +p). So for every neighborhood U of y, RK(x)∩RU (x)=E ∩RU (x) ≠ ∅. Thus, for every neighborhood U of y, U ∩ K ≠ ∅. Since K ⊆ X is closed, it follows that y ∈ K.

We will ﬁnd it convenient to use the following analogue of Lemma 1.2.6 for sets

E ⊆ G having positive upper Banach density.

Lemma 1.2.7. Let (X, G) be a compact metrizable topological dynamical system and let K ⊆ X be a Borel set. If there is some invariant measure µ on X with µ(K) > 0 then there is some essentially recurrent y ∈ K.

16 Proof. Let µ be an invariant measure with µ(K) > 0. By the ergodic decomposition, we may assume without loss of generality that µ is ergodic. Let K1 = K ∩ supp(µ).

∞ Let F =(Fn)n∈N be any Følner sequence in G, and let (fn)n=1 be a sequence of continuous functions on X so that {fn : n ∈ N} is dense in C(X). By the mean ergodic theorem, for each n ∈ N one has

1 lim fn ◦ Tg = fndµ, k→∞ |Fk| g∈F "k ' where the convergence above is with respect to the L2-norm. Hence there is some

(1) ∞ (1) (1) (1) subsequence (Fk )k=1 and some Y ⊆ X with µ(Y ) = 1 such that for all x ∈ Y , 1 lim f1(Tgx)= f1dµ. k→∞ |Fk| g∈Fk ' " (n) ∞ (n) Let n ∈ N and assume that a subsequence (Fk )k=1 and some set Y ⊆ X have been chosen such that µ(Y (n)) = 1 and for all 1 ≤ i ≤ n and all x ∈ Y (n), 1 lim fi(Tgx)= fidµ. Then, by the mean ergodic theorem, k→∞ |F (n)| k g F (n) ' ∈"k 1 lim fn+1 ◦ Tg = fn+1dµ, k→∞ |F (n)| k g∈F (n) ' "k

(n+1) ∞ (n) ∞ (n+1) so there is some subsequence (Fk )k=1 of (Fk )k=1 and some set X with

(n+1) (n+1) 1 µ(X ) = 1 so that for all x ∈ X , lim fn+1(Tgx)= fn+1dµ. k→∞ |F (n+1)| k g∈F (n+1) ' "k Put Y (n+1) = Y (n) ∩ X(n+1). Then for all 1 ≤ i ≤ n + 1 and all x ∈ Y (n+1), 1 lim fi(Tgx)= fidµ. k→∞ |F (n+1)| k g∈F (n+1) ' "k ∞ (n) ∗ (n) ∗ ∗ ∞ Now, put Z = Y ,andforn ∈ N put Fn = Fn .ThenF =(Fn )n=1 is a n=1 % ∞ Følner sequence, µ(Z) = 1, and since (fn)n=1 is dense in C(X)onehas

1 lim f(TgX)= fdµ n→∞ |F ∗| n g F ∗ "∈ n ' for every x ∈ Z and every f ∈ C(X). 17 By an approximation argument similar to the one used in the proof of Lemma 1.2.4, if x ∈ Z and U ⊆ X is an open set satisfying µ(∂U)=0,thenonehas

∗ |RU (x) ∩ Fn | 1 dF ∗ (RU (x)) = lim = lim χU (Tgx)=µ(U). n→∞ |F ∗| n→∞ |F ∗| n n g F ∗ "∈ n

Now, µ(Z ∩ K1)=µ(K1)=µ(K) > 0, so Z ∩ K1 ≠ ∅.Lety ∈ K1 ∩ Z, and let U be a neighborhood of y.Fort

{t ∈ (0,d(y,X \ U)) : µ(∂Ut) > 0}. Since µ is a probability measure, P is countable.

So there is some t ∈ (0,d(y,X \ U)) such that µ(∂Ut) = 0. Since y ∈ supp(µ),

∗ µ(Ut) > 0. So, by the remarks of the preceeding paragraph, dF (RUt (y)) > 0. Since U is an arbitrary neighborhood of y, y ∈ K is essentially recurrent.

We are now prepared to prove Theorem 1.1.10.

Proof of Theorem 1.1.10. i) ⇒ ii): Since any D-set E ⊆ G is in particular a set

E ⊆ G having positive upper Banach density, this is obvious.

ii) ⇒ iii): Let E ⊆ G be any set having positive upper Banach density which is not piecewise syndetic. Let Y = {0, 1}G be endowed with the product topology, and for g ∈ G let Tg : Y → Y be given by: Tgx(h)=x(g + h) for all x ∈ Y and all h ∈ G.ThenY is compact by Tychonoﬀ’s theorem, and since G is countable Y is metric. One checks easily that (Tg)g∈G is a collection of continuous maps, so (Y,G) is a compact metric topological dynamical system. Let x =1E ∈ Y , and let X = O¯(x). Then X ⊆ Y is closed and invariant, so (X, G) is also a compact metric topological dynamical system.

We ﬁrst show that X is proximal. Let 0¯ denote the constant function deﬁned by 0(¯ h) = 0 for all h ∈ G.Weshowthat0¯ is the only uniformly recurrent point in X.

Then, by Lemma 1.2.1, X is proximal.

Since there must be some uniformly recurrent point y ∈ X, it will suﬃce to show that there is no uniformly recurrent point 0¯ ≠ y ∈ X.Soassume,toward 18 ′ a contradiction, that 0¯ ≠ y ∈ X is uniformly recurrent. Let h0 ∈ G be such

′ ′ that y (h0) = 1. let y = Th0 y .Theny ∈ X is uniformly recurrent also, and

′ ′ y(0G)=Th0 y (0G)=y (h0)=1.LetU = {z ∈ X : z(0G)=1}.ThenU is a closed and open neighborhood of y. So, letting A = RU (y), A ⊆ G is syndetic.

Let H ⊆ A be ﬁnite. Then for all h ∈ H, U is a neighborhood of Th(y). By continuity of Th, there is some neighborhood Vh of y so that Th(Vh) ⊆ U.LetV =

∩h∈H Vh.ThenV is a neighborhood of y, and since y ∈ X = O¯(x), there is some g ∈ G so that Tg(x) ∈ V .Then,foreveryh ∈ H, Th+gx ∈ U,andx(h + g)=Th+gx(0) = 1. So H + g ⊆ E. Since H ⊆ A is an arbitrary ﬁnite set, E is piecewise syndetic by

Lemma 1.1.6. But, E was assumed to not be piecewise syndetic, so no such y′ ∈ X exists, and 0¯ ∈ X is the only uniformly recurrent point.

iii) ⇒ i): Let (X, G) be a compact metrizable topological dynamical system which is proximal but not uniquely ergodic. Then, by Lemma 1.2.1, there is some ﬁxed point x0 ∈ X so that x0 is the only uniformly recurrent point in X. Since (X, G) is not uniquely ergodic, there is some invariant Borel measure µ on X which is distinct from

the point mass δx0 at x0. So there is some open U ⊆ X with x0 ∈/ U and µ(U) > 0.

By Lemma 1.2.7, there is some essentially recurrent point x ∈ U.Notethatx ≠ x0.

By Theorem 1.1.8 there is some essential idempotent p ∈ βG so that p- lim Tgx = x. g

Let W, V be open subsets of X so that x ∈ W , x0 ∈ V ,andW ∩ V = ∅.Let

E = RW (x). Since p- lim Tgx = x, E ∈ p,andthusE is a D-set.

If E were piecewise syndetic, then RU¯ (x) would also be piecewise syndetic, so there would be some uniformly recurrent point y ∈ U¯, by Lemma 1.2.6. But then y ≠ x0, a contradiction. So E is not piecewise syndetic.

19 1.3 Some Examples

One may, naively, suspect that any set E ⊆ G which has positive upper Banach density and contains a ﬁnite sums set is a D-set. Our ﬁrst example shows that this is not the case.

n ∞ Example 1.3.1. Let E =(2N +1) FS((10 )n=1).ClearlyE has positive upper Banach density (in fact, E has natural$ density 1/2)andE contains the ﬁnite sums

n ∞ n ∞ set FS((10 )n=1).IfE were a D-set, then either 2N +1 or FS((10 )n=1) would also be a D-set. But the former is not an IP set (hence is not a D-set) and the latter has upper Banach density 0 (hence is not a D-set).

In Section 1.2 we showed, using dynamical methods, that any abelian group has a subset E which is a D-set but which is not piecewise syndetic (hence is not central). The following examples serve to illustrate some of the subtle diﬃculties which arise when one tries to give a more constructive proof of this result.

As noted in the introduction, the ﬁrst construction of a non-central D-set E ⊆ N

∞ was given by Adams in [1]. Adams constructed a sequence (xn)1 ⊆ N so that:

∞ i) For all m ∈ N,thesetEm =FS((xn)m )) has positive upper Banach density.

ii) The set E = E1 is not piecewise syndetic. Then it follows that E is a noncentral D-set6 Of course, the construction of such a sequence and the veriﬁcation that conditions i) and ii) are satisﬁed are no small

∞ tasks. To facilitate these tasks, Adams arranged for the sequence (xn)n=1 to have certain properties, among them the property of having unique ﬁnite sums: That is, if F, H ∈Pfin and xn = xn,thenF = H. n F n H "∈ "∈

6Condition ii) guarantees that E is not central. Condition i) guarantees that the semigroup S = ∞

Em intersects nontrivially with β(L). Then the existence of an essential idempotent with m=1 E%∈ p is an application of Ellis’ theorem (Theorem 2.1.1) to S ∩ β(L). 20 Deﬁnition 1.3.2 ([2], Deﬁnition 2.5). Let G be an abelian group. Then a sequence

∞ ∞ (xn)n=1 ⊆ G is called nice if (xn)n=1 has unique ﬁnite sums and is such that for every

∞ ∞ g ∈ G there is k ∈ N so that FS((xn)n=1) ∩ (g + F((xn)n=k)) ≠ ∅.

Nice sequences can actually be used to provide interesting examples in quite a few semigroups, and were studied rather extensively in [2]. So, if one wishes to prove that there are noncentral D-sets in an abelian group G, it seems most natural to try mimicking the construction of Adams. That is, one should look for a nicesequence

∞ (xn)n=1 ⊆ G which satisﬁes i) and ii) above. We now show that this method will not

∞ produce any noncentral D-set E ⊆⊕n=1Z/3Z. The following is a straightforward extension of [2, Theorem 3.12].

∞ ∞ ∞ Theorem 1.3.3. If (xn)n=1 ⊆⊕n=1Z/3Z has unique ﬁnite sums, then FS((xn)n=1) does not satisfy the conclusion of the Central Sets Theorem.

∞ Proof. By contradiction. Suppose that (xn)n=1 is a sequence which has unique ﬁnite

∞ sums such that E =FS((xn)n=1) satisﬁes the conclusion of the Central Sets Theorem.

∞ ∞ Let (yn)n=1 be any sequence in ⊕n=1Z/3Z. Applying the conclusion of the Central

∞ ∞ ∞ Sets Theorem to the three sequences (0)n=1,(yn)n=1,and(2yn)n=1, we conclude that there are nonzero a, d ∈ G such that {a, a + d, a +2d}⊆E.Fori ∈{0, 1, 2},choose

ﬁnite Hi ⊆ N so that a + id = xn, and let n H "∈ i 3

Ki = {n ∈ Hj : n is in exactly i of the sets H1,H2, and H3}. j=1 $ Then

0=a +(a + d)+(a +2d)=( xn)+( 2xn)+( 3xn) n K n K n K "∈ 1 "∈ 2 "∈ 3 =( xn) − ( xn). n∈K n∈K "1 "2

21 So, by uniqueness of ﬁnite sums, K1 = K2. However, one clearly has K1 ∩ K2 = ∅, hence K1 = K2 = ∅.ButthenH1 = H2 = H3 and a = a + d = a +2d. Since d =0,̸ this is a contradiction.

Example 1.3.4. Since any D-set satisﬁes the conclusion of the Central Sets Theorem,

∞ ∞ the above theorem shows that given any nice sequence (xn)n=1 ⊆⊕n=1Z/3Z,theset

∞ E = FS((xn)n=1) is not a D-set.

∞ The condition that the sequence (xn)n=1 be nice served only to ease the veriﬁcation of conditions i) and ii) above. If one managed to construct any sequence which

∞ satisﬁed i) and ii), then FS((xn)n=1) would be a noncentral D-set. Of course, this method produces only D-sets E having the additional property that there is some

∞ ﬁnite sum set FS((xn)n=1) ⊆ E of positive density. We now give examples showing that this additional assumption is in fact nontrivial.

∞ Example 1.3.5. There exists a central set E ⊆ Z with the property that if (xn)n=1

∞ ∗ ∞ is any sequence with FS((xn)n=1) ⊆ E then d (FS((xn)n=1)) = 0.

Sketch of proof. We outline an example which was attributed to Vitaly Bergelson in ∞ ∞ [35]. The idea is as follows: Let E = In,where(In)n=1 is a sequence of intervals n=1 with lengths |In| tending to inﬁnity.$ Then E is thick, and hence central. If the

In’s are chosen carefully enough, they will also have the property that for all n ∈ N,

∞ ∞ (In +In)∩E = ∅.Then,anysequence(xn)n=1 such that FS((xn)n=1) ⊆ E will consist

∗ ∞ of at most one element of In for each n ∈ N, and it follows that d (FS((xn)n=1)) = 0.

3n 3n+1 For example, one may choose In = [2 , 2 ). The remaining details are left as an exercise.

Example 1.3.5 shows that there are D-sets in Z which cannot be obtained from

∞ some sequence (xn)n=1 ⊆ Z satisfying i) and ii). Of course, this observation doesn’t preclude the possibility that, in any abelian group G, one may obtain some noncentral 22 D-set from the construction of such a sequence. Our next example shows that, in fact, there is an abelian group G in which every D-set E ⊆ G is as in Example 1.3.5.

∞ ∞ ∗ Example 1.3.6. If E = FS((gn)n=1) ⊆⊕n=1Z/2Z then either d (E)=0or E is syndetic.

∞ Proof. Let G = ⊕n=1Z/2Z. Since every element of G has order 2, any IP set E ⊆ G is either equal to a subgroup H of G or is of the form H \{0G} for some subgroup H ≤ G.ForasubgroupH ≤ G, H is syndetic if and only if H has ﬁnite index.

∞ If H has inﬁnite index, then there is an injective sequence (xn)n=1 ⊆ G so that the ∗ collection {gn + H : n ∈ N} is pairwise disjoint. This implies that d (H)=0.

Note the following corollary:

∞ Corollary 1.3.7. If E ⊆⊕n=1Z/2Z is a D-set which is not piecewise syndetic, and

∞ ∞ ∗ ∞ (gn)n=1 is a sequence in G such that FS((gn)n=1) ⊆ E,thend(FS((gn)n=1)) = 0.

1.4 Further Investigations

The above examples serve to illustrate the care one must take when dealing with

D-sets, and are interesting in their own right. They furthermore show that our dynamical approach is not merely a matter of style, but rather that it overcomes certain obstructions one encounters when trying to adapt the constructive technique used in [1] to the more general setting of countable abelian groups. Unfortunately, however, the argument presented in Section 1.2 cannot be extended to amenable groups.

As mentioned in Section 1.3, Theorem 1.2.5 was proven by constructingaset

E ⊆ G which has positive upper Banach density, but no shift of which is an IP set.

The ﬁrst example of such a set E ⊆ N was given by Ernst Straus in answer to a

23 question asked by Erd˝os. We therefore call any such set a Straus example7. It is natural to ask if a Straus example exists for any amenable group. Alas, the answer is “No”. In Chapter 4 we obtain a full characterization of those amenable groups for which a Straus example exists. This is exactly the class of amenable groups in which an extension of the techniques of Section 1.2 will produce a non-piecewise syndetic

D-set.

The reader may have noticed the formal similarity between Lemma 1.2.6 and Lemma 1.2.7. In fact, they are both examples of the phenomenon known as “forcing

F-recurrence”, for a family F of subsets of G. This notion was introduced in [22] and was also studied in [47]. In Chapter 5 we establish a connection between this notion and certain algebraic properties of the Stone-Cechˇ compactiﬁcation βG of G.

Our techniques allow us to work in the very general setting of a discrete semigroup

S, and we recover many results of [22] and [47] as special cases. In Chapter 6 we supply, using the language developed in chapter 5, a generalization to countable amenable groups of Theorem 1.1.10. We then give a proof (which, unlike the above proof of Theorem 1.2.5, is valid for all amenable groups) of the existence of a set E ⊆ G which has positive upper Banach density but is not piecewise syndetic. This will, finally, establish the existence of noncentral D-sets in any countable amenable group. In Chapter 7 we use the results of Chapters 5 and 6 to characterize certain properties of a topological dynamical system via ultrafilters. We then illustrate the usefulness of this approach by providing a unified treatment of diverse topics in dynamics.

7See Chapter 4 for further discussion. 24 CHAPTER 2

DEFINITIONS AND BACKGROUND

In this chapter we establish notation and record the basic facts concerning those notions which will be the main objects of study in Chapters 3-7. Speciﬁcally, we review

1) the semigroup structure on the Stone-Cechˇ compactiﬁcation βS of a semigroup

S, 2) topological dynamical systems (X, S), where X is a compact metric space and S is a countable semigroup, 3) the notion of amenability for discrete groups G,and

4) measurable dynamical systems (X, B,µ,(Tg)g∈G), where (X, B,µ) is a probability space and G is a countable amenable group.

2.1 Algebra in the Stone-Cechˇ Compactiﬁcation

We record here the basic theory of algebra in the Stone-Cechˇ compactiﬁcation βS of a discrete semigroup S, as developed in [38]. Although we will later limit our discussion to countable semigroups, the results of this section hold for any discrete semigroup. The semigroup (βS,·) will play a central role in the remainder of the thesis. We realize βS as the set of all ultraﬁlters1 on S endowed with the topology determined by the base

B = {A¯ : A ⊆ S},

1 Recall that an ultrafilter on a set S is a collection q ⊆P(S) such that i) ∅ ∈/ q, ii) for any finite H ⊆ q one has H ∈ q, and iii) for all E ⊆ S either E ∈ q or (S \ E) ∈ q.Aprincipal ultrafilter on S is an ultrafilter% of the form q = {E ⊆ S : s ∈ E} for some s ∈ S. 25 where for A ⊆ S we have by definition A¯ = {p ∈ βS : A ∈ p}.TheneachA¯ ⊆ βS is both open and closed. Identifying s ∈ S with the principal ultrafilter containing the set {s}, one easily sees that the set A¯ is the topological closure of the set A ⊆ βS.

With this topology, βS is a totally disconnected compact space, and the principal ultraﬁlters are precisely the isolated points. If p is an ultraﬁlter on S, X is a compact

Hausdorﬀspace, and f : S → X is any map, deﬁne the limit of f along p by:

p- lim f(s)= f(E). (2.1.1) s E∈p % One easily checks that Equation (2.1.1) defines a unique point y ∈ X, which is also characterized by the condition that f −1(U) ∈ p for every neighborhood U of y. Given amapf : S → X, with X acompactHausdorffspace,themapf¯ : βS → X defined by f¯(p)=p- lim f(s)extendsf uniquely to a continuous map f¯ on βS, showing that s βS actually satisfies the defining property of the Stone-Cechˇ compactification (and justifying our terminology and notation).

We will consider a function f : S → X to be a “sequence in X, indexed by S”, and we accordingly write (xs)s∈S. then Equation 2.1.1 translates to:

p- lim xs = {xs : s ∈ E}. s E p %∈ For E ⊆ S and x ∈ S, let x−1E = {y ∈ S : xy ∈ E} and let Ex−1 = {y ∈ S : yx ∈ E}. There are two diﬀerent ways that the operation on a discrete semigroup S can be extended to an operation on βS:

p · q = {E ⊆ S : {x ∈ S : x−1E ∈ q}∈p} (2.1.2) or

p ◦ q = {E ⊆ G : {x ∈ G : Ex−1 ∈ p}∈q}. (2.1.3)

For each p ∈ βG,themapq 0→ p ◦ q and the map q 0→ q · p are continuous. The maps q 0→ q ◦ p and q 0→ p · q are continuous if q is principal ([38, Theorem 4.1]). We 26 will have reason to consider both the right-topological semigroup (βS,·) and the left- topological semigroup (βS,◦). Perhaps the most important aspect of the operations

· and ◦ is expressed by the following formulas: For a sequence (xs)s∈S in a compact Hausdorﬀspace X and for any p, q ∈ βS,

p · q- lim xr = p- lim(q- lim xst) (2.1.4) r s t and

p ◦ q- lim xr = q- lim(p- lim xts). (2.1.5) r s t

As we will see in Section 2.2, Equations (2.1.4) and (2.1.5) are at the heart of the connection between the semigroup structures on βS and the topological dynamical systems with acting semigroup S.

A left (right) ideal of a semigroup S is a nonempty set I ⊆ S with the property that SI ⊆ I (respectively IS ⊆ I). An ideal of S is a set which is both a left and a right ideal. A left(right) ideal I is minimal if no proper subset of I is a left(right) ideal. By [38, Theorem 1.51], any compact right or left topological semigroup T has a unique smallest ideal K(T ), and in either case one has the equality

K(T )={L ⊆ T : L is a minimal left ideal} $ = {R ⊆ T : R is a minimal right ideal}. $ For a discrete semigroup S, a minimal left ideal in (βS,·) is closed and a minimal right ideal in (βS,◦) is closed ([38, Corollary 2.6]). As will become clear, the idempotents in (βS,·)and(βS,◦) have importance for both dynamics and combinatorics. So the following famous theorem of Ellis will be used throughout our work.

Theorem 2.1.1 (Ellis’ Theorem). If (T,·) is any compact semitopological semigroup, then there is some idempotent t = t · t ∈ T .

27 For a compact semitopological semigroup (T,·), we denote the set of all idempotents in T by I(T,·)2. The following two theorems establish connections between the algebraic properties of an ultraﬁlter p ∈ βS and the combinatorial properties enjoyed by the sets E ∈ p. The establishing of such connections will become an important theme in the following chapters.

∞ Deﬁnition 2.1.2. Let Pfin = {α ⊆ N : α is ﬁnite}.Fixasequence(xn)n=1 ⊆ S.For

∞ {k1

∞ in increasing order of indices and let Dα((xn)n=1)=gkn gkn−1 ...gk1 be the product taken in decreasing order of indices. Then

∞ ∞ FPi((xn)n=1)={Iα((xn)n=1):α ∈Pfin}

∞ is the increasing ﬁnite products set generated by (xn)n=1 and

∞ ∞ FPd((xn)n=1)={Dα((xn)n=1):α ∈Pfin}

∞ is the decreasing ﬁnite products set generated by (xn)n=1.AsetE ⊆ S is called an

∞ increasing (decreasing) IP set if there is some sequence (xn)n=1 of distinct elements of S so that E contains the increasing (decreasing) ﬁnite products set generated by

∞ ∞ (xn)n=1.Ifthesequence(xn)n=1 is clear from context, we will write only Iα and Dα

∞ ∞ rather than Iα((xn)n=1) and Dα((xn)n=1).Wedenotethecollectionofallincreasing (decreasing) IP sets in S by IPl(S) (respectively IPr(S))3.IfthesemigroupS is clear from context, we will simply write IPl and IPr.IfS is commutative, then

2In [38], the collection of all idempotents in a semigroup T is denoted by E(T ). We will use E(X, S) to denote the enveloping semigroup of a topological dynamical system (X, S) (see Section 2.2) and we often use E to denote an arbitrary subset of the semigroup S. To avoid confusion, therefore, we use the notation I(T ) for the set of all idempotent elements of the semigroup T.

3The subscripts l and r stand for “left” and “right”. This notation reﬂects the fact that the increasing IP sets arise when considering left translations in the semigroup S, and the decreasing IP sets arise when considering right translations in S. This notation also ﬁts our conventions regarding other classes of subsets of S which have both a “right version” and a “left version”. 28 IPl = IPr.Inthiscasewereferonlyto“IPsets”andthecollectionofall IP sets will be denoted by IP.

Theorem 2.1.3 ([38, Theorem 5.12]). Let S be a semigroup, let E ⊆ S.Then

I((βS,·)) ∩ E¯ ≠ ∅ if and only if E ∈IPl and I((βS,◦)) ∩ E¯ ≠ ∅ if and only if

E ∈IPr.

We remark that Theorem 2.1.3 gives Hindman’s theorem4 on ﬁnite sums sets as an immediate corollary. Galvin and Glazer established (one implication of)Theorem

2.1.3, and thereby gave a greatly simpliﬁed proof of Hindman’s theorem. See the notes to Chapter 5 of [38] for further discussion.

We now turn our attention to a combinatorial characterization of the elements of ultraﬁlters in the minimal ideals K((βS,·)) and K((βS,◦)).

Deﬁnition 2.1.4. AsetE ⊆ S is left (right) syndetic if there is some ﬁnite H ⊆ G so that G = t−1A (G = At−1). We denote the class of all left (right) syndetic t∈H t∈H $l r $ sets in S by S (S) (S (S)). If the semigroup S is clear from context, then we only write Sl and Sr.IfS is commutative, then Sl = Sr,andwewillonlywriteS for the collection of all syndetic sets.

AsetE ⊆ S is right (left) thick if for every ﬁnite H ⊆ S there is x ∈ S such that

Fx ⊆ E (respectively, xF ⊆ E). The class of all right (left) thick sets in S will be denoted by T r(S) (respectively, T l(S)). Again, if the semigroup S is clear, we only write T l and T r,andifS is commutative then T l = T r and this set is denoted by T .

r 4 Hindman’s Theorem states: If N = Ci is a ﬁnite coloring of the natural numbers, then there is i=1 ∞ $ ∞ some increasing sequence (xn)n=1 of natural numbers and some 1 ≤ i ≤ r such that FS((xn)n=1) ⊆ Ci.

29 A set is A ⊆ S is left syndetic if and only if it intersects every right thick set B ⊆ S nontrivially ([19, Theorem 2.4]). Of course, the analogous statement for right syndetic (and left thick) sets also holds.

Deﬁnition 2.1.5. AsetE ⊆ S is left (right) piecewise syndetic if there is some

ﬁnite H ⊆ S so that the set t−1E is right thick (respecively, there is some H ⊆ S t H $∈ so that the set Et−1 is left thick). Equivalently, E ⊆ S is left piecewise syndetic t∈H if there is some$ ﬁnite set H ⊆ S so that the collection

{( t−1E)x−1 : x ∈ S} t H $∈ has the ﬁnite intersection property5,andtheanalogousstatementholdsalsoforright piecewise syndeticity.

The collection of all left (right) piecewise syndetic sets in S will be denoted by

PSl(S) (by PSr(S)). If S is clear from context, we only write PSl and PSr.IfS is commutative, then PSr = PSl,andwedenotethissetbyPS.

AsetE ⊆ S is left (right) piecewise syndetic if and only if there is some left

(right) syndetic set A and some right (left) thick set B such that E = A ∩ B ([19], Theorem 2.4). The following theorem relates the classes PSl and PSr to the minimal ideals K((βS,·)) and K((βS,◦)).

Theorem 2.1.6 ([38, Theorem 4.40]). AsetE ⊆ S is left (right) piecewise syndetic if and only if K((βS,·)) ∩ E¯ ≠ ∅ (respectively, K((βS,◦)) ∩ E¯ ≠ ∅).

We conclude this section with another characterization of piecewise syndetic sets.

This characterization will be used in Chapter 3 and will serve as a motivating example for Deﬁnition 5.2.1.

5A collection C of sets has the ﬁnite intersection property if for every ﬁnite H ⊆Cone has H ̸= ∅. % 30 Theorem 2.1.7. AsetE ⊆ S is left (right) piecewise syndetic if and only if there is aleft(right)syndeticsetA ⊆ S such that the collection {a−1E : a ∈ A} (respectively,

{Ea−1 : a ∈ A})hastheﬁniteintersectionproperty.

Proof. We prove the statement for left piecewise syndetic sets, the proof for right piecewise syndetic sets is obtained by making routine modiﬁcations.

So let E ⊆ S be left piecewise syndetic. Let q ∈ K((βS,·)) be such that E ∈ q

(some such q exists by Theorem 2.1.6). By Theorem 4.39 of [38], the set A = {x ∈ S : x−1E ∈ q} is left syndetic.

If H ⊆ A is ﬁnite, then the collection {x−1E : x ∈ H} is contained in q,so

∅̸= {x−1E : x ∈ H}. Since H is an arbitrary ﬁnite subseteq of A, the collection

{a−1%E : a ∈ A} has the ﬁnite intersection property.

Conversely, if E ⊆ S and there is some left syndetic set A ⊆ S such that the collection B = {a−1E : a ∈ A} has the ﬁnite intersection property, then there is some q ∈ βS so that {a−1E : a ∈ A}⊆q (by Zorn’s lemma). Since A is left piecewise syndetic, there is some p ∈ K((βS,·)) with E ∈ p by Theorem 2.1.6.

So A ⊆{x ∈ S : x−1E ∈ q},and{x ∈ S : x−1E ∈ q}∈p.SoE ∈ p · q. Since

K((βS,·)) is an ideal, p · q ∈ K((βS,·)). Thus E ∈PSl by Theorem 2.1.6.

2.2 Topological Dynamics

Let S be a semigroup. A left (right) topological dynamical system, written (X, S), is a compact metric space X together with an action (anti-action) of S on X by homeomorphisms. Explicitly, for every s ∈ S there is a continuous map Ts : X → X, and for all s, t ∈ S one has Tst = TsTt (Tst = TtTs). Given a topological dynamical system one can form the product system (X × X, S) by setting Ts(x, y)=(Tsx, Tsy).

31 For x ∈ X, the orbit closure of x is given by O¯(x)={Tsx : s ∈ S}. We will also ﬁnd the following notation useful. For x ∈ X and A ⊆ X,put

RA(x)={s ∈ S : Tsx ∈ A}.

Recurrence is an important theme in topological dynamics. A point x ∈ (X, G) is a recurrent point if for every neighborhood U of x there is nonidentity s ∈ S such that s ∈ RU (x). The point x is uniformly recurrent if for every neighborhood U of

l r x we have RU (x) ∈S (for right actions, we require RU (x) ∈S). The following propositions are fundamental results in topological dynamics, and they will be used freely.

Proposition 2.2.1. Let (X, S) be a left (right) topological system. The point x ∈ X

l is recurrent if and only if for every neighborhood U of x we have RU (x) ∈IP

r (RU (x) ∈IP ).

Proposition 2.2.2. Let (X, S) be a left (right) topological system. The point x ∈ X is uniformly recurrent if and only if the orbit-closure O¯(x) of x is a minimal system.

We will study recurrence in much more detail in Chapter 5. For now we review the enveloping semigroup of a system, and its relation to βS.

Deﬁnition 2.2.3. Let (X, S) be a left (right) topological dynamical system. Give XX the product topology. Let

E(X, S)={Ts : s ∈ S}.

For e, f ∈ E = E(X, S) and x ∈ X,let(ef)(x)=e(f(x)) (respectively (ef)(x)= f(e(x))). Then E is a compact semigroup. For e ∈ E,deﬁneλe : E → E and

ρe : E → E by λe(f)=ef and ρe(f)=fe for all f ∈ E.Thenforalle ∈ E,

ρe is continuous (respectively λe is continuous), so E is a compact right-topological semigroup (respectively E is a compact left-topological semigroup). 32 For s ∈ S and e ∈ E, Ts ∈ E and λTs (ρTs )iscontinuous.SettingTs(e)=λTs (e)

(setting Ts(e)=ρTs (e))makes(E,S) aleft(right)topologicaldynamicalsystem. For x ∈ X,themapπ : E → O¯(x) given by π(e)=e(x) is a surjective homomorphism of left (right) topological dynamical systems.

One of the most powerful techniques for understanding a topological dynamical system (X, S) is to study the S-invariant Borel measures on X. We will investigate this topic more in Section 2.4, after establishing some basic facts about measure- preserving dynamics and Følner sequences.

2.3 Amenability and Density

As discussed in Section 1.1.2, the notion of natural density of a set E ⊆ N can be generalized to a very large class of semigroups. If the semigroup S is not commutative, then there are two natural ways (a left version and a right version) to deﬁne the density of a set E ⊆ S. A semigroup S such that the notion of left (right) density of a set

E ⊆ S can be deﬁned is called left (right) amenable. Unfortunately, the situation becomes rather delicate in such generality. To avoid unnecessary technicalities we restrict our attention to countable groups G whenever discussing notions related to density. This setting is quite adequate for any applications we have in mind.

While some of our discussion can be readily extended to the more general setting of semigroups, the reader is warned that some fundamental results on countable amenable groups are no longer valid if one replaces the word “group” by the word “semigroup”.

Deﬁnition 2.3.1. Let G be a countable group. Let B(G) denote the Banach space of all bounded functions f : G → C.Amean on B(G) is a positive linear functional l : B(G) → C such that l(1) = 1.Ameanl is left (right) invariant if for every

33 f ∈ B(G) and every g ∈ G one has l(f)=l(xf) (respectively, l(f)=l(fx)), where xf(y)=f(xy) and fx(y)=f(yx) for all y ∈ G.

Deﬁnition 2.3.2. Let S be a countable semigroup. A left (right) Følner sequence in

S is a sequence F =(Fn)n∈N of ﬁnite subsets of S with the property that for all s ∈ S |sF ∩ F | |F s ∩ F | one has lim n n =1(respectively, one has lim n n =1). n→∞ |Fn| n→∞ |Fn|

Any Følner sequence F =(Fn)n∈N can be used to deﬁne the density of a set E ⊆ S in a way analogous to the deﬁnition given in the Introduction of natural density.

The translation-invariance of the natural density of a set E ⊆ N (that is, the fact that if d(E) exists then for every n ∈ N one has d(−n + E)=d(n + E)= d(E)) is an all-important aspect of the notion. Indeed, the very deﬁnition of Følner sequence is intended to guarantee this property. However, if the semigroup S is not left cancellative, then it is possible that for a left Følner sequence F =(Fn)n∈N in S, asetE ⊆ S,andsomes ∈ S,onehasdF (sE) =d̸ F (E). Furthermore, Theorem 2.3.3 below fails for noncancellative countable semigroups.

Theorem 2.3.3. Let G be a countable group. Then the following are equivalent:

i) There is a left invariant mean l on B(G).

ii) There is a right invariant mean l on B(G).

iii) There is a left Følner sequence F =(Fn)n∈N in G.

iv) There is a right Følner sequence F =(Fn)n∈N in G.

Deﬁnition 2.3.4. AcountablegroupG is amenable if G satisﬁes any (hence all) of the conditions of Theorem 2.3.3.

Thus, the countable amenable groups are precisely those for which the notion of positive upper Banach density of a set E ⊆ G can be properly deﬁned.

34 Deﬁnition 2.3.5. Let G be a countable group. For any (left or right) Følner sequence

F =(Fn)n∈N,anysetE ⊆ G,andanyn ∈ N deﬁne

|E ∩ Fn| An(E,F) := |Fn| and deﬁne the upper density of E with respect to F as

d¯F (E) := lim sup An(E,F), n→∞ the lower density of E with respect to F as

dF (E) := lim inf An(E,F), n→∞ and, if the limit exists, we deﬁne the density of E with respect to F as

dF (E) := lim An(E,F). n→∞

Now, deﬁne the left upper Banach density of a set E ⊆ G by:

∗ ¯ dL(E)=sup{dF (E):(Fn)n∈N is a left Følner sequence in G}

∗ and deﬁne the right upper Banach density dR(E) similarly. The class of all sets having positive left (right) upper Banach density will be denoted by Ll (respectively

Lr). If G is abelian, then Ll = Lr,andwewillwriteonlyL.

If E ⊆ G is such that there is some α ∈ [0, 1] such that for every left (right)

Følner sequence F =(Fn)n∈N one has dF (E)=α,thenα is the left (right) Banach density of E.

2.4 Measure Preserving Dynamics and Ergodic Theory

Let G be a countable group. If (X, B,µ) is a separable probability space, a map

T : X → X is measure preserving if for all A ∈Bone has µ(T −1A)=µ(A). If

35 (Tg)g∈G is a collection of measure preserving maps Tg : X → X such that for all g,h ∈ G one has Tg ◦ Th = Tgh then X¯ =(X, B,µ,(Tg)g∈G) is called a measure preserving system6. The following theorem, known as the von Neumann Ergodic

Theorem, will be used freely in the sequel.

Theorem 2.4.1 (Von Neumann ergodic theorem). Let G be an amenable group, let

((Ug)g∈G, H) be a unitary anti-representation of G on H (that is, each Ug : H→H is unitary, and Ug ◦ Uh = Uhg). Let (Fn)n∈N be a left Følner sequence in G.Let

I = {f ∈H: ∀g∈GUgf = f}.Forn ∈ N,deﬁne: 1 Pnf = Ugf |Φn| g∈Φ "n and let P : H→Hbe orthogonal projection onto the closed subspace I.Then limn→∞Pnf = Pf (limit taken in H).

Proof. Clearly I is a closed subspace of H.LetM = {Ugf − f : f ∈H,g ∈ G},and let W = span(M). We now show that H = I ⊕ W .

First, note that for f = Ugh − h ∈ M and k ∈ I,wehave:

⟨f,k⟩ = ⟨Ugh, k⟩ - ⟨h, k⟩ = ⟨Ugh, Ugk⟩−⟨h, k⟩ =0

So I ⊆ M ⊥ = W ⊥. It only remains to show that W ⊥ ⊆ I. So let h ∈ W ⊥, let g ∈ G, and let f ∈H.Wehave:

⟨h, f⟩ - ⟨Ugh, f⟩ = ⟨h, f⟩ - ⟨h, Ug−1 f⟩

= ⟨h, f − Ug−1 f⟩

=0.

6Technically, what we have deﬁned is a left measure preserving system, and we should give an analogous deﬁnition for right measure preserving systems. However, unlike the situation with topological dynamical systems, we will not need to concern ourselves with right measure preserving systems. 36 So h = Ugh. Since g ∈ G is arbitrary, h ∈ I.WehavethusshownthatH = I ⊕ W .

Now, deﬁne Pn as above. It is clear that for all i ∈ I and all n ∈ N we have

Pni = i. Also, each Pn is clearly linear, bounded. Hence it only remains to show that

lim Pnf = 0 for all f ∈ W . It suﬃces to show that lim Pnf = 0 for all f ∈ M: it n→∞ n→∞ then follows for all f ∈ W by linearity and a standard approximation argument.

So let f = Ug0 h − h ∈ M.Wehave:

1 ∥ lim Pnf∥ = lim Ugf n→∞ (n→∞ |Φn| ( ( g∈Φn ( ( " ( ( 1 ( = ( lim Ug((Ug0 h) − Ugh (n→∞ |Φn| ! #( ( g∈Φn g∈Φn ( ( " " ( ( 1 ( = ( lim Ug0gh − Ugh ( (n→∞ |Φn| ! #( ( g∈Φn g∈Φn ( ( " " ( ( ( ( 1 ( = lim Ugf − Ugh (n→∞ |Φn| ⎛ ⎞( ( g∈(g0Φn\Φn) g∈(Φn\g0Φn) ( ( " " ( ( ⎝ ⎠( ( |Φn△g0Φn| ( ≤∥(h∥·lim sup ( n→∞ |Φn| =0

Hence lim Pnf = 0, and the proof is complete. n→∞

There is a rich interplay between measure preserving dynamics and topological dynamics. For example, a topological dynamical system is sometimes best understood by looking for invariant measures.

Definition 2.4.2. Let X be a compact left (right) topological space. Let M(X) denote the set of all Borel probability measures on X.ThenM(X) is identified, via the Reisz representation theorem, with a weak∗ compact and convex subset of C(X)∗.If(X, G) is a topological dynamical system, g ∈ G,andµ ∈ M(X),defineTgµ ∈ M(X) by the 37 formula fd(Tgµ)= f ◦ Tgdµ for all f ∈ C(X).Thismakes(M(X),G) into a ' ' compact left (right) topological dynamical system.

−1 If µ ∈ M(X) satisﬁes µ(Tg A)=µ(A) for all g ∈ G and all Borel sets A ⊆ X,thenµ is an invariant measure,andthesetofallsuchmeasuresisdenotedby

M(X, G).

M(X, G) is a closed convex subset of M(X), hence M(X, G) is weak∗ compact.

Deﬁnition 2.4.3. AmeasurepreservingsystemX¯ =(X, B,µ,(Tg)g∈G) is ergodic if

−1 every A ∈Bwith µ(Tg A△A)=0for all g ∈ G satisﬁes µ(A) ∈{0, 1}.

Thus, X¯ is ergodic if every invariant set has either full measure or measure zero.

If (X, G) is a topological dynamical system, then a measure µ ∈ M(X, G) is ergodic if the measure preserving system X¯ =(X, B,µ,(Tg)g∈G) is ergodic (where here B denotes the class of all Borel subsets of X). The ergodic measures are precisely the extreme points of the (compact and convex) set M(X, G) (see, e.g., [29, Proposition

3.4]). Thus, the following may be seen as an application of Choquet’s theorem (see, e.g., [49]).

Theorem 2.4.4 (The Ergodic Decomposition). Let (X, G) be any left toopological dynamical system, and let µ ∈ M(X, G).Thenthereisameasureν on M(X, G) which is supported on the set of ergodic measures so that for every f ∈ C(X) one has

fdµ = fdm dν(m). ' ' -' . The following theorem, which is of fundamental importance, guarantees the existence of invariant measures on a topological dynamical system (X, G), in the event that G is amenable.

38 Theorem 2.4.5 (Krylov-Bugolioboﬀtheorem, cf [43]). Let G be an amenable group, let (X, G) be a t.d.s., and let (Fn)n∈N be a left Følner sequence in G.Letν ∈ M(X), and for n ∈ N deﬁne

1 1 νn = Tgν = ν ◦ Tg. |Fn| |Fn| g F g F "∈ n "∈ n ∗ ∞ and let µ be any weak -limitof(νn)n=1.Thenµ ∈ M(X, G).

∞ Proof. Let (ni)i=1 be a sequence of natural numbers such that for all f ∈ C(X),

fdµ = lim fdνni . Fix f ∈ C(X), g0 ∈ G.Wehave: i→∞ ' '

fdµ− f ◦ Tg0 dµ /' ' / / / / / / 1 / = lim f ◦ Tgdν − (f ◦ Tgo ) ◦ Tgdν i→∞ / |Φni | ⎛ ⎞/ / g∈Φni ' ' / / " / / ⎝ ⎠/ / 1 / = / lim f ◦ Tgdµ − f/◦ Tgdµ i→∞ / |Φni | ⎛ ⎞/ / g∈Φni \g0Φni ' g∈g0Φni \Φni ' / / " " / / ⎝ |Φ △g Φ | ⎠/ / ni o ni / ≤∥/ f∥1 lim sup / i→∞ |Φni | =0

Thus for countable amenable groups G, every topological dynamical system (X, G) has at least one invariant measure. Those systems (X, G) for which there are exactly one such measure are called uniquely ergodic.Let(X, G) be any topological dynamical system, with G amenable. Let ν ∈ M(X), and let (Fn)n∈N be any left Følner sequence in G.Forn ∈ N, let 1 νn = Tnν. |Fn| g∈F "n

39 Let p ∈ βN be any nonprincipal ultraﬁlter on N, and let µ = p- lim νn ∈ M(X). n One readily checks that µ ∈ M(X, G).

If (X, G) is uniquely ergodic with unique invariant measure m, it follows that p- lim νn = m for every nonprincipal ultraﬁlter p ∈ βN, and hence lim νn = m n n→∞ weakly. Applying this to the point-masses δx, x ∈ X, we conclude: For every Følner sequence (Fn)n∈N,everyx ∈ X,andeveryf ∈ C(X), one has 1 lim f(Tgx)= fdm. n→∞ |Fn| g F "∈ n ' Conversely, for every ergodic ν ∈ M(X, G) and every Følner sequence (Fn)n∈N

∞ 7 there exists some subsequence (Fni )i=1 and some x ∈ X such that for all f ∈ C(X) one has 1 lim f(Tgx)= fdν, i→∞ |Fn i| g∈F "ni ' So if there is some function Φ: C(X) → C such that for every Følner sequence

(Fn)n∈N and every x ∈ X one has 1 lim f(Tgx)=Φ(f), n→∞ |Fn| g∈F "n then (X, G) is uniquely ergodic, with unique invariant measure m given by fdm = ' Φ(f). To summarize, we have proven:

Proposition 2.4.6. The system (X, G) is uniquely ergodic with unique invariant measure m if and only if for every Følner sequence (Fn)n∈N in G,everyx ∈ X,and every f ∈ C(X),onehas

1 lim f(Tgx)= fdm. n→∞ |Fn| g∈F "n '

7 ∞ Such points are called “generic” for ν and the Følner sequence (Fni )i=1. The existence of such ∞ x and (Fni )i=1 can be derived from the von Neumann ergodic theorem, using the fact that 2 convergence in L (µ) implies almost-everywhere convergence along some subsequence along with a diagonal argument. This argument was given for abelian G as part of the proof of Lemma 1.2.7. 40 The notions discussed in this Chapter are all active and rich topics in mathematics, and our treatment is by no means complete. However, the facts outlined above do provide suﬃcient background for understanding the remainder of this thesis.

41 CHAPTER 3

βS AND TRANSITIVE SUBSHIFTS

In this chapter we establish a lemma that will be used often in the sequel. It is the foundation for an in-depth analysis of the relationship between the semigroup

(βS,·) and the topological dynamical system commonly referred to as “the shift space” or “the full shift”. We then use this relationship to provide, for a countable amenable group G, a metrizable rendering of a proof due to Beiglb¨ock which utilized the nonmetrizable system (βG,G) of the so-called “sumsets phenomenon”.

3.1 Shift Space

Let S be a countable semigroup. If S has an identity element e then let Se = S.

Otherwise let Se = S ∪{e} be the result of formally adding an identity e to S.

Se 1 Let Xs = {0, 1} , endowed with the product topology .ThenXs is compact by

Tychonoﬀ’s Theorem, and Xs is metrizable because Se is countable. An element of

Xs is a function x : Se →{0, 1}.Fors ∈ S,deﬁneTs : Xs → Xs by:

Tsx(t)=x(ts)

for all t ∈ Se.Then(Ts)s∈S is a left action of S on Xs by continuous maps, and

1The “s” in the subscript is intended to stand for “shift”. The semigroup S will always be clear from context.

42 (Xs,S) is a compact metrizable topological dynamical system. This system will arise naturally throughout our work, and we will refer to it as “the shift space”.

−1 For x ∈ Xs, let Ex = x (1) ⊆ Se.Thenx =1Ex is the characteristic function of

Ex, and in this way we identify Xs with the power set P(Se). Note that for E ⊆ Se and s ∈ S,

Ts(1E)=1Es−1 .

One now asks the following question: Given a set E ⊆ Se,whatcanonesayabout those sets B ⊆ Se with the property that 1B ∈ O¯(1E)? This question is answered rather neatly by our next lemma.

Deﬁnition 3.1.1. For E ⊆ Se and p ∈ βS,let

−1 −1 p E = {x ∈ Se : x E ∈ p}.

The reader undoubtedly recognizes the righthand side of the above line from the definition of the semigroup operation on (βS,·). This set appears prominently throughout the work [38], where it is denoted by A⋆(p) ([38, Definition 4.13]). The notation p−1A was introduced by Beiglböck in [7].

Lemma 3.1.2. For all p ∈ βS and all E ⊆ S,onehas

p- lim Ts(1E)=1p−1E. s

−1 Proof. Let y = p- lim Ts(1E), and let B = y (1). Fix t ∈ S. s If t ∈ B,thenV = {z ∈ X : z(h)=1} is a neighborhood of y,so

−1 t E = {s ∈ S : ts ∈ E} = {s ∈ S :1E(ts)=1}

= {s ∈ S : Ts(1E) ∈ V }∈p.

43 On the other hand, if t/∈ B,thenW = {z ∈ X : z(h)=0} is a neighborhood of y,and

−1 S \ (t E)={s ∈ S : ts∈ / E} = {s ∈ S :1E(ts)=0}

= {s ∈ S : Ts(1E) ∈ W }∈p, so t−1E/∈ p.

Thus, given t ∈ S, t ∈ B if and only if t−1E ∈ p.SoB = p−1E.

Lemma 3.1.2 is a straightforward extension of an argument due to B. Weiss which

ﬁrst appeared in the proof of Theorem 6.11 of [17]. There it is only shown that if

E ∈ p and y = p- lim Tg(1E), then y(e) = 1 (This suﬃced to prove Theorem 6.11 of g [17]). That same argument appears in the proofs of [14] Theorems 2.1 and 2.8, [47] Theorem 4.11, and many other places in the literature. So Lemma 3.1.2 is an easy extension of an argument which has been known for over twenty years.

Nevertheless, Lemma 3.1.2 is a powerful tool. It will be used often in Chapter 5.

To illustrate the usefulness of Lemma 3.1.2, note the following consequence: If p ≠ q ∈

βS, then there is some E ⊆ S such that E ∈ p and S \E ∈ q.Thenp- lim Ts(1E)(e)= s

1andq- lim Ts(1E)(e) = 0. Thus, the canonical map π : βS → E(Y,S) is injective. s Since π is also surjective and continuous, it follows from compactness that π is a homeomorphism. We have proven the following proposition.

Proposition 3.1.3. The semigroup (βS,·) is homeomorphically isomorphic to the enveloping semigroup E(Xs,S).

Proposition 3.1.3 is well-known. Our proof, via Lemma 3.1.2, seems to be new. Note furthermore that Lemma 3.1.2 is actually a strengthening of Proposition 3.1.3, in that it gives the map π : βS → E(Xs,S) explicitly.

As further evidence of Lemma 3.1.2’s value, let E ⊆ S, let x =1E ∈ Xs, let X = O¯(x), and let U = {z ∈ X : z(e)=1}.Letπ : βS → X be the canonical map 44 π(p)=p- lim Tsx.ThenE = RU (x), so for p ∈ βS one has π(p) ∈ U if and only if s E ∈ p.Inotherwords,π−1(U)=E¯ ⊆ βS.

Consider now a countable amenable group G, and let µ be an invariant measure on βG.Thenthepushforwardmeasureπ∗(µ)deﬁnedbyπ∗(µ)(A)=µ(π−1(A)) for all Borel sets A ⊆ X is an invariant measure on X,andπ∗(µ)(U)=µ(E¯). An immediate consequence is the following.

Proposition 3.1.4. Let G be a countable amenable group, let E ⊆ G.Thenthe following are equivalent:

i) E ∈Ll(G) (Recall that Ll(G) is deﬁned in Chapter 2, Section 3).

ii) There is an invariant measure µ on (βG,·) with µ(E¯) > 0.

iii) There is an invariant measure µ on (Xs,G) which is supported on X = O¯(1E) and such that, setting U = {x ∈ X : x(e)=1},onehasµ(U) > 0.

Thus, Lemma 3.1.2 establishes a dictionary between the topological dynamical system (βS,S)andthesystem(Xs,S) which, in the case that S is an amenable group, includes a correspondence between invariant measures. As nonmetrizable topological dynamical systems are generally considered somewhat pathological, there is some value in the observation that any argument which uses dynamical techniques on the system (βS,S) can be translated in a semi-algorithmic way to an argument which is based instead on the metrizable system (Xs,S). To illustrate, we now present a metrizable rendering of Beiglböck’s simple and elegant proof of Jin’s theorem on sumsets ([7]). It should be stressed that no new ideas are presented here, we’ve simply used Lemma 3.1.2 and its consequences to rewrite Beiglböck’s proof in different language. Although Beiglböck’s proof is valid in the more general context of left amenable semigroups, we adhere to our stated conventions regarding density and restrict our attention to countable amenable groups.

45 3.2 A Metrizable Rendering of Beiglb¨ock’s Proof of Jin’s

Theorem

The following result was ﬁrst obtained via model-theoretic methods in [40].

Theorem 3.2.1 (Jin). If A, B ⊆ Z both have positive upper Banach density, then

A + B is piecewise syndetic.

Theorem 3.2.1 was generalized in [41]. Further generalizations and reﬁnements are obtained, via dynamical methods, in [16] and [5]. We will now prove the following extension of Theorem 3.2.1.

Theorem 3.2.2 ([7], Theorem 3.). Let G be a countable amenable group. Let A, B ⊆

∗ ∗ G be such that min{dL(A), dL(B)} > 0.ThenAB is left piecewise syndetic.

The proof of Theorem 3.2.2 will hinge on the following lemma. The use of Fatou’s lemma below is an argument originally due to Vitaly Bergelson (see [10], Theorem

1.1).

∞ Lemma 3.2.3. Let G be a countable amenable group and let A, B ⊆ G.Let(Hn)n=1 |A ∩ H | be any sequence of ﬁnite subsets of G such that a = lim n exists. Then there n→∞ |Hn| ¯ |A ∩ C ∩ Hn| ∗ is some C ⊆ G with 1C ∈ O(1B) ⊆ Xs and such that lim sup ≥ dL(B)·a. n→∞ |Hn|

Proof. Let y =1B ∈ Y and let X = O¯(y). Let U = {x ∈ X : x(e)=1}.Choose

∗ |B ∩ Fn| some left Følner sequence F =(Fn)n∈N such that dL(B) = lim . n→∞ |Fn| Let δ denote the point mass at y,andforn ∈ N, let

1 ∗ µn = Tg (δ). |Fn| g F "∈ n ∗ ∞ Let µ be any weak -limit point of the sequence (µn)n=1. Then there is a subsequence

∗ −1 So there is some z ∈ X such that f(z) ≥ dL(B) · a.PutC = z (1), so z =1C . Finally, note that

1 |Hn ∩ A ∩ C| f(z) = lim sup F (Tgz) = lim sup , n→∞ |Hn| n→∞ |Hn| g∈A∩H " n and the proof is complete.

The following fact will be used in our proof of Theorem 3.2.2 and also in later chapters. It is a “right-hand version” of [19, Theorem 1.5], and our terminology diﬀers slightly from the terminology adopted in [19]. So, for the reader’s convenience, we include a proof here.

Proposition 3.2.4 ([19, Theorem 1.5]). Let G be a countable amenable group, and let E ⊆ G have positive right upper Banach density. Then E−1E is a right IP∗ set.

∞ ∗ Proof. Let (xn)n=1 be any sequence in G,choosek ∈ N so that 1/k < dR(E), and for

1 ≤ i ≤ k deﬁne yi by:

yi = x1 · x2 · ...· xi.

47 ∗ Fix some right Følner sequence F =(Fn)n∈N such that dR(E)=dF (E). Then for

−1 ∗ all 1 ≤ i ≤ k one has dF (Eyi )=dR(E) so there must exist some 1 ≤ i 1). Let i=1 i=1 −1 −1 $ " g ∈ Eyi ∩ Eyj .

Then gyi ∈ E and gyj ∈ E,so

−1 −1 −1 −1 −1 xi+1 · ...· xj =(xi · xi−1 · ...· x1 )(x1 · ...· xj)=(yi g )(gyj)

∈ E−1E,

∞ −1 ∞ and FPi((xn)n=1) ∩ E E ≠ ∅. Since (xn)n=1 is an arbitrary sequence in G, E−1E ∈ (IPr)∗.

Corollary 3.2.5. If G is a countable amenable group and E ⊆ G has positive right upper Banach density, then E−1E is left syndetic.

Proof. It suﬃces to show that every right thick set is a right IP set. So let B ⊆ G be right thick. Choose x1 ∈ B.

n Having chosen x1,...,xn ∈ B so that FPi((xj)j=1) ⊆ B, there is yn+1 ∈ G so that n FPi((xj)j=1)yn+1 ⊆ B, and now there is zn+1 ∈ G so that

n (FPi((xj)j=1)yn+1 ∪{yn+1})zn+1 ⊆ B.

n Putting xn+1 = yn+1zn+1,weseethat(FPi((xj)j=1)xn+1 ∪{xn+1}) ⊆ B.So

n+1 n n FPi((xj)j=1 )=FPi((xj)j=1) ∪ FPi((xj)j=1)xn+1 ∪{xn+1}⊆B.

∞ We have thus constructed a sequence (xn)n=1 so that

∞ ∞ n FPi((xn)n=1)= FPi((xj)j=1) ⊆ B. n=1 $

We are now prepared to prove Theorem 3.2.2. 48 ∗ −1 ∗ Proof of Theorem 3.2.2. Note that dR(A )=dL(A). Choose some right Følner

∞ −1 ∗ −1 sequence H =(Hn)n=1 in G such that dH (A )=dR(A ).

By Lemma 3.2.3, there is some C ⊆ G with 1C ∈ O¯(1B) ⊆ Xs so that

−1 ∗ −1 |C ∩ A ∩ Hn| −1 ∗ dR(C ∩ A ) ≥ lim sup ≥ dH (A ) · dL(B) n→∞ |Hn| ∗ ∗ =dL(A) · dL(B)

> 0.

Put E = C ∩ A−1. By Proposition 3.2.4 and Corollary 3.2.5, E−1E is left syndetic.

Since E−1E ⊆ AC, AC is also left syndetic.

By Theorem 2.1.7, it only remains to show that the collection

{t−1(AB):t ∈ AC} has the ﬁnite intersection property.

Let {aici :1≤ i ≤ n} be any ﬁnite subset of AC.Let

U = {z ∈ Xs : for all 1 ≤ i ≤ n,onehasz(ci)=1}.

Then U ⊆ Xs is a neighborhood of 1C , so there is some g ∈ G such that Tg(1B) ∈ U.

So, for all 1 ≤ i ≤ n,1B(cig)=1,andcig ∈ B. Thus, for all 1 ≤ i ≤ n, aicig ∈ AB,and n −1 g ∈ (aici) AB. i=1 %

49 CHAPTER 4

STRAUS EXAMPLES IN AMENABLE GROUPS

4.1 Introduction

The starting point for this chapter is an example of E. Straus which provided a (negative) answer to a question posed by Erd˝os in connection with Hindman’s theorem. We begin with the necessary deﬁnitions and historical background.

∞ ∞ Given a sequence (xn)n=1 of positive integers, the ﬁnite sums set FS((xn)n=1)

∞ generated by (xn)n=1 is deﬁned by

∞ FS((xn)n=1)= xn : H ⊆ N is ﬁnite and nonempty . 0n H 1 "∈ The following theorem, proved by N. Hindman in 1974, conﬁrmed a conjecture made independently by Rado, Sanders, and Graham and Rothschild.

Theorem 4.1.1 (Hindman, [33]). For any ﬁnite partition N = Ci of the natural i=1 $∞ numbers, there is some i ∈{1,...,r} and some sequence (xn)n=1 in N such that

∞ FS((xn)n=1) ⊆ Ci.

Assume now that E ⊆ N is a set of postitive natural density, meaning that

|E ∩{1, 2,...,n}| d(E) = lim n→∞ n exists and is postitive. By Szemer´edi’s theorem [54], E contains arbitrarily long arithmetic progressions. This is a “density version” of Van der Waerden’s theorem 50 r asserting that, given a ﬁnite partition N = Ci, some cell Ci contains arbitrarily i=1 long arithmetic progressions ([57]). $

In an attempt to ﬁgure out if there exists a sort of density version of Hindman’s theorem, Erd˝os asked whether d(E) > 0 implies that E contains a shift of some

ﬁnite sums set1. That is, is it true that for every set E ⊆ N satisfying d(E) > 0

∞ there is some sequence (xn)n=1 of natural numbers and a natural number t such that

∞ t +FS((xn)n=1) ⊆ E? The following theorem of E. Straus provided, in the words of Erd˝os, “a counterexample to all such attempts” ([27], page 105).

Theorem 4.1.2 (Ernst Straus, unpublished; see, e.g., [6, Theorem 2.20] or [34,

Theorem 11.6]). Let ϵ>0.ThenthereisasetE ⊆ N having natural density d(E) > 1 − ϵ so that no shift of E contains a ﬁnite sums set.

The story about the connections between large sets in groups and ﬁnite sums sets does not, however, end with Straus’ example. Indeed, as we have seen, the notion of sets of positive density naturally extends to general amenable groups. Since

Hindman’s theorem is valid in any inﬁnite semigroup (see Theorem 2.1.3 andthe following remarks), one may wonder whether the general version of Erd˝os’ question also has a negative answer.

As we will see, the answer depends on the richness of the class of almost periodic functions on our group G. Moreover, we will be able to completely characterize countable amenable groups for which the “Straus phenomenon” takes place.

In [15] it is shown that in certain amenable groups Theorem 4.1.2 fails spectacu- larly.

1As the condition “E contains a finite sums set” is not shift-invariant, but the condition d(E) > 0 is, it is evident that this shift is necessary. Indeed, the set E =2N + 1 satisfies d(E)=1/2, but contains no finite sums set.

51 Deﬁnition 4.1.3. Let G be a discrete group. Then G is called a WM group if G has no nontrivial ﬁnite-dimensional bounded representations.

Contrast the following theorem with Theorem 4.1.2.

Theorem 4.1.4 ([15, Theorem 2.4]). If G is a countable amenable group, then G is aWMgroupifandonlyifeveryE ⊂ G having positive (left) upper Banach density is a right IP set.

This motivates the following deﬁnition.

Deﬁnition 4.1.5. AdiscretecountableamenablegroupissaidtohavetheStraus property if for every ϵ>0 and for every left Følner sequence (Fn)n∈N there is some set E ⊆ G having lower density dF (E) > 1 − ϵ such that no left shift of E is a right IP set.

We will obtain a complete characterization of those discrete countable amenable groups which have the Straus property. Furthermore, we will show that if G has the

Straus property then for every ϵ>0 and every left (right) Følner sequence (Fn)n∈N

2 there is some E ⊆ G with density dF (E) > 1 − ϵ which satisﬁes an even stronger combinatorial property than the property of containing no shift of any (increasing or decreasing) IP set.

Deﬁnition 4.1.6. AsetE ⊆ G is an increasing ∆0 set if for every n ∈ N there are

−1 x1,...,xn ∈ G so that {xj xi :1≤ j

AsetE ⊆ G is a decreasing ∆0 set if for every n ∈ N there are x1,...,xn ∈ G

−1 so that {xixj :1≤ j

2It is significant that Straus’ original example has natural density, as defined in the paragraph following Theorem 4.1.1. That is, the set E ⊆ N constructed by Straus has well-defined density with respect to the Følner sequence (Fn)n∈N given by Fn = {1, 2,...,n}. Here we find, given any left or right Følner sequence (Fn)n∈N, a set which realizes the Straus phenomenon and which has well-defined density dF (E). 52 ∗ AsetE ⊆ G is a ∆0 set if for every increasing ∆0 set A ⊆ G and every decreasing

∆0 set B ⊆ G one has E ∩ A ≠ ∅̸= E ∩ B.

We will ﬁnd the following notation useful.

∞ Deﬁnition 4.1.7. For a sequence (xn)n=1 in a countable amenable group G,andfor

α = {n1,...,nk}∈Pfin(N),whereforeach1 ≤ i

∞ Iα((xn)n=1)=xn1 · xn2 · ...· xnk and

∞ Dα((xn)n=1)=xnk · xnk−1 · ...· xn1 .

AsetE ⊆ G is an increasing (decreasing) ∆ set if there is an inﬁnite sequence

∞ −1 −1 ∞ (xn)n=1 in G so that {xj xi : j

∞ ∞ sequence in G and for n ∈ N set yn = I{1,2,...,n}((xn)n=1), zn = D{1,2,...,n}((xn)n=1).

−1 ∞ Then {yj yi : j

−1 ∞ {zizj : j

(decreasing) ∆set, which is clearly an increasing (decreasing) ∆0 set. Thus the conclusion stated in case iii) of the following theorem implies that G has the Straus property.

Theorem 4.1.8. Let G be a countable amenable group. Then exactly one of the following holds:

i) G is a WM group. Then for every E ⊆ G with positive left upper Banach

∞ ∞ density there is some sequence (xn)n=1 of elements of G such that FPi((xn)n=1) ⊆ E. ii) G is not a WM group, but G is a virtually WM group. Then there is some

E ⊆ G with positive left upper Banach density such that E contains no increasing

ﬁnite products set. On the other hand, if B ⊆ G with d¯(B) > 0 then there is some

∞ ∞ sequence (xn)n=1 of elements of G and some h ∈ G so that FPi((xn)n=1) ⊆ Bh. 53 iii) G is not a virtually WM group. Then for every ϵ>0 and every left (right)

Følner sequence (Fn)n∈N there is a set E ⊆ G so that dF (E) is well-deﬁned, dF (E) >

1 − ϵ and no left (right) shift of A contains any ∆0 set.

Note that Theorem 4.1.8 characterizes those countable amenable groups having the Straus property:

Corollary 4.1.9. Let G be a countable amenable group. Then G has the Straus property if and only if G is not a virtually WM group.

We conclude this introduction with a brief discussion of the combinatorial richness of sets E ⊆ G having positive upper Banach density. It is a theorem of Hilbert

([32]) that, given any E ⊆ N with d(E) > 0andanyn ∈ N, one can ﬁnd distinct x1,...,xn ∈ N and t ∈ N so that

n t +FS(xi)i=1 = {t + xn : ∅̸= H ⊆{1, 2,...,n}}⊆ E. n∈H " See [11, Section 3] for more on Hilbert’s theorem. In fact, Hilbert’s theorem readily generalizes to any countable amenable group G.Thus,whereasasetE ⊆ G of positive upper Banach density may fail to contain a shift of an IP set, E nevertheless

“comes close” to containing the desired conﬁguration. Also, as noted in [15], an amenable group G is a WM group if and only if for any E ⊆ G with positive left upper Banach density one can ﬁnd x, y ∈ G so that

{x, y, xy}⊆E (this remark strengthens, in one direction, Theorem 4.1.4 above).

Considering this fact, one may suspect that an amenable group G is virtually WM if and only if any set E ⊆ G having positive left upper Banach density contains a left shift of some conﬁguration of the form {x, y, xy}, yet by the above-mentioned generalization of Hilbert’s theorem this is not true. The true characterization of virtually WM amenable groups is that an amenable group G is virtually WM if and

54 only if any set E ⊆ G having positive left upper Banach density contains some left shift of an increasing ﬁnite products set. Hence, the generalization of Erd˝os’ question to an amenable group G has a positive answer if and only if G is a virtually WM group.

In the remainder of the chapter, we develop the necessary machinery and prove

Theorem 4.1.8. This entails a brief review of some relevant notions regarding almost periodic functions, WM groups, and dynamics.

4.2 Main Results

Let G be any countable and discrete group. Let B(G)denotetheBanachspace of all bounded functions f : G → C, with the supremum norm given by ∥f∥∞ = sup{|f(x)| : x ∈ G}.Forg ∈ G let lg : B(G) → B(G)andrg : B(G) → B(G)

−1 be given by lgf(x)=f(g x)andrgf(x)=f(xg). A function f ∈ B(X) is almost periodic if the set {lgf : g ∈ G}⊆B(G) is compact. It is a classical result of W. Maak that f ∈ B(X) is almost periodic if and only if the set {rgf : g ∈ G} is compact ([48]). For k ∈ N,wedenotebyAut(Ck) the group of all invertible linear maps L : Ck → Ck.

A representation φ : G → Aut(Ck)ofG is bounded if there is some M ∈ R so that

∥φ(g)∥

k x,y x,y and for x, y ∈ C deﬁne fφ ∈ B(G)byfφ (g)=⟨φ(g)x, y⟩,where⟨·, ·⟩ denotes the standard inner product on Ck. The following characterization of the almost periodic functions on G was obtained by von Neumann in [55].

Theorem 4.2.1 ([55, Theorem 30]). Afunctionf ∈ B(G) is almost periodic if and only if for every ϵ>0 there exist n ∈ N, k1,...,kn ∈ N,boundedrepresentations

ki 2ki {φi : G → Aut(C ):1≤ i ≤ n},pairsofpoints{(xi,yi) ∈ C :1≤ i ≤ n},and n xi,yi constants {ci ∈ C :1≤ i ≤ n} such that ∥f − cifφi ∥∞ ≤ ϵ. i=1 " 55 It should be noted that there exist groups G with the property that the set AP(G) of all almost periodic functions on G contains only constants. Examples of such groups were ﬁrst given by von Neumann and Wigner in [56], where such groups are called “minimally almost periodic”. By Theorem 4.2.1, a group G is minimally almost periodic if and only if G is a WM group.

Now let

G0 = {g ∈ G : for every almost periodic function f on G we have f(g)=f(e)}.

G0 is the largest subgroup of G which is a WM group, and if H ≤ G is a normal subgroup such that the almost periodic functions on G/H separate points, then G0 ≤ H.

Von Neumann ﬁrst considered the set G0 in [55], and he proved via elementary methods that G0 is a normal subgroup of G. Note that by Theorem 4.2.1 G0 is the intersection of the kernels of all bounded ﬁnite-dimensional representations of G.The fact that G0 is a normal subgroup is now an immediate consequence. Note that a representation φ : G → Aut(Ck) is bounded if and only if there is some inner product ⟨·, ·⟩ on Ck with respect to which each φ(g) is unitary ([55,

Theorem 19]).

Theorem 4.2.2. The following conditions on a group G are equivalent:

i) [G : G0] < ∞. ii) There is some WM group H ≤ G with [G : H] < ∞.

iii) There is n ∈ N so that for every ﬁnite-dimensional bounded representation

φ : G → Aut(Ck) we have |Image(φ)|≤n.

Proof. i) ⇒ ii): Assume [G : G0]=n<∞.WeshowthatthenG0 is a WM group.

k If φ : G0 → Aut(C ) were a nontrivial ﬁnite-dimensional bounded representation of G0 then by a classical result in the theory of group representations, φ induces a 56 nontrivial bounded representation φ˜ : G → Aut(Ckn). More precisely, there is an

k kn k isometric embedding j : C → C so that for all g ∈ G0 and all x ∈ C we have j(φ(g)x)=φ˜(g)j(x) (see, e.g., Theorem 7.3, chapter XVIII of Lang’s Algebra [46]).

k Now since φ is nontrivial, there is some g0 ∈ G0 and there are x, y ∈ C so that

⟨φ(g0)x, y⟩̸= ⟨x, y⟩.Letf : G → C be given by f(g)= φ˜(g)j(x),j(y) .Thenf is 2 3 almost periodic and f(g0) ≠ f(e). As g0 ∈ G0, this is a contradiction. So there is no such φ,andG0 is a WM group.

ii) ⇒ i): Obviously for any WM subgroup H

k i) ⇒ iii) is clear. If n =[G : G0]andφ : G → Aut(C ) is any ﬁnite-dimensional bounded representation of G,thenG0 ≤ Kernel(φ)so

n =[G : G0] ≥ [G :Kernel(φ)] = |Image(φ)|.

iii) ⇒ i): We will proceed by contradiction. So assume that there are g1,...,gn+1

−1 so that for all 1 ≤ i

nij there is some bounded ﬁnite-dimensional representation φij : G → Aut(C ) with

φij(gi) ≠ φij(gj).

nij Let φ : G → Aut(⊕1≤i

φ(g)(v12,v13,...,vn−1n)=(φ12(g)(v12),φ13(g)(v13),...,φn−1n(g)(vn−1n))

Then φ is a bounded ﬁnite-dimensional representation of G,andfor

1 ≤ in, which contradicts iii).

Deﬁnition 4.2.3. If G is a group that satisﬁes any (and hence all) of the conditions i)-iii) of Theorem 4.2.2 then we say that G is a virtually WM group.

Example 4.2.4. Let

S(N)={σ : N → N : σ is a bijection and {n ∈ N : σ(n) ≠ n} is finite} 57 be the finite symmetric group of N and let A(N)={σ ∈ S(N):σ is even}.Then A(N) is a WM group3 and [S(N):A(N)] = 2 so S(N) is a virtually WM group. On the other hand, φ(σ)=(−1)sgn(σ) defines a nontrivial bounded representation φ of

S(N),soS(N) is not a WM group.

Obviously any WM group group is a virtually WM group.

We now proceed to the proof of Theorem 4.1.8. First a few lemmas:

Lemma 4.2.5. Let G be any discrete group. If H

−1 [G : H]=n − 1 then for every g1,...gn there are 1 ≤ i

−1 and there are 1 ≤ s

′ −1 Proof. Note that the second assertion follows from the ﬁrst by putting gi = gi , n−1

1 ≤ i ≤ n. Under the assumptions, there are k1,...kn−1 in G so that G = krH. r=1 Then there are 1 ≤ i

−1 −1 −1 −1 −1 gi(gj) ∈ (krH)(krH) =(krH)(H kr )=krHkr = H.

Note that Lemma 4.2.5 may be rephrased as follows: If H

∗ ﬁnite index, then H is a ∆0 set.

Lemma 4.2.6. If H

(Fn)n∈N we have dF (H)=1/n.

Proof. Let (Fn)n∈N be a left Følner sequence, and ﬁx g1,...gn ∈ G so that G = n ¯ ¯ giH. Then for all 1 ≤ i ≤ n we have dF (giH)=dF (H)anddF (giH)=dF (H). i=1 $ ∞ 3 One way to see that A(N) is a WM group is to derive this from the fact that A(N)= An, n=1 where An is the group of all even elements of the symmetric group Sn on n elements. Then$ one notes that the minimal nontrivial representation of An has dimension cn, where lim cn = ∞. n→∞

58 Also, for any E ⊆ G and any x1,...,xk ∈ G such that xiE ∩xj E = ∅ whenever i ≠ j, k k ¯ ¯ one has dF ( xiE)=kdF (E)anddF ( xiE)=kdF (E). Thus i=1 i=1 $ $ n ¯ ndF (H)=dF ( giH)=dF (G)=1=dF (G) i=1 $ n

= d¯F ( giH)=nd¯F (H) i=1 $ and dF (H)=1/n.If(Fn)n∈N is a right Følner sequence, then

−1 −1 ∞ F =(Fn )n=1 is a left Følner sequence, and by what was just shown we have

−1 dF (H)=dF −1 (H )=dF −1 (H)=1/n.

Lemma 4.2.7. Let φ : G → Aut(Ck) be a bounded ﬁnite-dimensional representation of G.If|Image(φ)| = ∞ then there is some x ∈ Ck with inﬁnite orbit.

k Proof. Let e1,...,ek be any basis for C . Assume that for all 1 ≤ i ≤ k the orbit

O(ei)ofei is ﬁnite. Let

k F = {f ∈ Aut(C ) : for all 1 ≤ i ≤ k,wehavef(ei) ∈ O(ei)}.

Then F is ﬁnite and Image(φ) ⊆ F .

We now recall some deﬁnitions from the theory of topological dynamical systems.

A topological dynamical system (X, G) is a metric space (X, d) together with an action j : G → Homeo(X)ofG on X by homeomorphisms. For g ∈ G and x ∈ X,wedenote j(g)(x) ∈ X simply by gx.Thesystem(X, G) is topologically transitive if there is some x ∈ X so that {gx : g ∈ G} = X, and is minimal if {gx : g ∈ G} = X for all x ∈ X.ABorelmeasureµ on X is invariant if for all Borel sets A ⊆ X and all g ∈ G one has µ(g−1(A)) = µ(A). If G is amenable and X is compact, then there is always some invariant Borel measure. The system (X, G) is uniquely ergodic if there is only one invariant Borel measure µ on X.Thesystem(X, G) is equicontinuous if the collection {j(g):g ∈ G} is an equicontinuous set of functions. 59 The following result is well-known to ergodic theorists, and a proof maybefound in [53], Theorem 7.

Lemma 4.2.8. Let (X, d) be a compact metric space, and assume that a countable amenable group G acts on X by homeomorphisms in such a way that the topological dynamical system (X, G) is equicontinuous and topologically transitive. Then (X, G) is minimal and uniquely ergodic. If X is inﬁnite and µ is the unique G-invariant measure, then µ is nonatomic.

We will ﬁnd the following notation useful. For a dynamical system (X, G), x ∈ X, and Borel subset A ⊆ X,thesetRA(x) ⊆ G is given by:

RA(x)={g ∈ G : gx ∈ A}.

Lemma 4.2.9. Let (X, G) be a compact metrizable topological dynamical system.

i) If (X, G) is equicontinuous and minimal, U ⊆ X is nonempty and open, and

∗ x ∈ U,thenRU is a ∆0 set. ii) If (X, G) is uniquely ergodic with invariant measure µ, A ⊆ X is a Borel set with µ(∂A)=0, x ∈ X,and(Fn)n∈N is any left Følner sequence, then dF (RA(x)) = µ(A).

Proof. i) Let d be any metric on X which is compatible with the topology. Let

ϵ>0besuchthat{y ∈ X : d(x, y) <ϵ}⊆U. By equicontinuity there is some

δ>0 such that, for all y,z ∈ Z with d(y,z) <δand all g ∈ G, d(gy,gz) <ϵ.Let

V = {y ∈ X : d(x, y) <δ/2}. By minimality, X = g−1(V ). So, by compactness g∈G $ n −1 of X there is n ∈ N and there are g1,...,gn ∈ G so that X = gi (V ). i=1 $ Now let h1 ...hn+1 ∈ G be any (n + 1) elements. Then there are

60 −1 1 ≤ i

{gshix, gshjx}⊆V ,andd(gshix, gshjx) <δ.So

−1 −1 −1 −1 −1 d(x, hi hjx)=d((hi gs )(gshix), (hi gs )(gshjx)) <ϵ.

−1 −1 Thus hi hjx ∈ U,andhi hj ∈ RU (x). Similarly there are 1 ≤ l

−1 −1 −1 −1 there is 1 ≤ t ≤ n so that {hl x, hk x}⊆gt (V ), and if follows that hkhl ∈ RU (x).

∗ Since h1,...,hn+1 is an arbitrary collection of (n + 1) elements of G, RU (x) ∈ ∆0.

ii) Let ϵ>0. Since µ(∂A)=0,therearef1,f2 ∈ C(X)sothatf1 ≤ 1A ≤ f2 and

(f2 − f1)dµ < ϵ.Now,forn ∈ N and x ∈ X,onehas: ' 1 1 1 f1(gx) ≤ 1A(gx) ≤ f2(gx), |Fn| |Fn| |Fn| g F g F g F "∈ n "∈ n "∈ n so 1 µ(A) − ϵ ≤ f1dµ = lim f1(gx) n→∞ |Fn| g∈F ' "n 1 1 = lim inf 1A(gx) ≤ lim sup 1A(gx) n→∞ |Fn| n→∞ |Fn| g F g F "∈ n "∈ n 1 ≤ lim f2(gx)= F2dµ n→∞ |Fn| g∈F "n ' ≤ µ(A)+ϵ.

Since ϵ>0 is arbitrary, we conclude that

1 dF (RA(x)) = lim 1A(gx)=µ(A). n→∞ |Fn| g∈F "n

We are now ready to prove the main result.

Proof of Theorem 4.1.8. That the cases i), ii), and iii) are mutually exclusive and exhaustive is obvious. So we need only prove that in each case the remaining claims hold. 61 i) Assume that G is a WM group. Let E ⊆ G satisfy d¯F (E) > 0 for some left

Følner sequence (Fn)n∈N. Then by [15, Theorem 2.4] (Theorem 4.1.4 above), there is

∞ ∞ some sequence (xn)n=1 in G so that FPi((xn)n=1) ⊆ E. ii) Assume that G is not a WM group but G is a virtually WM group. Again by [15, Theorem 2.4], there is some E ⊆ G with d(¯ E) > 0 but so that for all

∞ ∞ sequences (xn)n=1 in G we have FPi((xn)n=1) \ E ≠ ∅.Ontheotherhand,there n are h1,...,hn ∈ G so that G0hi = G. Furthermore, as shown in the proof of i=1 $ ¯ Theorem 4.2.2, G0 is a WM group. Let B ⊆ G be any set with d(B) > 0. For n ¯ 1 ≤ i ≤ n put Bi = B ∩ G0hi. Since B = Bi, there is 1 ≤ i0 ≤ n with d(Bi0 ) > 0. i=1 −1 ¯ $ Put C = Bi0 hi0 .ThenC ⊆ G0,andd(C) > 0(asasubgroupofG0). So, by the ∞ above-mentioned [15] Theorem 2.4, there is some sequence (xn)n=1 ⊆ G0 ⊆ G with

∞ −1 FPi((xn)n=1) ⊆ C ⊆ B(hi0 ). ∞ iii) Assume that G is not a virtually WM group. Let (gi)i=1 be any sequence such that {gn : n ∈ N} = G. Fix any left Følner sequence (Fn)n∈N.Letϵ>0. By Theorem 4.2.2, for every n ∈ N there is some bounded ﬁnite-dimensional

kn representation φn : G → Aut(C )sothat[G :Kernel(φn)] ≥ n. Either there is some bounded ﬁnite-dimensional representation φ with [G :Kernel(φ)] = ∞ or there is no

∞ such representation. We now show that, in either case, there is a sequence (En)n=1 n ∗ of subsets of G such that, for all n ∈ N, En is a ∆0 set and the set Bn = giEi has i=1 −n $ well-deﬁned density DF (Bn) <ϵ(1 − 2 ). Case 1: There is no bounded ﬁnite-dimensional representation φ with

[G :Kernel(φ)] = ∞.Then,forn ∈ N,putHn =Kernel(φn). Then each Hn is anormalsubgroupofG having ﬁnite index kn =[G : Hn] ≥ n. By Lemma 4.2.6,

∗ dF (Hn)=1/kn ≤ 1/n and by Lemma 4.2.5, Hn is a ∆0 set.

62 n For n ∈ N,chooseNn > 2 /ϵ,andputEn = HNn .Wemustshowthatforevery n −n n ∈ N, Bn = giEi has density dF (Bn) <ϵ(1 − 2 ). i=1 $ n So fix n ∈ N.LetK = En.ThenK is an intersection of finitely many i=1 subgroups of G, each subgroup% in the intersection having finite index. It follows that

K has ﬁnite index [G : K]=k ≥ max{ki :1≤ i ≤ n},andthatforeach1≤ i ≤ n,

Ei is a union of finitely many cosets of K.ThusBn is also a union of finitely many cosets of K. That is, there are r ≤ k and x1,...,xr ∈ G so that Bn is the disjoint r union Bn = xiK. By Lemma 4.2.6 and left shift-invariance of dF ,eachxiK has i=1 well-defined density$ with respect to F ,andthusBn does also. Finally, note that

n n n n −i n dF (Bn)=dF ( giEi) ≤ dF (giEi)) = dF (Ei) < 2 ϵ = ϵ(1 − 2 ). i=1 i=1 i=1 i=1 $ " " " Case 2: There is some bounded ﬁnite-dimensional representation φ : G → Aut(Ck) with [G : Kernel(φ)] = ∞. We may assume without loss of generality that each

φ(g), g ∈ G, is unitary. Then by Lemma 4.2.7, there is some x ∈ Ck such that X = {φ(g)x : g ∈ G} is inﬁnite. Since each φ(g) is unitary, X is compact. Thus

(X, G) is an equicontinuous and transitive compact topological dynamical system.

By Lemma 4.2.8, (X, G) is minimal and uniquely ergodic. Let µ be the unique invariant mean. Since X is inﬁnite, µ is nonatomic. So, for all n ∈ N there is an open neighborhood Vn of x with µ(Vn) < 1/n.Letd(·, ·) denote the standard metric

k k k on C .Fory ∈ C and ϵ>0, let Bϵ(y)={z ∈ C : d(y,z) <ϵ}.Foreachn ∈ N,

there is ϵn > 0sothatBϵn (x) ⊆ Vn.Then{∂(Bs(x)) : 0

is sn with 0

µ has full support. So µ(Un) > 0. Also, since Un ⊆ Vn, µ(Un) < 1/n.

63 ∗ So by Lemma 4.2.9, Rn = RUn (x) is a ∆0 set having well-deﬁned density dF (Rn)=

µ(Un) < 1/n.

n For n ∈ N,chooseNn > 2 /ϵ,andputEn = RNn .Wemustshowthatforevery n −n n ∈ N, Bn = giEi has density dF (Bn) <ϵ(1 − 2 ). i=1 $n

Put Y = φ(gi)(UNi ). Since each φ(gi) is a µ-preserving homeomorphism, for i=1 every 1 ≤ i ≤$n one has

µ (∂ (φ(gi)(UNi ))) = µ(φ(gi)(∂(UNi ))) = µ(∂(UNi )) = 0.

Furthermore, ∂(Y ) ⊆ ∂(φ(gi)(UNi )), so µ(∂(Y )) = 0. Thus i=1 $

n n

Bn = giRi = {gig : φ(g)x ∈ UNi } i=1 i=1 n $ $n = {g : φ(g)x ∈ φ(g )(U )} = R (x) i Ni φ(gi)(UNi ) i=1 i=1 $ $ = RY (x),

So Bn has well-deﬁned density with respect to F by Lemma 4.2.9. Finally note that

n n n n −i n dF (Bn)=dF ( giEi) ≤ dF (giEi)) = dF (Ei) < 2 ϵ = ϵ(1 − 2 ). i=1 i=1 i=1 i=1 $ " " " ∞ Thus, in any case, we obtain a seqence (En)n=1 as promised.

−n Let α = lim dF (Bn) ≤ lim sup ϵ(1 − 2 )=ϵ.Foreachn ∈ N, there is some n→∞ n→∞ 4 −n sn ∈ N so that for all m ≥ sn one has |Am(En,F) − dF (En)| < 2 .LetAn = sn n −1 −1 ∗ gnEn \ ( Fi)andCn = Ai.ThenEn \ (gn An) is ﬁnite, so gn An is a ∆0 set. i=1 i=1 $ $ Also Cn ⊆ Cn+1 and Bn \ Cn is ﬁnite, so dF (Cn)=dF (Bn).

4 Recall that the symbol An(E,F), for n ∈ N, F =(Fn)n∈N a Følner sequence, and E ⊆ G,was deﬁned in Chapter 2, Section 3.

64 ∞ ¯ Let C = Cn.dF (C) ≥ lim dF (Cn)=α. We now show that also dF (C) ≤ α. n→∞ n=1 Let t>αand$ put δ = t − α. Fix N1 ∈ N so that

∞ ∞ i max 1/2 , dF (Ai) <δ/3. 0i=N +1 i=N +1 1 "1 "1

Choose N2 ∈ N so that for all n ≥ N2, |An(CN1 ,F) − dF (CN1 )| <δ/3. Then, for n>N2,onehas:

∞

An(C, F) ≤ An(CN1 ,F)+An Ai,F

!i=N1+1 # $∞

< (dF (CN1 )+δ/3) + An(Ai,F) i=N +1 "1 −i < (α + δ/3) + (dF (Ai)+2 ) ≤ α + δ = t. i=N +1 "1 So d¯F (C) ≤ t. Since t>αis arbitrary, d¯F (C) ≤ α.ThusC has well-deﬁned density dF (C)=α ≤ ϵ.

Let E = G \ C.ThenE has density dF (E)=1− dF (C) ≥ 1 − ϵ.Foranyg ∈ G

−1 −1 ∗ there is i ∈ N so that g = gi.Thusgi Ai ⊆ G \ (g E) is a ∆0 set, and it follows

−1 that g E contains no left or right δ0 set.

−1 Finally, if (Fn)n∈N is a right følner sequence, then Gn = Fn deﬁnes a left Følner

∞ sequence G =(Gn)n=1.Forﬁxedϵ>0, there is a set A as constructed above so that

−1 no left shift of A contains any left or right ∆0 set and dG(A) >ϵ.ThenE = A satisﬁes dF (E)=dG(A) and no right shift of E contains any left or right ∆0 set.

4.3 Concluding Remarks

In Chapter 1 we proved the existence, for any countable abelian group G,ofaset

E ⊆ G which has positive upper Banach density but which is not piecewise syndetic.

The proof proceeded by examining an inﬁnite minimal and equicontinuous system 65 (X, G)5,andproducedasetE ⊆ G which has positive upper Banach density and contains no shift of any ∆0 set. Since every piecewise syndetic set contains a shift of a central set (and all central sets are ∆0 sets), E is not piecewise syndetic. In Section 4.2 we found that such sets E ⊆ G exist in a countable amenable group

G if and only if G is not a virtually WM group. Furthermore, if G is not a virtually

WM group then either for all n ∈ N there is a subgroup H ≤ G with n<[g : H] < ∞ or G admits an inﬁnite minimal equicontinuous topological dynamical system(X, G). We could now derive the existence of a D-set E ⊆ G which is not piecewise syndetic for any countable group which is not a virtually WM group. This would require a generalization of Theorem 1.1.10. Instead, we turn our attention toastudyofthe phenomenon underlying the proof of Theorem 1.1.10. This phenomenon, known as

“forcing recurrence”, is studied in the next chapter. In Chapter 6 we will generalize

Theorem 1.1.10 and prove the existence, for any countable amenable group G,ofa D-set E ⊆ G which is not piecewise syndetic.

5 Such systems are uniquely ergodic. 66 CHAPTER 5

IDEALS IN βS AND FORCING RECURRENCE

Let S be a discrete semigroup. In this chapter we give a detailed analysis of the closed ideals in (βS,·). We ﬁrst establish a one to one correspondence between closed subsets of βS and certain collections of subsets of S. We then proceed to characterize, in terms of combinatorial properties of the associated collections ofsubsetsofS,those closed subsets of βS which are semigroups and those which are ideals. Then the notion of “forcing recurrence”, which was introduced in [22], is shown to be connected to algebraic properties of ideals in βS. Finally, some applications are given to topological dynamical systems.

5.1 Closed Ideals and Semigroups of βS

Our starting point is the following theorem, which establishes a one-to-one correspondence between the upward hereditary families F⊆P(S) possessing the Ramsey property and closed subsets of βS. A collection F⊆P(S) is upward hereditary if whenever A ∈Fand A ⊆ B ⊆ S then it follows that B ∈F. For brevity, a nonempty and upward hereditary collection F⊆P(S) will be called a family.IfF is a family, the dual family F ∗ is given by

F ∗ = {E ⊆ S : for all A ∈F one has E ∩ A ≠ ∅}.

67 A family F has the Ramsey property if whenever A ∈Fand A = A1 ∪ A2 there is some i ∈{1, 2} such that Ai ∈F. It is an easy exercise to see that the family F has the Ramsey property if and only if the family F ∗ is a ﬁlter.

For a family F with the Ramsey property, let

β(F)={p ∈ βS : p ⊆F}.

Theorem 5.1.1 was ﬁrst obtained by Schmidt ([52]) and was rediscovered by Glas- ner.

Theorem 5.1.1 ( [30, Proposition 1.1] ). Let S be a discrete set. For every family

F⊆P(S) with the Ramsey property, β(F) ⊆ βS is closed. Furthermore, F =

β(F).Also,ifK ⊆ βS is closed, then FK = {E ⊆ S : E¯ ∩ K ≠ ∅} is a family with$ the Ramsey property, and K¯ = β(FK).

In [30], the Stone-Cechˇ compactiﬁcation is realized as the maximal ideal space of the Banach algebra B(S) of all bounded functions on S. The following proof, which is more suited to our methods, should be compared with the discussion following

Theorem 4.4 of [20].

Proof. Let q ∈ βS \ β(F). Then there is E ⊆ S with E/∈Fand E ∈ q.SoE¯ is a neighborhood of q with the property that E¯ ⊆ (βS \ β(F)). Since q ∈ βS \ β(F) is arbitrary, βS \ β(F) is open, hence β(F) is closed.

If K ⊂ βS is closed, then FK = {q : q ∈ K} is a union of ultraﬁlters, which is easily seen to be an upward hereditary$ family with the Ramsey property (since every ultraﬁlter is such a collection). We now show that F = β(F). Let E ∈F.LetC = {E ∩ A : A ∈F∗}. Fix

∗ any A ∈F, and let E1 = $E ∩ A and E2 = E \ A. Then, since E2 ∩ A = ∅ and

∗ A ∈F , it follows that E2 ∈/ F. Since F has the Ramsey property, we conclude that

E ∩ A = E1 ∈F. 68 ∗ Now, if A1,A2 ∈F , then by the preceeding paragraph (E ∩ A1) ∈F,so(E ∩

A1) ∩ (E ∩ A2)=(E ∩ A1) ∩ A2 ≠ ∅. By a simple induction, the collection C has the ﬁnite intersection property. So by Zorn’s lemma there is some q ∈ βS with C⊆q.

It remains to show that q ∈ β(F). If B ∈P(S) \F then, by upward heredity,

S \ B ∈F∗.SoE \ B ∈C,andE \ B ∈ q. Since B ∩ (E \ B)=∅, it follows that

B/∈ q. Since B ∈P(S) \F is arbitrary, q ⊆F.

Finally, note that since K ⊆ β(FK) is closed, K¯ ⊆ β(FK). If q ∈ β(FK)and

U ⊆ βS is open then there is some A ⊆ S with q ∈ A¯ ⊆ U. Since A ∈FK,there is some p ∈ K with A ∈ p,andthenp ∈ U ∩ β(FK). Since U is an arbitrary open neighborhood or q, q ∈ K¯ . Since q ∈ β(FK) is arbitrary, we have shown that

β(FK) ⊆ K¯ .

It is natural to now ask the questions: What condition on F coresponds to the assumption that β(F) ⊆ (βS,·) is a left ideal? A right ideal? Similarly for

β(F) ⊆ (βS,◦). The ﬁrst question has a very simple answer. The family F is called left (right) shift-invariant if for all s ∈ S and all E ∈F one has sE ∈F(Es ∈F).

The family F is called left (right) inverse shift-invariant if for all s ∈ S and all E ∈F one has s−1E ∈F (Es−1 ∈F). To avoid confusion, we will distinguish between the semigroup element s ∈ S and the principal ultraﬁlter on S containing the set {s}. So, for the remainder of the present chapter, we will denote the latter bys ¯.

Theorem 5.1.2. If F is a family having the Ramsey property then β(F) ⊆ (βS,·) is a left ideal if and only if F is left shift-invariant. Similarly, β(F) ⊆ (βS,◦) is a right ideal if and only if F is right shift-invariant.

69 Proof. We prove the ﬁrst statement only, the proof of the second statement being entirely analogous. For p ∈ βS and s ∈ S note that

s¯ · p = {E ⊆ S : {x : x−1E ∈ p}∈s¯} = {E ⊆ S : s−1E ∈ p}.

Since E ⊆ s−1(sE), if E ∈ p then sE ∈ s¯· p. Thus, if β(F) is a left ideal and E ∈F,

S ∈ S, then there is some p ∈ β(F) with E ∈ p,andsE ∈ s¯ · p ∈ β(F). So sE ∈F.

Since E ∈F and s ∈ S are arbitrary, F is left shift-invariant.

On the other hand, if F is left shift-invariant, then for every p ∈ β(F), every s ∈ S,andeveryE ∈ s¯ · p, s−1E ∈ p ⊆F,soE ⊇ s(s−1E) ∈F.So¯s · p ∈ β(F).

Now, for every p ∈ βS the map q 0→ q · p is continuous. Thus, for every p ∈ β(F), we have:

βS · p = {s¯ : s ∈ S}·p ⊆ {s¯ · p : s ∈ S}⊆β(F)=β(F).

So βS · β(F)= {βS · p : p ∈ β(F)}⊆β(F), and β(F) ⊆ (βS,·) is a left ideal. $

In order to characterize those F for which β(F) is a closed right ideal in (βS,·), we will first have to consider the following family of subsets of S which is derived from F in a natural way. Call a set B ⊆ S a left (right) “broken F set” if there is some E ∈Fsuch that the collection {x−1B : x ∈ E} has the finite intersection property ({Bx−1 : x ∈ E} has the finite intersection property). Equivalently, B is a left (right) broken F set if there is some E ∈Fso that for every finite H ⊆ E there is s ∈ S with Hs ⊆ B (sH ⊆ B). The collection of all broken F sets forms a family which has the Ramsey property if F does (see Corollary 5.1.4 below). This construction was first defined for subsets of N in [22], and was also considered (for

S = Z) more recently in [39] and in [47]. We will denote the family of left (right) broken F sets by brl(F)(brr(F)).

70 The following theorem characterizes the closed right ideals in (βS,·). In S = Z, this theorem is implicit in Jian Li’s work [47], but only one implication is stated explicitly ([47, Lemma 3.5]). An alternative condition characterizing closed left ideals in (βS,◦) was obtained by Davenport ([24, Theorem 2.7]).

Theorem 5.1.3. AsetB ⊆ S is a left broken F set if and only if

B¯ ∩ β(F) · βS ≠ ∅.Equivalently,β(brl(F)) = β(F) · βS.Thus,β(F) is a right ideal in (βS,·) if and only if brl(F) ⊆F.Similarly,B ⊆ S is a right broken F set if and only if and only if B¯ ∩ βS ◦ β(F) ≠ ∅,andβ(F) ⊆ (βS,◦) is a left ideal if and only if brr(F) ⊆F.

Proof. Again, we only prove the statements about (βS,·). Let B ∈ brl(F). Let E ∈F be so that {x−1B : x ∈ E} has the ﬁnite intersection property. Let p ∈ β(F) with E ∈ p, and let q ∈ βS with {x−1B : x ∈ E}⊆q.ThenE ⊆{x ∈ S : x−1B ∈ q}, so {x ∈ S : x−1B ∈ q}∈p.SoB ∈ p · q and p · q ∈ B¯ ∩ β(F) · βS.

Now let B ⊆ S be such that B¯ ∩ β(F) · βS ≠ ∅, and let p ∈ β(F), q ∈ βG be such that B ∈ p · q.LetE = {x ∈ S : x−1B ∈ q}.ThenE ∈ p (by the deﬁnition of multiplication in (βS,·)), and {x−1B : x ∈ E}⊆q,hence{x−1B : x ∈ E} has the

ﬁnite intersection property. Since p ∈ β(F), E ∈F and thus B ∈ brl(F).

We now record some easy corollaries of Theorem 5.1.3. Corollary 5.1.4 was recorded for families F⊆Z in [39].

Corollary 5.1.4. If F is a family with the Ramsey property then brl(F) and brr(F) also have the Ramsey property.

Proof. By Theorem 5.1.3,

brl(F)={E ⊆ S :therearep ∈ β(F)andq ∈ βS such that E ∈ p · q}

71 and

brr(F)={E ⊆ S :therearep ∈ β(F)andq ∈ βS such that E ∈ q ◦ p}.

So each collection is a union of ultraﬁlters.

Now note that brl(F) is right shift-invariant, and that brr(F) is left shift-invariant. This gives us an alternative proof of the following fact, which is an application of [38,

Theorem 2.19]: If S is commutative then every closed right ideal I ⊆ (βS,·) is a two-sided ideal. In fact, we obtain the following result relating closed ideals in (βS,·) to closed ideals in (βS,◦).

Corollary 5.1.5. If I ⊆ (βS,·) is a closed right ideal, then I is a closed right ideal of (βS,◦).IfI ⊆ (βS,◦) is a closed left ideal, then I is a closed left ideal of (βS,·).

If S is commutative, then every closed right ideal of (βS,·) is a two-sided ideal, and similarly for every closed left ideal of (βS,◦).

Proof. Let I be a closed right ideal of (βS,·) and let F be such that I = β(F). By

Theorem 5.1.3, brl(F) ⊆F. Also, if E ∈Fand s ∈ S,thenEs ∈ brl(F). So, by Theorem 5.1.2, I ⊆ (βS,◦) is a right ideal.

If I is a closed left ideal of (βS,◦)andF is the family determined by I = β(F), then by Theorem 5.1.3, brr(F) ⊆F and for E ∈F and s ∈ S one has sE ∈ brr(F). So I ⊆ (βS,·) is a left ideal by Theorem 5.1.2. If S is commutative, then the notion of right-shift invariance coincides with the notion of left-shift invariance.

Example 5.1.6. Let F = {S}. F is upward hereditary, but clearly does not have the

l r Ramsey property. It is immediate that brl(F)=T and brr(F)=T .

l l Example 5.1.7. brl(S )=PS .ThisisareformulationofTheorem2.1.7,andmay serve as a motivating example. Of course, Sl does not have the Ramsey property. This 72 fact, along with the fact that K((βS,·)) may not be closed, complicate the discussion somewhat. Any set E ⊆ G such that there is an idempotent p ∈ K((βS,·)) with

E ∈ p is called (left) quasicentral.Wedenotethefamilyofallleftquasicentralsets in S by QCl(S) or, if the semigroup S is clear from context, simply by QCl.

Example 5.1.8. Let G be a countable amenable group and ﬁx some left Følner sequence (Fn)n∈N.ForE ⊆ G let

|E ∩ Fn| d¯F (E) = lim sup n→∞ |Fn| and let LF = {E ⊆ G : d¯F (E) > 0}.ThenLF is a left shift-invariant family with

∞ ∞ the Ramsey property. For any sequence (gn)n=1 in G,thesequenceΦ=(Fngn)n=1 is again a left Følner sequence, and an easy calculation shows that

∞ |E ∩ Fngn| brl(LF )={E ⊆ G : ∃ asequence(gn)n=1, lim sup > 0}. n→∞ |Fngn| l Now, by [5, Lemma 3.3], brl(LF )=L .SobyTheorems5.1.2and5.1.3,β(LF ) ⊆ βG is a closed left ideal, but not in general a right ideal. Indeed, if G = Z and Fn = [1,n]

l for n ∈ N,thenE = {n ∈ Z : n<0} satisﬁes dF (E)=0and satisﬁes E ∈ brl(LF ).

l So β(LF ) ⊆ (βZ, ·) is not a right ideal.

l l l l On the other hand, brl(L )=brl(brl(LF )) = brl(LF )=L ,where(Fn)n∈N is any left Følner sequence. So β(Ll) ⊆ G is a closed two-sided ideal.

l Example 5.1.9. The collection brl(IP ) will turn out to be of importance. If E ∈

l l L (G),thenE ∈ brl(IP ).Aproofvalidforanycountableamenablegroupcanbe obtained via the Furstenberg Correspondence Principle (See, eg, [9, Theorem 2.1] for astatementoftheFurstenbergCorrespondencePrinciple). However, this fact is an immediate consequence of Theorem 6.1.2 below.

The final result of this section identifies those families F having the Ramsey property for which β(F) ⊆ (βS,·) is a semigroup. The condition is a rather technical weakening of left shift-invariance. The corresponding characterization of closed 73 semigroups in (βS,◦) is obtained with routine modifications. Davenport obtained an alternative characterization of subsemigroups of (βS,◦) in [24, Theorem 2.6].

Theorem 5.1.10. Let S be any semigroup, and let F be a family of subsets of S having the Ramsey property. Then the following are equivalent:

1. β(F) ⊆ (βS,·) is a semigroup.

2. F has the following property: If E ⊆ S is any set, and if there is A ∈F such

that for all finite H ⊆ A one has ( x−1E) ∈F,thenE ∈F. x∈H % Proof. Assume that β(F) ⊆ (βS,·) is a semigroup. Let E ⊆ S and assume that there is some A ∈F such that for every finite H ⊆ A one has ( x−1E) ∈F.Forx ∈ A, x∈H −1 % let Ix = β(F) ∩ x E.TheneachIx ⊆ βS is closed, and by assumption {Ix : x ∈ A} has the finite intersection property. So, by compactness,

Ix ≠ ∅. x∈A %

Let q ∈ Ix.Thenq ∈ β(F). Since A ∈F, there is p ∈ β(F) ∩ A¯.Nownote x∈A that A ⊆{x%: x−1E ∈ q},so{x : x−1E ∈ q}∈p.HenceE ∈ p · q. Since β(F) is a semigroup, p · q ∈ β(F), and thus E ∈F.

Now assume that F has the property stated in condition ii) of the theorem. Let p, q ∈ β(F), and let E ∈ p · q.LetA = {x : x−1E ∈ q}. Since E ∈ p · q, A ∈ p.

So A ∈F. Furthermore, for any ﬁnite H ⊆ A, {x−1E : x ∈ H}⊆q,andhence ( x−1E) ∈ q ⊆F. So, by our assumption on F, E ∈F. Since E ∈ p·q is arbitrary, x∈H it% follows that p · q ∈ β(F). Since p, q ∈ β(F) are arbitrary, β(F) ⊆ (βS,·).

74 5.2 An Algebraic Characterization of Sets that Force

F-Recurrence

We now turn our attention to a study of the idempotent elements of closed subsemigroups of (βS,·). It turns out that this topic is intimately related to recurrence in left topological dynamical systems. Fix some semigroup S.

Deﬁnition 5.2.1. Let (X, S) be a left topological dynamical system, let x ∈ X,and let F⊆P(S) be a family. Then x is F-recurrent if for every neighborhood U of x one has RU (x) ∈F. AsetE ⊆ S forces F-recurrence if whenever (X, S) is a left topological dynamical system, K ⊆ X is closed, and there is some x ∈ X such that E = RK(x) then there is some F-recurrent y ∈ K.

Lemma 5.2.2. Let F be a family of subsets of S such that β(F) be a closed semigroup in (βS,·).Let(X, S) be a left topological dynamical system and let x ∈ X.Then x is F-recurrent if and only if there is some idempotent p ∈ (β(F), ·) such that p- lim Tsx = x. s

Proof. Assume ﬁrst that there is some idempotent p ∈ (β(F), ·)suchthatp- lim Tsx = s x. Fix any neighborhood U of x. Since p- lim Tsx = x ∈ U,onemusthaveRU (x) ∈ p, s hence RU (x) ∈F. Since U is an arbitrary neighborhood of x, it follows that x is F-recurrent.

Now assume that x ∈ X is F-recurrent. Deﬁne π : βS → X by: π(p)=p- lim Tsx. s Then π is continuous. Let N be any neighborhood base at x.ForN ∈N, let

−1 KN = π (N¯) ∩ β(F).

We now show that for all N ∈N, KN is a closed nonempty subset of βS.Soﬁx N ∈N. Since N¯ is closed and π is continuous, π−1(N¯) is also closed. Since β(F) is closed, KN is closed (as the intersection of two closed sets). Also, by assumption, 75 one has RN (x) ∈F. So there is q ∈ I so that RN (x) ∈ q. It immediately follows that

π(q) ∈ N¯,soq ∈ KN .

Given any ﬁnite collection {N1,...,Nr}⊆N, there is some N0 ∈N so that N0 ⊆

r r −1 ∩i=1Ni,andhence∅̸= KN0 ⊆∩i=1KNi .Sobycompactness,J = π ({x}) ∩ β(F)=

−1 ∩N∈N KN ≠ ∅. As the intersection of closed sets, J is also closed. Now, π ({x} is easily seen to be a subsemigroup of βG, and since β(F) is also a semigroup, it follows that J is a semigroup. As a closed semigroup, J contains idempotents by Ellis’ theorem (Theorem 2.1.1).

The following theorem characterizes those E ⊆ S for which there is some idempotent p ∈ β(F)suchthatE ∈ p.ForG = Z, it was obtained using diﬀerent methods by Li in [47, Proposition 4.10]. It was also obtained by Johnson in [42, Theorem 3.3].

Theorem 5.2.3. Let F be a family with the Ramsey property such that β(F) be a semigroup in (βS,·).ForE ⊆ S,thefollowingareequivalent:

i) There is an idempotent p ∈ (β(F), ·) such that E ∈ p.

ii) There is some compact t.d.s. (X, S),somepair(x, y) of points such that y is

F-recurrent and (y,y) is in the orbit closure O¯(x, y) of (x, y) in the product system

(X×X, S) and some neighborhood U of (y,y) such that E = {s ∈ S :(Tsx, Tsy) ∈ U}.

Proof. i)⇒ii): Let (Xs,S) be the shift of chapter 3. Let x =1E ∈ Y , and let X = O¯(x). Then (X, S) is a compact metrizable topological dynamical system. Let

U = {z ∈ X : z(e)=1} and let y = p- lim Tsx.ThenU is closed and E = RU (x), so s y ∈ U. Since p is an idempotent, p- lim Tsy = y. Since p ∈ β(F), y is F-recurrent. s

Now, (p- lim Tsx, p- lim Tsy)=(y,y), so in the product system X × X one has s s

(y,y) ∈ O¯(x, y). Finally, U × X is open and E = RU×X (y,y).

ii)⇒i): Let π : βS → O¯(x, y) ⊆ X ×X be given by: π(p)=(p- lim Tsx, p- lim Tsy). s s

76 By assumption, π−1(y,y) ≠ ∅, let q ∈ π−1(y,y). By Theorem 5.2.2, there is some p1 ∈ I with p1- lim Tsy = y.Thenp1q ∈ β(F)and s

π(p1q)=(p1q- lim Tsx, p1q- lim Tsy) s s

=(p1- lim Tr(q- lim Ttx),p1- lim Tr(q- lim Ttx)) r t r t

=(p1- lim Try,p1- lim Try) r r =(y,y)

−1 −1 So p1q ∈ β(F) ∩ π (y,y). Thus I = β(F) ∩ π (y,y) ≠ ∅ is a closed subsemigroup of βS (one checks easily that π−1(y,y) is a closed semigroup, so I is the intersection of two closed semigroups), and there is some idempotent p ∈ I.

If E/∈ p,thenRX×X\U (x, y)=G\E ∈ p,andπ(p) ∈ X ×X \U, which contradicts p ∈ π−1(y,y). So E ∈ p, and the proof is complete.

Example 5.2.4. Letting S = Z and F = L,werecoverTheorem2.8of[14].Letting

S be any semigroup and F = PSl we recover Theorem 3.4 of [23]. Letting S be any semigroup and letting F be the class of all J sets1 we recover [42, Theorem 4.8].

Note that, since K((βS,·)) is not closed, Theorem 5.2.3 does not yield a statement concerning central sets as a corollary.

As Theorem 5.2.3 illustrates, the members of idempotents p ∈ β(F) are closely related to F recurrence. So we adopt the following notation: Let

Fi = {E ∈F: there is idempotent p ∈ β(F)suchthatE ∈ p}.

Then β(brl(Fi)) is the smallest closed right ideal in (βS,·) which contains every

l l l l l idempotent p ∈ I.NotethatIi = IP , PS i = QC ,andLi = D .Wearenow

1 We will not be overly concerned with the class of J sets. It suﬃces to note that the class J of all J sets in S is a family with the Ramsey property such that brl(J ) ⊆J.FormoreonJsetssee, e.g., [42] or [37].

77 prepared to characterize those sets E ⊆ S which force F-recurrence. For S = Z, this was obtained by Li in [47, Theorem 5.5].

Theorem 5.2.5. Let S be a discrete semigroup. Let F be a family of subsets of S which has the Ramsey property and such that β(F) ⊆ (βS,·) is a semigroup. Then, for E ⊆ S,thefollowingareequivalent:

i) The set E forces F recurrence.

ii) One has E ∈ brl(Fi). iii) There is some idempotent p ∈ (β(F), ·) and some q ∈ βS such that E ∈ p · q.

Proof. iii) ⇔ ii) is Theorem 5.1.3, applied to the family Fi.

i) ⇒ iii): Again, let (Xs,S) be the shift space. Let x =1E ∈ X, let X = O¯(x) ⊆ Y , and let U = {z ∈ X : x(e)=1}.ThenU is both open and closed. Also,

E = RU (x), so by assumption there is some F-recurrent y ∈ U. By Theorem 5.2.2 there is some idempotent p ∈ I with p- lim Tgy = y. Since y ∈ O¯(x), there is some g q ∈ βS such that q- lim Tsx = y. s

Now, (pq)- lim Tsx = p- lim Tr(q- lim Ttx)=p- lim Try = y,andU is a neighbor- s r t r hood of y,soE = RU (x) ∈ (pq). iii) ⇒ i): Let p = p · p ∈ I, q ∈ βS be such that E ∈ pq.Let(X, G)beany compact topological dynamical system, let K ⊆ X be closed, and assume that there is some x ∈ X so that E = RK(x). Let y =(pq)- lim Tsx. Since K is closed and s E ∈ pq,onehasy ∈ K. Also,

p- lim Tsy = p- lim Tr(pq- lim Ttx)=p · pq- lim Tsx = pq − lim Tsx = y, s r t s s so y is F-recurrent by Theorem 5.2.2. Since (X, S)andK ⊆ X are arbitrary, E forces F-recurrence.

Corollary 5.2.6. [[22, Theorem 5], in N]IfS is any semigroup such that the class

78 I of all inﬁnite subsets is left shift invariant (in particular, if S is left cancellative),

l then a set E ⊆ S forces recurrence if and only if E ∈ brl(IP ).

Theorem 5.2.7. [[22, Theorem 7], in N]IfS is any semigroup, then a set E ⊆ S forces uniform recurrence if and only if E ∈PSl.

Proof. This is not an immediate consequence of Theorem 5.2.5, because the minimal two-sided ideal K(βS,·) is not necessarily closed. However, we may proceed as follows:

Assume ﬁrst that E ⊆ S forces uniform recurrence. Let (Y,S) be the shift space, x =1E, X = O¯(x), and U = {z ∈ X : z(e)=1}.ThenE = RU (x)soby assumption there is a uniformly recurrent y ∈ U. Since y ∈ O¯(x) there is some q ∈ βS with q- lim Tsx = y. Since y is uniformly recurrent, there is some minimal s idempotent p with p- lim Tsy = y ([38, Theorem 19.23]). Now, (pq)- lim Tsx = y ∈ U, s l l l so E = RU (x) ∈ pq. So, by Theorem 5.1.3, E ∈ brl(Cen ) ⊆ brl(PS )=PS . Now assume that E ∈PSl. Then there is some minimal right ideal R ⊆ (βS,·) and some r ∈ R with E ∈ r ([38, Theorem 4.40]). Pick some idempotent p ∈ R. Then p · βS ⊆ R, so by minimality p · βS = R and there is some q ∈ βS such that r = pq.

If (X, G) is any compact system, K ⊆ X is closed, and there is x ∈ X such that

RU (x)=E,theny =(pq)- lim Tsx ∈ K.Now, s

p- lim Tsy = p- lim Ts (pq)- lim Ttx =(p · (pq)) - lim Tsx =(pq)- lim Tsx = y, s s t s s 4 5 and since p ∈ K(βS,·), this implies that y is uniformly recurrent by [38, Theorem

19.23]. Thus E forces uniform recurrence if and only if E ∈PSl.

An immediate corollary of Theorem 5.2.5 is that every E ∈Fforces F-recurrence

l if and only if F =brl(Fi). This is satisﬁed by PS (See, eg, Theorem 4.43 in [38]).

79 We show in the next chapter that, if S = G is an amenable group, then this is also satisﬁed by Ll. This will, via Theorem 5.2.5, yield some combinatorial consequences for sets having positive left upper Banach density.

5.3 Anti-F-recurrence

In this section we develop some results concerning F-recurrent points which will be useful in the ﬁnal two chapters.

Our starting point in this section is the following result: If S is commutative then the system (X, S) is proximal if and only if there is a ﬁxed point m ∈ X such that m is the only minimal (i.e. uniformly recurrent) point. As this does not hold for noncommutative groups S, and as it is the latter condition which will be used in the sequel, we now make the following deﬁnition.

Deﬁnition 5.3.1. Let S be a discrete semigroup and (X, S) acompacttopological dynamical system. Then (X, S) is anti-minimal if there is a unique minimal2 point m ∈ X.

Note that if (X, S) is anti-minimal then the minimal point m is in fact a ﬁxed point. One can generalize Deﬁnition 5.3.9 in the following natural way.

Deﬁnition 5.3.2. Let S be a discrete semigroup and let F be a family of subsets of

S.Thet.d.s.(X, S) is anti-F-recurrent if there is m ∈ X so that m is the only

F-recurrent point in X.

If the system (X, S) has a ﬁxed point m as the only recurrent point, we say that (X, S) is antirecurrent.

2The point x ∈ X is minimal if (O¯(x),S) is a minimal system. It is well known that x ∈ X is a minimal point if and only if x is uniformly recurrent.

80 Remark 5.3.3. As noted above, if S is commutative then the system (X, S) is antiminimal if and only if (X, S) is proximal. By part iv) of the following theorem, antiminimality implies proximality even for noncommutative semigroups S.Furthermore, if β(F) ⊆ (βS,·) is an twp-sided ideal, then K(βS) ⊆ β(F),soanti-F-recurrence implies antiminimality. Thus, for those families F which we consider, anti-F-recurrence of the topological dynamical system is a stronger condition than proximality. Further- more, if one assumes that the system (X, S) has a ﬁxed point, then the antiminimality of the system is equivalent to the proximality of the system.

After our previous work, the following theorem is quite trivial to prove. Neverthe- less, as will become apparent, it has a number of interesting consequences for both topological and measurable dynamics. It should be compared with Li’s Corollary 5.6

([47]).

Theorem 5.3.4. The following are equivalent: i) The system (X, S) is anti-F-recurrent.

ii) There is m ∈ X so that, for every for every idempotent p ∈ β(F) and every x ∈ X, p- lim Tsx = m. s

iii) There is m ∈ X so that, for every x ∈ X and every q ∈ β(brl(Fi)), q- lim Tsx = s m.

iv) There is m ∈ X so that for every neighborhood U of m and every x ∈ X one

∗ has RU (x) ∈ (brl(Fi)) .

Proof. i) ⇔ ii) and iii) ⇔ iv) are trivial. Also, clearly iii) ⇒ ii). So we need only show that i) ⇒ iii). We proceed by contradiction.

So assume that m ∈ X is ﬁxed and that there is x ∈ X and q ∈ β(brl(Fi)) such that q- lim Tsx = y ≠ m. Choose a neighborhood V of y so that m/∈ V¯ . Since y ∈ V¯ , s one must have RV¯ (x) ∈ q.SoRV¯ ∈ brl(Fi), and by Theorem 5.2.5 there is some F-recurrent z ∈ V¯ .Thenz ≠ m,andm is not the only F-recurrent point in X. 81 The following corollary gives (for G = Z) a slight strengthening of [29, Lemma 9.20].

Corollary 5.3.5. For a t.d.s. (X, S),thefollowingareequivalent:

i) There is a unique recurrent point m ∈ X.

ii) There is m ∈ X so that, for every for every nonprincipal idempotent p ∈ (βS,·) and every x ∈ X, p- lim Tsx = m. s l iii) There is m ∈ X so that, for every x ∈ X and every q ∈ β(brl(IP )), q- lim Tsx = m. s iv) There is m ∈ X so that for every neighborhood U of m and every x ∈ X one

l ∗ has RU (x) ∈ (brl(IP )) .

We have an analogous characterization of antiminimal systems.

Corollary 5.3.6. For a t.d.s. (X, S),thefollowingareequivalent:

i) (X, S) is antiminimal with unique minimal point m.

ii) For every minimal idempotent p ∈ (βS,·) and every x ∈ X one has p- lim Tsx = s m.

l iii) For q ∈ K(β(S)) = β(PS ) and every x ∈ X one has q- lim Tsx = m. s l ∗ iv) For every neighborhood U of m and every x ∈ X one has RU (x) ∈ (PS ) .

Proof. For every minimal idempotent p ∈ βS and every x ∈ X the point z = p- lim Tsx is a minimal point. Furthermore, if z ∈ X is minimal, then there is some s minimal idempotent p ∈ βS with p- lim Tsz = z. Thus i) is equivalent to ii). Now, s by Theorem 5.3.4, iii) is equivalent to iv). Also, clearly iii) implies ii). So it only remains to prove that i) implies iii).

So assume that there is some q ∈ K(β(S)) = β(PSl)andsomex ∈ X such that q- lim Tsx = z ≠ m. Then there is some pair of open sets U and V such that s l U ∩ V = ∅, z ∈ U,andm ∈ V . Since z in U,onehasRU (x) ∈ q,henceRU (x) ∈PS. 82 So by Corollary 5.2.7 there is some uniformly recurrent point y ∈ U.Theny ≠ m, contradicting i).

We can now provide a short and simple proof of Lemma 1.2.1.

Proof of Lemma 1.2.1. Assume ﬁrst that G is an abelian group, and that (X, G) is proximal. If Y,Z ⊆ X are both nonempty minimal closed invariant sets, then either

Y ∩Z = ∅ or Y ∩Z is a closed invariant set contained in Y and in Z, so by minimality Y = Y ∩ Z = Z. Thus there is at most one minimal closed and invariant subset of

X, and it only remains to show that there is a ﬁxed point x ∈ X.

Let Z be any minimal nonempty closed invariant subset of X,andﬁxx ∈ Z .Let

∞ g ∈ G. Then by proximality there is a sequence (gn)n=1 in G and some z ∈ Z such that lim Tgn x = lim Tgn (Tgx)=z. Then by the continuity of Tg we have: n→∞ n→∞

Tgz = Tg( lim Tgn x) = lim Tg(Tgn x) = lim Tgn (Tgx)=z. n→∞ n→∞ n→∞

And thus the set Y = {z ∈ Z : Tgz = z} is nonempty. Y is closed by the continuity of Tg, and if y ∈ Y and h ∈ G then Tg(Thy)=Th(Tgy)=Thy,soY is invariant. Thus, by minimality, Y = Z, and in particular Tgx = x. Since g ∈ G is arbitrary, x is a ﬁxed point.

Assume, on the other hand, that (X, G) is antiminimal, and let m ∈ X be the unique minimal point in X.Letx, y ∈ X.Letϵ>0, and let

U = {z ∈ X : d(z, m) <ϵ/2}.

l ∗ l ∗ By Corollary 5.3.6, {RU (x),RU (y}⊆(PS ) . Since (PS ) is a ﬁlter, there is g ∈

RU (x) ∩ RU (y). Then by the triangle inequality d(Tgx, Tgy) <ϵ. Since x, y ∈ X and ϵ>0 are arbitrary, (X, G) is proximal.

Theorem 5.3.4 and its corollaries can be used to investigate transitive subshifts. 83 Theorem 5.3.7. Let S be a countable semigroup, and let F be a family of subsets of S which has the Ramsey property and is such that β(F) ⊆ (βS,·) is a semigroup.

Denote, as always, the shift space by (Xs,S).LetE ⊆ S,letx =1E ∈ Xs,and let X = O¯(x).Thenthesystem(X, S) is anti-F-recurrent if and only if either

−1 −1 t E/∈ brl(Fi) for all t ∈ S or (S \ t E) ∈/ brl(Fi)for all t ∈ S.

Proof. Let 0=1¯ ∅,so0(¯ s) = 0 for all s ∈ Se.Letφ : Xs → Xs be deﬁned by: φx(s) = 0 if and only if x(s)=1. We will show that (X, S) is anti-F-recurrent with unique F-recurrent point 0¯ if

−1 and only if for all t ∈ S one has t E/∈ brl(Fi). The theorem then follows by noting that φ is an automorphism of the system (Xs,S). Assume that (X, S) is anti-F-recurrent, and that 0¯ is the only F-recurrent point.

−1 Let U = {z ∈ X : z(e)=1}.Lett ∈ Se.Thent E = RU (Ttx) (if we agree to

−1 −1 interpret Te : Xs → Xs as the identity map). If t E ∈ brl(Fi)thent E forces F- recurrence, and there would be some F-recurrent point y ∈ U. Since, by assumption,

−1 there is no such y, it follows that t E/∈ brl(Fi).

−1 Assume now that t E/∈ br(Fi) for all t ∈ Se.Lety ∈ X be F-recurrent, and let A = y−1(1). We must show that A = ∅. If there is some t ∈ A,then

V = {z ∈ X : z(t)=1} is a neighborhood of y. Since y is F-recurrent, there is some idempotent p ∈ β(F)suchthatp- lim Tsy = y. Since y ∈ X = O¯(x), there is some s ultraﬁlter q ∈ βS so that q- lim Tsx = y. s −1 Then (pq)- lim Tsx = y,andt E = RV (x) ∈ pq. By Theorem 5.1.3, then, s −1 t E ∈ brl(Fi). This contradicts our assumption, so there is no such t.

Corollary 5.3.8. If S is any semigroup and E ⊆ S,thenO¯(1E) ⊆ (Xs,S) is an-

−1 l tirecurrent if and only if either for all t ∈ St E/∈ brl(IP ) or for all t ∈ S

−1 l (S \ t E) ∈/ brl(IP ).

84 Corollary 5.3.9. If S is any semigroup and E ⊆ S,thenO¯(1E) ⊆ (Xs,S) is antiminimal if and only if either E/∈PSl or (S \ E) ∈/ PSl.

l Proof. By Corollary 5.3.6, O¯(1E) is antiminimal if and only if it is anti-PS -recurrent. The corollary now follows from Theorem 5.3.7 by noting that E ∈PSl if and only if

−1 l for every t ∈ Se one has t E ∈PS.

85 CHAPTER 6

SETS THAT FORCE ESSENTIAL RECURRENCE

In this chapter we establish the fact that if G is a countable amenable group then every set having positive left upper Banach density forces essential recurrence. That is, every such set forces Ll-recurrence. This fact will yield some combinatorial consequences, as well as help us to ﬁnally establish the existence in any countable amenable group of noncentral D-sets.

6.1 The Main Result

For the remainder of this chapter, ﬁx some countable amenable group G. We remind the reader here of Lemma 1.2.7. Note that the proof supplied in Chapter 1 of Lemma

1.2.7 involves no group operations, and is therefore valid for any amenable G.

Lemma (Lemma 1.2.7). Let (X, G) be a compact metrizable topological dynamical system and let K ⊆ X be a Borel set. If there is some invariant measure µ on X with µ(K) > 0 then there is some essentially recurrent y ∈ K.

Theorem 6.1.1. Let G be a countable amenable group. Then β(Ll) is the smallest

l l closed right ideal containing the essential idempotents. Thus brl(D )=L ,andE ⊆ G forces essential recurrence if and only if E ∈Ll.

Proof. By Theorem 5.2.5, only the ﬁrst statement must be proven. We must show

86 that if E ∈Ll then there is some idempotent p ∈ β(Ll)andsomeq ∈ βG with E ∈ p · q.

l So assume that E ∈L.Let(Xs,G) be the shift space of Chapter 3. Let x =1E ∈

Xs, and let X = O¯(x). Then (X, G) is a compact topological dynamical system. Let U = {z ∈ X : z(e)=1}. By Proposition 3.1.4, there is invariant µ ∈ M(X)such that µ(U) > 0.

So by Lemma 1.2.7 there is some essentially recurrent point y ∈ U.ByLemma

l 5.2.2, there is some idempotent p ∈ β(L ) with p- lim Tgy = y. Since y ∈ X = O¯(x), g there is some q ∈ βG with q- lim Tgx = y.Thus g

(p · q)- lim Tgx = p- lim Ts(q- lim Ttx)=p- lim Tsy = y. Since U is a closed neighbor- g s t s hood of y,onemusthaveE = RU (x) ∈ p · q.

6.2 Consequences

As was proven in Chapter 3, if G is not a virtually WM group, then for every ϵ>0

∗ there exists some E ⊆ G with dL(E) > 1 − ϵ and such that no shift of E contains any IP set. In particular, no shift of E is a D-set. This result should be contrasted with the relation between piecewise syndetic sets and central sets: Any left piecewise syndetic set is a shift of some central set. Thus, the two-sided ideal β(Ll) fails to be as “nice” as the ideal K((βG,·)). By Theorem 6.1.1, though, this failure is not complete.

Proposition 6.2.1. If E ⊆ G has positive left upper Banach density, then there is some left D-set B ⊆ G so that for all ﬁnite H ⊆ B there is some g ∈ G with Hg ⊆ E.

It was shown in [4] that any D-set E ⊆ Z contains solutions to all Rado systems.

Recall that a Rado system is a p × q matrix A ∈ Zp×q (for some p, q)suchthatfor

87 r every ﬁnite partition N = Ci of the positive integers there is some 1 ≤ i ≤ r and i=1 $ t some x1,...,xq ∈ Ci so that A(x1,...,xq) =(0...,0). The following corollary is now immediate from Proposition 6.2.1.

Corollary 6.2.2. If A ∈ Zp×q is any Rado system and E ⊆ Z has positive upper

Banach density, then there is some t ∈ Z and there are some x1,...,xq ∈ E so that

t A(x1 + t, . . . , xq + t) =(0,...,0).Inwords,E contains some shift of a solution to any Rado system.

If (X, G) is any left topological dynamical system, we say that (X, G) is anti- essentially-recurrent if (X, G) is anti-Ll-recurrent. The importance of the following two results will become more apparent in Chapter 7, where we show that anti- essential-recurrence is equivalent to a more familiar condition on the system (X, G).

Corollary 6.2.3. The following are equivalent:

i) The system (X, G) is anti-essentially-recurrent.

ii) The system (X, G) has a ﬁxed point and is uniquely ergodic.

iii) There is m ∈ X so that, for every for every essential idempotent p ∈ (βG,·) and every x ∈ X, p- lim Tsx = m. s l iv) There is m ∈ X so that, for every x ∈ X and every q ∈ β(L ), q- lim Tsx = m. s v) There is m ∈ X so that for every neighborhood U of m and every x ∈ X,one has RU (x) has left Banach density 1.

Proof. Noting that the collection of all sets having left Banach density 1 is equal to

(Ll)∗, the equivalence of i) with iii)-v) is an immediate consequence of Theorems 6.1.1 and 5.3.4. The equivalence of i) and ii) follows immediately from Lemma 6.1.

Corollary 6.2.4. Let (Xs,G) denote the shift space. Let E ⊆ G,letx =1E ∈ Xs, and let X = O¯(x).Thenthesubshift(X, G) is anti-essentially-recurrent if and only

∗ if either dL(E)=0or E has left Banach density 1. 88 Proof. This follows immediately from Theorems 5.2.3 and 5.3.4.

6.3 Noncentral D-sets

In this section we ﬁnally establish the existence of noncentral D-sets in amenable groups. We begin by generalizing Theorem 1.1.10.

Theorem 6.3.1. The following are equivalent:

i) There is some set E ⊆ G having positive left upper Banach density which is not left piecewise syndetic (succinctly, Ll(G) ≠ PSl(G)).

ii) There is some left D-set E ⊆ G which is not left piecewise syndetic (and hence not left central).

iii) There is some antiminimal compact metrizable topological dynamical system

(X, G) with unique minimal point m ∈ X and some G-invariant probability Borel measure µ on X with µ ≠ δm.

Proof. i) =⇒ iii): Let (Xs,S) denote the shift space, let E be as assumed in i), and let x =1E ∈ Xs.LetX = O¯(x). By Corollary 5.3.9, 0¯ is the only minimal point in X.LetU = {z ∈ X : z(e)=1}.ThenE = RU (x), so there is some invariant measure µ with µ(U) > 0. Clearly, then, µ ≠ δ0¯.

iii) =⇒ ii): Let (X, G) be as in iii). Since µ ≠ δm, there is some open U ⊆ X with m/∈ U¯ and µ(U) > 0. Then, by Lemma 6.1 there is some essentially recurrent point

l x ∈ U.LetE = RU (x). There is some idempotent p ∈ β(L ) with p- lim Tgx = x.It g follows that E ∈ p,soE ∈D. It remains to show that E is not left piecewise syndetic. If E were left piecewise syndetic, then by Theorem 5.2.7, there would be some uniformly recurrent point y ∈ U¯.Butm/∈ U¯ is the only uniformly recurrent point in X,soE is not left piecewise syndetic.

89 ii) =⇒ i) is trivial.

Thus, just as in Chapter 1, it will suﬃce to show that there is a set E ⊆ G which has positive left upper Banach density but which is not left piecewise syndetic.

Note that if G is not a virtually WM group then this has already been established in

Chapter 4, but if G is a virtually WM group then the techniques of Chapter 4 will fail to produce such a set. The existence, for any countable amenable group G,ofa left D-set E ⊆ G which is not left piecewise syndetic was recently established in [13].

Theorem 6.3.2 ([13, Theorem 1.9]). Let G be a countable amenable group and let

ϵ>0.LetF =(Fn)n∈N be any left Følner sequence in G.thenthereissomeE ⊆ G with dF (E) > 1 − ϵ such that E is not left piecewise syndetic.

Corollary 6.3.3. For any countable amenable group G there is some left D-set E ⊆ G which is not left piecewise syndetic. Furthermore, there is anontrivialantiminimal topological dynamical system (X, G) which admits an invariant measure of full support.

Thus, despite the failure of the techniques of the Introduction to produce non- piecewise syndetic D-sets in virtually WM amenable countable groups, there is a non-piecewise syndetic D-set E ⊆ G for any countable amenable group G.

90 CHAPTER 7

AFFINE ACTIONS OF AMENABLE GROUPS

Let V be a locally convex topological linear space, let X ⊆ V be compact and convex.

If a group G acts on X by continuous aﬃne maps Tg : X → X, g ∈ G, then the left topological dynamical system (X, G) is called an aﬃne system.

If V is a reﬂexive Banach space then the weak and weak∗ topologies on V coincide. Let

H = {T : V → V : T is linear and bounded and ∥T ∥≤1}.

Let

B1 = {v ∈ V : ∥v∥≤1}.

Then B1 is weakly compact by the Banach-Alaoglu theorem and every T ∈ H is (weak-to-weak) continuous. So if G is any group, and φ : G → H is a homomorphism, then (B1,G) is an aﬃne action of G on the (weakly) compact convex set B1. Such systems arise naturally in dynamics, as many properties of a (topological or measure-preserving) dynamical system are best understood (or are even deﬁned) in terms of some “induced” representation. The Koopman representation associated with a measure-preserving dynamical system (X, B,µ,(Tg)g∈G) is one famous example of this phenomenon. Another example is provided by considering, for a topological dynamical system (X, G), the system induced on the set M(X) of all Borel probability measures on X, which is a weakly compact convex subset of C(X)∗.

91 We will apply the results of Chapters 5 and 6 to the study of aﬃne systems (X, G), where G is a countable amenable group.

7.1 Proximality and Strong Proximality

In this section we record some relevant facts about proximal and strongly proximal topological dynamical systems. We follow [31]. Recall that a topological dynamical system is proximal if every pair (x, y) ∈ X × X is a proximal pair, and that if G is abelian then a system (X, G) is proximal if and only if (X, G) is anti-uniformly- recurrent1. In fact, this property is shared by a much larger class of groups.

Deﬁnition 7.1.1. AcountablegroupG is strongly amenable if every minimal and proximal topological dynamical system (X, G) is necessarily trivial.

One easily veriﬁes that a countable group G is strongly amenable if and only if the notions of proximality and anti-uniform-recurrence coincide for topological dynamical systems (X, G).

Proposition 7.1.2. If G is a countable group, and if there is a nilpotent subgroup H ≤ G with ﬁnite index, then G is strongly amenable.

Proof. This is a special case of [31, Theorem 3.4].

It is worth noting that there exist countable amenable groups which aren’t strongly amenable. Note that proving the existence of a countable amenable group G which is not strongly amenable is equivalent to proving the existence of a topological dynamical system (X, G) with G countable and amenable such that (X, G) is proximal but not antiminimal.

1This is the content of Lemma 1.2.1, which we proved in Chapter 5 (see the discussion following Corollary 5.3.6).

92 The following example is given in [31] and is attributed there to Furstenberg. Let V be the character group of the discrete group (Q, +). We denote the identity in V by 0, even though it is actually the constant function 0(q) = 1 for all q ∈ Q.

Since Q is countable and discrete, V is compact metrizable, and the topology on

V is the topology of pointwise convergence. For q ∈ Q and v ∈ V ,deﬁneqv ∈ V by qv(s)=v(qs) for all s ∈ Q. This makes V into a vector space over Q,andthe map v 0→ qv is continuous for all q ∈ Q.LetU ⊆ V be a countable dense set (every compact metric space second countable, and therefore it is ﬁrst countable).

We will ﬁnd matrix notation useful. Let

v X = ⎧⎛ ⎞ : v ∈ V ⎫ ⎨⎪ 1 ⎬⎪ ⎜ ⎟ ⎝ ⎠ and ⎩⎪ ⎭⎪ qu S = ⎧⎛ ⎞ : q ∈ Q \{0},u∈ U⎫ . ⎨⎪ 01 ⎬⎪ ⎜ ⎟ ⎝ ⎠ ⎪ ⎪ ⎩ v ⎭ The map Φ: V → X given by Φ(v)=⎛ ⎞ is a bijection. Endow X with the 1 ⎜ ⎟ topology it inherits from V . That is, a⎝ set A⎠⊆ X is open if and only if the set v {v ∈ V : ⎛ ⎞ ∈ A} is open. 1 ⎜ ⎟ S is a⎝ (countable)⎠ group under matrix multiplication, and since S is solvable, S is amenable. qu v For s = ⎛ ⎞ ∈ S and x = ⎛ ⎞ ∈ X, let 01 1 ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ qv + u Tsx = Sx = ⎛ ⎞ ∈ X. 1 ⎜ ⎟ ⎝ ⎠ 93 This makes (X, S) into a compact topological dynamical system. Certainly the v system (X, S) is nontrivial. Furthermore, given ⎛ ⎞ ∈ X and an open subset W of 1 ⎜ ⎟ ⎝ ⎠ v v u ⎛ ⎞, there is some u ∈ U and there is some q ∈ Q such that q ⎛ ⎞ + ⎛ ⎞ ∈ 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ W⎝ .Then⎠ ⎝ ⎠ ⎝ ⎠ qu v ⎛ ⎞ ⎛ ⎞ ∈ W. 01 1 ⎜ ⎟ ⎜ ⎟ So, every orbit is dense in X,and⎝ X is⎠ minimal.⎝ ⎠ n 0 Now, for n ∈ N, let sn = ⎛ ⎞ ∈ S.Thensn ·sm = snm,andthemapn 0→ sn 01 ⎜ ⎟ is an injective homomorphism⎝ of the⎠ semigroup (N, ·) into S. We now consider the topological dynamical system (X, (Tsn )n∈N) with acting semigroup (N, ·). Note that, in this system, 0 x0 = ⎛ ⎞ 1 ⎜ ⎟ is a ﬁxed point. ⎝ ⎠ v N Fix any x = ⎛ ⎞ ∈ X.LetY = {Tsn x : n ∈ } be the orbit-closure of x under 1 ⎜ ⎟ ⎝ ⎠ v (Tsn )n∈N.Themapv 0→ ⎛ ⎞ identiﬁes Y with the set 1 ⎜ ⎟ ⎝ ⎠ W = {nv : n ∈ N}⊆V.

Note that W is a closed subsemigroup of the compact group V . So, by Ellis’ theorem on idempotents, there is some idempotent in W . Since V is a group, this idempotent is equal to 0. So 0 ∈ W and x0 ∈ Y .

Thus the ﬁxed point x0 is in the orbit closure of every x ∈ X (under (Tsn )n∈N),

94 and it follows that (X, (Tsn )n∈N) is antiminimal with unique ﬁxed point x0.So

(X, (Tsn )n∈N) is proximal. Thus the system (X, S) is also proximal, and since it is minimal and nontrivial it is not antiminimal.

To summarize: We start with a proximal (and necessarily antiminimal) action of the commutative semigroup (N, ·), then extend the acting semigroup to the group S.

Since the original action was proximal, this extended action is also proximal. But the extended action now “has enough transformations to make the action minimal”.

Furthermore, S is solvable, hence amenable.

The relationship between strongly amenable groups and proximal dynamical systems is mirrored by the relationship between amenable groups and “strongly proximal” dynamical systems.

Deﬁnition 7.1.3. Atopologicaldynamicalsystem(X, G) is strongly proximal if the induced system (M(X),G) is proximal.

Identifying X with the extreme points of M(X), and noting that a subsystem of a proximal system is proximal, we see that every strongly proximal topological dynamical system is proximal. The term “strongly amenable” seems to imply that any such group is amenable. This is indeed the case, as is seen most readily by noting the following characterization of amenable groups.

Theorem 7.1.4 (see, e.g., [31, Theorem 3.1]). Let G be a countable group. Then the following are equivalent:

1. G is amenable.

2. Every topological dynamical system (X, G) has an invariant measure.

3. Every aﬃne system (X, G) has a ﬁxed point.

95 4. Every minimal and strongly proximal topological dynamical system (X, G) is trivial.

Note that condition 4 of the above theorem, which clearly implies that every strongly amenable group is amenable, also establishes an analogy between, on the one hand, strongly proximal actions of amenable groups, and on theotherhand proximal actions of strongly amenable groups.

Note also that the example constructed above of a nontrivial and proximal dynamical system (X, G) with G amenable is also, by Theorem 7.1.4, an example of a proximal but not strongly proximal system.

We record here the following basic observations relating proximality to anti- uniform-recurrence.

Fact 7.1.5. If G is any countable group, then a topological dynamical system (X, G) is anti-uniformly-recurrent if and only if (X, G) is proximal and has a ﬁxed point.

Fact 7.1.6. If G is a strongly amenable countable group, then a topological dynamical system (X, G) is anti-uniformly-recurrent if and only if (X, G) is proximal (this is in fact a rephrasing of the deﬁnition of strong amenability).

Fact 7.1.7. If G is an amenable countable group, and (X, G) is an affine system, then (X, G) is anti-uniformly-recurrent if and only if (X, G) is proximal. This follows from Fact 7.1.6 and condition 3) of Theorem 7.1.4. Thus, the difference between the condition that G be amenable and the condition that G be strongly amenable is not witnessed by an affine action (X, G).

We now turn our attention to establishing similar, but less trivial, results relating anti-essential-recurrence to strong proximality.

96 7.2 Strong Proximality and Anti-essential-recurrence

In this section we prove the equivalence, for an amenable countable group G,ofthe strong proximality of a topological dynamical system (X, G) with the anti-essential- recurrence of (X, G). We will also prove that for any aﬃne system (X, G) with G amenable, the notions of proximality and strong proximality coincide. This fact will be exploited in the remaining sections of this chapter to study the analogy between mixing properties of a measure preserving dynamical system X¯ =(X, B,µ,(Tg)g∈G) and “twisting” properties of a topological dynamical system (X, G), as introduced in

[50]. We begin with a couple of lemmas. The proof of our ﬁrst lemma will require the use of the “barycenter map”.

Proposition 7.2.1 ([49, Proposition 1.1] and [49, Proposition 1.4]). Let X be a compact convex subset of a locally convex topological linearspaceV .Foreachµ ∈

M(X),thereisauniqueelementB(µ) of X such that for all l ∈ V ∗ one has

l(x)dµ(x)=l(B(µ)). 'X Furthermore, the map B : M(X) → X is aﬃne and continuous (when M(X) is given the weak∗ topology).

−1 For x ∈ X, B ({x})={δx} if and only if x ∈ Ext(X).

Lemma 7.2.2. If G is a countable group and (X, G) is an aﬃne system which has an extreme point as the unique ﬁxed point, then (X, G) is anti-essentially-recurrent.

Proof. Let E =Ext(X) denote the extreme points of X, and let x ∈ E be the unique

ﬁxed point of (X, G). We must show that δx is the only ﬁxed point of the system (M(X),G). Let B : M(X) → X denote the barycenter map of proposition 7.2.1.

Then B :(M(X),G) → (X, G) is a homomorphism of topological dynamical systems,

−1 and for any e ∈ E, B (e)={δe}. 97 Let m ∈ M(X) be a ﬁxed point of the system (M(X),G). Then B(m) is a ﬁxed point of the system (X, G), so B(m)=x. Since x ∈ E, it follows that m = δx.

Lemma 7.2.3. If G is an amenable countable group and (X, G) is a proximal aﬃne system, then (X, G) is anti-essentially-recurrent.

Proof. As noted in Fact 7.1.7, (X, G) has a unique ﬁxed point, call it x. Since each

Tg : X → X is an aﬃne bijection, E =Ext(X) is an invariant set. By proximality, now, we conclude that x ∈ E. Hence, by Lemma 7.2.2, (X, G) is anti-essentially- recurrent.

We are now prepared to state the main result of this section. It is an analog of

Fact 7.1.6, but now concerning strongly proximal actions of amenable groups. Note, however, that while Fact 7.1.6 is an immediate consequence of the deﬁnitions, the following theorem is nontrivial. Furthermore, whereas the notion of anti-uniform- recurrence makes sense even for an acting group which is not strongly amenable (and for such groups is actually stronger than proximality), the notion of anti-essential- recurrence can only be deﬁned for amenable groups.

Theorem 7.2.4. If G is a countable amenable group, then a topological dynamical system (X, G) is strongly proximal if and only if it is anti-essentially-recurrent.

Proof. Assume (X, G) is strongly proximal. There is some minimal subsystem of

(X, G), which must be a ﬁxed point m ∈ X by Theorem 7.1.4. Thus δm ∈ M(X) is a ﬁxed point of (M(X),G). By proximality of (M(X),G), now, (X, G) is uniquely ergodic. Thus (X, G) is anti-essentially-recurrent by Corollary 6.2.3.

Now assume that (X, G) is anti-essentially-recurrent. Let x ∈ X be the unique

ﬁxed point of (X, G). Then by Corollary 6.2.3, δx is the only ﬁxed point of the system

(M(X),G). Since δx is an extreme point of M(X), it follows from Lemma 7.2.2 that

98 (M(X),G) is anti-essentially-recurrent. Let q ∈ β(Ll). Then for any m, n ∈ M(X), q- lim Tgm = q- lim Tgn = δx. This clearly implies that (M(X),G) is proximal. g g

Corollary 7.2.5. If G is a countable amenable group and if (X, G) is an aﬃne system, then the following are equivalent:

1. The system (X, G) is proximal.

2. The system (X, G) is strongly proximal.

3. The system (X, G) is uniquely ergodic.

4. The system (X, G) has a ﬁxed point x such that for every neighborhood U of x,

l ∗ and for every y ∈ X,thesetRU (y) is a (PS ) set.

5. The system (X, G) has a ﬁxed point x such that for every neighborhood U of x,

and for every y ∈ X,thesetRU (y) has (left) Banach density 1.

6. The system (X, G) has a ﬁxed point x such that for every minimal idempotent

ultraﬁlter p ∈ (βG,·) and every y ∈ X one has p- lim Tgy = x. g

7. The system (X, G) has a ﬁxed point x such that for every essential idempotent

ultraﬁlter p ∈ (βG,·) and every y ∈ X one has p- lim Tgy = x. g

8. The system (X, G) has a ﬁxed point x such that for every ultraﬁlter q ∈ K(βG,·)

and every y ∈ X one has q- lim Tgy = x. g

9. The system (X, G) has a ﬁxed point x such that for every ultraﬁlter q ∈ β(Ll)

and every y ∈ X one has q- lim Tgy = x. g

Proof. 1) =⇒ 2) is Lemma 7.2.3 in conjunction with Theorem 7.2.4. By the amenability of G there is some ﬁxed point x ∈ X, so unique ergodicity of (X, G) is equivalent to anti-essential-recurrence of (X, G). The rest now follows from Corollaries 6.2.3 and

5.3.6 99 Corollary 7.2.6. If G is a countable amenable group and (X, G) is a topological dynamical system, then the following are equivalent:

1. The system (X, G) is strongly proximal.

2. The system (X, G) has a ﬁxed point and is uniquely ergodic.

3. The system (X, G) has a ﬁxed point m such that for every essential idempotent

ultraﬁlter p ∈ (βG,·) and every x ∈ X one has p- lim Tgx = m. g

4. The system (X, G) has a ﬁxed point m such that for every ultraﬁlter q ∈ β(Ll)

and every x ∈ X one has q- lim Tgx = m. g

5. The system (X, G) has a ﬁxed point m such that for every neighborhood U of

m,andforeveryx ∈ X,thesetRU (x) has (left) Banach density 1.

6. The system (M(X),G) is strongly proximal.

7. The system (M(X),G) is uniquely ergodic.

8. There is some ﬁxed point m ∈ X such that for every µ ∈ M(X),everyminimal

idempotent p ∈ (βG,·),andeveryf ∈ C(X),onehas

p- lim f(Tgx)dµ(x)=f(m). g ' 9. There is some ﬁxed point m ∈ X such that for every µ ∈ M(X),everyq ∈

K((βG,·)),andeveryf ∈ C(X),onehas

q- lim f(Tgx)dµ(x)=f(m). g ' 10. There is some ﬁxed point m ∈ X such that for every µ ∈ M(X),everyϵ>0,

and every f ∈ C(X),onehas

l ∗ {g ∈ G : | f(Tgx)dµ(x) − f(m)| <ϵ}∈(PS ) . ' 100 11. There is some ﬁxed point m ∈ X such that for every µ ∈ M(X),everyessential idempotent p ∈ (βG,·),andeveryf ∈ C(X),onehas

p- lim f(Tgx)dµ(x)=f(m). g ' 12. There is some ﬁxed point m ∈ X such that for every µ ∈ M(X),everyq ∈

β(Ll),andeveryf ∈ C(X),onehas

q- lim f(Tgx)dµ(x)=f(m). g ' 13. There is some ﬁxed point m ∈ X such that for every µ ∈ M(X),everyϵ>0,

and every f ∈ C(X),theset

{g ∈ G : | f(Tgx)dµ(x) − f(m)| <ϵ} ' has left Banach density 1.

Proof. After Theorem 7.2.4, the equivalence of 1) with 2) - 5) is a restatement of

Corollary 6.2.3. The equivalence of 1) with 6) - 13) is now an application of Corollary

7.2.5 to the aﬃne system (M(X),G).

Example 7.2.7. Let E ⊆ G be a set having positive upper Banach density which is not piecewise syndetic. Let (Xs,G) be the shift space, and let x =1E ∈ Xs.LetX = O¯(x).Then,byCorollary5.3.6,Corollary6.2.4,andTheorem7.2.4, (X, G) is an example of a system which is antiminimal (hence proximal) butnotstronglyproximal.

Thus every countable amenable group admits an antiminimal but not strongly proximal topological dynamical system by Theorem 6.3.2.

7.3 Mixing Properties of Measure Preserving Dynamical Sys-

tems

Let G be a countable group. 101 We now apply our results to the notion of F-mixing for probability measure preserving systems X¯ =(X, B,µ,(Tg)g∈G). For ϵ>0andA, B ∈Bwith min(µ(A),µ(B)) > 0, put

ϵ −1 RA,B = {g ∈ G : µ(Tg A ∩ B) >µ(A)µ(B) − ϵ} and put

¯ ϵ R(X)={RA,B : ϵ>0and{A, B}⊆Bwith min(µ(A),µ(B)) > 0}.

The following notion was studied in [45], where it is called “F ∗-convergence ergodic”.

Deﬁnition 7.3.1. Let F be a family of subsets of G and let X¯ =(X, B,µ,(Tg)g∈G) be any measure preserving system. Then X¯ is F-mixing if R(X¯) ⊆F∗.

Now ﬁx some measure preserving system X¯ =(X, B,µ,(Tg)g∈G). For g ∈ G and

2 2 f ∈ L (µ), let Ugf = f ◦ Tg. This yields a unitary antirepresentation of G on L (µ). We will restrict our attention to the following subset of L2(µ). Let

L(X¯)={f ∈ L2(µ): fdµ =0and∥f∥≤1}. ' Equip L(X¯) with the topology induced by the weak∗ topology on L2(µ). Then

(L(X¯),G) is a compact right topological dynamical system, which is metrizable if (X, B,µ) is separable. Due to the fact that we are studying a right action of G on

L(X¯), the right families IPr, PSr, Lr will be most signiﬁcant. Also it is the operation ◦ in βG, along with the closed ideals of (βG,◦), which will correspond to notions of recurrence. So, for the remainder of this section, we will use the right versions (as opposed to the left versions) of results in Chapter 5. For a family F, let

r Fi = {E ⊆ G : There is some p = p ◦ p ∈ βF such that E ∈ p}.

102 Theorem 7.3.2. Let F be a family of subsets of G having the Ramsey property. Let

X¯ =(X, B,µ,(Tg)g∈G) be any measure preserving system. Then the following are equivalent:

i) The system X¯ is F-mixing.

2 ii) For all f1,f2 ∈ L (µ) and all p ∈ β(F) one has

p- lim ⟨Ugf1,f2⟩ = f1dµ f2dµ. g ' '

iii) For every p ∈ β(F) and every f ∈ L(X¯) one has p- lim Ugf =0(weakly). g If F is right shift invariant (equivalently, if β(F) ⊆ (βG,◦) is a left ideal) then i)-iii) are also equivalent to

iv) For every p ∈ β(F) and every f ∈ L(X¯) one has p- lim ⟨Ugf,f⟩ =0. g If β(F) is the smallest closed left ideal of (βG,◦) containing the idempotents in

(β(F), ◦),theni)-iii)abovearealsoequivalentto:

v) The right topological dynamical system (L(X¯),G) is anti-F-recurrent.

vi) For every idempotent p = p ◦ p ∈ β(F) and every f ∈ L(X¯) one has p- lim Ugf =0. g ¯ r ∗ vii) R(X) ⊆ (Fi ) .

Proof. i) =⇒ ii): Note ﬁrst that if A, B ∈Band X¯ is F-mixing, then

−1 ϵ ϵ {g ∈ G : |µ(Tg A ∩ B) − µ(A)µ(B)| <ϵ}⊇RA,B ∩ RA,X\B is in F ∗.

Fix p ∈ β(F). For f ∈ L2(µ), let

2 Hf = {f2 ∈ L (µ):p- lim ⟨Ugf,f2⟩ = fdµ f2dµ}. g ' '

103 2 Hf is a linear subspace of L (µ): if a, b ∈ C and f2,f3 ∈Hf then

p- lim ⟨Ugf,(af2 + bf3)⟩ =¯ap- lim(⟨Ugf,f2⟩)+¯bp- lim(⟨Ugf,f3⟩) g g g ¯ =¯a fdµ f2dµ + b fdµ f3dµ ' ' ' '

= fdµ (af2 + bf3)dµ. ' ' Furthermore, Hf is norm closed. Now let

2 2 H = {f ∈ L (µ):Hf =L(µ)}.

One similarly veriﬁes that H is also a closed linear subspace of L2(µ). By i), if A ∈B

2 satisﬁes µ(A) > 0, and f ∈ L (µ) is the equivalence class containing 1A,thenfor

2 every B ∈Bwith µ(B) > 0, the element fB of L (µ) which contains the function 1B

2 satisﬁes fB ∈Hf . Since the closed linear span of {fB : µ(B) > 0} is equal to L (µ), f ∈H.Now,forthesamereason,H =L2(µ).

2 ii) =⇒ iii): Assume ii). If f1 ∈ L(X¯), then for every f2 ∈ L (µ)andp ∈ β(F),

p- lim ⟨Ugf1,f2⟩ = f1dµ f2dµ =0. g ' ' iii) =⇒ i): Assume iii), let A, B ∈Beach have positive measure, and let ϵ>0. Let

2 fA be the equivalence class in L (µ) containing 1A, and similarly deﬁne fB. Finally, let f1 = fA − µ(A)andf2 = fB − µ(B). Then for any p ∈ β(F)onehas:

−1 0=p- lim ⟨Ugf1,f2⟩ = p- lim µ(Tg A ∩ B) − µ(A)µ(B). g g

−1 ϵ So p- lim µ(Tg A ∩ B)=µ(A)µ(B), and RA,B ∈ p. Since p ∈ β(F) is arbitrary, g ϵ ∗ RA,B ∈F . Assume now that F is right shift invariant. Clearly iii) =⇒ iv) in any event.

iv) ⇒ iii): Assume iv), and let f ∈ L(X¯). Fix q ∈ β(F), let z = q- lim Ugf,and g let

H = span{Ugf : g ∈ G}. 104 2 2 Let P : L (X, B,µ) → L (X, B,µ) be the orthogonal projection onto H. Fix g0 ∈ G.

∗ Let ϵ>0 and let E = {g ∈ G : |⟨Ugf,f⟩|<ϵ}. By assumption E ∈F ,so

∗ {g ∈ G : |⟨Ugf,Ug0 f⟩|<ϵ} = Eg0 ∈F .

Thus {g ∈ G : |⟨Ugf,Ug0 f⟩| <ϵ}∈q and, since ϵ>0 is arbitrary, ⟨z, Ug0 f⟩ = q- lim ⟨Ugf,Ug f⟩ = 0. By linearity, ⟨z, h⟩ = 0 for all h ∈ span{Ugf : g ∈ G}.Now, g 0 by continuity of the inner product, ⟨z, h⟩ = 0 for all h ∈H. But clearly z ∈H,so z =0.

r Finally, if F =brr(Fi ), then condition iii) is equivalent to v)-vii) by Theorem 5.3.4.

The following notion has been studied in [18].

Lemma 7.3.3. Let In : G → G be the inverse function: In(x)=x−1.ThenIn extends uniquely to a continuous function In : βG → βG which is given by:

In(p)={E−1 : E ∈ p}.

We will write In(p)=p−1.

Furthermore, the following relation holds for all p, q ∈ βG:

q−1 ◦ p−1 =(p · q)−1.

Proof. Let i : G → βG be the natural injection (i(g)=¯g). Then i ◦ In : G → βG is a compactification of G, hence extends uniquely to a continuous function defined on βG by the universal property of the Stone-Cechˇ compactification. Now note that p 0→ p−1 = {E−1 : E ∈ p} is a continuous extension of i ◦ In, hence is precisely the unique continuous extension.

Finally, compute:

105 q−1 ◦ p−1 = {E ⊆ G : {x : Ex−1 ∈ q−1}∈p−1}

= {E ⊆ G : {x : Ex ∈ q−1}∈p}

= {E ⊆ G : {x : x−1E−1 ∈ q}∈p}

= {E−1 : {x : x−1E ∈ q}∈p}

=(p · q)−1.

Lemma 7.3.4. Let X¯ =(X, B,µ,(Tg)g∈G) be any measure preserving system Let p ∈ βG.Thenthefollowingareequivalent:

i) For all f ∈ L(X¯), p- lim Ugf =0. g −1 ii) For all f ∈ L(X¯),p - lim Ugf =0. g

Proof. Since p =(p−1)−1, it suﬃces to show that i) implies ii). So let

X¯ =(X, B,µ,(Tg)g∈G) be any measure preserving system , and let p ∈ βG satisfy i). Let U be a basic neighborhood of 0 ∈ L(X¯). So there is ϵ>0andtherearen ∈ N

2 and h1,...,hn ∈ L (X, B,µ)sothat

U = {h ∈ L(X¯) : for all 1 ≤ i ≤ n, |⟨h, hi⟩|<ϵ}.

Now, ﬁx i ∈{1,...,n}. Since p- lim Ughi =0,theset g

Ei = {g ∈ G : |⟨Ughi,f⟩| <ϵ}

is an element of p.Forg ∈ Ei,

|⟨Ug−1 f,hi⟩|= |⟨f,Ughi⟩|= |⟨Ughi,f⟩| = |⟨Ughi,f⟩|<ϵ,

106 −1 and thus Ei ⊆{g ∈ G : |⟨Ugf,hi⟩|<ϵ}. By upward heredity,

−1 {g ∈ G : |⟨Ugf,hi⟩| <ϵ}∈p . Since i ∈{1,...,n} is arbitrary, we now conclude that n −1 {g ∈ G : Ugf ∈ U} = {g ∈ G : |⟨Ugf,hi⟩|<ϵ}∈p . i=1 % −1 Since U is an arbitrary basic neighborhood of 0, it follows that p - lim Ugf =0. g Since f ∈ L(X¯) is arbitrary, the proof is now complete.

Applying the above results to the families IPr, PSr, and if G is amenable to Lr gives a list of conditions equivalent to weak mixing and a list of conditions equivalent to mild mixing for a measure preserving system.

2 Fix a countable group G.Avectorf ∈ L (µ) is compact if O¯(f)={Ugf : g ∈ G} (where the closure is taken in the norm topology) is compact (in the norm topology).

A measure preserving system X¯ =(X, B,µ,(Tg)g∈G) is weak mixing if there are no nonconstant compact vectors f ∈ L2(µ). Weak mixing has many other characterizations, see [58] for G = Z, and [21] or [12] for more general groups G. The following lemma is [12, Theorem 4.3]. However, since we are considering an antirepresentation rather than a representation, the proof requires some translation, which we provide for the reader’s convenience.

Lemma 7.3.5 ([12, Theorem 4.3]). Let X¯ =(X, B,µ,(Tg)g∈G) be a measure preserving system and let f ∈ L2(µ).Thefollowingareequivalent:

1. f is a compact vector.

2. For every idempotent p = p ◦ p ∈ (βG,◦) one has p- lim Ugf = f. g

3. There is some minimal idempotent p = p ◦ p ∈ K(βG,◦) such that

p- lim Ugf = f. g

107 Proof. Clearly 2 =⇒ 3. We now show that 1 =⇒ 2.

Assume that the norm closure K = {Ugf : g ∈ G} is compact in the norm topology. Then (K, G) is an isometric topological dynamical system, and therefore any proximal pair (x, y) ∈ K × K satisﬁes x = y.Letp(βG,◦) be any idempotent, and let h = p- lim Ugf. Since p- lim Ugh = p- lim Ugf = h,(f,h) is a proximal pair, and g g g thus h = f.

3=⇒ 1: Assume that there is some p = p◦p ∈ K(βG,◦)suchthatp- lim Ugf = f. g

Let ϵ>0. Let E = {g ∈ G : ∥Ugf − f∥ <ϵ/4}.ThenE ∈ p, so by (the right-handed version of) [38, Theorem 4.39], the set B = {x ∈ G : Ex−1 ∈ p} is right syndetic.

−1 Note that for x ∈ B, E ∩ Ex ∈ p,andtherearee1,e2 ∈ E such that e1x = e2. So x ∈ E−1E.ThusB ⊆ E−1E,andE−1E is right syndetic. Note also that for

−1 −1 e1,e2 ∈ E, ∥Ue1 f − ue2 f∥ <ϵ/2. So for g = e1 e2 ∈ E E,onehas

−1 ∥Ugf − f∥ = ∥U (Ue1 f) − f∥ = ∥Ue1 f − Ue2 f∥ <ϵ/2, e2

−1 so Diam({Ugf : g ∈ E E}) <ϵ. n −1 −1 Finally, choose n ∈ N and g1,...,gn ∈ G such that G = (E E)gi .Let i=1 −1 −1$ −1 V = {Ugf : g ∈ E E} and for 1 ≤ i ≤ n let Vi = {Ugf : g ∈ (E E)g } = U −1 (V ). i gi

Then for each 1 ≤ i ≤ n one has Diam(Vi) <ϵ(since each U −1 is an isometry) and gi n

{Ugf : g ∈ G} = Vi. Since ϵ>0 is arbitrary, we conclude that K = {Ugf : g ∈ G} i=1 is compact. $

The following theorem gives a number of conditions characterizing weak mixing.

While most of these characterizations are already known (see, for example, [12, The- orem 4.7] and [14, Theorem 1.6]), our proof diﬀers from the traditional methods.

Corollary 7.3.6. Let G be a discrete countable group. Then, for a measure preserving system X¯ =(X, B,µ,(Tg)g∈G) the following are equivalent: i) The system X¯ is weakly mixing. 108 ii) For every minimal idempotent p ∈ (βG,◦) and every f ∈ L(X¯) one has p- lim Ugf =0. g iii) The right topological dynamical system (L(X¯),G) is proximal.

iv) For every q ∈ K(βG,◦) and every f ∈ L(X¯) one has q- lim Ugf =0. g

v) For every f1,f2 ∈ L(X¯) and every ϵ>0,theset

r ∗ l ∗ {g ∈ G : |⟨Ugf1,f2⟩|<ϵ} is in (Cen ) ∩ (Cen ) .

vi) For every f1,f2 ∈ L(X¯) and every ϵ>0,theset

r ∗ l ∗ {g ∈ G : |⟨Ugf1,f2⟩|<ϵ} is in (PS ) ∩ (PS ) . If also G is assumed amenable, then i)-vi) above are also equivalent tothefollow- ing:

vii) The right topological dynamical system (L(X¯),G) is uniquely ergodic.

viii) The right topological dynamical system (L(X¯),G) is strongly proximal.

ix) For every essential idempotent p ∈ (βG,◦) and every f ∈ L(X¯) one has p- lim Ugf =0(weakly). g r x) For every q ∈ β(L ) and every f ∈ L(X¯) one has q- lim Ugf =0(weakly). g

xi) For every f1,f2 ∈ L(X¯) and every ϵ>0,theset

{g ∈ G : |⟨Ugf1,f2⟩|<ϵ} has lower left and right Banach density 1.

xii) For every f1,f2 ∈ L(X¯) and every ϵ>0,theset

r ∗ l ∗ {g ∈ G : |⟨Ugf1,f2⟩|<ϵ} is in (D ) ∩ (D ) .

∞ Proof. i)⇔ii): If (xn)n is a bounded sequence in a Hilbert space, lim xn = x weakly, =1 n and lim ∥xn∥ = ∥x∥, then lim xn = x in the norm topology. Thus, if f ∈ L(X¯), n n p ∈ (βG,◦) is a minimal idempotent, and z = p- lim Ugf (weakly), then p- lim Ugz = z g g (weakly, and hence in norm by the preceeding comments), and z is a compact vector by Lemma 7.3.5. If X¯ is weakly mixing, we must have z =0.

On the other hand, if f ∈ L2(µ) is a nonconstant compact vector, then 1 f = (f − fdµ) is a nonzero compact vector in L(X¯). By Lemma 1 ∥f − fdµ∥ ' ? 109 7.3.5, there is some minimal idempotent p ∈ (βG,◦) with p- lim Ugf = f (in norm g and hence weakly).

Now the equivalence of ii)-vi) is an application of Theorem 7.3.2 and Lemma

7.3.4 to the family PSr, noting that (PSr)−1 = PSl and that by Lemma 7.3.3

(Cenr)−1 = Cenl.

If G is assumed to be amenable, then the equivalence of vii)-xii) is similarly an application of Theorem 7.3.2 and Lemma 7.3.4 to the family Lr, noting that (Lr)−1 = Ll and that by Lemma 7.3.3 (Dr)−1 = Dl. Finally, if G is amenable then the equivalence of viii) with iii) is Corollary 7.2.5.

Remark 7.3.7. The traditional proof (as in, for example, [14, Lemma 3.4]) that condition ii) of Corollary 7.3.6 is equivalent to condition xi) of the same corollary proceeds by noting that if E ∈Lr then E−1E is right syndetic. The unitary nature of the antirepresentation (Ug)g∈G plays a crucial role in this argument. The proof supplied above, which relied instead on Corollary 7.2.5, is valid in a more general setting. This will become important when we study “weak twisting” in the next section.

We can similarly derive from Corollary 5.3.5 a list of conditions equivalent to mild

2 ∞ mixing. A vector f ∈ L (µ) is rigid if there is some sequence (gn)n=1 ⊆ G \{e} with

2 lim Ugf = f weakly ,andX¯ is mild mixing if there are no nonconstant rigid vectors n→∞ f ∈ L2(µ).

Let us momentarily restrict our attention to Z.DenotebyIP+ the collection of all sets E ⊆ Z such that some shift of E is an IP set. The following characterizations of mild mixing systems X¯ =(X, B,µ,T) have been obtained in the literature.

2 If we instead required that lim Ugf = f strongly (or even almost everywhere, or in measure), n→∞ we’d obtain an equivalent definition (see for example the discussion following Definition 9.6 in [29]). We find that weak convergence works best for our purposes.

110 Theorem 7.3.8. Let X¯ =(X, B,µ,T) be a measure preserving system. Then each of the following conditions is equivalent to the statement that X¯ is mild mixing.

i) ([29], Proposition 9.22) One has R(X¯) ⊆ (IP)∗.

ii) ([44], Theorem 5.5) One has R(X¯) ⊆ (IP−IP)∗.

iii) ([14], Theorem 1.6) One has R(X¯) ⊆ (IP+)∗.

Note that IP ! (IP−IP) ! IP+, so conditions i) - iii) provide successively stronger statements about all mild mixing systems. Also, IP+ ! br(IP), so putting G = Z in the following theorem yields a strengthening of Theorem 7.3.8.

Corollary 7.3.9. Let G be a discrete countable group. Then, for a measure preserving system X¯ =(X, B,µ,(Tg)g∈G) the following are equivalent: i) The system X¯ is mild mixing.

ii) The right topological dynamical system (L(X¯),G) is antirecurrent.

iii) For every idempotent p ∈ (βG,◦) and every f ∈ L(X¯) one has p- lim Ugf =0. g r iv) For every q ∈ β(brr(IP )) and every f ∈ L(X¯) one has q- lim Ugf =0. g

v) For every f1,f2 ∈ L(X¯) and every ϵ>0,theset

r ∗ l ∗ {g ∈ G : |⟨Ugf1,f2⟩|<ϵ} is in (IP ) ∩ (IP ) .

vi) For every f1,f2 ∈ L(X¯) and every ϵ>0,theset

r ∗ l ∗ {g ∈ G : |⟨Ugf1,f2⟩|<ϵ} is in (brr(IP )) ∩ (brl(IP )) .

Proof. i)⇔ii): Any nonzero recurrent point of L(X¯) is a rigid vector. On the other 1 hand, if f ∈ L2(µ) is a nonconstant rigid vector, then f = (f − fdµ) 1 ∥f − fdµ∥ ' is a nonzero recurrent point in L(X¯). Thus X¯ is mild mixing if and only if (L(X¯),G) ? is antirecurrent. The remainder of the theorem now follows by applying Theorem

7.3.2 and Lemma 7.3.4 to the family IPr, noting that by Lemma 7.3.3 one has

(IPr)−1 = IPl.

111 We have established that weak and mild mixing are characterized by the condition that 0 ∈ L(X¯) is the unique F-recurrent point, for some family F.Wenownotethat ergodicity and mixing3 can similarly be characterized by conditions on L(X¯).

Proposition 7.3.10. Let X¯ =(X, B,µ,(Tg)g∈G) be a measure preserving dynamical system.

1. X¯ is ergodic if and only if 0 is the only ﬁxed point of (L(X¯),G).

2. X¯ is weak mixing if and only if 0 is the only minimal point of (L(X¯),G).

3. X¯ is mild mixing if and only if 0 is the only recurrent point of (L(X¯),G).

4. X¯ is mixing if and only if 0 is a universal attractor for the system (L(X¯),G).

Proof. 2 and 3 have already been proven. For 1, note that if f ∈ L2(µ) is a nonconstant invariant vector, then

1 f = f − fdµ 1 ∥f − fdµ∥ - ' . is a nonzero ﬁxed point of L(X¯). Applying? Theorem 7.3.6 to the family

F = {E ⊆ G : E is inﬁnite}, we see that 4 is a restatement of the deﬁnition of mixing.

7.4 Twisting and Weak Twisting

The notions of twisting and weak twisting for a topological dynamical system (X, T) were introduced by Fabrizio Polo in [50]. Here we observe that these notions can be characterized via ultraﬁlters. Since it requires no extra eﬀort, we state our results for any countable amenable group G.

3 The system X¯ =(X, B,µ,(Tg)g∈G) is mixing if for every A, B ∈Band every nonprincipal ultraﬁlter p ∈ βG one has p- lim µ(T −1A ∩ B)=µ(A)µ(B). g g 112 We will consider the maximal equicontinuous factor (K(X),G) of a topological dynamical system (X, G). Recall that for any homomorphism φ :(X, G) → (Y,G), the set

R = {(x, y) ∈ X : φ(x)=φ(y)}⊆X × X is a closed invariant equivalence relation, and that given any closed invariant equivalence relation R ⊆ X × X, G acts naturally on the quotient space X/R in such a way that the projection π : X → X/R is a homomorphism of dynamical systems (see, e.g.,[3, page 23]). The following proposition asserts that every system(X, G) admits a maximal equicontinuous factor.

Proposition 7.4.1 ([3, Chapter 9, Theorem 1]). If (X, G) is any topological dynamical system, there is a smallest closed invariant equivalence relation Seq on X such that the quotient system (X/Seq,G) is equicontinuous.

Example 7.4.2. Let X¯ =(X, B,µ,(Tg)g∈G) be a measure preserving system. Let

2 Hc = {f ∈ L (µ):f is a compact vector} and let

2 Hwm = {f ∈ L (µ):for every minimal idempotent p ∈ (βG,◦),

one has p- lim Ugf =0}. g

2 Then L (µ)=Hc ⊕Hwm (see [12, Theorem 4.5]). Let K = L(X¯) ∩Hc and let

2 P : L (µ) →Hc be the orthogonal projection. Then K is a (weakly) compact invariant set, P :(L(X¯),G) → (K, G) is a factor map, and (K, G) is the maximal equicontinuous factor of (L(X¯),G).

It will be important for us that the maximal equicontinuous factor be minimal.

Since every transitive and equicontinuous system is minimal, it would suﬃce to assume that (X, G) be transitive. Transitivity of the system is a standing assumption in 113 [50]. We will recover the main theorem of [50] under the weaker assumption that the maximal equicontinuous factor is minimal. The following example, communicated to us by Tomasz Downarowicz and Mariusz Lema´nczyk, shows that there are in fact nontransitive systems whose maximal equicontinuous factors are minimal.

Example 7.4.3. There exists a compact topological dynamical system (X, T) which is not transitive but is such that the maximal equicontinuousfactorisminimal.

Proof. In words, our system will consist of three concentric circles, with the same irrational rotation on each, plus a countable orbit “spiraling” between the two outermost circles and a countable orbit “spiraling between” the two innermost circles.

More precisely, let

C1 = {z ∈ C : |z| =1/3},

C2 = {z ∈ C : |z| =2/3}, and

C3 = {z ∈ C : |z| =1}.

Fix some α ∈ C3 which is not a root of unity. For n ∈ Z with n ≥ 0, let

n −n xn = α 5/6+(1− 2 )(1/6) @ A and

n −n yn = α 1/2+(1− 2 )(1/6) . @ A For n ∈ Z with n<0, let

n n xn = α (5/6 − (1 − 2 )(1/6)) and

n n yn = α (1/2 − (1 − 2 )(1/6)) .

114 Finally, let

X = C1 ∪ C2 ∪ C3 ∪{xn : n ∈ Z}∪{yn : n ∈ Z} and deﬁne T : X → X by:

xn+1 if x = xn,n∈ Z ⎧ T (x)=⎪ yn+1 if x = yn,n∈ Z ⎪ ⎨⎪ αx otherwise. ⎪ ⎪ X is compact and T : X → X is⎩⎪ a homeomorphism.

Note that 1 ∈ C3 and x0 =5/6 are proximal:

lim |T n(1) − T n(5/6)| = lim αn − αn[5/6+(1+2−n)(1/6)] n→∞ n→∞ /αn / = lim / (2−n) / n→∞ 6 / / / / =0. / / / /

Also, 2/3 ∈ C2 and x0 are proximal:

lim |T n(2/3) − T n(5/6)| = lim |αn(2/3) − αn[5/6(1 + 2n)(1/6)]| n→−∞ n→∞ αn = lim (−2n) n→−∞ 6 / / / / =0. / / / /

And similarly y0 =1/2 is proximal to both 2/3 ∈ C2 and 1/3 ∈ C1.

Let K = C3.ThenK is a compact invariant set, so (K, T) is a topological dynamical system. Deﬁne φ : X → K by:

n α if x = xn or x = yn,n∈ Z φ(x)=⎧ x otherwise. ⎨⎪ |x| Then φ :(X, T) → (K, T)⎩⎪ is a homomorphism of dynamical systems. We will show that (K, T) is the maximal equicontinuous factor of (X, T). Note that T : K → K is an isometry, so clearly (K, T) is equicontinuous. Let

RK = {(x, y):φ(x)=φ(y)}. 115 If R ⊆ X × X is any closed invariant equivalence relation such that the factor (X/R, T ) is equicontinuous, then since (1, 5/6) is a proximal pair, we must have

n (1, 5/6) ∈ R. By invariance, {(α ,xn):n ∈ Z}⊆R. Similarly, (2/3, 5/6) ∈ R and

n by invariance {(α (2/3),xn):n ∈ Z}⊆R. Now, since R is an equivalence relation,

{(αn,αn(2/3)) : n ∈ Z}⊆R and since R ⊆ X × X is closed

n n {(z, z(2/3)) : z ∈ C3} = {(α ,α (2/3)) : n ∈ Z}⊆R.

Similarly, (1/3, 1/2) ∈ R and (2/3, 1/2) ∈ R so {(z(2/3),z(1/3)) : z ∈ C3}⊆R.

We have shown that RK ⊆ R. Since R is arbitrary among closed invariant equivalence relations such that the factor (X/R, T ) is equicontinuous, (K, T) is the maximal equicontinuous factor of (X, T). Finally, note that (X, T) is clearly not transitive, and (K, T) is minimal.

Let G be any countable amenable group. Let (K(X),G) denote the maximal equicontinuous factor of (X, G), and assume that the system (K(X),G) is minimal.

Since (K(X),G) is minimal and equicontinuous, (K(X),G) is uniquely ergodic. De- note the unique invariant measure on K(X)byλ.Letπ : X → K(X)denotethe factor map. Then, for a Borel set A ⊆ K(X)andm ∈ M(X), π∗m(A)=m(π−1(A)) deﬁnes an aﬃne map π∗ : M(X) → M(K(X)).

Deﬁnition 7.4.4. Put

∗ −1 P1 =(π ) (λ) and for g ∈ G,letTg : P1 → P1 be deﬁned by fd(Tgµ)= f ◦ gdµ.Then,when

∗ ' ' P1 is endowed with the weak topology, (P1,G) is a compact left topological dynamical system. 116 Deﬁnition 7.4.5. The left topological dynamical system (X, G) is twisting if

(K(X),G) is minimal and there is some ν ∈ P1 so that, for every µ ∈ P1 and every neighborhood U of ν,theset{g ∈ G : Tgµ ∈ U} is coﬁnite. The system (X, G) is weak twisting if (K(X),G) is minimal and the topological dynamical system (P1,G) is uniquely ergodic.

In [50], many interesting examples of weak twisting systems (X, T) are given. In particular, it is shown that every transitive nilrotation (X, T) is weak twisting ([50,

Theorem 1.2]), and that a large class of skew products over an irrational rotation are also weak twisting ([50, Theorem 1.4]). Furthermore, weak twisting systems are shown to have strong equidistribution properties (see [50, Proposition 2.6] or Theorem

7.4.6 below).

It is noted in [50] that any twisting system (X, T) is weak twisting and any weak twisting system (X, T) is uniquely ergodic. Observe that the topological dynamical system (X, G) is twisting if and only if (K(X),G) is minimal and there is a ﬁxed point ν ∈ P1 such that for every µ ∈ P1 and every nonprincipal ultraﬁlter p ∈ βG one has p- lim Tgµ = ν. Thus by Corollary 7.2.5, items ii) and vi), every twisting system g (X, G) is weak twisting.

Furthermore, if µ ∈ M(X) is invariant, then π∗(µ) ∈ M(K) is invariant. Since

(K(X),G) is assumed to be uniquely ergodic, it follows that M(X, G) ⊆ P1. Clearly the uniquely ergodic system (P1,G) has at most one ﬁxed point, so every weak twisting system (X, G) is uniquely ergodic. In fact, Corollary 7.2.5 yields the following list of conditions equivalent toweak twisting. For G = Z, condition vi) below, which can be summarized by saying that “every µ ∈ P1 equidistributes in density”, is equivalent to [50, Proposition 2.6, statement 2].

Theorem 7.4.6. Let (X, G) be a left topological dynamical system, assume that 117 (K(X),G) is minimal, and let ν be a G-invariant measure on X.Thenthefollowing are equivalent:

i) The system (X, G) is weak twisting, with unique invariant measure ν.

ii) The system (P1,G) is proximal.

iii) The system (P1,G) is strongly proximal.

iv) For every µ ∈ P1 and every essential idempotent p ∈ (βG,·) one has p- lim Tgµ = g ν.

l v) For every µ ∈ P1 and every p ∈ β(L ) one has p- lim Tgµ = ν. g

vi) For every µ ∈ P1,everyϵ>0,andeveryf ∈ C(X),theset

{g ∈ G : | f(Tgx)dµ − fdν| <ϵ} ' ' has left Banach density 1.

vii) For every µ ∈ P1 and every minimal idempotent p ∈ (βG,·) one has p- lim Tgµ = g ν.

viii) For every µ ∈ P1 and every p ∈ K(βG,◦) one has p- lim Tgµ = ν. g

vi) For every µ ∈ P1,everyϵ>0,andeveryf ∈ C(X),theset

{g ∈ G : | f(Tgx)dµ − fdν| <ϵ} ' ' is in (PSl)∗.

Proof. By deﬁnition, (X, G) is weak twisting if and only if (P1,G) is uniquely ergodic. Now apply Corollary 7.2.5.

The analogy between ergodicity, weak mixing, and mixing on the one hand and, on the other hand, unique ergodicity, weak twisting, and twisting is one of the most interesting aspects of twisting properties. It is therefore natural to make the following deﬁnition.

118 Deﬁnition 7.4.7. Let (X, G) be a left topological dynamical system, assume that (K(X),G) is minimal, and let ν be a G-invariant probability measure on X.Then the measure-preserving system is mild twisting if the ﬁxed point ν is the only recurrent point of the system (P1,G).

Clearly twisting implies mild twisting and mild twisting implies weak twisting. As is the case for weak twisting, the results of Section 4 immediately give us a number of characterizations for mild twisting.

Theorem 7.4.8. Let (X, G) be a left topological dynamical system, assume that (K(X),G) is minimal, and let ν be a G-invariant measure on X.Thenthefollowing are equivalent:

i) The system (X, G) is mild twisting with unique invariant measure ν.

ii) For every µ ∈ P1 and every idempotent p ∈ (βG,·) one has p- lim Tgµ = ν. g l iii) For every µ ∈ P1 and every p ∈ β(brl(IP ) one has p- lim Tgµ = ν. g

iv) For every µ ∈ P1,everyϵ>0,andeveryf ∈ C(X),theset

{g ∈ G : | f(Tgx)dµ − fdν| <ϵ} ' ' l ∗ is in (brl(IP )) .

Weak twisting is related to strong proximality in the following simple way.

Proposition 7.4.9. Atopologicaldynamicalsystem(X, G) is strongly proximal if and only if (X, G) is weak twisting and has no nontrivial equicontinuous factors.

Proof. Assume that (X, G) is strongly proximal, and let φ :(X, G) → (Y,G)beany factor map. If x1 ≠ x2 ∈ X,then(x1,x2) is a proximal pair, and thus so is the pair

(φ(x1),φ(x2)). If Y is equicontinuous, then it follows that φ(x1)=φ(x2).

119 Thus (X, G) has no nontrivial equicontinuous factors, and the maximal equicontinuous factor (K, G) consists of a single ﬁxed point k.SoP1 = M(X), and (X, G) is weak twisting by Corollaries 7.4.6 and 7.2.6.

Assume now that (X, G) is weak twisting and has no nontrivial equicontinuous factors. Then the maximal eqiucontinuous factor (K, G) consists of a single ﬁxed point, and P1 = M(X). Thus, again by Corollaries 7.4.6 and 7.2.6, (X, G) is strongly proximal.

Corollary 7.4.10. If G is an amenable WM group, then a topological dynamical system (X, G) is weak twisting if and only if it is strongly proximal.

Proof. Assume that G is an amenable WM group. If (X, G) is an equicontinuous topological dynamical system, x ∈ X,andf ∈ C(X), then F (g)=f(Tgx)deﬁnes an almost periodic function F on G. By our assumption on G,then,F is constant.

Since f ∈ C(X) is arbitrary, it follows that x is a ﬁxed point. Thus no weak twisting system (X, G) has nontrivial equicontinuous factors, and the result now follows from Proposition 7.4.9.

We can similarly relate mild twisting to antirecurrence. The proof of the following proposition is entirely analogous to the proof of Proposition 7.4.9.

Proposition 7.4.11. Atopologicaldynamicalsystem(X, G) is antirecurrent if and only if (X, G) is mild twisting and has no nontrivial equicontinuous factors.

Remark 7.4.12. If we took “The system (X, G) is transitive” as a requirement in the deﬁnition of weak twisting, then Proposition 7.4.9 would read “A topological dynamical system is a transitive strongly proximal system if and only ifitisweaktwistingandhas no nontrivial equicontinuous factors”. Propostion 7.4.11 would need to be adjusted similarly.

120 It is important to note that the notions we have deﬁned do not turn out to all be equivalent. We now examine some examples.

Example 7.4.13. Let E ⊆ G be an inﬁnite set which has upper Banach density 0.

Let (Xs,G) denote the shift space, and let X = O¯(1E) ⊆ Xs.Then(X, G) is weak twisting but not twisting.

Proof. By Corollary 6.2.3, Theorem 7.2.4, and Proposition 7.4.9, (X, G) is weak twisting, with unique invariant measure δ0¯,where0=1¯ ∅ ∈ X.Letf ∈ C(X) be given by:

f(z)=z(e). Put x0 =1E.Thenforg ∈ E one has fd(Tgδx0 )=1=0≠ fdδ0¯, ' ' showing that (X, G) is not twisting (since P1 = M(X)).

Example 2.12 of [50] is exactly such a construction. It is noted in [50] that examples having trivial maximal equicontinuous factor are somewhat unsatisfying (and this assessment is supported by Proposition 7.4.9). Furthermore, as twisting notions are geometrically motivated, examples in which the phase space X is totally disconnected may also be viewed as unenlightening. We now provide examples of connected topological dynamical systems (X, T) which have nontrivial maximal equicontinuous factors and which distinguish between weak twisting, mild twisting, and twisting.

While such examples are nontrivial from a topological perspective, note that they are measure-theoretically isomorphic to their maximal equicontinuous factors, so they are still trivial from a measure-theoretic perspective. Furthermore, the systems we construct are not transitive. We may restrict our attention to a transitive subsystem, but such subsystems are not guaranteed to still be connected.

The question whether there exists a weak twisting but not twisting system (X, T) which admits an invariant measure of full support, asked in [50], is left open. Note that such a system is necessarily transitive.

121 Example 7.4.14. Let X¯ =(X, B,µ,(Tg)g∈G) be a weak mixing but not mild mixing measure preserving system. Let L(X¯) be deﬁned as in Section 7.3 (so for f ∈ L(X¯), one has Uf = f ◦ T ). Let S1 denote the unit circle in the complex plane and let

α ∈ S1 be any number which is not a root of unity. Finally, let Z = L(X¯) × S1 and let T : Z → Z be given by T (f,y)=(Uf,α· y).Then(Z, T) is weak twisting but not mild twisting (and not strongly proximal).

1 Proof. For y ∈ S , let Tαy = α · y.Then(S1,Tα) is a minimal group rotation, and is

1 equicontinuous. Let π2 : Z → S denote the projection onto the second coordinate: π(f,y)=y.Thenπ is clearly a factor of topological dynamical systems. We must

1 show that (S ,Tα) is the maximal equicontinuous factor of (Z, T).

Assume that (Z1,T) is equicontinuous and φ : Z → Z1 is a factor map. Let z1,z2 ∈

1 Z,andassumethatπ2(z1)=π2(z2). So there are f1,f2 ∈ L(X¯)andy ∈ S so that z1 =(f1,y)andz2 =(f2,y). Since (f1,f2) is a proximal pair in the system (L(X¯),U), it follows that (z1,z2) is a proximal pair, and hence so too is (φ(z1),φ(z2)). Since Z1 is equicontinuous, we conclude that φ(z1)=φ(z2). Since this is true for every z1,z2 ∈ Z ¯ 1 satisfying π2(z1)=π2(z2), we can fix f0 ∈ L(X¯)anddefine(φ):(S ,Tα) → (Z1,T) by phi¯ (y)=φ(f0,y). Then φ¯ is a factor map, and φ = phi¯ ◦ π2.Thusπ2 satisfies

1 the universal property deﬁning the maximal equicontinuous factor, and (S ,Tα) is (isomorphic to) the maximal equicontinuous factor (K(Z),T).

Let π1 : Z → L(X¯) denote projection onto the ﬁrst coordinate: π1(f,y)=f.Let

∗ 1 l µ ∈ P1.Soπ2(µ)=λ (where λ is Haar measure on the group S ). Let p = p·p ∈ β(L ) n n be any essential idempotent, let ν = p- lim T µ.Thenπ2(ν)=p- lim Tα (π2 ∗ (µ)) = n n n n p- lim Tα (λ)=λ,andπ1(ν)=p- lim U (π1 ∗ (µ)) = δ0 by Corollary 7.3.6. It follows n n

(since π1(ν) is a point mass), that ν = δ0 ⊗ λ is the product measure. Since µ ∈ P1

l and p ∈ β(D ) are arbitrary, we conclude that (P1,T) is uniquely ergodic with unique invariant measure δ0 ⊗ λ,andhence(Z, T) is weak twisting.

122 Since X¯ is not mild mixing, there is some nonzero f ∈ L(X¯) and some idempotent

n ultraﬁlter p ∈ βG such that p- lim U f = f.Letµ = δf ⊗ λ. Then clearly µ ∈ P1, n n but p lim T µ = δf ⊗ λ ≠ δ0 ⊗ λ,hence(Z, T) is not mild twisting.

1 Finally note that, since (Z, T) has the nontrivial equicontinuous factor (S ,Tα), (Z, T) is not strongly proximal.

The veriﬁcation of the following example is entirely analogous to the proof above. We omit the details.

Example 7.4.15. Let X¯ =(X, B,µ,(Tg)g∈G) be a mild mixing but not mixing measure preserving system. Let L(X¯) be deﬁned as in Section 7.3 (so for f ∈ L(X¯),one has Uf = f ◦ T ). Let S1 denote the unit circle in the complex plane and let α ∈ S1 be any number which is not a root of unity. Finally, let Z = L(X¯) × S1 and let

T : Z → Z be given by T (f,y)=(Uf,α · y).Then(Z, T) is mild twisting but not twisting (and not antirecurrent).

We conclude this chapter by returning our attention to the analogy between on the one hand ergodicity, weak mixing, and mixing of a measure preserving system and on the other hand unique ergodicity, weak twisting, and twisting of a topological dynamical system. Compare the following proposition to Proposition 7.3.10.

Proposition 7.4.16. Let (X, G) be a topological dynamical system, and assume that (K(X),G) is minimal. Let ν ∈ M(X, G).

1. The system (X, G) is uniquely ergodic if and only if ν is the unique ﬁxed point

of (P1,G).

2. The system (X, G) is weak twisting if and only if ν is the unique minimal point

of (P1,G).

123 3. The system (X, G) is mild twisting if and only if ν is the unique recurrent point

of (P1,G).

4. The system (X, G) is twisting if and only if ν is a universal attractor for the

system (P1,G).

Proof. As noted above, M(X, G) ⊆ P1. This proves 1. Statement 2 is Theorem 7.4.6. Statement 3 is the deﬁnition of mild twisting, and statement 4 is merely a rephrasing of the deﬁnition of twisting.

Thus, not only is there an analogy between the hierarchy of mixing properties for measure preserving systems and the hierarchy of twisting properties for topological systems, but they are both in fact characterized by a hierarchy of uniqueness properties of a ﬁxed point in some induced aﬃne system. Both weak twistingandweak mixing may be characterized by either uniqueness of limits along minimal idempotents or uniqueness of limits along essential idempotents, and the equivalence of these two conditions is an application of Corollary 7.2.5.

124 CHAPTER 8

CONCLUSION

We began our investigations with a single question: Is there, in any countable amenable group G,asetD-setE ⊆ G which is not piecewise syndetic? In the

Introduction, we established that, for G abelian, the existence of such a set E is equivalent to the existence of a set having positive upper Banach density which is not piecewise syndetic, and also to the existence of a topological dynamical system

(X, G) having certain properties. It was then proved, via dynamical methods, that there exists in any abelian group G asetE ⊆ G which has positive upper Banach density but which is not piecewise syndetic, thus answering the original question in the event that the group G is abelian.

In fact, the methods of the Introduction produced more than stated. It was shown in Chapter 4 that the methods of the Introduction can be extended to a large class of countable amenable groups (those which are not virtually WM), butnottoall amenable groups.

In Chapter 5 the notion of “forcing F-recurrence” was introduced, and was shown to be intimately related to algebraic properties of closed ideals in βG.Theabstract language developed in Chapter 5 proved to be a convenient framework with which to provide a uniﬁed treatment of various notions of recurrence in topological dynamical systems. In chapter 6 we use the methods of Chapter 5 to ﬁnally achieved our goal of

125 proving the existence, in any countable amenable group G,ofaD-setE ⊆ G which is not piecewise syndetic.

The usefulness of our techniques was further illustrated in Chapter 7, wherein a number of diverse results in dynamics were seen to be corollaries of a single result

(namely Theorem 5.3.4).

Throughout Chapters 4-7, the primary tool used was limits along ultraﬁlters. This notion, and its inherent connection to topological dynamical systems, was investigated in Chapter 3.

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