Uniqueness of a Furstenberg system
aa r X i v : . [ m a t h . D S ] M a y UNIQUENESS OF A FURSTENBERG SYSTEM
VITALY BERGELSON AND ANDREU FERRÉ MORAGUES
Abstract.
Given a countable amenable group G , a Følner sequence ( F N ) ⊆ G , and a set E ⊆ G with ¯ d ( F N ) ( E ) = lim sup N →∞ | E ∩ F N || F N | > , Furstenberg’s correspondence principleassociates with the pair ( E, ( F N )) a measure preserving system X = ( X, B , µ, ( T g ) g ∈ G ) anda set A ∈ B with µ ( A ) = ¯ d ( F N ) ( E ) , in such a way that for all r ∈ N and all g , . . . , g r ∈ G one has ¯ d ( F N ) ( g − E ∩ · · · ∩ g − r E ) ≥ µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) . We show that undersome natural assumptions, the system X is unique up to a measurable isomorphism. We alsoestablish variants of this uniqueness result for non-countable discrete amenable semigroupsas well as for a generalized correspondence principle which deals with a finite family ofbounded functions f , . . . , f ℓ : G → C . Introduction
Szemerédi’s celebrated theorem on arithmetic progressions states that any “large” set E ⊆ N in the sense that for some sequence of intervals ( I N ) with | I N | → ∞ as N → ∞ (1.1) ¯ d ( I N ) ( E ) = lim sup N →∞ | E ∩ I N || I N | > contains arbitrarily long arithmetic progressions, i.e. for all k ∈ N there is some n ∈ N suchthat E ∩ ( E − n ) ∩ . . . ( E − kn ) = ∅ . In the seminal paper [F1], Furstenberg derived Sze-merédi’s theorem from his multiple recurrence result, which states that for any probabilitymeasure preserving system ( X, B , µ, T ) , for any k ∈ N and for any A ∈ B with µ ( A ) > there is some n ∈ N such that µ ( A ∩ T − n A ∩ · · · ∩ T − kn A ) > .Furstenberg’s derivation of Szemerédi’s theorem from his multiple recurrence result hinges ona correspondence principle which allows one to associate with any “large” set E a measure pre-serving system ( X, B , µ, T ) and a set A ∈ B with µ ( A ) > such that A ∩ T − n A ∩ . . . T − kn A = ∅ implies E ∩ ( E − n ) ∩ · · · ∩ ( E − kn ) = ∅ .We now describe Furstenberg’s approach to creating such a measure preserving system.Given a “large” set E , he identifies E with a point ω = ( E ( n )) n ∈ Z in the symbolicspace { , } Z . Now let X = { T n ω : n ∈ Z } , where T is the restriction of the shift map σ (( x n ) n ∈ Z ) = ( x n +1 ) n ∈ Z to X , B X be the completion of the Borel σ -algebra and µ be aweak* limit point of a sequence of measures | I N | P n ∈ I N δ T n ω .The system ( X, B X , µ, T ) satisfies the following natural conditions:(1) There is a set A ∈ B X (namely, A = { x ∈ X : x (0) = 1 } ) such that µ ( A ) =¯ d ( I N ) ( E ) > .
2) There is a subsequence ( I N k ) such that µ = w*- lim k →∞ | I Nk | P n ∈ I Nk δ T n ω , whichimplies that for any r ∈ N and h , . . . , h r ∈ Z we have d ( I Nk ) ( E − h ∩ · · · ∩ E − h r ) = lim k →∞ | ( E − h ) ∩ · · · ∩ ( E − h r ) ∩ I N k || I N k | = µ ( T − h A ∩ · · · ∩ T − h r A ) . (3) The σ -algebra B X is generated by the family of sets { T n A : n ∈ Z } .In principle, there are other ways to create a probability measure preserving system ( X, B , µ, T ) satisfying the conditions (1), (2) and (3) above (see Section 2). We will call any such systema Furstenberg system associated with the pair ( E, ( I N )) .The goal of this short paper is to show that any system satisfying (1), (2) and (3) is (metri-cally) isomorphic to the symbolic Furstenberg system described above. Actually, we will dothis in the natural framework of amenable groups. A countable amenable group G can beconveniently defined via the notion of a Følner sequence. A sequence ( F N ) of finite non-emptysubsets of a countable group G is a (left) Følner sequence if for all g ∈ G lim N →∞ | F N ∆ gF N || F N | = 0 . A countable group G is amenable if it admits a (left) Følner sequence . Given a Følnersequence ( F N ) ⊆ G and a set E ⊆ G we write ¯ d ( F N ) ( E ) = lim sup N →∞ | E ∩ F N || F N | . Before continuing our discussion we want to formulate a general version of Furstenberg’scorrespondence principle (see for example [B]).
Theorem 1.1 (Furstenberg’s Correspondence Principle) . Let G be a countable amenablegroup. Let E ⊆ G with ¯ d ( F N ) ( E ) > for some Følner sequence ( F N ) ⊆ G . Then there exista probability measure preserving system ( X, B , µ, ( T g ) g ∈ G ) , a set A ∈ B with µ ( A ) = ¯ d ( F N ) ( E ) and a subsequence ( F N k ) such that for any r ∈ N and g , . . . , g r ∈ G one has: (1.2) d ( F Nk ) ( g − E ∩ · · · ∩ g − r E ) = lim k →∞ | g − E ∩ · · · ∩ g − r E ∩ F N k || F N k | = µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) . Theorem 1.1 justifies the following definition:
Definition 1.2.
Let G be a countable amenable group, let ( F N ) ⊆ G be a Følner sequence,and let E ⊆ G with ¯ d ( F N ) ( E ) > . We say that a standard probability measure preservingsystem ( X, B , µ, ( T g ) g ∈ G ) is a Furstenberg system associated with the pair ( E, ( F N )) if In this paper “isomorphism” means metric isomorphism. We will also be using in Section 4 a weaker,boolean, form of isomorphism which we will refer to as conjugacy . Every countable amenable group G admits also right- and indeed two-sided Følner sequences (see Corol-lary 5.3 in [N]). Throughout this paper we deal only with left Følner sequences and routinely omit theadjective "left". This means that the probability space ( X, B , µ ) is a Lebesgue space, so, in particular, it is a completemeasure space (see Definition 2.3 in [W]). There is a set A ∈ B such that µ ( A ) = ¯ d ( F N ) ( E ) > . (2) There is a subsequence ( F N k ) such that for all r ∈ N and all g , . . . , g r ∈ G we have d ( F Nk ) ( g − E ∩ · · · ∩ g − r E ) = µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) . (3) The σ -algebra B is (the completion of ) the σ -algebra generated by the family of sets { T g A : g ∈ G } . Theorem 1.3.
Let G be a countable amenable group and let ( F N ) ⊆ G be a Følner sequence.Let E ⊆ G with ¯ d ( F N ) ( E ) > and let X = ( X, B , µ, ( T g ) g ∈ G ) be a Furstenberg system for thepair ( E, ( F N )) . Then, there exists a subsequence ( F N k ) such that the system X is isomorphicto the symbolic system Y = ( Y, B Y , ν, ( S g ) g ∈ G ) , where Y = { S g ω : g ∈ G } , ω = ( E ( g )) g ∈ G ,the maps S g are given by ( S g x ) g = x gg for all g, g ∈ G , ν = w*- lim k →∞ | F Nk | P g ∈ F Nk δ S g ω ,and B Y is the completion of the Borel σ -algebra of Y While in Theorem 1.3 we choose to work, for the sake of simplicity, with countableamenable groups G , the proof extends without major modifications to countable cancellativeamenable semigroups (they possess Følner sequences). In particular, Theorem 1.3 is validfor, say, ( N , +) and ( N , · ) . One can actually extend the framework to an even more generalsetup. A discrete (not necessarily countable) group G is called amenable if there exists aleft-invariant mean on ℓ ∞ ( G ) . Call a set E ⊆ G large if, for some invariant mean m , wehave m ( E ) > . One can modify Definition 1.2, introduce a notion of Furstenberg systemassociated with the pair ( E, m ) and then establish a general version of Theorem 1.3 (seeSection 4).The method of constructing the symbolic Furstenberg system described above for the case G = Z works equally well if one replaces the indicator function E with any bounded func-tion f : G → C . Moreover, one can actually work with any finite family of bounded C -valuedfunctions (see for example [FrH]). We will show in the Appendix that Theorem 1.3 can benaturally extended to this setup as well.The structure of the paper is as follows. In Section 2 we describe four ways to construct aFurstenberg system. In Section 3 we prove Theorem 1.3. In Section 4 we obtain a counter-part for Theorem 1.3 for not necessarily countable groups G . We also establish necessary andsufficient conditions for two pairs ( E , ( F N )) and ( E , ( G N )) to admit isomorphic Fursten-berg systems. In Section 5 we prove that, given an ergodic measure preserving system ( X, B , µ, ( T g ) g ∈ G ) and a set A ∈ B with µ ( A ) > , there is a set E ⊆ G and a Følner se-quence ( F N ) such that ¯ d ( F N ) ( E ) = µ ( A ) and for which (1.2) holds. Finally, the Appendixdeals with Furstenberg systems associated with a finite family of bounded functions. Notation.
Throughout this paper, given a topological space X , we will routinely denote by B X the completion of the Borel σ -algebra of X . Constructing a Furstenberg system
The purpose of this section is to describe four natural approaches to constructing aFurstenberg system ( X, B , µ, ( T g ) g ∈ G ) associated with a “large” set E ⊆ G (i.e. a set with ¯ d ( F N ) ( E ) > for some Følner sequence ( F N ) ⊆ G ). The first of these approaches is a versionfor general amenable groups of Furstenberg’s original construction (for G = Z ) , based on thesymbolic space Y = { , } G , which was reviewed in the introduction; we present it in this ection for the sake of completeness. All four of the presented approaches have a unifyingthread, which involves the usage (either implicitly or explicitly) of Riesz’s representationtheorem for positive linear functionals.(i) (cf. [F1]). We consider the symbolic space Y = { , } G and proceed as follows. First,let ( F N k ) be a subsequence of ( F N ) such that d ( F Nk ) ( E ) = ¯ d ( F N ) ( E ) > . Then, letting ω = ( E ( g )) g ∈ G , we consider a sequence of measures(2.1) | F N k | X g ∈ F Nk δ S g ω . Now, the space of probability measures on { , } G is a compact, metric space withrespect to the weak* topology. Passing, if necessary, to a subsequence, let ν = lim k →∞ | F N k | X g ∈ F Nk δ S g ω . Let A = { x : x ( e ) = 1 } . We have ¯ d ( F N ) ( E ) = d ( F Nk ) ( E ) = ν ( A ) , and ν (( S g ) − A ∩ · · · ∩ ( S g r ) − A ) = d ( F Nk ) ( g − E ∩ · · · ∩ g − r E ) for all r ∈ N and all g , . . . , g r ∈ G . Observe now that the family { S g A : g ∈ G } generates the Borel σ -algebra of Y . Therefore, the completion of the σ -algebragenerated by A is equal to B Y , and thus the system Y = ( Y, B Y , ν, ( S g ) g ∈ G ) satisfies(1), (2) and (3).(ii) In the presentation of the second approach (which actually was hinted at in [F1]) wefollow [B]. Since the family of sets of the form T ri =1 g − i E, r ∈ N , g , . . . , g r ∈ G , iscountable, we can find a Følner subsequence ( F N k ) such that d ( F Nk ) ( E ) = ¯ d ( F N ) ( E ) > and(2.2) lim k →∞ (cid:12)(cid:12)(cid:12)T ri =1 g − i E ∩ F N k (cid:12)(cid:12)(cid:12) | F N k | = d ( F Nk ) r \ i =1 g − i E ! exists for all r ∈ N and g , . . . , g r ∈ G .Let A = Span C { , Q ri =1 g − i E : r ∈ N , g , . . . , g r ∈ G } , a vector subspace of ℓ ∞ ( G ) .Equation (2.2) allows us to define a functional L : A → C by setting L ( ) = 1 and L (cid:16)Q ri =1 g − i E (cid:17) = d ( F Nk ) Ä T ri =1 g − i E ä and extending by linearity.One can easily check that | L ( f ) | ≤ || f || ℓ ∞ ( G ) for all f ∈ A , so we can in fact ex-tend L by continuity to A E , the closure of A with respect to the ℓ ∞ ( G ) -norm. Noticethat A E is a separable, unital C ∗ -algebra. By Gelfand’s representation theorem,there is a compact metric space X so that A E is isometrically isomorphic to C ( X ) .Let Γ : A E −→ C ( X ) be the corresponding isomorphism.The functional L induces a positive linear functional ˜ L : C ( X ) → C . By Riesz’srepresentation theorem, there exists a regular Borel probability measure µ on theBorel σ -algebra of X such that for any ϕ ∈ A E L ( ϕ ) = ˜ L (Γ( ϕ )) = Z X Γ( ϕ ) dµ, ince E is an idempotent, and since Γ is, in particular, an algebraic isomorphism,then Γ( E ) = A , where A ∈ Borel ( X ) and satisfies µ ( A ) = ˜ L ( A ) = ˜ L (Γ( E )) =¯ d ( F N ) ( E ) .The shift operators given by ϕ ( h ) ϕ ( gh ) , ϕ ∈ A E , g ∈ G induce a G -antiaction on C ( X ) , and are, by a theorem of Banach, induced by homeomorphisms T g : X → X .Using again the fact that Γ is an algebraic isomorphism we have for all r ∈ N , and g , . . . , g r ∈ G (2.3) µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) = d ( F Nk ) ( g − E ∩ · · · ∩ g − r E ) . (Note that by (2.3) the homeomorphisms T g , g ∈ G preserve the measure µ ). Clearly X = ( X, B X , µ, ( T g ) g ∈ G ) satisfies (1), (2) and (3).In order to describe the third approach we need some facts on left invariant means. Recallthat a discrete semigroup G is left amenable if there exists a left invariant mean m : ℓ ∞ ( G ) → C . Note that, for discrete countable groups, this is equivalent to the definition of leftamenability given in the Introduction.(iii) We follow [BL] and work with G -invariant means directly. Namely, let m : ℓ ∞ ( G ) → C be a left-invariant mean on G . Take X = βG , the Stone-Čech compactification of G .Use Riesz’s representation theorem to get the unique measure µ corresponding to m ,so that m ( f ) = R X ˆ f dµ , where the function ˆ f is the extension by continuity to βG of f ∈ ℓ ∞ ( G ) . The maps h gh , h ∈ G have a unique continuous extension to βG which we denote by T g . Since m is a G -invariant mean, we have that the maps T g are measure preserving homeomorphisms of βG .Let A = E , where ¯ E denotes the closure of E in βG . Observe that µ ( A ∩ ( T g ) − A ∩ · · · ∩ ( T g r ) − A ) = m ( E · r Y i =1 g − i E ) , Let C A be the restriction of the σ -algebra B X to the completion of the σ -algebragenerated by A . Then, the system X = ( X, C A , µ, ( T g ) g ∈ G ) satisfies (1), (2) and (3).(iv) The last approach we present follows [BMc] and has elements in common with (i)and (iii). Let Y = { , } G and consider the cylinders of the form(2.4) { x ∈ Y : x ( h ) = ε , . . . , x ( h r ) = ε r } , where r ∈ N , and h , . . . , h r ∈ G are distinct and ε i ∈ { , } . For any cylinder C ofthe form (2.4), put λ ( C ) = m r Y i =1 h − i E i ! , where E i is equal to E or E c according to whether ε i = 1 or ε i = 0 , respectively.Since Y is a compact metric space, the premeasure λ extends to a measure µ on the We say that m ∈ ℓ ∞ ( G ) ∗ is a left invariant mean if it is a continuous linear functional from ℓ ∞ ( G ) to C such that (i) for every f ∈ ℓ ∞ ( G ) and for every g ∈ G we have m ( g f ) = m ( f ) , where g f ( x ) := f ( gx ) for all x ∈ G , (ii) m ( f ) ≥ for any non-negative function f : G → C , and (iii) m (1) = 1 . orel σ -algebra of Y . One can easily check that µ is invariant under the shifts S g asin (i). Letting A = { x : x ( e ) = 1 } we get(2.5) µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) = m r Y i =1 g − i E ! for all r ∈ N and g , . . . , g r ∈ G . Moreover, the measure µ is determined by theintersections appearing in (2.5) above, so the completion of the σ -algebra generatedby A is equal to B Y , and the system Y = ( Y, B Y , µ, ( S g ) g ∈ G ) satisfies (1), (2) and (3).3. Theorem 1.3 and a combinatorial corollary.
The main purpose of this section is to prove Theorem 1.3 that characterizes Fursten-berg systems for countable groups up to a measurable isomorphism. In particular, we showthat any Furstenberg system is isomorphic to the symbolic measure preserving system Y described in item (i) of Section 2. As a corollary we obtain a criterion which determineswhen the Furstenberg systems of two pairs ( E, ( F N )) , ( D, ( G N )) are isomorphic (here E, D are subsets of G and ( F N ) , ( G N ) are Følner sequences).In the following two sections and in the Appendix, we will make repeated use of the fol-lowing notation: if A is a commutative, unital C ∗ -algebra, then ˆ A denotes the space ofcharacters of A , i.e. the space of algebra homomorphisms χ : A → C . Recall that ˆ A is acompact Hausdorff space with respect to the weak* topology (in fact, a metric space if thealgebra A is separable).We start with the following technical result which will be needed in the sequel: Theorem 3.1 (cf. Satz 1 in [Neu] and Remark 4.1 in [P]) . Consider two standard mea-sure preserving systems X = ( X, B X , µ, ( T g ) g ∈ G ) and Y = ( Y, B Y , ν, ( S g ) g ∈ G ) . Let Φ : L ∞ ( X, B X , µ ) → L ∞ ( Y, B Y , ν ) be a Banach algebra isomorphism satisfying Φ( f ◦ T g ) = Φ( f ) ◦ S g and Z Y Φ( f ) dν = Z X f dµ for all g ∈ G, f ∈ L ∞ ( X, B X , µ ) . Then, there exists an isomorphism between X and Y .Proof. Arguing as in the proof of Remark 4.1 in [P] one deduces that the measure algebras ( B X , µ ) and ( B Y , ν ) are isomorphic (as Boolean algebras). Now Satz 1 in [Neu] (see Theorem1.4.6 of [P]) implies that Φ arises from a point isomorphism, say ˜Φ : ( X, B X , µ ) −→ ( Y, B Y , ν ) .Finally, the condition Φ( f ◦ T g ) = Φ( f ) ◦ S g , g ∈ G, f ∈ L ∞ ( X, B X , µ ) implies that for every g ∈ G , there is a set X g with µ ( X g ) = 1 such that S g ˜Φ( x ) = ˜Φ T g ( x ) , for x ∈ X g . It followsthat the restriction of ˜Φ to the set T g ∈ G X g is the desired point isomorphism (observe thatsince G is countable, this intersection is a measurable set with full measure). (cid:3) We now give a proof of Theorem 1.3 which we state here again for the convenience of thereader:
Theorem 3.2.
Let G be a countable amenable group and let ( F N ) ⊆ G be a Følner sequence.Let E ⊆ G with ¯ d ( F N ) ( E ) > and let X = ( X, B , µ, ( T g ) g ∈ G ) be a Furstenberg system for thepair ( E, ( F N )) . Then, there exists a subsequence ( F N k ) such that the system X is isomorphicto the symbolic system Y = ( Y, B Y , ν, ( S g ) g ∈ G ) , where Y = { S g ω : g ∈ G } , ω = ( E ( g )) g ∈ G , he maps S g are given by ( S g x ) g = x gg for all g, g ∈ G , ν = w*- lim k →∞ | F Nk | P g ∈ F Nk δ S g ω ,and B Y is the completion of the Borel σ -algebra of Y .Proof. Let f = A , and put A f = Span C { Q ri =1 T g i f, : r ∈ N , g , . . . , g r ∈ G } L ∞ ( µ ) , the uni-tal C ∗ -algebra generated by products of shifts of f . Since X is a Furstenberg system, itsatisfies condition (3) in Definition 1.2, which implies that A f is dense in L ∞ ( B , µ ) withrespect to the L ( µ ) -norm.Let Z = ˆ A f . By Gelfand’s representation theorem, there is an isometric isomorphism Γ : A f → C ( Z ) . Notice Z is compact and metric given that A f is separable. The measure µ induces a positive linear functional F : C ( Z ) → C via F ( ϕ ) = µ (Γ − ( ϕ )) , where ϕ ∈ C ( Z ) .Let ˜ µ be the measure obtained from F via Riesz’s representation theorem. Each measurepreserving transformation T g , g ∈ G , induces an algebra isomorphism of A f (via precompo-sition), which in turn, induces an isomorphism of C ( Z ) via Γ . A theorem of Banach ensuresthat this isomorphism of C ( Z ) is given by a homeomorphism of Z , which we will denote by ˜ T g , g ∈ G . Observe that the homeomorphism ˜ T g , g ∈ G is ˜ µ -preserving. We claim that thesystem X = ( X, B , µ, ( T g ) g ∈ G ) is isomorphic to the system Z = ( Z, B Z , ˜ µ, ( ˜ T g ) g ∈ G ) .Note that since A f is dense, with respect to the L ( µ ) -norm, in L ∞ ( B , µ ) , and since C ( Z ) isdense, with respect to the L (˜ µ ) -norm, in L ∞ ( Z, B Z , ˜ µ ) , we can extend Γ by continuity to aBanach algebra isomorphism ˜Γ between L ∞ ( X, B , µ ) and L ∞ ( Z, B Z , ˜ µ ) . Moreover, it is nothard to see that the map ˜Γ satisfies ˜Γ( f ◦ T g ) = ˜Γ( f ) ◦ ˜ T g and Z Z ˜Γ( f ) d ˜ µ = Z X f dµ for all g ∈ G, f ∈ L ∞ ( X, B , µ ) . (Indeed, it is enough to check that the above equalities hold for functions of the form Q ri =1 T g i f , with r ∈ N and g , . . . , g r ∈ G ).Thus, by Theorem 3.1, there exists an isomorphism between the measure preserving sys-tems X and Z . From this point on we are going to keep working with the system Z .Define a map Φ from Z to the compact space { , } G by Φ( χ ) = ( χ ( T g f )) g ∈ G . Note that Φ is well defined. Indeed, since χ ∈ Z , and T g f is idempotent for all g ∈ G , it follows that χ ( T g f ) ∈ { , } .Next, we show that Φ is injective. Suppose that Φ( χ ) = Φ( χ ) . Then, χ ( T g f ) = χ ( T g f ) for all g ∈ G . Since χ , χ are multiplicative, we have χ ( Q ri =1 T g i f ) = χ ( Q ri =1 T g i f ) forall r ∈ N , and all g , . . . , g r ∈ G . Since χ , χ are linear, we see that χ ( g ) = χ ( g ) for any g ∈ Span C { Q ri =1 T g i f : r ∈ N , g i ∈ G } . Finally, since χ , χ are continuous, it follows that χ ( g ) = χ ( g ) for any g ∈ A f , whence χ = χ , which implies that Φ is injective.Finally, the map Φ is continuous. Indeed, for each g ∈ G , the map χ χ ( T g f ) is anevaluation map, and as such, is continuous with respect to the weak* topology for all g ∈ G .This implies that Φ is continuous (we consider { , } G with respect to its natural product Note that since supp ( ν ) ⊆ { S g ω : g ∈ G } , we could take Y = { , } G . opology). Notice that, in particular, this means that Φ is measurable.Now, let ν = Φ ∗ µ . The map Φ provides a measurable isomorphism between the mea-sure preserving systems Z and ( { , } G , B { , } G , ν, ( S g ) g ∈ G ) , where S g , g ∈ G denote the shiftmaps. Letting B = { x ∈ { , } G : x ( e ) = 1 } , we see that the probability measure ν isdetermined by the values(3.1) ν (( T g ) − B ∩ . . . ∩ ( T g r ) − B ) , for all r ∈ N , g , . . . , g r ∈ G. We have(3.2) ν (( T g ) − B ∩ . . . ∩ ( T g r ) − B ) = Z Y ( T g ) − B ∩ ... ∩ ( T gr ) − B d Φ ∗ ˜ µ = Z Z ( T g ) − B ∩ ... ∩ ( T gr ) − B (Φ( χ )) d ˜ µ ( χ ) = Z Z ( T g ) − B ∩ ... ∩ ( T gr ) − B (( χ ( T g f )) g ∈ G ) d ˜ µ ( χ ) = Z Z χ ( T g f · . . . · T g r f ) d ˜ µ ( χ ) = Z Z Γ( T g f · . . . · T g r f ) d ˜ µ ( χ ) = Z X T g f · . . . · T g r f dµ = µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) = d ( F Nk ) ( g − E ∩ . . . ∩ g − r E ) , where we used the definition of the Gelfand transform in the second to last step, and thefact that X is a Furstenberg system in the last one. It follows that, for ω = ( E ( g )) g ∈ G , onehas(3.3) ν = w* lim k →∞ | F N k | X g ∈ F Nk δ S g ω , whence supp ( ν ) ⊆ { S g ω : g ∈ G } . Notice that a reasoning as in (i) of Section 2 shows thatthe right hand side in (3.3) is well defined as a weak* limit. We are done. (cid:3) Next, we make some comments on ergodicity of Furstenberg systems. In general, a Fursten-berg system associated with a pair ( E, ( F N )) will not be ergodic (this is for example the casefor ( E = S n ∈ N [2 n , n +1 ) , F N = [1 , N ]) .) Nonetheless, one can show (see [BF]) that if forsome Følner sequence ( F N ) the set E satisfies d ( F N ) ( E ) > , then there is a Følner sequence ( G N ) such that the Furstenberg system associated to ( E, ( G N )) is ergodic.The following corollary of Theorem 3.2 provides a characterization of pairs ( E, ( F N )) , ( D, ( G N )) that have isomorphic Furstenberg systems: Theorem 3.3.
Let
E, D be two subsets of G and let ( F N ) , ( G N ) be Følner sequences suchthat d ( F N ) ( E ) > and d ( G N ) ( D ) > . (1) If the Furstenberg systems associated with ( E, ( F N )) and ( D, ( G N )) are isomorphic,then, for all r ∈ N and all g , . . . , g r ∈ G , (3.4) d ( F N ) ( g − E ∩ · · · ∩ g − r E ) = d ( G N ) ( g − D ∩ · · · ∩ g − r D ) . (2) Suppose that the pairs ( E, ( F N )) and ( D, ( G N )) satisfy (3.4) . Then, ( E, ( F N )) and ( D, ( G N )) admit isomorphic Furstenberg systems. roof. (1) Let us assume that the pairs ( E, ( F N )) and ( D, ( G N )) satisfy (3.4). By Theorem3.2 the sets E and D have each a Furstenberg system of the form ( Y, B Y , ν , ( S g ) g ∈ G ) and ( Y, B Y , ν , ( S g ) g ∈ G ) , where Y = { , } G and where, for ω = ( E ( g )) g ∈ G and ω = ( D ( g )) g ∈ G ,we put ν = w*- lim N →∞ | F N | P g ∈ F N δ S g ω and ν = w*- lim N →∞ | G N | P g ∈ G N δ S g ω (we use Y = { , } G instead of the corresponding orbital closures for convenience -see footnote 5).As we know from (i) in Section 2, the measures ν and ν are determined by their valuesat sets of the form S − g A ∩ · · · ∩ S − g r A for r ∈ N , g , . . . , g r ∈ G , where A = { x : x ( e ) = 1 } .The result in question follows now from (3.4).(2) If the pairs ( E, ( F N ) and ( D, ( G N )) admit isomorphic Furstenberg systems, say Y i =( Y, B Y , ν i , ( S g ) g ∈ G ) , for i = 1 , , it follows that ν (( S g ) − A ∩ · · · ∩ ( S g r ) − A ) = ν (( S g ) − A ∩· · · ∩ ( S g r ) − A ) , for all r ∈ N and g , . . . , g r ∈ G , which implies, given the definition of ν , ν ,that we can find Følner sequences ( F N ) and ( G N ) such that (3.4) holds, as desired. (cid:3) A version of Theorem 1.3 when G is uncountable Since there are quite a few results of Ramsey-theoretical nature which are valid for un-countable amenable semigroups (see for example [BL], [HS] and [DL]), it makes sense toconsider a variant of Theorem 1.3 for general (discrete) amenable semigroups. In whatfollows, we will be assuming (without loss of generality) that G has a neutral element. Definition 4.1.
Let G be a left amenable semigroup. Let m : ℓ ∞ ( G ) → C be a left-invariantmean and let E ⊆ G be such that m ( E ) > . We say that a measure preserving system ( X, B , µ, ( T g )) is a Furstenberg system associated with the pair ( E, m ) if (1) There is a set A ∈ B such that µ ( A ) = m ( E ) > . (2) For all r ∈ N and all g , . . . , g r ∈ G we have m ( g − E · . . . · g − r E ) = µ (( T g ) − A ∩· · · ∩ ( T g r ) − A ) . (3) The σ -algebra B is (the completion of ) the σ -algebra generated by the family of sets { T g A : g ∈ G } . Next we describe a “symbolic” Furstenberg system associated to a pair ( E, m ) . Lemma 4.2.
Let G be a left amenable semigroup and let E ⊆ G be such that for some left-invariant mean m on G we have m ( E ) > . Put X = { , } G and let A = { x : x ( e ) = 1 } .Then, the system X = ( X, B X , µ, ( S g ) g ∈ G ) , where S g is given by ( S g x ) g = x gg for all g, g ∈ G , and µ is a (unique) measure satisfying (4.1) µ (( S g ) − A ∩ · · · ∩ ( S g r ) − A ) = m ( g − E · . . . · g − r E ) for all r ∈ N and g , . . . , g r ∈ G , satisfies (1), (2) and (3) of Definition 4.1.Proof. We start by noting that µ is well defined (this follows, for example, from the proof ofTheorem 2.1 in [BMc], given that countability of G is not used there). Now we observe thatuniqueness of µ follows from the Stone-Weierstrass theorem. Indeed, the unital C -algebragenerated by the functions { ( S g ) − A : g ∈ G } clearly separates points of X and is closedunder conjugation. Thus, by Stone-Weierstrass, it is dense in C ( X ) , and it follows that thevalues in (4.1) determine µ . The system X clearly satisfies (1) and (2) of Definition 4.1 by(4.1). Moreover, the G -invariant σ -algebra generated by A contains Borel ( X ) , whence X satisfies (3), completing the proof. (cid:3) he following variant of Theorem 3.1 will be needed for the proof of Theorem 4.4 below. Theorem 4.3 (cf. Remark 4.1 [P]) . Let ( X, B , µ, ( T g ) g ∈ G ) and ( Y, C , ν, ( S g ) g ∈ G ) be twomeasure preserving systems. Suppose that we can find a Banach algebra isomorphism Φ : L ∞ ( X, B , µ ) → L ∞ ( Y, C , ν ) satisfying Φ( f ◦ T g ) = Φ( f ) ◦ S g and Z Y Φ( f ) dν = Z X f dµ for all g ∈ G, f ∈ L ∞ ( X, B , µ ) . Then, ( T g ) g ∈ G and ( S g ) g ∈ G are conjugate (see Definition 2.5 in [W] ).Proof. The argument goes along the lines of the first part of the proof of Theorem 3.1.Notice that we do not assume that ( X, B , µ ) or ( Y, C , ν ) are standard. This generality iscompensated by the fact that the map Φ is just an algebra isomorphism, rather than apointwise map. (cid:3) Here is finally a version of Theorem 1.3 for general amenable semigroups.
Theorem 4.4.
Let G be a left amenable semigroup and let m be a left-invariant mean. Let E ⊆ G with m ( E ) > and let X = ( X, B X , µ, ( T g ) g ∈ G ) be a Furstenberg system for thepair ( E, m ) . Let Y be the symbolic Furstenberg system for ( E, m ) constructed in Lemma 4.2.Then, ( T g ) g ∈ G and ( S g ) g ∈ G are conjugate.Proof. The proof is essentially the same as that of Theorem 3.2. Only two changes haveto be made. First, instead of applying Theorem 3.1, we need to make use of Theorem 4.3,since G is not assumed to be countable. Second, we have to replace d ( F Nk ) in the formula µ (( T g ) − A ∩ · · · ∩ ( T g r ) − A ) = d ( F Nk ) ( g − E ∩ . . . ∩ g − r E ) (see equation (3.2)) with the mean m and the measure ν in (3.3) by the measure ν obtained in Lemma 4.2. (cid:3) We conclude this section with the observation that an analog of Theorem 3.3 for pairs ( E, m ) and ( D, m ) holds for a general amenable semigroup G . We omit the details.5. From a dynamical system back to the group
The purpose of this short section is to prove the following partial converse to Theorem1.3.
Theorem 5.1.
Let G be a countable amenable group. Let X = ( X, B , µ, ( T g ) g ∈ G ) be anergodic measure preserving system. Let A ∈ B with µ ( A ) > and let ( F N ) be a Følnersequence in G . Then there exists a subsequence ( F N k ) and a set E ⊆ G such that (5.1) µ (( T g ) − A ∩ · · · ∩ ( T g k ) − A ) = d ( F Nk ) ( g − E ∩ · · · ∩ g − k E ) for all k ∈ N and g , . . . , g k ∈ G (in particular, µ ( A ) = d ( F Nk ) ( E ) ).Proof. Let A ∈ B with µ ( A ) > . Let P f ( G ) denote the set of finite subsets of G . Put F = A , and for every G ∈ P f ( G ) , let F G = Q g ∈G T g F . Note that the set { F G : G ∈ P f ( G ) } is countable. By von Neumann’s mean ergodic theorem we have(5.2) lim N →∞ | F N | X g ∈ F N T g F G = Z X F G dµ, or all G ∈ P f ( G ) , where convergence takes place in the L ( µ ) -norm. Since P f ( G ) is count-able, we can extract a subsequence ( F N k ) via a diagonal process and a subset X ⊆ X with µ ( X ) = 1 such that for all x ∈ X and all G ∈ P f ( G ) we have(5.3) lim k →∞ | F N k | X g ∈ F Nk F G ( T g x ) = Z X F G dµ. Let x ∈ X , and let E ( x ) = { g ∈ G : T g x ∈ A } . Clearly, the set E ( x ) satisfies (5.1). (cid:3) Corollary 5.2.
Let X be the subset of X obtained in the proof of Theorem 5.1. For each x ∈ X , the Furstenberg system associated with ( E ( x ) , ( F N k )) is isomorphic to a factor Z x of X which can be identified with the completion of the G -invariant σ -algebra generated bythe set A . Moreover, for any x, y ∈ X , the factors Z x and Z y are isomorphic. Appendix A. Furstenberg systems that arise from finitely many functions
Let G be a countable amenable group. The purpose of this section is to introduce Fursten-berg systems associated with finite families of functions f , . . . , f ℓ : G → D , where D is theclosed unit disk in C , and to establish results analogous to those in Section 3. Definition A.1.
Let G be a countable amenable group. Let ℓ ∈ N and f , . . . , f ℓ : G → D .Assume that there exists a Følner sequence ( F N ) ⊆ G and functions a : G → R ≥ and b : N → R ≥ satisfying b ( N ) | F N | P g ∈ F N a ( g ) → as N → ∞ . We will call such a triple (( F N ) , a, b ) an averaging scheme. We say that the family { f , . . . , f ℓ } is accordant with theaveraging scheme (( F N ) , a, b ) if for any choice of ˜ f i ∈ { f , . . . , f ℓ , ¯ f , . . . , ¯ f ℓ } the limit (A.1) lim N →∞ b ( N ) | F N | X g ∈ F N a ( g ) r Y i =1 ˜ f i ( g i g ) exists for all r ∈ N , g , . . . , g r ∈ G . We are in a position to define Furstenberg systems associated to a family of functions { f , . . . , f ℓ } accordant with the averaging scheme (( F N ) , a, b ) : Definition A.2.
Let G be a countable amenable group. Let (( F N ) , a, b ) be an averagingscheme. Let ℓ ∈ N and let f , . . . , f ℓ : G → D be a family of functions accordant with (( F N ) , a, b ) . We say that a standard measure preserving system ( X, B , µ, ( T g )) is a Fursten-berg system for the tuple ( { f , . . . , f ℓ } , ( F N ) , a, b ) if there are functions F , . . . , F ℓ ∈ L ∞ ( µ ) such that for any r ∈ N and any g , . . . , g r ∈ G and for any choice of ˜ f i ∈ { f , . . . , f ℓ , ¯ f , . . . , ¯ f ℓ } and ˜ F i ∈ { F , . . . , F ℓ , ¯ F , . . . , ¯ F ℓ } (where the choice of ˜ f i and ˜ F i is made in a “simultaneous”manner) the limit (A.2) lim N →∞ b ( N ) | F N | X g ∈ F N a ( g ) r Y i =1 ˜ f i ( g i g ) = Z X T g ˜ F · . . . · T g r ˜ F r dµ exists, and moreover, the σ -algebra B is equal to the completion of the G -invariant σ -algebragenerated by the measurable functions F , . . . , F ℓ . As in Section 2, one can construct a symbolic Furstenberg system for the tuple ( { f , . . . , f ℓ } , ( F N ) , a, b ) : emma A.3. Let G be a countable amenable group. Let (( F N ) , a, b ) be an averaging scheme.Let { f , . . . , f ℓ } be a family of functions accordant with (( F N ) , a, b ) . Let Y = ( D ℓ ) G . Put Y =( Y, B Y , ν, ( S g ) g ∈ G ) , where for ω = ( f ( g ) , . . . , f ℓ ( g )) g ∈ G , ν = w*- lim N →∞ b ( N ) | F N | P g ∈ F N a ( g ) δ S g ω ,and ( S g x ) g = ( x ( gg ) , . . . , x ℓ ( gg )) for all g, g ∈ G (we will show that this weak* limit iswell defined). Then, Y is a Furstenberg system for the tuple ( { f , . . . , f ℓ } , ( F N ) , a, b ) .Proof. Let Y = ( D ℓ ) G and put F i = x i (0) ∈ C ( Y ) , so F i is measurable with respect to B Y . Then one easily checks that C ( F , . . . , F ℓ ) , the σ -algebra generated by the functions F , . . . , F ℓ , ¯ F , . . . , ¯ F ℓ , is equal to B Y (since we tacitly consider Y with the product topologyand because, by the Stone-Weierstrass theorem, the subalgebra generated by these functionsis dense in C ( Y ) ). Invoking the Stone-Weierstrass theorem again, we see that the measure ν is determined by the values Z Y r Y i =1 T g i G i dρ for all r ∈ N , g , . . . , g r ∈ G and G i ∈ { F , . . . , F ℓ , ¯ F , . . . , ¯ F ℓ } . Let ν = w*- lim N →∞ b ( N ) | F N | P g ∈ F N a ( g ) δ S g ω , where ω = ( f ( g ) , . . . , f ℓ ( g )) g ∈ G . Observe that ν is well defined because Z Y r Y i =1 T g i ˜ F i dν = lim N →∞ b ( N ) | F N | X g ∈ F N a ( g ) r Y i =1 ˜ f i ( g i g ) for all r ∈ N , g , . . . , g r ∈ G, and ˜ F i ∈ { F , . . . , F ℓ , ¯ F , . . . , ¯ F ℓ } and ˜ f i ∈ { f , . . . , f ℓ , ¯ f , . . . , ¯ f ℓ } , respectively. Thus, Y isa Furstenberg system for the tuple ( { f , . . . , f ℓ } , ( F N ) , a, b ) , as desired. One can also checkthat ν satisfies supp ( ν ) ⊆ { T g ω : g ∈ G } . (cid:3) We are now in a position to establish the “functional” analog of Theorem 3.2:
Theorem A.4.
Let G be a countable amenable group and let (( F N ) , a, b ) be an averag-ing scheme. Let { f , . . . , f ℓ } be a family of bounded functions on G that is accordantwith (( F N ) , a, b ) , and let X = ( X, B X , µ, ( T g ) g ∈ G ) be a Furstenberg system for the tuple ( { f , . . . , f ℓ } , ( F N ) , a, b ) . Then, X is isomorphic to the Furstenberg system Y = ( Y, B Y , ν, ( S g ) g ∈ G ) constructed in Lemma A.3.Proof. The proof of this Theorem is very similar to the proof of Theorem 3.2, and so we areonly going to point out the major changes that have to be implemented. The first majorstep of the proof is done in the same way, but for the algebra A = Span { Q ri =1 T g i ˜ F i , r ∈ N , g , . . . , g r ∈ G, ˜ F i ∈ { F , . . . , F ℓ }} L ∞ ( µ ) . Arguing as in the proofof Theorem 3.2 one obtains an isomorphism X ∼ = Z , where Z = ( Z, B Z , ˜ µ, ( ˜ T g ) g ∈ G ) isconstructed in the same way as in the proof of Theorem 1.3 (but now for the algebra A ). Let Y = ( D ℓ ) G . Define a map Φ from Z to the compact space Y by Φ( χ ) =(( χ ( T g F ) , . . . , χ ( T g F ℓ )) g ∈ G . Note that Φ is well defined. Indeed, since χ ∈ Z , and A isa unital C ∗ -algebra, we have that χ is a positive linear functional that respects conjugation.Now, F i ( X ) ⊆ D , whence − | F i | is a positive element (of L ∞ (˜ µ ) and χ (1) = 1 , whichimplies that | χ ( F i ) | ≤ , so χ ( F i ) ∈ D , as claimed. Finally, arguing again as in the proof ofTheorem 3.2 we can see that Φ is injective and continuous. Notice that, in particular, thismeans that Φ is measurable. Now, let ν = Φ ∗ µ . The map Φ provides a measurable isomor-phism between the measure preserving system Z and the system Y which was constructed n Lemma A.3. Letting H i = x i ( e ) , we see (again by Stone-Weierstrass) that the probabilitymeasure ν is determined by the values Z Y r Y j =1 S g j ˜ H j dν, for all r ∈ N , g , . . . , g r ∈ G, ˜ H j ∈ { H , . . . , H ℓ , ¯ H , . . . , ¯ H ℓ } . We have: Z Y r Y j =1 S g j ˜ H j dν = Z Z r Y j =1 S g j ˜ H j (Φ( χ )) d ˜ µ ( χ ) = Z Z r Y j =1 S g j ˜ H j (( χ ( S g F ) , . . . , χ ( S g F ℓ )) g ∈ G ) d ˜ µ ( χ ) = Z Z χ r Y i =1 T g i ˜ F i ! d ˜ µ ( χ ) = Z Z Γ r Y i =1 T g i ˜ F i ! d ˜ µ ( χ ) = Z X r Y i =1 T g i ˜ F i dµ = lim N →∞ b ( N ) | F N | X g ∈ F N a ( g ) r Y i =1 ˜ f i ( g i g ) , where we used the definition of the Gelfand isomorphism Γ :
A → C ( Z ) in the second tolast step, and the fact that X is a Furstenberg system for the tuple ( { f , . . . , f ℓ } , ( F N ) , a, b ) in the last. It follows that for ω = (( f ( g ) , . . . , f ℓ ( g ))) g ∈ G ,(A.3) ν = w* lim k →∞ b ( N ) | F N | X g ∈ F N a ( g ) δ S g ω , whence supp ( ν ) ⊆ { S g ω : g ∈ G } . Notice that the right hand side in (A.3) is well defineddue to the assumption that { f , . . . , f ℓ } is accordant with the averaging scheme (( F N ) , a, b ) .We are done. (cid:3) Let (( F N ) , a, b ) be an averaging scheme. As was done in Section 5, we can also go back froma system an ergodic system ( X, B , µ, ( T g ) g ∈ G ) and bounded measurable functions F , . . . , F ℓ : X → C to families of bounded functions { f , . . . , f ℓ } that are accordant with (( F N ) , a, b ) : Theorem A.5.
Let G be a countable amenable group. Let ( X, B , µ, ( T g ) g ∈ G ) be an ergodicmeasure preserving system. Let F , . . . , F ℓ ∈ L ∞ ( µ ) and let (( F N ) , a, b ) be an averagingscheme. Then there exists a subsequence ( F N k ) and a family of C -valued bounded functions { f , . . . , f ℓ } such that (A.4) Z X r Y i =1 T g i ˜ F i dµ = lim k →∞ b ( N k ) | F N k | X g ∈ F Nk a ( g ) r Y i =1 ˜ f i ( g i g ) , for all r ∈ N and g , . . . , g r ∈ G , where ˜ F i ∈ { F , . . . , F ℓ , ¯ F , . . . , ¯ F ℓ } and ˜ f i ∈ { f , . . . , f ℓ , ¯ f , . . . , ¯ f ℓ } respectively.Proof. We argue as in the proof of Theorem 5.1 for F := { Q mi =1 T g i ˜ F i : r ∈ N , g , . . . , g r ∈ G, ˜ F i ∈ { F , . . . , F ℓ , ¯ F , . . . , ¯ F ℓ }} to obtain a full measure subset X such that for all x ∈ X and H ∈ F we have(A.5) lim k →∞ b ( N k ) | F N k | X g ∈ F Nk a ( g ) H ( T g x ) = Z X H dµ.
Now, for each x ∈ X we can let f i,x ( g ) = F i ( T g x ) , i = 1 , . . . , ℓ , and one easily checks thatfor all x ∈ X the family { f ,x , . . . , f ℓ,x } satisfies (A.4) as desired. (cid:3) eferences [BBF] M. Beiglböck, V. Bergelson and A. Fish, Sumset phenomenon in countable amenable groups, Adv.Math. (2010), no. 2, 416–432.[B] V. Bergelson, Ergodic theory and Diophantine problems, in Topics in symbolic dynamics and applica-tions (Temuco, 1997) , 167–205, London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press,Cambridge.[BF] V. Bergelson and A. Ferré Moragues, Juxtaposing d ∗ and ¯ d , Preprint: https://arxiv.org/abs/2003.03029 [BL] V. Bergelson and A. Leibman, Cubic averages and large intersections, in Recent trends in ergodic theoryand dynamical systems , 5–19, Contemp. Math., 631, Amer. Math. Soc., Providence, RI.[BMc] V. Bergelson and R. McCutcheon, Recurrence for semigroup actions and a non-commutative Schurtheorem, in
Topological dynamics and applications (Minneapolis, MN, 1995) , 205–222, Contemp. Math.,215, Amer. Math. Soc., Providence, RI.[DL] M. Di Nasso and M. Lupini, Nonstandard analysis and the sumset phenomenon in arbitrary amenablegroups, Illinois J. Math. (2014), no. 1, 11–25[FrH] N. Frantzikinakis and B. Host, Furstenberg systems of bounded multiplicative functions and applica-tions. Preprint, URL: https://arxiv.org/abs/1804.08556 .[F1] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmeticprogressions, J. Analyse Math. (1977), 204–256.[F2] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory , Princeton UniversityPress, Princeton, NJ, 1981.[HS] N. Hindman and D. Strauss, Density and invariant means in left amenable semigroups, Topology Appl. (2009), no. 16, 2614–2628.[N] I. Namioka, Følner’s conditions for amenable semi-groups, Math. Scand. (1964), 18–28.[Neu] J. von Neumann, Einige Sätze über messbare Abbildungen, Ann. of Math. (2) (1932), no. 3,574–586.[P] K. Petersen, Ergodic theory , Cambridge Studies in Advanced Mathematics, 2, Cambridge UniversityPress, Cambridge, 1983.[W] P. Walters,
An introduction to ergodic theory , Graduate Texts in Mathematics, 79, Springer-Verlag,New York, 1982.
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
E-mail address : [email protected] Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
E-mail address : [email protected]@osu.edu