Mixing rank-one actions of locally compact abelian groups

Mixing rank-one actions of locally compact Abelian groups

Alexandre I. Danilenko

, Cesar E. Silva

aMax Planck Institute of Mathematics, Vivatsgasse 7, D53111 Bonn, Germany bDepartment of Mathematics, Williams College, Williamstown, MA 01267, USA

Received 26 April 2005; received in revised form 3 April 2006; accepted 30 May 2006 Available online 15 December 2006

Abstract

Using techniques related to the (C, F)-actions we construct explicitly mixing rank-one (by cubes) actionsT ofG=R^d1×Z^d2 for any pair of non-negative integersd₁,d₂. It is also shown thath(T_g)=0 for eachg∈G.

©2006 Elsevier Masson SAS. All rights reserved.

Résumé

En utilisant des techniques liées aux action(C, F ), nous construisons explicitement des actions mélangeantes de rang un (par cubes),T deG=R^d1×Z^d2pour toute paire de nombres entiersd₁, d₂0. On prouve aussi queh(T_g)=0 pour chaqueg∈G.

©2006 Elsevier Masson SAS. All rights reserved.

MSC:37A25; 37A15

Keywords:Ergodic action; Mixing; Rank-one action; Entropy

0. Introduction

Mixing rank-one transformations (and actions of more general groups) have been of interest in ergodic theory since 1970 when Ornstein constructed an example of mixing transformation without square root [18]. His method was used later as the core of a number of other remarkable constructions (see [20,21,13,10,17,19,8], etc.) Since then the dynamical properties of mixing rank-one transformations have been deeply investigated. It is now well known that such transformations are mixing of all orders [14,22] and have minimal self-joinings of all orders [15,22]. This implies in turn that they are prime and have trivial centralizer [21]. The results on multiple mixing were extended to rank-one mixing actions ofR^d andZ^d[22–24] and to rank-one mixing actions of a wide class of discrete countable Abelian groups having an element of infinite order [11].

✩ This project was supported in part by the NSF under the COBASE program, contract INT-0002341. Also, the first named author was supported in part by CRDF, grant UM1-2546-KH-03.

* Corresponding author.

E-mail addresses:danilenko@ilt.kharkov.ua (A.I. Danilenko), csilva@williams.edu (C.E. Silva).

1 Permanent address: Institute for Low Temperature Physics & Engineering of Ukrainian National Academy of Sciences, 47 Lenin Ave., Kharkov, 61164, Ukraine.

0246-0203/$ – see front matter ©2006 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.05.002

Despite this progress, there are not many concrete examples of rank-one mixing actions that are known. Most of them were obtained via stochastic cutting-and-stacking techniques using “random spacers”. Ornstein initiated this technique in [18], and more recent generalizations include the constructions ofR-actions in [19] and actions of infinite sums of finite groups in [8], as well as the del Junco–Madore actions of Abelian extensions ofZ^d by locally finite groups [10,17]. The latter actions were only shown to be weakly mixing but conjectured to be mixing in [17]. While demonstrating the existence of mixing rank-one actions (which is a non-trivial problem!), these works do not exhibit a specific such transformation or action. In 1992 Adams and Friedman [2] gave a non-random algorithm that leads to a mixing rank-one construction. Using the ideas from that manuscript Adams [1] proved in 1998 the old conjecture that the classical staircase is mixing. That gave the first explicit example of mixing cutting-and-stacking transformation.

Higher dimensional mixing staircaseZ^d-actions were later constructed in [3]. We note that the complete proof of the fact that they are mixing was given there only in dimensiond=2. As one of the consequences of our work, we complete the proof for alld >2 (see Remark 4.12 below). Recently, a more general family of mixing “polynomial”

staircaseZ-actions was constructed in [5]. Another interesting non-random construction appears in a recent work [12]

devoted to smooth realizations of mixing rank-one flows on the 3-torus.

Our main purpose here isto construct explicitly a family of mixing rank-one actions of R^d1 ×Z^d2 for all non- negatived₁andd₂. It seems plausible that Orntein’s stochastic method also can be adapted to produce mixing rank- one actions of these groups. We note however that our construction is more general and the ‘randomness’ can be incorporated into it (see [5] and [8] for a detailed discussion on that forZ-actions and actions of infinite sums of finite groups respectively). Moreover, the main advantage of our approach is that the examples in our family are ‘absolutely concrete’, i.e., the parameters in the construction are all explicitly specified—the ‘spacer mappings’ are polynomials with known coefficients.

As a corollary we show that this family includes all the examples of mixing rank-one Z^d-actions constructed previously in [1,3] and [5]. Our approach is based on ideas that first appeared in those three works. However, in this paper we proceed entirely in the framework of (C, F)-actions for locally compact second countable (l.c.s.c.) Abelian groups, and in fact we develop a large part of the theory in the more general context of these actions. In particular, we encounter here some new problems that are specific to higher dimensions and the continuity of the groups. Recall that the (C, F)-construction of finite measure-preserving actions of discrete countable amenable groups appeared in [10]

as an algebraic counterpart of the “geometrical” cutting-and-stacking method developed forZ-actions. Later it was used (in a modified form) by the authors in the framework of infinite measure-preserving and non-singular countable Abelian group actions, as a convenient tool for modeling examples and counterexamples with various properties of weak mixing and multiple recurrence (see [6,7,9]).

LetGbe a non-compact l.c.s.c. Abelian group andT =(T_g)_g∈Ga measurable action ofGon a standard probability space(X,B, μ).

Definition 0.1.T is said to bemixingif for all subsetsA, B∈Bwe have

glim→∞μ(TgA∩B)=μ(A)μ(B). (0.1)

A sequencegn→ ∞inGis calledmixingif(0.1)holds alonggnasn→ ∞.

Notice that an action is mixing whenever each sequence converging to infinity inGcontains a mixing subsequence.

Definition 0.2.

(i) ARokhlin tower or columnforT is a triple(Y, f, F ), whereY ∈B,F is a relatively compact subset ofGand f:Y→Fis a measurable mapping such that for any Borel subsetH⊂Fand an elementg∈Gwithg+H⊂F, one hasf⁻¹(g+H )=Tgf⁻¹(H ).

(ii) We say that T is of funny rank-one if there exists a sequence of Rokhlin towers (Y_n, f_n, F_n) such that limn→∞μ(Y_n)=1 and for any subsetB∈B, there is a sequence of Borel subsetsH_n⊂F_nsuch that

nlim→∞μ

Bf_n⁻¹(H_n)

=0.

(iii) IfG=R^d1×Z^d2,T is of funny rank-one and, in addition, the subsetsF_nfrom (ii) are as follows F_n=

(t₁, . . . , t_d1+t2)∈G|0t_i< a_nfor alli=1, . . . , d1+d₂

for somean∈R,n=1,2, . . ., then we say thatT is ofrank-one(orrank-one by cubes).

It is easy to see that any funny rank-one action is ergodic.

Note that what we call funny rank-one is called rank-one by del Junco and Yassawi in case G is discrete and countable and G=Z[11]; also del Junco and Yassawi require in addition that the sequence of sets(F_n)^∞_n=1 is a Følner sequence.

The paper is organized as follows. In Section 1 we extend the concept of (C, F)-action introduced for countable discrete groups (see [10,6]) to the class of l.c.s.c. Abelian ones. A special family of actions, whose mixing properties will be under investigation in subsequent sections, is defined. In Section 2 we introduce a concept of uniformly mixing sequence and prove a fundamental lemma (Lemma 2.2) linking the uniform mixing along some special sequences with Cesàro means for the ‘spacer mappings’. Then we find a sufficient condition for the total ergodicity of the actions under considerations. We also start to check the uniform mixing property for some special sequences. In particular, we show that if a sequence is of ‘moderate growth’ relative to a fixed Følner sequence inGthen (under some extra conditions onGand the action) it is uniformly mixing (Lemma 2.9). Section 3 is devoted to the actions with restricted growth—

the property which was phrased explicitly in [5] forG=Zbut used already in [1] and [3] in an implicit form. We note that our definition of restricted growth differs from that introduced in [5] (the latter does not extend fromZto arbitrary l.c.s.c. Abelian group actions). However, they are equivalent for polynomial staircase actions. Theorem 3.5 provides a sufficient condition for the (C, F)-actions with restricted growth to be mixing. We also include here a couple of statements (Lemmas 3.9–3.11) facilitating verification of this condition for theR^d1×Z^d2-actions to be constructed in the next section. Section 4 contains the main results of the paper: Theorems 4.9–4.11 and 4.13 which provide families of mixing rank-one actions ofR^d withd >1,R,Z^d andR^d1 ×Z^d2 respectively. Every such action is determined completely by a sequence of positive integers (r_n)^∞_n=1 (corresponding to the sequence of ‘cuts’ in the cutting-and- stacking construction) and a sequence (s_n)^∞_n=1of ‘monotonic’ polynomials of d1+d2 variables (corresponding to the sequence of ‘spacer’s maps’ on then-th step). The sequences are chosen in the following way:(r_n)^∞_n=1is any sequence of sub-exponential growth with limn→∞rn= ∞and(sn)^∞_n=1consists of some specially selected quadratic polynomials from Example 4.2. Moreover, ifd1=1 (and only in this case) then(sn)^∞_n=1can be chosen constant. If d1=1 then(sn)^∞_n=1can be chosen consisting of two alternating polynomials. Furthermore, using our techniques plus the Hilbertian van der Corput trick we can also treat a more complicated case where(sn)^∞_n=1consists of polynomials of degree>2 (see Proposition 4.14). Example 4.15 provides a family of rank-one mixing transformations including the polynomial staircases from [5]. In the final section (Section 5) we show that the actions constructed in Section 4 have ‘very weak’ stochastic properties—the entropy of any individual transformation from such actions is zero. This fact holds for any rank-one (by cubes) action. However, it is no longer true for a more general class of actions of rank-one ‘by rectangles’ (see [20] for a counterexample).

1. (C, F)-actions of locally compact Abelian groups

In this section we introduce the (C, F)-actions of l.c.s.c. Abelian groups and specify a subclass of them (see Definitions 1.2 and 1.4). We explain how the classical cutting-and-stacking transformations are included into this subclass (Remark 1.6). The aim of the paper is to show that this subclass contains mixing actions.

LetGbe a l.c.s.c. Abelian group. Denote by λG a (σ-finite) Haar measure on it. Given two subsets E, F⊂G, byE+F we mean their algebraic sum, i.e.,E+F = {e+f |e∈E, f ∈F}. The algebraic differenceE−F is defined in a similar way. We hope that the reader will not confuse it with the set theoretical differenceE\F. IfEis a singleton, sayE= {e}, then we will writee+F forE+F. If(E−E)∩(F−F )= {0}thenEandF are called independent. For an elementg∈Gand a subsetE⊂G, we setE(g)=E∩(E−g).

To define a (C, F)-action ofGwe need two sequences(Fn)n0and(Cn)n>0of subsets inGsuch that the following hold

F0⊂F1⊂F2⊂ · · · is a Følner sequence inG, (1.1)

C_nis finite and #Cn>1, (1.2)

Fn+Cn+1⊂Fn+1, (1.3)

F_nandC_n+1are independent. (1.4)

We putXn:=Fn×

k>nCk, endowXnwith the standard product Borelσ-algebra and define a Borel embedding X_n→X_n+1by setting

(f_n, c_n+1, c_n+2, . . .) →(f_n+c_n+1, c_n+2, . . .). (1.5) Then we haveX₁⊂X₂⊂ · · ·. HenceX:=

nX_n endowed with the natural Borelσ-algebra, sayB, is a standard Borel space. Given a Borel subsetA⊂F_n, we denote the set

x∈X|x=(fn, cn+1, cn+2. . .)∈Xnandfn∈A

by[A]nand call it ann-cylinder. It is clear that theσ-algebraBis generated by the family of all cylinders.

Now we are going to define a measure on (X,B). Let κ_n stand for the equidistribution on C_n and ν_n:=

(#C1· · ·#Cn)⁻¹λ_GF_nonF_n. We define a product measureμ_nonX_nby setting μ_n=ν_n×κ_n+1×κ_n+2× · · ·,

n∈N. Then the embeddings (1.5) are all measure preserving. Hence aσ-finite measureμonXis well defined by the restrictionsμX_n=μ_n,n∈N. To put it in another way,(X, μ)=inj limn(X_n, μ_n). Since

μ_n+1(X_n+1)= ν_n+1(F_n+1)

νn+1(Fn+Cn+1)μ_n(X_n)= λ_G(F_n+1) λG(Fn)#Cn+1

μ_n(X_n), it follows thatμis finite if and only if

∞ n=0

λG(Fn+1) λG(Fn)#Cn+1

<∞, i.e.,

∞ n=0

λG(Fn+1\(Fn+Cn+1)) λG(Fn)#Cn+1

<∞. (1.6)

For the rest of the paper we will assume that (1.6) is satisfied. Moreover, we choose (i.e., normalize)λ_Gin such a way thatμ(X)=1.

To construct a measure-preserving action ofG on(X, μ), we fix a filtrationK1⊂K2⊂ · · ·of Gby compact subsets. Thus_∞

m=1K_m=G. Givenn, m∈N, we set D⁽ⁿ⁾_m :=

k∈Km

(Fn−k)∩Fn

×

k>n

Ck⊂Xn

and

R_m⁽ⁿ⁾:=

k∈Km

(F_n+k)∩F_n

×

k>n

C_k⊂X_n. It is easy to verify that

D⁽ⁿ⁾_m+1⊂D⁽ⁿ⁾_m ⊂D_m⁽ⁿ⁺¹⁾ and R⁽ⁿ⁾_m+1⊂R_m⁽ⁿ⁾⊂R⁽ⁿ_m⁺¹⁾. We define a Borel mapping

K_m×D_m⁽ⁿ⁾(g, x) →T_m,g⁽ⁿ⁾x∈R_m⁽ⁿ⁾ by setting forx=(f_n, c_n+1, c_n+2, . . .),

T_m,g⁽ⁿ⁾(fn, cn+1, cn+2, . . .):=(g+fn, cn+1, cn+2, . . .).

Now letD_m:=_∞

n=1D_m⁽ⁿ⁾andR_m:=_∞

n=1R⁽ⁿ⁾_m . Then a Borel mapping K_m×D_m(g, x) →T_m,gx∈R_m

is well defined by the restrictions T_m,gD_m⁽ⁿ⁾=T_m,g⁽ⁿ⁾ for g∈K_m andn1. It is easy to see that D_m⊃D_m+1, R_m⊃R_m+1 andT_m,gD_m+1=T_m+1,g for allm. It follows from (1.1) thatμ_n(D⁽ⁿ⁾_m )→1 andμ_n(R⁽ⁿ⁾_m )→1 as n→ ∞. Henceμ(D_m)=μ(R_m)=1 for allm∈N. Finally we setX:=_∞

m=1D_m∩_∞

m=1R_mand define a Borel mapping

T :G×X(g, x)→Tgx∈X

by settingT_gx:=T_m,gxfor some (and hence any)msuch thatg∈K_m. It is clear thatμ(X)=1.

Proposition 1.1. T =(Tg)g∈G is a free Borel measure preserving action of Gon a conull subset of the standard probability space(X,B, μ). It is of funny rank-one.

Proof. It suffices to verify only the latter claim. According to Definition 0.2 we have to find a sequence of Rokhlin towers ‘appoximating’ the dynamical system. Let s_n denote the projection of X_n=F_n×C_n+1× · · ·onto the first coordinate. It is easy to see that the sequence(X_n, s_n, F_n)is as desired. 2

Throughout the paper we will not distinguish between two measurable sets (or mappings) which agree almost everywhere. It is easy to see thatT does not depend on the choice of filtration(K_m)^∞_m=1.

Definition 1.2.T is called the (C, F)-action of Gassociated with(C_n, F_n)_n.

We will often use the following simple properties of(X, μ, T ): for Borel subsetsA, B⊂F_n,

[A∩B]n= [A]n∩ [B]n, (1.7)

[A]n= [A+C_n+1]n+1=

c∈Cn+1

[A+c]n+1, (1.8)

Tg[A]n= [A+g]n ifA+g⊂Fn, (1.9)

μ [A]n

=#Cn+1·μ

[A+c]n+1

for everyc∈Cn+1, (1.10)

μ [A]n

λ_G(A)

λ_G(F_n), (1.11)

where the signmeans the union of mutually disjoint sets.

Recall that an actionT ofGon(X,B, μ)ispartially rigidif there existsδ >0 with lim inf

g→∞ μ(T_gB∩B)δμ(B) for allB∈B.

It is clear that partial rigidity is incompatible with the mixing. For the (C, F)-actions, there is a simple condition that implies the partial rigidity.

Proposition 1.3.Iflim infn→∞#Cn<∞thenT is partially rigid and hence is not mixing.

Proof. Letn_i< n₂<· · ·be a sequence of indices with #Cn1=#Cn2= · · ·. Selectc_i =c_i inC_i and setg_i:=c_i−c_i. Theng_i∈/F_ni−F_ni by (1.4). On the other hand, it follows from (1.1) that_∞

i=1(F_ni−F_ni)=G. Henceg_i→ ∞as i→ ∞. Take a cylinderB∈B. We can represent it eventually (i.e., for all large enoughi) asB= [B_i]ni, whereB_i is a Borel subset ofFn_i. If follows from (1.7)–(1.10) that

μ(T_giB∩B)=μ

T_gi[B_i−1+C_ni]ni∩ [B_i−1+C_ni]ni

μ

T_gi[B_i−1+c_i]ni∩ [B_i−1+c_i]ni

=μ

[B_i−1+c_i]ni

= 1

#Cn_i

μ

[B_i−1]ni−1

= 1

#Cn1

μ(B).

Since the cylinders generate a dense subalgebra inB, we are done. 2

Now we isolate a special subfamily of (C, F)-actions to show in the sequel that it contains mixing actions.

LetH be a discrete countable group, and letφ_n, s_n andc_n+1 be three mappings fromH toGsuch thatφ_n is a homomorphism,s_n(0)=0 andc_n+1:=φ_n+s_n,n∈N. Suppose that

(H_n)_n0is a Følner sequence inH, 0∈H_n (1.12)

and

φ_n(H )is a lattice inG. (1.13)

Now we defineFn⊂Gto be a Borel fundamental domain forφn(H )(i.e., a subset which meets everyφn(H )-coset exactly once) and putCn+1:=cn+1(Hn),n0. Assume that (1.1)–(1.4) are all satisfied.

Definition 1.4.We call the corresponding (C, F)-actionT ofGon the probability space(X,B, μ)the action associ- ated with(H_n, φ_n, s_n, F_n)_n.

In view of Proposition 1.3, we will always assume that limn→∞#Hn= ∞.

Notice also that ifs_nare all trivial, i.e.s_n(h)=0 for allh∈H_nthen the action ofGassociated with(H_n, φ_n, s_n, F_n) has pure point spectrum with rational eigenvalues only. This simple fact will not be used in this paper. We leave its proof to the reader.

In the statements of our main results here it will be assumed that the mappingssnare polynomials of degree>1.

Definition 1.5.[16] For anyh∈H, theh-derivativeofsis a mapping∂_hs:H→Ggiven by∂_hs(k)=s(k+h)−s(k).

Letd be a non-negative integer. Thens is called apolynomial of degreed if for anyh1, . . . , hd+1∈H\ {0}, we have∂h₁· · ·∂h_d+1s=0. The minimaldwith this property is calledthe degreeofs.

It is easy to see that every polynomial of degree 0 is constant. As was shown in [16], a polynomial of degree one is a non-constant affine mapping (i.e., a homomorphism plus a constant). A polynomial fromZ^dtoR^lis anl-tuple of usual polynomials indvariables with real coefficients. A polynomial fromZ^dtoZ^lis anl-tuple(p1, . . . , pl)of usual polynomials indvariables with rational coefficients such thatpi(Z^d)⊂Zfor alli=1, . . . , l.

Remark 1.6.Here we are going to explain how the (C, F)-construction forZ-actions is related to the classical cutting- and-stacking construction. Recall that the latter one defines ergodic measure-preserving transformations on intervals inR(or on[a,+∞)) furnished with Lebesgue measure via an inductive procedure. Acolumnis an ordered collection of intervals, calledlevels, of the same length. The number of levels is called theheightof the column. The associated column mappingis defined by translation of each level to the level above it (i.e., next in the order). Hence the column mapping is defined from all but the top level onto all but the bottom level. Suppose now that we are given a sequence (r_n)^∞_n=1of positive integers and a sequence of arrays of non-negative integers(σ_n(j ), j=0,1, . . . , rn−1)^∞_n=1. Then we define inductively a sequence of columns as follows. Let the initial columnY₀consists of one level of length 1.

Suppose that on then-th step we have a columnY_n consisting of levelsI (i, n), 0i < a_n. Cut everyI (i, n) into rnsublevelsIk(i, n), 0k < rn, numbered from left to right. Then we obtainrnsubcolumnsYn,k:= {Ik(i, n)|i= 0, . . . , an−1}, 0k < rn, ofYnof the same height. Now placeσn(k)spacers(i.e., the intervals of the same length as I_k(i, n)) aboveY_n,kand stack the resulting subcolumns with spacers right to the top of left. This yields a new column Y_n+1of heighta_n+1=a_nr_n+r_n−1

k=0 σ_n(k)and a natural inclusion ofY_nintoY_n+1. Notice that the associated(n+1)- column mapping restricted toY_ncoincides with then-th column mapping. Hence the associated sequence of column mappings approaches a transformation defined on all but a measure zero subset of the union of the initial level and the spacers added at each column. It is easy to see that this transformation corresponds exactly to the (C, F)-action of Z associated with(Hn, φn, sn, Fn)n if we putHn:= {0,1, . . . , rn−1}, φn(t ):=ant,sn(t ):=t

k=0σn(k) and F_n= {0,1, . . . , an−1}. If we setσ_n(k)=kfor all 0k < r_nandn∈Nthen the corresponding cutting-and-stacking transformation is called astaircase. If, moreover,r_n=n for alln∈N, we obtain theclassical staircasewhich is finite measure-preserving. In case the sequence(σ_n)^∞_n=1consists of polynomials, the corresponding transformations are calledpolynomial staircases[5].

2. Uniformly mixing sequences

For the remaining of the paper (X,B, μ, T ) will stand for the (C, F)-action of G associated to a sequence (Hn, φn, sn, Fn)n.

In this section we prove a fundamental Lemma 2.2 and use it to show that some special sequences in G are uniformly mixing. As an auxiliary result for that we exhibit a sufficient condition for the total ergodicity ofT. A con- nection between the uniform mixing and total ergodicity is established in Corollary 2.6.

Definition 2.1.A sequence(gn)^∞_n=1of elements fromGis calleduniformly mixingforT if sup

A⊂Fn

μ

Tg_n[A]n∩B

−μ [A]n

μ(B)→0 asn→ ∞ (2.1)

for every subsetB∈B.

It is easy to see that if a sequence is uniformly mixing then it is mixing.

Notice that a somewhat different definition for uniform mixing was given in [5] in caseG=Z. To state it precisely we assume that the sequence (H_n, φ_n, s_n, F_n)_n is chosen as described in Remark 1.6. Then a sequence of positive integers(g_n)^∞_n=1was called uniformly mixing in [5] if

f∈F_pn

μ

T_gn[f]pn∩B

−μ [f]pn

μ(B)→0

for every subsetB∈B, wherepnis the unique positive integer such thatap_ngn< ap_n+1. We only observe that this implies the uniform mixing in the sense of Definition 2.1 ifpnnfor alln.

LetLbe a finite set anda:L→Ga mapping. We define a linear operatorMa,LinL²(X, μ)by setting Ma,L(f ):= 1

#Ll∈L

f◦T_a(l).

LetP₀stand for the projection onto the subspace of constant functions, i.e.P₀(f )=

Xfdμ. The inner product in L²(X, μ)will be denoted by·,·.

Fix a sequence(hn)_n1of elements fromH. For brevity, we will denote∂h_nsnbys_n,n∈N.

Lemma 2.2.If the following conditions are satisfied

#Hn(h_n)

#Hn →1, (2.2)

1

#Hn

h∈Hn(hn)

λ_G(F_n\F_n(s_n(h)))

λ_G(F_n) →0 (2.3)

then for alln-cylindersA, B⊂X, we have μ(Tφ_n(h_n)A∩B)= 1

#Hn

h∈Hn(hn)

μ(A∩T_sn(h)B)+o(1)

=

χA,Ms_n,Hn(hn)(χB) +o(1),

whereχ_A andχ_B are the indicators ofAandB respectively ando(1)denotes a sequence that tends to0and that does not depend onAandB. The same formula holds as well for

(i) an arbitrary subsetB∈BandAas above witho(1)depending onBonly and (ii) arbitrary subsetsA, B∈Bwitho(1)depending on bothAandB.

Proof. LetA_n andB_n be the Borel subsets of F_n such that A= [A_n]n and B= [B_n]n. Forh∈H_n(h_n), we put A_n,h:=A_n∩F_n(−s_n(h)). Then

A_n,h−s_n(h)⊂F_n. (2.4)

We also make a simple but important observation that φ_n(h_n)+c_n+1(h)=φ_n(h_n)+φ_n(h)+s_n(h)

=φ_n(h+h_n)+s_n(h+h_n)−s_n(h)

=c_n+1(h+h_n)−s_n(h). (2.5)

It follows from (1.8), (1.9), (2.4), (1.3), (2.5) and (1.10) that

μ(Tφ_n(h_n)A∩B)=

h∈H_n

μ Tφ_n(h_n)

An+cn+1(h)

n+1∩ [Bn]n

=

h∈Hn(hn)

μ T_φn(hn)

A_n,h+c_n+1(h)

n+1∩ [B_n]n

±μ

(An\An,h)+cn+1(h)

n+1

±

h∈Hn\Hn(hn)

μ

Fn+cn+1(h)

n+1

=

h∈H_n(h_n)

μ

A_n,h−s_n(h)+c_n+1(h+h_n)

n+1∩ [B_n]n

± 1

#Hn

μ

[A_n\A_n,h]n

±

1−#Hn(h_n)

#Hn

r

.

Notice that for allc∈C_n+1andh∈H_n(h_n), we have by (1.7), (1.8) and (1.11), [An,h−s_n(h)+c]n+1∩ [Bn]n=

An,h−s_n(h)

∩Bn

+c

n+1 (2.6)

and μ

(A_n\A_n,h)

n

λ_G(A_n\A_n,h)

λ_G(F_n) λ_G(F_n\F_n(s_n(h)))

λ_G(F_n) . (2.7)

Hence it follows from (1.10), (2.2), (2.3), (2.6) and (2.7) that μ(T_φn(hn)A∩B)= 1

#Hn

h∈H_n(h_n)

μ

A_n,h−s_n(h)

∩B_n

n

+o(1).

Applying (1.7), (1.9) and (2.4) we obtain μ(T_φn(hn)A∩B)= 1

#Hnh∈H_n(h_n)

μ T_−s

n(h)[A_n,h]n∩ [B_n]n

+o(1)

= 1

#Hn

h∈Hn(hn)

μ(A∩T_sn(h)B)±μ

[A_n\A_n,h]n

+o(1).

It remains to make use of (2.7), (2.3) and (2.2).

The final claim of Lemma 2.2 follows from the fact that the cylinders generate a dense subalgebra inB. 2 Corollary 2.3.Let(2.2)and(2.3)hold. Then the following are satisfied:

(i) The sequence(φn(hn))^∞_n=1is mixing forT if and only ifMs_n,Hn(hn)→P0in the weak operator topology.

(ii) IfMs_n,Hn(hn)→P0in the strong operator topology then(φn(hn))^∞_n=1is uniformly mixing.

We now examine whenT is totally ergodic. Recall some standard definitions.

Definition 2.4.

(i) Given a subsetB∈B, we denote byG_BthestabilizerofB, i.e.,G_B:= {g∈G|T_gB=B}.

(ii) T is calledtotally ergodicif for any co-compact subgroupK⊂G, the action(T_g)_g∈K is ergodic.

(iii) T is calledweakly mixing if the diagonal action (T_g×T_g)_g∈G of Gis ergodic. Equivalently, if there exist a functionf ∈L²(X, μ)and a continuous characterχofGsuch thatf ◦T_g=χ (g)f a.e. thenf is constant.

It is easy to see thatT is totally ergodic if and only if the stabilizer of any subsetB∈Bwith 0< μ(B) <1 is not co-compact. Moreover, if an action is weakly mixing then it is totally ergodic. The converse is true forG=Rbut it does not hold for general groups.

Proposition 2.5.Let(2.2)and (2.3)hold. LetK be a co-compact subgroup of Gandπ:G→G/K stand for the corresponding quotient map. Denote byκ_nthe image of the equidistributed probability onH_n(h_n)under the mapping (π◦s_n)_∗,n∈N. Ifκ_ndoes not∗-weakly converge to a Diracδ-measure onG/KthenKis not the stabilizer of any measurable subsetB∈Bwith0< μ(B) <1.

Proof. Suppose that the contrary holds, i.e., there existsB∈B with 0< μ(B) <1 and K=GB. Then the quo- tient compact groupG/K acts naturally on the sub-σ-algebraF of (Tg)g∈K-invariant subsets. Denote this action by T. Then Tπ(g)A:=TgA for all g∈Gand A∈F. It is clear thatTis free. We setan:=π(φn(hn)). Passing to a subsequence, if necessary, we may assume without loss of generality that an converges to somea∈G. Then μ(Ta_nBTaB)→0 asn→ ∞. Hence

μ(B)=μ(T_φn(hn)B)=μ

T_φn(hn)B∩T_anB

=μ

T_φn(h)B∩T_aB +o(1).

We then deduce from this formula and Lemma 2.2(ii) that μ(B)= 1

#Hn

h∈Hn(hn)

μ

B∩T_sn(h)T_aB +o(1)

= 1

#Hn(h_n)

h∈Hn(hn)

μT_π(sn(h))+aB∩B +o(1)

=

G/K

μTa+bB∩B

dκn(b)+o(1). (2.8)

Since G/K is compact, we may assume (passing to a subsequence, if necessary) thatκ_n converges∗-weakly to a probabilityκwhich is not a Diracδ-measure by the condition of the proposition. Hence passing to a limit in (2.8) we obtain

μ(B)=

G/H

μT_a+bB∩B dκ(b)

Henceμ(B)=μ(T_a+bB∩B), i.e.,B=T_a+bB, forκ-a.a.b∈G/K. SinceT is free, we deduce that Suppκ= {−a}.

Hence Suppκis a singleton, a contradiction. 2

Now we are interested in the following particular case. There exist a non-zerok∈Hand a polynomials:H→G of degree 2 such thathn=kandsn=sfor alln∈N. Then, of course, (2.2) is satisfied. Moreover, for any non-zero t∈H, we have

∂_ts(h)=ψ_t(h)+a_t, at allh∈H

for some non-trivial homomorphismψ_t:H→Gand an elementa_t∈G(see the text following Definition 1.5). Hence s_n(h)=ψ_k(h)+a_kfor allh∈H andn∈N.

Corollary 2.6.Let the above assumptions and(2.3)hold. Then the following are satisfied:

(i) If the action(T_ψk(h))_h∈H is ergodic then the sequence(φ_n(k))^∞_n=1is uniformly mixing.

(ii) If the subgroup generated by

t∈Hψ_t(H )is dense inGthenT is totally ergodic.

(iii) If the subgroup generated by

t∈Hψ_t(H )is dense inGandψ_k(H )is a lattice inGthen the sequence(φ_n(k))^∞_n=1 is uniformly mixing.

Proof. (i) follows from Corollary 2.3 and the von Neumann mean ergodic theorem for(T_ψk(h))_h∈H.

(ii) Suppose that the contrary holds, i.e., there exists a co-compact subgroupK⊂Gand a subsetB⊂Xsuch that 0< μ(B) <1 andK=G_B. Fixt∈H,t=0. Thenκ_nis the translation of(π◦ψ_t)_∗ν_nbyπ(a_t), whereν_nstands for the equidistribution onH_n(t ),n∈N. Denote byG_t the closure of the subgroup(π◦ψ_t)(H )inG/K. SinceH_n(t )is a Følner sequence inH andπ◦ψ_t:H→G/Kis a homomorphism, it is easy to verify that(π◦ψ_t)_∗ν_nconverges