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MIXING RANK-ONE ACTIONS FOR INFINITE

SUMS OF FINITE GROUPS

Alexandre I. Danilenko

Abstract. Let Gbe a countable direct sum of ﬁnite groups. We construct an

uncountable family of pairwise disjoint mixing (of any order) rank-one strictly ergodic

free actions of Gon a Cantor set. All of them possess the property of minimal self-

joinings (of any order). Moreover, an example of rigid weakly mixing rank-one strictly

ergodic free G-action is given.

0. Introduction and definitions

This paper was inspired by the following question of D. Rudolph:

Question. Which countable discrete amenable groups Ghave mixing (funny) rank

one free actions?

Recall that a measure preserving action T= (Tg)g∈Gof Gon a standard prob-

ability space (X, B, µ) is called

—mixing if limg→∞ µ(A∩TgB) = µ(A)µ(B) for all A, B ∈B,

—mixing of order lif for any ² > 0 and A0, . . . , Al∈B, there exists a ﬁnite

subset K⊂Gsuch that

|µ(Tg0A0∩ · ·· ∩ TglAl)−µ(A0)· · · µ(Al)|< ²

for each collection g0, . . . , gl∈Gwith gig−1

j/∈Kif i6=j,

—weakly mixing if the diagonal action T×T:= (Tg×Tg)g∈Gof Gon the

product space (X×X, B⊗B, µ ×µ) is ergodic,

—totally ergodic if every co-ﬁnite subgroup in Gacts ergodically,

—rigid if there exists a sequence gn→ ∞ in Gsuch that limn→∞ µ(A∩

TgnB)→µ(A∩B) for all A, B ∈B.

We say that Thas funny rank one if there exist a sequence of measurable subsets

(An)∞

n=1 in Xand a sequence of ﬁnite subsets (Fn)∞

n=1 in Gsuch that the subsets

TgFn,g∈Fn, are pairwise disjoint for any nand

lim

n→∞ min

H⊂Fn

µµB4G

g∈H

TgAn¶= 0 for every B∈B.

1991 Mathematics Subject Classiﬁcation. 37A40.

Key words and phrases. Joining, mixing, rank-one action.

The work was supported in part by CRDF, grant UM1-2546-KH-03.

Typeset by A

M

S-T

E

X

1

If, moreover, (Fn)∞

n=1 is a subsequence of some ‘natural’ Følner sequence in G, we

say that Thas rank one. For instance, if G=Zd, this ‘natural sequence’ is just

the sequence of cubes; if G=P∞

i=1 Giwith every Gia ﬁnite group, the sequence

Pn

i=1 Giis ‘natural’, etc.

Up to now various examples of mixing rank-one actions were constructed for

—G=Zin [Or], [Ru], [Ad], [CrS], etc.,

—G=Z2in [AdS],

—G=Rin [Pr], [Fa],

—G=Rd1×Zd2in [DaS].

We also mention two more constructions of rank-one actions for

—G=Z⊕L∞

n=1 Z/2Zin [Ju], where it was claimed that the Z-subaction is

mixing but it was only shown that it is weakly mixing, and

—Gis a countable Abelian group with a subgroup Zdsuch that the quotient

G/Zdis locally ﬁnite in [Ma], where it was proved that a Z-subaction is

mixing and it was asked whether the whole action is mixing.

Notice that in all of these examples Gis Abelian and has elements of inﬁnite

order. In contrast to that we provide a diﬀerent class of groups for which the answer

to the question of D. Rudolph is aﬃrmative.

Theorem 0.1. Let G=L∞

i=1 Gi, where Giis a non-trivial ﬁnite group for every i.

(i) There exist uncountably many pairwise disjoint (and hence pairwise non-

isomorphic) mixing rank-one strictly ergodic actions of Gon a Cantor set.

Moreover, these actions are mixing of any order.

(ii) There exists a weakly mixing rigid (and hence non-mixing) rank-one strictly

ergodic action of Gon a Cantor set.

Concerning (i), it is worth to note that any mixing rank-one Z-action is mixing

of any order by [Ka] and [Ry] (see also an extension of that to actions of some

Abelian groups with elements of inﬁnite order in [JuY]). We do not know whether

this fact holds for all mixing rank-one action of countable sums of ﬁnite groups.

To prove the theorem, we combine the original Ornstein’s idea of ‘random spacer’

(in the cutting-and-stacking construction process) [Or] and the more recent (C, F )-

construction developed in [Ju], [Da1], [Da2], [DaS1], [DaS2] to produce funny rank-

one actions with various dynamical properties. However, unlike all of the known

examples of (C, F )-actions, the actions in this paper are constructed without adding

any spacer (cf. with [Ju], where all the spacers relate to Z-subaction only). Instead

of that on the n-th step we just cut the n-‘column’ into ‘subcolumns’ and then

rotate each ‘subcolumn’ in a ‘random way’. In the limit we obtain a topological

G-action on a compact Cantor space.

Our next concern is to describe all ergodic self-joinings of the G-actions con-

structed in Theorem 0.1. Recall a couple of deﬁnitions.

Given two ergodic G-actions Tand T0on (X, B, µ) and (X0,B0, µ0) respectively,

we denote by J(T, T 0) the set of joinings of Tand T0, i.e. the set of (Tg×T0

g)g∈G-

invariant measures on B⊗B0whose marginals on Band B0are µand µ0re-

spectively. The corresponding dynamical system (X×X0,B⊗B0, µ ×µ0) is also

called a joining of Tand T0. By Je(T , T 0)⊂J(T, T ) we denote the subset of

ergodic joinings of Tand T0(it is never empty). In a similar way one can de-

ﬁne the joininings J(T1, . . . , Tl) for any ﬁnite family T1, . . . , Tlof G-actions. If

2

J(T1, . . . , Tl) = {µ1× · · · × µl}then the family T1, . . . , Tlis called disjoint. If

T1=· · · =Tlwe speak about l-fold self-joinings of T1and use notation Jl(T) for

J(T,...,T

| {z }

ltimes

). For g∈G, we denote by g•the conjugacy class of g. We also let

FC(G) := {g∈G|g•is ﬁnite}.

Clearly, FC(G) is a normal subgroup of G. If Gis Abelian or Gis a sum of

ﬁnite groups then FC(G) = G. For any g∈FC(G), we deﬁne a measure µg•on

(X×X, B⊗B) by setting

µg•(A×B) := 1

#g•X

h∈g•

µ(A∩ThB).

It is easy to verify that µg•is a self-joining of T. Moreover, the map (x, T −1

hx)7→

(x, h) is an isomorphism of (X×X, µg•, T ×T) onto (X×g•, µ ×ν, e

T), where νis

the equidistribution on g•and the G-action e

T= ( e

Tt)t∈Gis given by

e

Tt(x, h) = (Ttx, tht−1), x ∈X, h ∈g•.

It follows that e

T(and hence the self-joining µg•of T) is ergodic if and only if

the action (Tt)t∈C(g)is ergodic, where C(g) = {t∈G|tg =gt}stands for the

centralizer of gin G. Notice also that C(g) is a co-ﬁnite subgroup of Gbecause of

g∈FC(G). Hence {µg•|g∈FC(G)} ⊂ Je

2(T) whenever Tis totally ergodic.

Deﬁnition 0.2. If Je

2(T)⊂ {µg•|g∈FC(G)}∪{µ×µ}then we say that Thas

2-fold minimal self-joinings (MSJ2).

This deﬁnition extends naturally to higher order self-joinings as follows. Given

l≥1 and g∈Gl+1, we denote by g•lthe orbit of gunder the G-action on Gl+1 by

conjugations:

h·(g0, . . . , gl) := (hg0h−1, . . . , hglh−1).

Let Pbe a partition of {0, . . . , l}. For an atom p∈P, we denote by ipthe

minimal element in p. We say that an element g= (g0, . . . , gl)∈FC(G)l+1 is

P-subordinated if gip= 1Gfor all p∈P. For any such g, we deﬁne a measure µg•l

on (Xl+1,B⊗(l+1) ) by setting

µg•l(A0× · ·· × Al) := 1

#g•lX

(h0,...,hl)∈g•lY

p∈P

µµ\

i∈p

ThiAi¶.

It is easy to verify that µg•lis an (l+ 1)-fold self-joining of T. Reasoning as above

one can check that µg•lis ergodic whenever Tis weakly mixing.

Deﬁnition 0.3. We say that Thas (l+ 1)-fold minimal self-joinings (MSJl+1) if

Je

l+1(T)⊂ {µg•l|gis P-subordinated for a partition Pof {0, . . . , l}}.

If Thas MSJlfor any l > 1, we say that Thas MSJ.

In case Gis Abelian, these deﬁnitions agree with the—common now—deﬁnitions

of MSJl+1 and MSJ by A. del Junco and D. Rudolph [JuR] who considered self-

joinings µg•lonly when gbelongs to the center of Gl+1. However we ﬁnd their deﬁni-

tion somewhat restrictive for non-commutative groups since, for instance, countable

sums of non-commutative ﬁnite groups can never have actions with MSJ2in their

sense.

Now we record the second main result of this paper.

3

Theorem 0.4. The actions constructed in Theorem 0.1(i) all have MSJ.

We notice that a part of the analysis from [Ru] can be carried over to the case

of G-actions with MSJ. In this paper we only show that such actions have trivial

product centralizer. Moreover, as follows from [Da3], every G-action with MSJ2is

eﬀectively prime, i.e. has no factors except for the obvious ones: the sub-σ-algebras

of subsets ﬁxed by ﬁnite normal subgroups in G. In particular, there exist no free

factors.

We now brieﬂy summarize the organization of the paper. In Section 1 we outline

the (C, F )-construction of rank-one actions as it appeared in [Da1]. In Section 2,

for any countable sum Gof ﬁnite groups, we construct a (C, F )-action Tof Gwhich

is mixing of any order. A rigid weakly mixing action of Galso appears there. In

Section 3 we demonstrate that Thas MSJ. In Section 4 we show how to perturb

the construction of Tto obtain an uncountable family of pairwise disjoint mixing

rank-one G-actions with MSJ. In the ﬁnal Section 5 we discuss some implications

of MSJ: trivial centralizer, trivial product centralizer and eﬀective primality.

Acknowledgement. The author thanks the referee for the useful suggestions that

improved the paper. In particular, in the present proof of Theorem 0.4 we deduce

MSJlfrom the l-fold mixing (as J. King does for Z-actions in [Ki]). Our original

proof (independent of multiple mixing) was longer and noticeably more complicated.

1. (C, F )-construction

In this section we recall the (C, F )-construction of rank-one actions.

From now on G=P∞

i=1 Gi, where Giis a non-trivial ﬁnite group for each i≥1.

To construct a probability preserving (C, F )-action of G(see [Ju], [Da1], [DaS2])

we need to deﬁne two sequences (Fn)n≥0and (Cn)n≥1of ﬁnite subsets in Gsuch

that the following are satisﬁed:

(Fn)n≥0is a Folner sequence in G, F0={1G},(1-1)

FnCn+1 ⊂Fn+1, Cn+1 >1,(1-2)

Fnc∩Fnc0=∅for all c6=c0∈Cn+1,(1-3)

lim

n→∞

#Fn

#C1· · · #Cn

<∞.(1-4)

Suppose that an increasing sequence of integers 0 < k1< k2<··· is given.

Then we deﬁne (Fn)n≥0by setting F0:= {1G}and Fn:= Pkn

i=1 Gifor n≥1.

Clearly, (1-1) is satisﬁed. Suppose now that we are also given a sequence of maps

sn:Hn→Fn, where H0:= Pk1

i=1 Giand Hn:= Pkn+1

i=kn+1 Gifor n≥1. Then we

deﬁne two sequences of maps cn+1, φn:Hn→Fn+1 by setting φn(h) := (0, h) and

cn+1(h) := (sn(h), h). Finally, we let Cn+1 := cn+1(Hn) for all n≥0. It is easy to

verify that (1-2)–(1-4) are all fulﬁlled. Moreover, a stronger version of (1-2) holds:

(1-5) FnCn+1 =Fn+1.

We now put Xn:= Fn×Cn+1 ×Cn+2 × · · · and deﬁne a map in:Xn→Xn+1 by

setting

in(fn, dn+1, dn+2 , . . . ) := (fndn+1, dn+2 , . . . ).

Clearly, Xnis a compact Cantor space. It follows from (1-5) and (1-3) that in

is well deﬁned and it is a homeomorphism of Xnonto Xn+1. Denote by Xthe

4

topological inductive limit of the sequence (Xn, in)∞

n=1. As a topological space X

is canonically homeomorphic to any Xnand in the sequel we will often identify X

with Xnsuppressing the canonical identiﬁcation maps. We need the structure of

inductive limit to deﬁne the (C, F )-action Ton Xas follows. Given g∈G, consider

any n≥0 such that g∈Fn. Every x∈Xcan be written as an inﬁnite sequence

x= (fn, dn+1, dn+2 , . . . ) with fn∈Fnand dm∈Cmfor m > n (i.e. we identify X

with Xn). Now we put

Tgx:= (gfn, dn+1, dn+2, . . . )∈Xn.

It is easy to verify that Tgis a well deﬁned homeomorphism of X. Moreover,

TgTg0=Tgg0, i.e. T:= (Tg)g∈Gis a topological action of Gon X.

Deﬁnition 1.1. We call Tthe (C, F )-action of Gassociated with (kn, sn−1)∞

n=1.

We list without proof several properties of T. They can be veriﬁed easily by the

reader (see also [Da1]).

—Tis a minimal uniquely ergodic (i.e. strictly ergodic) free action of G.

— Two points x= (fn, dn+1, dn+2, . . . ) and x= (f0

n, d0

n+1, d0

n+2, . . . )∈Xn

are T-orbit equivalent if and only if di=d0

ieventually (i.e. for all large

enough i). Moreover, x0=Tgxif and only if

g= lim

i→∞ f0

nd0

n+1 · · · d0

n+id−1

n+i· · · d−1

n+1f−1

n.

— The only T-invariant probability measure µon Xis the product of the

equidistributions on Fnand Cn+i,i∈N(if Xis identiﬁed with Xn).

For each A⊂Fn, we let [A]n:= {x= (fn, dn+1, . . . )∈Xn|fn∈A}and call it an

n-cylinder. The following holds:

[A]n∩[B]n= [A∩B]n,and [A]n∪[B]n= [A∪B]n,

[A]n=G

d∈Cn+1

[Ad]n+1,

Tg[A]n= [gA]nif g∈Fn,

µ([Ad]n+1) = 1

#Cn+1

µ([A]n) for any d∈Cn+1,

µ([A]n) = λFn(A),

where λFnis the normalized Haar measure on Fn. Moreover, for each measurable

subset B⊂X,

(1-6) lim

n→∞ min

A⊂Fn

µ(B4[A]n) = 0.

Hence Thas rank one.

2. Mixing (C, F )-actions

Our purpose in this section is to construct a rank-one action of Gwhich is

mixing of any order. This action will appear as a (C, F )-action associated with

some specially selected sequence (kn, sn−1)n≥1. We ﬁrst state several preliminary

results.

Given ﬁnite sets Aand Band a map x∈AB, we denote by dist xor distb∈Bx(b)

the measure (#B)−1Pb∈Bχx(b)on A. Here χx(b)stands for the probability sup-

ported at the point x(b).

5

Lemma 2.1. Let Abe a ﬁnite set and let λbe the equidistribution on A. Then for

any ² > 0there exist c > 0and m∈Nsuch that for any ﬁnite set Bwith #B > m,

λB({x∈AB| kdist x−λk> ²})< e−c#B.

For the proof we refer to [Or] or [Ru]. We will also use the following combinatorial

lemma.

Lemma 2.2. For any l∈N, let Nl:= 3l(l−1)/2and δl:= 5−l(l−1)/2Let Hbe a

ﬁnite group. Then for any family h1, . . . , hlof mutually diﬀerent elements of H

and any subset B⊂Hwith #B > 3/δl, there exists a partition of Binto subsets

Bi,1≤i≤Nl, such that the subsets h1Bi, h2Bi, . . . , hlBiare mutually disjoint

and #Bi≥δl#Bfor any i.

Proof. We leave to the reader the simplest case when l= 2. Hint: assume that

h1= 1Hand consider the partition of Hinto the right cosets by the cyclic group

generated by h2.

Suppose that we already proved the assertion of the lemma for some land we

want to prove it for l+ 1. Take any h16=h26=· · · 6=hl+1 ∈H(in such a

way we denote mutually diﬀerent elements of H). Given a subset B⊂Hwith

#B > 3/δl, we ﬁrst partition Binto subsets Bi, 1 ≤i≤Nl, such that the subsets

h2Bi, h3Bi, . . . , hl+1Biare mutually disjoint and #Bi≥δl#B≥3·5l. For every

i, there exists a partition Bi=F3

i1=1 Bi,i1such that h1Bi,i1∩h2Bi,i1=∅and

#Bi,i1≥0.2#Bi, 1 ≤i1≤3.Next, we partition every Bi,i1into 3 subsets Bi,i1,i2

such that h1Bi,i1,i2∩h3Bi,i1,i2=∅and #Bi,i1,i2≥0.2#Bi,i1, 1 ≤i2≤3, and so

on. Finally, we obtain a partition

B=

Nl

G

i=1

3

G

i1,...,il=1

Bi,i1,...,il

which is as desired. ¤

Given a ﬁnite set A, a ﬁnite group Hand elements h1, . . . , hl∈H, we denote

by πh1,...,hlthe map AH→(Al)Hgiven by

(πh1,...,hlx)(k) = (x(h1k), . . . , x(hlk)).

For x∈AH, we deﬁne x∗∈AHby setting x∗(h) := x(h−1), h∈H.

Lemma 2.3. Given l∈Nand ² > 0, there exists m∈Nsuch that for any ﬁnite

group Hwith #H > m, one can ﬁnd s∈AHsuch that

(2-1) kdist πh1,...,hls−λlk< ² and kdist πh1,...,hls∗−λlk< ²

for all h16=h26=· · · 6=hl∈H.

Proof. Take any ﬁnite group Hand set

BH:= [

h16=···6=hl∈H

{x∈AH| kdist πh1,...,hlx−λlk> ²}.

6

To prove the left hand side inequality in (2-1) it suﬃces to show that λH(BH)<1

whenever #His large enough. Moreover, since the map AH3x7→ x∗∈AH

preserves the measure λH, the right hand side inequality in (2-1) will follow from

the left hand side one if we prove that λH(BH)<0.5.

Fix h16=· · · 6=hl∈Hand apply Lemma 2.2 to partition Hinto subsets Hi,

1≤i≤Nl, such that

#Hi≥δl#Hand(2-2)

the subsets h1Hi, . . . , hlHiare mutually disjoint(2-3)

for every i. Denote by ri: (Al)H→(Al)Hithe natural restriction map. Then we

deduce from (2-3) that ri◦πh1,··· ,hlmaps λHonto (λl)Hi. Since dist πh1,...,hlx=

Pi(#Hi/#H)·dist(ri◦πh1,...,hl)x, it follows that

λH({x∈AH| kdist πh1,...,hlx−λlk> ²})

≤X

i

λH({x∈AH| kdist (ri◦πh1,...,hl)x−λlk> ²})

=X

i

(λl)Hi({y∈(Al)Hi| kdist y−λlk> ²}).

By Lemma 2.2 and (2-2), there exists c > 0 such that if #His large enough then

the i-th term in the latter sum is less then e−c#Hi< e−cδl#H. Hence

λH(BH)≤Nlµ#H

l¶e−cδl#H

and the assertion of the lemma follows. ¤

Now we are ready to deﬁne the sequence (kn, sn−1)n≥1. Fix a sequence of positive

reals ²n→0. On the ﬁrst step one can take arbitrary k1and s0. Suppose now—on

the n-th step—we already have knand sn−1and we want to deﬁne kn+1 and sn.

For this, we apply Lemma 2.3 with A:= Fn,l:= nand ²:= ²nto ﬁnd kn+1 large

so that there exists sn∈AHnsatisfying

(2-4) kdist πh1,...,hnsn−(λFn)nk< ²nfor all h16=· · · 6=hn∈Hn.

Recall that Hn:= Pkn+1

i=kn+1 Giand Fn:= Pkn

i=1 Gifor n≥1. Without loss of

generality we may also assume that kn+1 −kn≥nand hence P∞

n=1(#Hn)−1<∞.

Denote by Tthe (C, F )-action of Gon (X, B, µ) associated with (kn, sn−1)∞

n=1.

Theorem 2.4. Tis mixing of any order.

Proof. (I) We ﬁrst show that Tis mixing (of order 1).

Recall that a sequence gn→ ∞ in Gis called mixing for Tif

lim

n→∞ µ(TgnB1∩B2) = µ(B1)µ(B2) for all B1, B2∈B.

Clearly, Tis mixing if and only if any sequence going to inﬁnity in Gcontains a

mixing subsequence. Since every subsequence of a mixing sequence is mixing itself,

to prove (I) it suﬃces to show that every sequence (gn)∞

n=1 in Gwith gn∈Fn+1 \Fn

7

for all nis mixing. Notice ﬁrst that there exist (unique) fn∈Fnand hn∈Hn\ {1}

with gn=fnφn(hn). Fix any two subsets A, B ⊂Fn. We notice that for each

h∈Hn,

gnAcn+1(h) = fnAsn(h)φn(hnh) = fnAsn(h)sn(hnh)−1cn+1 (hnh)

and fnAsn(h)sn(hnh)−1⊂Fn. Hence

(2-5)

µ(Tgn[A]n∩[B]n) = X

h∈Hn

µ(Tgn[Acn+1(h)]n+1 ∩[B]n)

=X

h∈Hn

µ([fnAsn(h)sn(hnh)−1cn+1(hnh)]n+1 ∩[B]n)

=X

h∈Hn

µ([(fnAsn(h)sn(hnh)−1∩B)cn+1(hnh)]n+1 )

=1

#HnX

h∈Hn

µ([fnAsn(h)sn(hnh)−1∩B]n)

=1

#HnX

h∈Hn

λFn(fnAsn(h)∩Bsn(hnh)).

We deﬁne a map rA,B :Fn×Fn→Rby setting

rA,B(g, g0) := λFn(fnAg ∩Bg0).

Then it follows from (2-5) and (2-4) that

µ(Tgn[A]n∩[B]n) = ZFn×Fn

rA,B d(dist π1,hnsn)

=ZFn×Fn

rA,B dλFn×Fn±²n

=ZFn×Fn

λFn(fnAg ∩Bg0)dλFn(g)dλFn(g0)±²n

=λFn(A)λFn(B)±²n

=µ([A]n)µ([B]n)±²n.

Hence we have

(2-6) maxA,B⊂Fn|µ(Tgn[A]n∩[B]n)−µ([A]n)µ([B]n)|< ²n.

This and (1-6) imply that the sequence (gn)∞

n=1 is mixing.

(II) Now we ﬁx l > 1 and prove that Tis mixing of order l. To this end it is

suﬃcient to show the following: given l+ 1 sequences (g0,n)∞

n=1, . . . , (gl,n )∞

n=1 in G

such that gi,n ∈Fn+1 and gi,ng−1

j,n /∈Fnwhenever i6=j,

max

A0,...,Al

|µ(Tg0,n [A0]n∩ · ·· ∩ Tgl,n [Al]n)−µ([A0]n)· · · µ([Al]n)|< ²n

8

for all n > l. Notice that for every n∈Nand 0 ≤j≤l, there exist unique fj,n ∈Fn

and hj,n ∈Hnwith gj,n =fj,nφn(hj,n ). Moreover, h0,n 6=h2,n · · · 6=h1,n. Then

slightly modifying the argument in (I), we compute

(2-7)

µ(Tg0,n [A0]n∩ · · · ∩ Tgl,n [Al]n)

=ZFl

n

λFn(f0,nA0g0∩ · · · ∩ fl,n Algl)d(λFn)l+1(g0, . . . , gl)±²n

=λFn(A0)···λFn(Al)±²n=µ([A0]n)· · · µ([Al]n)±²n.

¤

To construct a weakly mixing rigid action of Gwe deﬁne another sequence

(e

kn,esn−1)n≥1. When nis odd, we choose e

knand esn−1to satisfy the following

weaker version of (2-4):

(2-8) max

16=h∈Hn

kdist π1,hsn−λFn×λFnk< ²n.

When nis even, we just set e

kn:= e

kn−1+ 1 and esn≡1G. Denote by e

Tthe

(C, F )-action of Gon ( e

X, e

B,eµ) associated with (e

kn,esn−1)∞

n=1.

Theorem 2.5. e

Tis weakly mixing and rigid.

Proof. Take any sequence hn∈H2n\ {1}. It follows from the part (I) of the proof

of Theorem 2.4 and (2-8) that the sequence (φ2n(hn))∞

n=1 is mixing for e

T. Clearly,

it is also mixing for e

T×e

T. Hence e

T×e

Tis ergodic, i.e. e

Tis weakly mixing.

Now take any sequence hn∈H2n+1 \ {1}. Notice that (2-5) holds for any choice

of (kn, sn−1)n≥1. Hence we deduce from (2-5) and the deﬁnition of es2n+1 that

µ(e

Tφ2n+1(hn)[A]2n+1 ∩[B]2n+1 ) = λF2n+1 (A∩B) = µ([A∩B]2n+1)

for all subsets A, B ⊂F2n+1. This plus (1-6) yield

lim

n→∞ µ(e

Tφ2n+1(hn)e

A∩e

B) = µ(e

A∩e

B)

for all e

A, e

B∈e

B. This means that e

Tis rigid. ¤

3. Self-joinings of T

This section is devoted entirely to the proof of the following theorem.

Theorem 3.1. The action Tconstructed in the previous section has MSJ.

Proof. (I) We ﬁrst show that Thas MSJ2. Since Tis weakly mixing, we need to

establish that

Je

2(T) = {µg•|g∈G}∪{µ×µ}.

Take any ν∈Je

2(T). Let Fndenote the sub-σ-algebra of (Tg×Tg)g∈Fn-invariant

subsets. Then F1⊃F2⊃ · · · and TnFn={∅, X ×X}(mod ν). Since there

are only countably many cylinders, we deduce from the martingale convergence

theorem that for ν-a.a. (x, x0),

(3-1) E(χB×B0|Fn−1)(x, x0) = 1

#Fn−1X

g∈Fn−1

χB×B0(Tgx, Tgx0)→ν(B×B0)

9

as n→ ∞ for any pair of cylinders B, B0⊂X. Fix such a point (x, x0). It is called

generic for (T×T, ν). Given any n > 0, we can write xand x0as inﬁnite sequences

x= (fn, dn+1, dn+2 , . . . ) and x0= (f0

n, d0

n+1, d0

n+2, . . . )

with fn, f 0

n∈Fnand di, d0

i∈Cifor all i > n. Recall that fn:= f0d1· · · dnand

f0

n:= f0

0d0

1· · · d0

n. We set tn:= f0

nf−1

n,n > 0. Fix a pair of cylinders, say m-

cylinders, Band B0. If n>mand g∈Fnthen Tgx0= (gf 0

n, d0

n+1, d0

n+2, . . . ).

Hence Tgx0∈B0if and only if TgTtnx∈B0. Therefore

χB×B0(Tgx, Tgx0) = χT−1

gB∩T−1

tnT−1

gB0(x).

Since xis generic for (T, µ), it follows that

lim

l→∞

1

#FlX

a∈Fl

χT−1

gB∩T−1

tnT−1

gB0(Tax) = µ(T−1

gB∩T−1

tnT−1

gB0).

Therefore (3-1) yields

(3-2) lim

n→∞

1

#Fn−1X

g∈Fn−1

µ(T−1

gB∩T−1

tnT−1

gB0) = ν(B×B0).

Consider now two cases. If tn/∈Fn−1for inﬁnitely many nthen passing to the limit

in (3-2) along this subsequence and making use of (2-6) we obtain that µ(B)µ(B0) =

ν(B×B0). Hence µ×µ=ν. If, otherwise, there exists N > 0 such that tn∈Fn−1,

i.e. dn=d0

n, for all n > N then xand x0are T-orbit equivalent, tn=tNand

1

#Fn−1X

g∈Fn−1

µ(T−1

gB∩T−1

tnT−1

gB0) = 1

#FNX

g∈FN

µ(B∩TgT−1

tNT−1

gB0)

=µ(t−1

N)•(B×B0)

Passing to the limit in (3-1) we obtain that ν=µ(t−1

N)•.

(II) Now we ﬁx l > 1 and show that Thas MSJl+1. Take any joining ν∈Je

l+1(T)

and ﬁx a generic point (x0, . . . , xl) for (T× · · · × T, ν). Deﬁne a partition Pof

{0, . . . , l}by setting: i1and i2are in the same atom of Pif xi1and xi2are T-orbit

equivalent. As in (I), for any n, we can write

xj= (fj,n−1, dj,n, dj,n+1 , . . . )∈Xn−1, j = 0, . . . , l.

Suppose ﬁrst that #P=l+ 1, i.e. Pis the ﬁnest possible. Then by the proof

of (I), each 2-dimensional marginal of νis µ×µ. Since P∞

i=1(#Ci)−1<∞and

µ=λF0×λC1×λC2× · · · , it follows from the Borel-Cantelli lemma that for ν-a.a.

(y0, . . . , yl)∈Xl+1,

∃N > 0 such that y0,i 6=y1,i 6=· · · 6=yl,i whenever i > N,

where yj,i ∈Ciis the i-th coordinate of yj∈F0×C1×C2× · · · . Hence without

loss of generality we may assume that this condition is satisﬁed for (x0, . . . , xl).

Thus, if we set tj,n := fj,nf0,n−1=fj,n−1dj,n d0,n−1f0,n−1−1then tj,nt−1

i,n /∈Fn−1

10

whenever i6=j. Slightly modifying our reasoning in (I) and making use of (2-7)

instead of (2-6) we now obtain

ν(B0× · · · × Bl) = lim

n→∞ X

g∈Fn−1

χB0×···×Bl(Tgx0, . . . .Tgxl)

= lim

n→∞ X

g∈Fn−1

χB0×···×Bl(Tgx0, TgTt1,n x0, . . . , TgTtl,n x0)

= lim

n→∞ X

g∈Fn−1

µ(TgB0∩T−1

t1,n TgB1∩ · ·· ∩ T−1

tl,n TgBl)

=µ(B0)· · · µ(Bl)

for any (l+ 1)-tuple of cylinders B0, . . . , Bl. Hence ν=µ× · · · × µ.

Consider now the general case and put tj,n := fj,nf−1

ip,n for each j∈p,p∈P.

Recall that ip= minj∈pj. Then

χB0×···×Bl(Tgx0, . . . , Tgxl) = Y

p∈P

χAp(xip),

where Ap:= Tj∈pT−1

tj,n T−1

gBj. Notice that the point (xip)p∈P∈X{ip|p∈P}is

generic for (T× · · · × T(#Ptimes), κ), where κstands for the projection of νonto

X{ip|p∈P}. By the ﬁrst part of (II), κ=µ× · · · × µ(#Ptimes). Hence

ν(B0× · ·· × Bl) = lim

n→∞

1

#Fn−1X

g∈Fn−1

χB0×···×Bl(Tgx0, . . . .Tgxl)

= lim

n→∞

1

#Fn−1X

g∈Fn−1Y

p∈P

µ(Ap).

As in (I), a ‘stabilization’ property holds: there exists M > 0 such that tj,n =tj,M

for all n > M. We now set g:= (t−1

0,M , . . . , t−1

l,M ). Clearly, gis P-subordinated.

Hence

ν(B0× · · · × Bl) = 1

#FMX

g∈FMY

p∈P

µµ\

j∈p

TgTtj,M T−1

gBj¶=µg•l(B0× · · · × Bl).

¤

4. Uncountably many mixing actions with MSJ

In this section the proof of Theorems 0.1(i) and 0.4 will be completed. We ﬁrst

apply Lemma 2.3 to construct kn+1 and sn,bsn∈FHn

nin such a way that (2-4) is

satisﬁed for both snand bsnand, in addition,

(4-1) kdisth∈Hn(sn(hk),bsn(hk0)) −λFn×λFnk< ²n

for all k, k0∈Hn. Next, given σ∈ {0,1}Nand n∈N, we deﬁne sσ

n:Hn→Fnby

setting

sσ

n=½snif σ(n) = 0,

bsnif σ(n) = 1.

Now we denote by Tσthe (C, F )-action of Gassociated with (kn, sσ

n−1)∞

n=1. Let

Σ be an uncountable subset of {0,1}Nsuch that for any σ, σ0∈Σ, the subset

{n∈N|σ(n)6=σ0(n)}is inﬁnite.

11

Theorem 4.1.

(i) For any σ∈ {0,1}N, the action Tσis mixing and has MSJ.

(ii) If σ, σ0∈Σand σ6=σ0then Tσand Tσ0are disjoint.

Proof. (i) follows from the proof of Theorem 3.1, since (2-4) is satisﬁed for sσ

nfor

all σ∈ {0,1}Nand n∈N.

(ii) Let ν∈Je(Tσ, T σ0). Take a generic point (x, x0) for (Tσ×Tσ0, ν). Consider

any nsuch that σ(n)6=σ0(n). Then we can write xand x0as inﬁnite sequences x=

(fn, dn+1, dn+2 , . . . ) and x0= (f0

n, d0

n+1, d0

n+2, . . . ) with fn, f 0

n∈Fnand dm, d0

m∈

Cmfor all m > n. Take any g∈Fn+1. Then we have the following expansions

g=aφn(h), dn+1 =sσ

n(hn)φn(hn) and d0

n+1 =sσ0

n(h0

n)φn(h0

n)

for some uniquely determined a∈Fnand h, hn, h0

n∈Hn. Since

gfndn+1 =afnsσ

n(hn)sσ

n(hhn)−1cn+1(hhn) and

gf 0

nd0

n+1 =af0

nsσ0

n(h0

n)sσ0

n(hh0

n)−1cn+1(hh0

n),

the following holds for any pair of subsets A, A0⊂Fn:

#{g∈Fn+1 |(Tσ

gx, T σ0

gx0)∈[A]n×[A0]n}

#Fn+1

=1

#FnX

a∈Fn

#{h∈Hn|afnsσ

n(hn)sσ

n(hhn)−1∈A, af0

nsσ0

n(h0

n)sσ0

n(hh0

n)−1∈A0}

#Hn

=1

#FnX

a∈Fn

ξn(A−1afnsσ

n(hn)×A0−1af0

nsσ0

n(h0

n)),

where ξn:= disth∈Hn(sσ

n(hhn), sσ0

n(hh0

n)). This and (4-1) yield

(4-2)

#{g∈Fn+1 |(Tσ

gx, T σ0

gx0)∈[A]n×[A0]n}

#Fn+1

=λFn(A)λFn(A0)±²n

=µ([A]n)µ([A0]n)±²n.

Since (x, x0) is generic for (Tσ×Tσ0, ν) and (4-2) holds for inﬁnitely many n, we

deduce that ν=µ×µ.¤

By reﬁning the above argument the reader can strengthen Theorem 0.1(i) as

follows: there exists an uncountable family of mixing (of any order) rank-one G-

actions with MSJ such that any ﬁnite subfamily of it is disjoint.

5. On G-actions with MSJ

It follows immediately from Deﬁnition 0.2 that if Thas MSJ2then the centralizer

C(T) of Tis ‘trivial’, i.e. C(T) = {Tg|g∈C(G)}, where C(G) denotes the center

of G. Moreover, we will show that Thas trivial product centralizer (as D. Rudolph

did in [Ru] for Z-actions).

Let (Xl,B⊗l, µl, T (l)) denote the l-fold Cartesian product of (X, B, µ, T ). Given

a permutation σof {1, . . . , l}and g1, . . . , gn∈C(T), we deﬁne a transformation

Uσ,g1,...,glof (Xl,B⊗l, µl, T (l)) by setting

Uσ,g1,...,gl(x1, . . . , xl) := (Tg1xσ(1), . . . , Tglxσ(l)).

Of course, Uσ,g1,...,gl∈C(T(l)). We show that for the actions with MSJ, the

converse also holds.

12

Proposition 5.1. If Thas MSJ then for any l∈N, each element of C(T(l))equals

to Uσ,g1,...,glfor some permutation σand elements g1, . . . , gl∈C(G).

Proof. Let S∈C(T(l)). We deﬁne an ergodic 2-fold self-joining νof T(l)by setting

ν(A×B) := µl(A∩S−1B) for all A, B ∈B⊗l. Notice that ν∈Je

2l(T). Since T

has MSJ2l, there exists a partition Pof {1,...,2l}and a P-subordinated element

g= (g1, . . . , g2l)∈FC(G)2lsuch that

(5-1) ν(A1× · · · × A2l) = 1

#g•2lX

(h1,...,h2l)∈g•2lY

p∈P

µµ\

i∈p

ThiAi¶.

for all subsets A1, . . . , A2l∈B. Substituting at ﬁrst A1=· · · =Al=Xand then

Al+1 =· · · =A2l=Xin (5-1) we derive that #P=l, #p= 2 for all p∈Pand

#g•2l= 1. Hence g1, . . . , g2l∈C(G) and there exists a bijection σof {1, . . . , l}

such that P={{i, σ(i) + l} | i= 1, . . . , l}. Therefore in follows from (5-1) that

S−1(Al+1 × · · · × A2l) = Tgl+1 Al+σ(1) × · · · × Tg2lAl+σ(l).

¤

As a simple corollary we derive that if Thas MSJ then the G-actions T, T (2) , . . .

and T×T× · · · are pairwise non-isomorphic.

After this paper was submitted the author introduced a companion to MSJ

concept of simplicity for actions of locally compact second countable groups [Da3].

As appeared, this concept is more general that the simplicity in the sense of A. del

Junco and D. Rudolph [JuR] even for Z-actions. For instance, there exist simple

transformations which are disjoint from all 2-fold del Junco-Rudolph’s-simple ones.

It is shown in [Da3] that an analogue of Veech theorem on the structure of factors

holds for this extended class of simple actions. In particular, if Thas MSJ2then

for every non-trivial factor Fof Tthere exists a compact normal subgroup Kof G

such that

F= Fix K:= {A∈B|µ(TkA4A) = 0 for all k∈K}.

Thus if Thas MSJ2then Tis eﬀectively prime, i.e. Thas no eﬀective factors.

(Recall that a G-action Qis called eﬀective if Qg6= Id for each g6= 1G.)

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Institute for Low Temperature Physics & Engineering of Ukrainian National

Academy of Sciences, 47 Lenin Ave., Kharkov, 61164, UKRAINE

E-mail address:danilenko@ilt.kharkov.ua

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