Mixing rank-one actions for infinite sums of finite groups

Abstract

LetG be 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 ofG on 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 freeG-action is given.

Full text

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
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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
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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.
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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
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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).
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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}.
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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
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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
14