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Mixing rank-one actions for infinite sums of finite groups



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.
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)gGof Gon a standard prob-
ability space (X, B, µ) is called
mixing if limg→∞ µ(ATgB) = µ(A)µ(B) for all A, B B,
mixing of order lif for any ² > 0 and A0, . . . , AlB, there exists a finite
subset KGsuch that
|µ(Tg0A0∩ · ·· ∩ TglAl)µ(A0)· · · µ(Al)|< ²
for each collection g0, . . . , glGwith gig1
j/Kif i6=j,
weakly mixing if the diagonal action T×T:= (Tg×Tg)gGof Gon the
product space (X×X, BB, 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(AB) for all A, B B.
We say that Thas funny rank one if there exist a sequence of measurable subsets
n=1 in Xand a sequence of finite subsets (Fn)
n=1 in Gsuch that the subsets
TgFn,gFn, are pairwise disjoint for any nand
n→∞ min
TgAn= 0 for every BB.
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
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
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
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
invariant measures on BB0whose marginals on Band B0are µand µ0re-
spectively. The corresponding dynamical system (X×X0,BB0, 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
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
| {z }
). For gG, we denote by gthe conjugacy class of g. We also let
FC(G) := {gG|gis 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 gFC(G), we define a measure µgon
(X×X, BB) by setting
µg(A×B) := 1
It is easy to verify that µgis a self-joining of T. Moreover, the map (x, T 1
(x, h) is an isomorphism of (X×X, µg, T ×T) onto (X×g, µ ×ν, e
T), where νis
the equidistribution on gand the G-action e
T= ( e
Tt)tGis given by
Tt(x, h) = (Ttx, tht1), x X, h g.
It follows that e
T(and hence the self-joining µgof T) is ergodic if and only if
the action (Tt)tC(g)is ergodic, where C(g) = {tG|tg =gt}stands for the
centralizer of gin G. Notice also that C(g) is a co-finite subgroup of Gbecause of
gFC(G). Hence {µg|gFC(G)} ⊂ Je
2(T) whenever Tis totally ergodic.
Definition 0.2. If Je
2(T)⊂ {µg|gFC(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
l1 and gGl+1, we denote by glthe orbit of gunder the G-action on Gl+1 by
h·(g0, . . . , gl) := (hg0h1, . . . , hglh1).
Let Pbe a partition of {0, . . . , l}. For an atom pP, 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 pP. For any such g, we define a measure µgl
on (Xl+1,B(l+1) ) by setting
µgl(A0× · ·· × Al) := 1
It is easy to verify that µglis an (l+ 1)-fold self-joining of T. Reasoning as above
one can check that µglis ergodic whenever Tis weakly mixing.
Definition 0.3. We say that Thas (l+ 1)-fold minimal self-joinings (MSJl+1) if
l+1(T)⊂ {µgl|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 µglonly 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
Now we record the second main result of this paper.
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
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 i1.
To construct a probability preserving (C, F )-action of G(see [Ju], [Da1], [DaS2])
we need to define two sequences (Fn)n0and (Cn)n1of finite subsets in Gsuch
that the following are satisfied:
(Fn)n0is a Folner sequence in G, F0={1G},(1-1)
FnCn+1 Fn+1, Cn+1 >1,(1-2)
FncFnc0=for all c6=c0Cn+1,(1-3)
#C1· · · #Cn
Suppose that an increasing sequence of integers 0 < k1< k2<··· is given.
Then we define (Fn)n0by setting F0:= {1G}and Fn:= Pkn
i=1 Gifor n1.
Clearly, (1-1) is satisfied. Suppose now that we are also given a sequence of maps
sn:HnFn, where H0:= Pk1
i=1 Giand Hn:= Pkn+1
i=kn+1 Gifor n1. Then we
define two sequences of maps cn+1, φn:HnFn+1 by setting φn(h) := (0, h) and
cn+1(h) := (sn(h), h). Finally, we let Cn+1 := cn+1(Hn) for all n0. 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:XnXn+1 by
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
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 gG, consider
any n0 such that gFn. Every xXcan be written as an infinite sequence
x= (fn, dn+1, dn+2 , . . . ) with fnFnand dmCmfor 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)gGis a topological action of Gon X.
Definition 1.1. We call Tthe (C, F )-action of Gassociated with (kn, sn1)
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
n+1 · · · d0
n+i· · · d1
— The only T-invariant probability measure µon Xis the product of the
equidistributions on Fnand Cn+i,iN(if Xis identified with Xn).
For each AFn, we let [A]n:= {x= (fn, dn+1, . . . )Xn|fnA}and call it an
n-cylinder. The following holds:
[A]n[B]n= [AB]n,and [A]n[B]n= [AB]n,
Tg[A]n= [gA]nif gFn,
µ([Ad]n+1) = 1
µ([A]n) for any dCn+1,
µ([A]n) = λFn(A),
where λFnis the normalized Haar measure on Fn. Moreover, for each measurable
subset BX,
(1-6) lim
n→∞ min
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, sn1)n1. We first state several preliminary
Given finite sets Aand Band a map xAB, we denote by dist xor distbBx(b)
the measure (#B)1PbBχx(b)on A. Here χx(b)stands for the probability sup-
ported at the point x(b).
Lemma 2.1. Let Abe a finite set and let λbe the equidistribution on A. Then for
any ² > 0there exist c > 0and mNsuch that for any finite set Bwith #B > m,
λB({xAB| kdist xλk> ²})< ec#B.
For the proof we refer to [Or] or [Ru]. We will also use the following combinatorial
Lemma 2.2. For any lN, let Nl:= 3l(l1)/2and δl:= 5l(l1)/2Let Hbe a
finite group. Then for any family h1, . . . , hlof mutually different elements of H
and any subset BHwith #B > 3l, there exists a partition of Binto subsets
Bi,1iNl, such that the subsets h1Bi, h2Bi, . . . , hlBiare mutually disjoint
and #Biδl#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 BHwith
#B > 3l, we first partition Binto subsets Bi, 1 iNl, such that the subsets
h2Bi, h3Bi, . . . , hl+1Biare mutually disjoint and #Biδl#B3·5l. For every
i, there exists a partition Bi=F3
i1=1 Bi,i1such that h1Bi,i1h2Bi,i1=and
#Bi,i10.2#Bi, 1 i13.Next, we partition every Bi,i1into 3 subsets Bi,i1,i2
such that h1Bi,i1,i2h3Bi,i1,i2=and #Bi,i1,i20.2#Bi,i1, 1 i23, and so
on. Finally, we obtain a partition
which is as desired. ¤
Given a finite set A, a finite group Hand elements h1, . . . , hlH, we denote
by πh1,...,hlthe map AH(Al)Hgiven by
(πh1,...,hlx)(k) = (x(h1k), . . . , x(hlk)).
For xAH, we define xAHby setting x(h) := x(h1), hH.
Lemma 2.3. Given lNand ² > 0, there exists mNsuch that for any finite
group Hwith #H > m, one can find sAHsuch that
(2-1) kdist πh1,...,hlsλlk< ² and kdist πh1,...,hlsλlk< ²
for all h16=h26=· · · 6=hlH.
Proof. Take any finite group Hand set
BH:= [
{xAH| kdist πh1,...,hlxλlk> ²}.
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→ xAH
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=hlHand apply Lemma 2.2 to partition Hinto subsets Hi,
1iNl, such that
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 riπh1,··· ,hlmaps λHonto (λl)Hi. Since dist πh1,...,hlx=
Pi(#Hi/#H)·dist(riπh1,...,hl)x, it follows that
λH({xAH| kdist πh1,...,hlxλlk> ²})
λH({xAH| kdist (riπh1,...,hl)xλlk> ²})
(λl)Hi({y(Al)Hi| kdist yλlk> 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 ec#Hi< el#H. Hence
and the assertion of the lemma follows. ¤
Now we are ready to define the sequence (kn, sn1)n1. Fix a sequence of positive
reals ²n0. On the first step one can take arbitrary k1and s0. Suppose now—on
the n-th step—we already have knand sn1and 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 snAHnsatisfying
(2-4) kdist πh1,...,hnsn(λFn)nk< ²nfor all h16=· · · 6=hnHn.
Recall that Hn:= Pkn+1
i=kn+1 Giand Fn:= Pkn
i=1 Gifor n1. Without loss of
generality we may also assume that kn+1 knnand hence P
Denote by Tthe (C, F )-action of Gon (X, B, µ) associated with (kn, sn1)
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
n→∞ µ(TgnB1B2) = µ(B1)µ(B2) for all B1, B2B.
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 gnFn+1 \Fn
for all nis mixing. Notice first that there exist (unique) fnFnand hnHn\ {1}
with gn=fnφn(hn). Fix any two subsets A, B Fn. We notice that for each
gnAcn+1(h) = fnAsn(h)φn(hnh) = fnAsn(h)sn(hnh)1cn+1 (hnh)
and fnAsn(h)sn(hnh)1Fn. Hence
µ(Tgn[A]n[B]n) = X
µ(Tgn[Acn+1(h)]n+1 [B]n)
µ([fnAsn(h)sn(hnh)1cn+1(hnh)]n+1 [B]n)
µ([(fnAsn(h)sn(hnh)1B)cn+1(hnh)]n+1 )
We define a map rA,B :Fn×FnRby 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)
rA,B Fn×Fn±²n
λFn(fnAg Bg0)Fn(g)Fn(g0)±²n
Hence we have
(2-6) maxA,BFn|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,ng1
j,n /Fnwhenever i6=j,
|µ(Tg0,n [A0]n∩ · ·· ∩ Tgl,n [Al]n)µ([A0]n)· · · µ([Al]n)|< ²n
for all n > l. Notice that for every nNand 0 jl, 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
µ(Tg0,n [A0]n∩ · · · ∩ Tgl,n [Al]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
kn,esn1)n1. When nis odd, we choose e
knand esn1to satisfy the following
weaker version of (2-4):
(2-8) max
kdist π1,hsnλFn×λFnk< ²n.
When nis even, we just set e
kn:= e
kn1+ 1 and esn1G. Denote by e
(C, F )-action of Gon ( e
X, e
B,eµ) associated with (e
Theorem 2.5. e
Tis weakly mixing and rigid.
Proof. Take any sequence hnH2n\ {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. Hence e
Tis ergodic, i.e. e
Tis weakly mixing.
Now take any sequence hnH2n+1 \ {1}. Notice that (2-5) holds for any choice
of (kn, sn1)n1. Hence we deduce from (2-5) and the definition of es2n+1 that
Tφ2n+1(hn)[A]2n+1 [B]2n+1 ) = λF2n+1 (AB) = µ([AB]2n+1)
for all subsets A, B F2n+1. This plus (1-6) yield
n→∞ µ(e
B) = µ(e
for all e
A, 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
2(T) = {µg|gG}∪{µ×µ}.
Take any νJe
2(T). Let Fndenote the sub-σ-algebra of (Tg×Tg)gFn-invariant
subsets. Then F1F2 · · · 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|Fn1)(x, x0) = 1
χB×B0(Tgx, Tgx0)ν(B×B0)
as n→ ∞ for any pair of cylinders B, B0X. 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
nFnand di, d0
iCifor all i > n. Recall that fn:= f0d1· · · dnand
n:= f0
1· · · d0
n. We set tn:= f0
n,n > 0. Fix a pair of cylinders, say m-
cylinders, Band B0. If n>mand gFnthen Tgx0= (gf 0
n, d0
n+1, d0
n+2, . . . ).
Hence Tgx0B0if and only if TgTtnxB0. Therefore
χB×B0(Tgx, Tgx0) = χT1
Since xis generic for (T, µ), it follows that
gB0(Tax) = µ(T1
Therefore (3-1) yields
(3-2) lim
gB0) = ν(B×B0).
Consider now two cases. If tn/Fn1for 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 tnFn1,
i.e. dn=d0
n, for all n > N then xand x0are T-orbit equivalent, tn=tNand
gB0) = 1
Passing to the limit in (3-1) we obtain that ν=µ(t1
(II) Now we fix l > 1 and show that Thas MSJl+1. Take any joining νJe
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,n1, dj,n, dj,n+1 , . . . )Xn1, 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
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 yjF0×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,n1=fj,n1dj,n d0,n1f0,n11then tj,nt1
i,n /Fn1
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
χB0×···×Bl(Tgx0, . . . .Tgxl)
= lim
n→∞ X
χB0×···×Bl(Tgx0, TgTt1,n x0, . . . , TgTtl,n x0)
= lim
n→∞ X
t1,n TgB1∩ · ·· ∩ T1
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,nf1
ip,n for each jp,pP.
Recall that ip= minjpj. Then
χB0×···×Bl(Tgx0, . . . , Tgxl) = Y
where Ap:= TjpT1
tj,n T1
gBj. Notice that the point (xip)pPX{ip|pP}is
generic for (T× · · · × T(#Ptimes), κ), where κstands for the projection of νonto
X{ip|pP}. By the first part of (II), κ=µ× · · · × µ(#Ptimes). Hence
ν(B0× · ·· × Bl) = lim
χB0×···×Bl(Tgx0, . . . .Tgxl)
= lim
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:= (t1
0,M , . . . , t1
l,M ). Clearly, gis P-subordinated.
ν(B0× · · · × Bl) = 1
TgTtj,M T1
gBj=µgl(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,bsnFHn
nin such a way that (2-4) is
satisfied for both snand bsnand, in addition,
(4-1) kdisthHn(sn(hk),bsn(hk0)) λFn×λFnk< ²n
for all k, k0Hn. Next, given σ∈ {0,1}Nand nN, we define 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. Let
Σ be an uncountable subset of {0,1}Nsuch that for any σ, σ0Σ, the subset
{nN|σ(n)6=σ0(n)}is infinite.
Theorem 4.1.
(i) For any σ∈ {0,1}N, the action Tσis mixing and has MSJ.
(ii) If σ, σ0Σand σ6=σ0then Tσand Tσ0are disjoint.
Proof. (i) follows from the proof of Theorem 3.1, since (2-4) is satisfied for sσ
all σ∈ {0,1}Nand nN.
(ii) Let νJe(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
nFnand dm, d0
Cmfor all m > n. Take any gFn+1. Then we have the following expansions
g=n(h), dn+1 =sσ
n(hn)φn(hn) and d0
n+1 =sσ0
for some uniquely determined aFnand h, hn, h0
nHn. Since
gfndn+1 =afnsσ
n(hhn)1cn+1(hhn) and
gf 0
n+1 =af0
the following holds for any pair of subsets A, A0Fn:
#{gFn+1 |(Tσ
gx, T σ0
n(hhn)1A, af0
where ξn:= disthHn(sσ
n(hhn), sσ0
n)). This and (4-1) yield
#{gFn+1 |(Tσ
gx, T σ0
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|gC(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,Bl, µl, T (l)) denote the l-fold Cartesian product of (X, B, µ, T ). Given
a permutation σof {1, . . . , l}and g1, . . . , gnC(T), we define a transformation
Uσ,g1,...,glof (Xl,Bl, µl, T (l)) by setting
Uσ,g1,...,gl(x1, . . . , xl) := (Tg1xσ(1), . . . , Tglxσ(l)).
Of course, Uσ,g1,...,glC(T(l)). We show that for the actions with MSJ, the
converse also holds.
Proposition 5.1. If Thas MSJ then for any lN, each element of C(T(l))equals
to Uσ,g1,...,glfor some permutation σand elements g1, . . . , glC(G).
Proof. Let SC(T(l)). We define an ergodic 2-fold self-joining νof T(l)by setting
ν(A×B) := µl(AS1B) for all A, B Bl. Notice that νJe
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
for all subsets A1, . . . , A2lB. Substituting at first A1=· · · =Al=Xand then
Al+1 =· · · =A2l=Xin (5-1) we derive that #P=l, #p= 2 for all pPand
#g2l= 1. Hence g1, . . . , g2lC(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
S1(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:= {AB|µ(TkA4A) = 0 for all kK}.
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
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