S é m i n a i re s
& Cong rès
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SELF-JOININGS OF RANK-ONE ACTIONS
AND APPLICATIONS
V. V. Ryzhikov
ÉCOLE DE THÉORIE ERGODIQUE
Numéro 20
Y. Lacroix, P. Liardet, J.-P. Thouvenot, éds.
SOCIÉTE MATHÉMATIQUE DE FRANCE
Séminaires & Congrès
20, 2010, p. 193–206
SELF-JOININGS OF RANK-ONE ACTIONS AND
APPLICATIONS
by
V. V. Ryzhikov
Abstract. — This paper contains short lecture notes on joinings of rank-one transformations. Using the approximation of self-joinings by off-diagonal measures we
prove King’s theorems on two-fold minimal self-joinings (§1) and the weak closure
theorem for rank-one Z-actions (and flows) (§2). Higher order self-joinings are also
approximated by off-diagonal measures, and this gives the connection between multiple mixing and minimal self-joinings (§3). Blum-Hanson’s ergodic theorem for mixing
transformations and Kalikow’s lemma on microreturns of the blocks (§4) are used in
the joining proof of Kalikow’s theorem on 3-fold mixing for rank-one transformations
(§5).
Résumé (Auto-couplages d’actions de rang 1 et applications). — Cet article est constitué
de courtes notes sur l’auto-couplage des transformations de rang un. En utilisant
l’approximation des auto-couplages par les mesures hors diagonale nous démontrons
les théorèmes de King sur les auto-couplages minimaux à deux feuilles (§1) et le
théorème de fermeture faible pour les Z-actions de rang un (et les flots) (§2). Des
auto-couplages d’ordre plus Ãl’levÃl’s sont aussi approchÃl’s par des mesures hors
diagonale ce qui donne un lien entre le mixage multiple et les autocouplages mininaux (§3). Le thÃl’orÃĺme ergodique de Blum-Hanson pour les transformations
mÃl’langeantes et le lemme de Kalikow sur les micro-retours des blocks (§4) sont utilisÃl’s dans la dÃl’monstration du thÃl’orÃĺme de Kalikow sur le 3-mÃl’lange pour
les transformations de rang un (§5).
The first examples of rank-one transformations appeared in the well-known works
by R. Chacon (geometric constructions), A. Katok, V. Oseledets, A. Stepin (theory
of periodic approximation), D. Ornstein (stochastic constructions). Rank one mixing
transformations have no factors and commute only with their powers (D. Ornstein).
D. Rudolph gave an example of rank-one mixing transformation with extreme properties (minimal self-joinings) for his machinery of counterexamples. Rank one mixing
transformations possess multiple mixing [7, 13] and they have to have the property
2000 Mathematics Subject Classification. — 28D05.
Key words and phrases. — Rank-one transformation, self-joining.
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V. V. RYZHIKOV
of minimal self-joinings (J. King, see [9]). Let us remark that all known examples of
Z-actions with minimal self-joinings (MSJ) are of rank one. In [12] A. Prikhod’ko
announced the existence of infinite rank transformations with MSJ. Note also that
some (infinite rank) horocyclic flows have MSJ (M. Ratner, see [15]). Some spectral properties of rank-one mixing transformations are studied in [1, 4] (stochastic
constructions) and [10, 14] (staircase constructions). We refer a reader to the bibliography presented in the articles [2, 9, 13] and the surveys [5, 6, 15].
Examples of rank-one actions of unusual commutative (and noncommutative)
groups were built first by A. del Junco, then by A.I. Danilenko and C. Silva (see [2]
and references therein). Several theorems on rank-one transformations can be lost
below the horizon of Z, R-actions. There is a loss even in the case of Z2 -actions
[3]. In addition, T. Downarowicz recently constructed a partially mixing rank-one
Z2 -action with non-trivial factors. However, all partially mixing rank-one Z-actions
have no factors [8], and moreover, have minimal self-joinings [9]. It seems that the
zoo of examples in [2] anticipate some “modern” theory which could be in contrast
with “classical” one. Let us speak about the latter.
1. Rank one transformation. The approximation of self-joinings by off-diagonal
measures. Mixing and two-fold minimal self-joinings
We consider probability Lebesgue space (X, µ). An automorphism (a measurepreserving invertible transformation) T : X → X is said to be of rank one, if there is
a sequence ξj of measurable partitions of X in the form
ξj = {Ej , T Ej , T 2 Ej , . . . , T hj Ej , Ẽj }.
such that ξj converges to the partition onto points. The collection
Ej , T Ej , T 2 Ej , . . . , T hj Ej
Fhj i
is called Rokhlin’s tower (Ẽj is the set X \ i=0
T Ej ).
A self-joining (of order 2) is defined to be a T × T -invariant measure ν on X × X
with the marginals equal to µ:
ν(A × X) = ν(X × A) = µ(A).
The joining ν is called ergodic if the dynamical system (T × T, X × X, ν) is ergodic.
The measures ∆i = (Id × T i )∆ (so called off-diagonals measures) are defined by
the formula
∆i (A × B) = µ(A ∩ T i B).
If T is ergodic, then ∆i are ergodic self-joinings. We say that T has minimal selfjoinings of order 2 (and we write T ∈ M SJ(2)) if T has no ergodic joinings except of
µ × µ and ∆i .
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We say that T is mixing, if
∆i −→ µ × µ, i −→ ∞,
i.e., for all measurable A, B
∆i (A × B) = µ(A ∩ T i B) −→ µ × µ(A × B) = µ(A)µ(B).
Theorem 1.1. — The mixing rank-one transformation T has minimal self-joinings of
order two.
Corollary 1.2. — A mixing rank-one transformation commutes only with its powers
and has no factors ( i.e., no non-trivial T -invariant σ-algebras).
Proof of Corollary. — Suppose the automorphism S commutes with T . The joining
∆S = (Id × S)∆ is ergodic: the system (T × T, X × X, ∆S ) is isomorphic to the
ergodic system (T, X, µ). Then for some i we get
(Id × S)∆ = (Id × T i )∆,
this implies S = T i .
Let P be the orthoprojection operator onto the space L2 (X, A, µ), where A is a
factor algebra (a T -invariant σ-subalgebra). Let us define a measure ν on X × X by
setting
Z
ν(A × B) =
P χA χB dµ.
X
Since P commutes with T we obtain ν(A × B) = ν(T A × T B), so ν is a self-joining.
From T ∈ M SJ(2) we see that
X
P = cΘ +
ck T k ,
where Θ, the orthoprojection onto the space of the constants, corresponds to the
measure µ × µ; the operators T k correspond to the off-diagonal measures (Id × T k )∆.
From P 2 = P we see that P = Θ or P = Id. Thus the factor algebra A must be
trivial.
Theorem 1.3. — Let T be of rank-one and ν an ergodic self-joining. Then there is
a sequence kj such that (Id × T ki )∆ → 21 ν + 12 ν 0 for some self-joining ν 0 : for all
measurable A, B
1
1
µ(A ∩ T ki B) −→ ν(A × B) + ν 0 (A × B).
2
2
Corollary 1.4. — Theorem 1.1.
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Proof of Corollary. — Let ν be an ergodic self-joining. T is mixing, hence, (Id ×
T ki )∆ → µ × µ as kj → ∞. From Theorem 1.3 we have: either ν = µ × µ, or
ν = (Id × T k )∆. Thus T ∈ M SJ(2).
Proof of Theorem 1.3. — Our strategy is following: first we prove that joining can
be approximated by sums of parts of the off-diagonal measures, then applying the
Choice Lemma we find a sequence of parts tending to ν.
Given δ > 0, 0 ≤ k ≤ δhj , we define the sets Cjk (called columns):
hj −k
Cjk
=
G
T i T k Ej × T i Ej .
i=0
For negative k (−δhj ≤ k ≤ 0), we put
hj +k
Cjk =
G
T i Ej × T i T −k Ej .
i=0
Let us consider the set
[δhj ]
Djδ =
G
Cjk .
(1)
k=−[δhj ]
For δ >
1
2
we have
ν(Djδ ) > 1 − 2(1 − δ) = 2δ − 1 > 0.
The sets Djδ are asymptotically T × T -invariant, this implies
ν( | Djδ ) −→ ν,
since the limit measure is invariant and absolutely continuous with respect to ergodic ν.
Now we have
X
ν(Cjk |Djδ )ν( |Cjk ) −→ ν.
k
If Aj , Bj are ξj -measurable, then
ν(Aj × Bj |Cjk ) = ∆k (Aj × Bj |Cjk ).
The density of the projections of the measures ∆k ( |Cjk ) and ν( |Cjk ) are bounded
by (1−δ)−1 . For arbitrary measurable sets A, B we can find ξj -measurable sets Aj , Bj
such that
εj = µ(A∆Aj ) + µ(B∆Bj ) −→ 0.
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Then we have
ν(A × B|Cjk ) − ∆k (A × B|Cjk )
≤ |ν(A × B|Cjk ) − ν(Aj × Bj |Cjk ) + ∆k (Aj × Bj |Cjk ) − ∆k (A × B|Cjk )|
≤ |ν((A∆Aj ) × X|Cjk ) + ν(X × (B∆Bj )|Cjk ) +
+ ∆k ((A∆Aj ) × X|Cjk ) + ∆k (X × (B∆Bj )|Cjk )|
≤ 2(1 − δ)−1 εj
(here we use the projection properties of the measures). Thus, we get
X
ν(Cjk |Djδ )∆k ( |Cjk ) −→ ν.
k
We rewrite the expression
P
k
ν(Cjk |Djδ )∆k ( |Cjk ) in the form
P
k
akj ∆kj .
In fact most (with respect to the distribution akj ) of the measures ∆kj are close to
ergodic ν.
Lemma 1.5 (Choice Lemma). — Let ν be an ergodic measure of T × T . Let a sequence
of measures νjk satisfy the conditions: for all A, B
k
νj (T A × T B) − νjk (A × B) < d(j) −→ 0
and
X
akj νjk −→ ν (
k
X
akj = 1 ).
k
Then there is a sequence kj such that
k
νj j
→ ν.
Proof. — Given sets A, B and ε > 0 we consider the sets Kj ⊂ [−δhj , δhj ] of all
integers k such that
ν(A × B) − νjk (T A × T B) > ε
(or νjk (A × B) − ν(A × B) > ε). Suppose that the (sub)sequence Kj satisfies the
condition
X
akj ≥ a > 0.
k∈Kj
P
P
Let λ be a limit point for the sequence of the measures ( k akj )−1 k akj νjk . The
measure λ is invariant, and λ 6= ν. However we represent the ergodic measure ν as
ν = aλ + (1 − a)λ0 ,
for some invariant measure λ0 . From ergodicity of ν we get λ = ν. The contradiction
shows that
X
akj −→ 0.
k∈Kj
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Thus, for most of the measures νjk we have
|νjk (A × B) − ν(A × B)| < ε.
For a family {A1 , A2 , . . .}, which is dense in the algebra of all µ-measurable sets, using
the diagonal method we find a sequence kj such that
k
|νj j (Am × An ) − ν(Am × An )| −→ 0.
This implies
k
|νj j (A × B) − ν(A × B)| −→ 0
for all A, B.
We return to the proof of Theorem 1.3. Let
Yjk =
hj
G
T i Ej , hj ≥ k ≥ 0,
i=k
hj +k
G
Yjk =
T i Ej , −hj ≤ k ≤ 0.
i=0
We apply the Choice Lemma for the measures νjk = ∆kj and obtain
k
k
∆j j = ∆kj ( | Yj j × X) −→ ν,
k
∆kj (A × B | Yj j × X) =
k
Since µ(Yj j ) ≥ 1 − δ and δ >
1
2
1
k
µ(Yj j )
k
µ(Yj j ∩ A ∩ T kj B) −→ ν(A × B).
we have
∆kj −→ (1 − δ)ν + δν 0 .
Let δ → 21 , we get
∆kj −→
1
1
ν + ν0.
2
2
Theorem 1.3 is proved.
2. Weak Closure Theorems
Let us formulate the main result of King’s work [8] (for new results in this direction
see [11]).
Theorem 2.1 (Weak closure theorem). — If the automorphism T ∈ Rank 1 commutes
with the automorphism S, then there is a sequence kj such that T kj → S.
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Lemma 2.2 (Weak closure lemma). — If the automorphism T ∈ Rank 1 commutes
with the automorphism S, then there is a sequence kj and a sequence Yj such that for
all measurable sets A, B
µ(A ∩ T kj B ∩ Yj ) −→ dµ(A ∩ SB),
where Yj has the form
G
Yjd =
G
T i Ej or Yjd =
i, i<dhj
T i Ej
i, (1−d)hj <i<hj
for some number d ≥ 12 .
k
Lemma 2.2 is proved in Section 1 (put ν = ∆S and Yj = Yj j ).
Proof of Theorem 2.1. — We fix T and consider the maximal number d for which the
statement of Lemma 2.2 is true. (If d = 1, the theorem is proved).
So we start from the following statement: for all measurable A, B
µ(A ∩ T kj B ∩ Yjd ) −→ dµ(A ∩ SB).
F
Let us consider the first case: Yjd = i, i<dhj T i Ej .
We have
µ(A ∩ T kj B ∩ SYjd ) = µ(S j A ∩ T kj S j B ∩ Yjd )
and
µ(S j A ∩ T kj S j B ∩ Yjd ) −→ dµ(S j A ∩ SS j B) = dµ(A ∩ SB).
Thus,
µ(A ∩ T kj B ∩ SYj ) −→ µ(A ∩ SB).
From the ergodicity of the joining ∆S we obtain
µ A ∩ T kj B ∩ (Yj ∪ SYj ) −→ uµ(A ∩ SB),
where u = limj µ(Yj ∪ SYj ) (if the limit does not exist, then we consider some subsequence of {j}).
Our aim is to show that
u=1
(this implies the weak closure theorem). We denote
Wjd = (X \ Yjd ).
Let us prove that the maximality of d implies
µ(Wjd ∩ SWjd ) −→ 0.
Indeed, let
lim sup ν(Wjd × Wjd ) = lim sup µ(Wjd ∩ SWjd ) = c > 0.
j
j
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Then for some d0 > d we have
0
0
lim sup µ(Wjd ∩ SWjd ) > c − 2(δ − δ 0 ) >
j
where δ = 1 − d, δ 0 = 1 − d0 and δ −
c
3
c
> 0,
3
< δ 0 < δ. Now for
[δ 0 hj ]
0
Djδ
G
=
Cjk
k=−[δ 0 hj ]
(see (1) for notation) we obtain
0
lim sup ν(Djδ ) >
j
0
0
c
3
0
since Djδ contains the set Wjd × Wjd .
Hence, for some d00 ≥ 1 − δ 0 we can find (as in the proof of Theorem 1.3) the
00
sequence Yjd such that
00
µ(A ∩ T kj B ∩ Yjd ) −→ d00 µ(A ∩ SB).
But d00 ≥ 1 − δ 0 = d0 > d. This contradicts the maximality of d.
Thus, maximality of d implies µ(Wjd ∩ SWjd ) → 0 and so
µ(Yjd ∪ SYjd ) −→ 1.
We get u = 1 for the sequence Yj = Yjd . WCT is proved.
2.1. Flows. — The above “joining proof” of WCT was proposed in [13]. The author
pointed out therein that the same result can be established for flows by word-for-word
repetition of the arguments. To check this a reader needs only the definition of the
rank-one flows.
A flow {Tt } has rank one, if there exists a sequence ξj of partitions of the form
ξj = {Ej , Tsj Ej , Tsj Ej , . . . , Tsj hj Ej , Ẽj }
such that ξj converges to the partition onto points, and sj hj → ∞.
Theorem 2.3. — If a rank-one flow {Tt } commutes with the automorphism S, then
there is a sequence {tj } such that Ttj → S.
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2.2. Some applications of WCT. — Let us see that WCT gives some information about
the rank of automorphisms and flows.
Example 2.4. — If for p > 1 the power T p is not rigid, then T p cannot be of rank-one.
Suppose T p is of rank one. Since T p commutes with T , the automorphism T is a
limit points for powers T pk , k ∈ Z. But this implies the rigidity of T p :
T pkj+1 −pkj −→ Id.
Example 2.5 (1) . — The product T q × T p cannot be of rank-one.
Suppose that the product T q × T p has the weak closure property. Since Id × T
commutes with T q × T p we get
(T q × T p )kj −→ Id × T,
T qkj −→ Id,
T pkj −→ T.
But
T qkj −→ Id implies T qpkj −→ Id
and
T pkj −→ T
implies T qpkj −→ T q .
Thus,
T q = Id.
If T q × T p is of rank 1, then it must be ergodic, hence, the equality T q = Id does not
hold. So T q × T p cannot be of rank 1.
Remark 2.6. — E. Roy asked about the weak closure property for rank-one transformations defined on infinite measure spaces. An attempt to extend WCT to this case
allows us to define some classes of “infinite” transformations with the weak closure
property. For examples, J.-P. Thouvenot and the author have shown that “mixing”
transformations have such properties.
(1) The problem on the rank of T × T p , p ≥ 2, is proposed by A.I. Danilenko. If S and T commute,
it is possible for S × T to be of rank-one.
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3. Approximations of higher order self-joinings by the off-diagonal measures. Multiple
mixing and minimal self-joinings
The order n self-joining ν is called off-diagonal if it can be represented in the form
ν(A1 × · · · × An ) = µ(T k1 A1 ∩ · · · ∩ T kn An ).
The property M SJ(n) means that each order n ergodic self-joining is a product of
off-diagonals measures. One can see that M SJ(n) is equivalent to M SJ(2) with the
property: the product measure µ(1) × · · · × µ(n) is a unique pairwise independent
self-joinings of order n. Pairwise independent joinings arise naturally in the study of
the multiple mixing property.
The automorphism T of the probability space is said to be n-fold mixing if for any
measurable sets A0 , . . . , Ak and sequences m1 , . . . , mn such that |ki − ki0 | → ∞ as
1 ≤ i 6= i0 ≤ n we have:
µ(T k1 A1 · · · ∩ T kn An ) −→ µ(A1 ) . . . µ(An ).
Rokhlin multiple mixing problem: Does 2-fold mixing imply n-fold mixing? Let T be
2-mixing. Let λ be a limit measure for the sets of off-diagonals measures
(T k1 × · · · × T kn )∆,
and suppose that λ is pairwise independent. If such a measure is always equal to
µn = µ(1) × · · · × µ(n) , then T is n-mixing. Thus, the mixing transformation T , which
has no pairwise independent self-joinings except product measures, must be multiple
mixing!
Let us consider again the rank-one transformations. What we can say about its
self-joinings of order n = 3? The same arguments as in the case n = 2 with “3dimension” modifications show that the ergodic self-joining ν can be approximated
by parts of off-diagonal measures ∆kj . But now k = (k1, k2), |k1|, |k2| ≤ (1 − δ)hj ,
δ > 32 ,
∆k (A × B × C) = µ(T k1 A1 ∩ T k2 A2 ∩ A3 ),
∆kj = ∆k ( | Cjk ),
where
H(k)
Cjk =
G
T i+k1 Ej × T i+k2 Ej × T i Ej ,
H(k) = min{hj − k1, hj − k2}.
i=0
From (“3-dimensional”) Choice Lemma we get
Theorem 3.1. — Let T be a rank-one transformation and ν be an ergodic self-joining of
order 3. Then there is a sequence (k1j , k2j ) such that (Id×T k1j ×T k2j )∆ → 31 ν + 23 ν 0
for some self-joining ν 0 .
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Corollary 3.2. — If T is a 3-fold mixing and rank-one transformation, then T ∈
M SJ(3).
4. Blum-Hanson’s ergodic theorem for mixing transformations. Kalikow’s lemma on
microreturns of the blocks for rank-one mixing transformations
Theorem 4.1 (Blum-Hanson’s theorem). — Let T be mixing. Suppose the sequence
{azj }, z ∈ Z, j ∈ N, satisfies the conditions:
X
azj = 1, a ≥ 0, lim max{azj } = 0.
j→∞
z
Then
z
X
z z
lim aj T f − Θf = 0.
j z
2
z z
z aj T .
Let us show that kPj f k2 → 0. One has
X
w
Pj∗ Pj =
bw
j T ,
P
Proof. — Let Θf = 0. Put Pj =
w
where the sequence
{bw
j }
satisfies
X
bjw ≤
aw−z
azj ≤ max azj −→ 0.
j
z
z
Since T is mixing, one has
w w
w bj T f
P
→ 0 (weakly). Thus,
kPj f k = (Pj∗ Pj f, f ) → 0.
2
If Θf 6= 0, we get
kPj (f − Θf )k2 → 0,
kPj f − Θf k2 −→ 0.
Lemma 4.2 (Kalikow’s lemma). — Let T be a mixing rank-one transformation. Let
azj = µ(T z Ej |Ej ). Then limj maxz>0 {azj } = 0.
Proof. — We have maxz>0 {azj } = maxz>hj {azj }. Suppose limj max{azj : z > hj } =
a > 0, µ(T zj Ej |Ej ) → a, hence,
µ(T zj T k Ej |T k Ej ) −→ a, (0 ≤ k ≤ hj ).
We can approximate the measurable set A by ξj -measurable sets Aj (Aj is a union
of some T k Ej ). We have
lim sup µ(T zj Aj |Aj ) ≥ a,
j
this implies
lim sup µ(T zj A|A) ≥ a.
j
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Since µ(A) < a we obtain the contradiction with the property of mixing:
µ(T zj A|A) → µ(A). Thus,
lim max{azj : z > 0} = 0.
j
5. Pairwise independent joinings. Heavy blocks generate light blocks. Approximation by
the light columns. Joining proof of Kalikow’s theorem on 3-fold mixing
Theorem 5.1. — Let T be a mixing rank-one transformation. Let ν be a pairwise
independent self-joining of order n ≥ 3. Then ν = µn .
Corollary 5.2. — If T is mixing rank-one transformation, then T is mixing of all
orders.
We consider the case n = 3. The treatment by n > 3 is analogous. For each ε > 0
we define the sets of ε-light blocks:
Dε,j = {z ∈ [0, hj ]3 : z = (z1, z2, z3), Ējz = T z1 Ej ×T z2 Ej ×T z3 Ej ν(Ējz ) < εµ(Ej )2 }.
Now we calculate the total mass Di(ν) of “infinite light blocks”:
X
Di(ν) = lim lim sup
ν(Ējz ) .
ε→0
j→∞
z∈Dε,j
Lemma 5.3. — If ν is a pairwise independent self-joining, then Di(ν) > 0.
Hint: use Kalikow’s lemma to prove that the heavy blocks under the action of some
powers of T × T × T generate many light blocks, hence, Di(ν) > 0.
Theorem 5.1 follows from the next lemma.
Lemma 5.4. — If ν is a pairwise independent ergodic self-joining, and Di(ν) > 0,
then ν = µ ⊗ µ ⊗ µ.
Proof. — Now we define columns in the following way:
δhj
Cjw,s
=
G
T w+i Ej × T i Ej × T s+i Ej .
i=0
Given small δ > 0 we want to find a sequence of sets Fj such that
ν( |Fj ) → ν
and the sets Fj (so-called fibers) have the form
G
Fj =:
(Id × Id × T h )Cj ,
h∈Sj
for some sequences
wj ,sj
Sj ⊂ {0, 1, . . . (1 − δ)hj }, sj , wj ∈ {0, 1, . . . (1 − δ)hj }, Cj = Cj
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Since the sets as Fj are almost invariant with respect to T × T × T and ν is ergodic,
from the Choice Lemma we get some sequence Fj such that ν( |Fj ) → ν.
Moreover we have Di(ν) > 0, so one can find a sequence of fibers Fj with such sets
Sj that numerate only light columns. The sets Sj will satisfy the condition:
X
max{ajs } −→ 0, j −→ ∞,
ajs = 1,
s∈Sj
s∈Sj
where
ajs = ν((Id × Id × T s )Cj | Fj ).
Let Yj be the projection of Cj into the factor X(2) of the product X(1) ×X(2) ×X(3) .
The measure ν( |Fj ) is close to λj (a part of off-diagonal measure in X × X × X),
which is defined by the equality
Z
1 X j
λj (A × B × C) =:
as
χYj χT wj A χB χT s C dµ.
aµ(Yj ) s
X
From the Blum-Hanson theorem we get
X
ajs χT s C −→L2 Const ≡ aµ(C) (j → ∞).
s∈Sj
Using pairwise independence of the self-joining ν we obtain
ν(A × B × C) = lim ν(A × B × C | Fj ) = lim λj (A × B × C)
j→∞
j→∞
Z
ÅX
ã
1
j
w
s
χYj χT j A χB
as χT C dµ
= lim
j→∞ aµ(Yj ) X
s∈Sj
= µ(C) ν(A × B × X)
= µ(A)µ(B)µ(C).
The author is thankful to Y. Lacroix and P. Liardet for their hospitality during
École de théorie ergodique II (April, 2006), to A.I. Danilenko for discussions and to
J.-P. Thouvenot for support. He is also indebted to the Referee for encouragement
and useful remarks.
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V. V. Ryzhikov, Department of Mathematics, Moscow State University, Moscow 119922
E-mail : [email protected]
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