Élise Janvresse
email: elise.janvresse@univ-rouen.fr

Thierry De

AND L A Rue

Valery Ryzhikov

AROUND KING'S RANK-ONE THEOREMS: FLOWS AND Z n -ACTIONS

Keywords: 2000 Mathematics Subject Classification. 37A10,37A15,37A35 Rank-one actions, Weak Closure Theorem, Factors, Joinings

We study the generalizations of Jonathan King's rank-one theorems (Weak-Closure Theorem and rigidity of factors) to the case of rank-one R-actions (flows) and rank-one Z n -actions. We prove that these results remain valid in the case of rank-one flows. In the case of rank-one Z n actions, where counterexamples have already been given, we prove partial Weak-Closure Theorem and partial rigidity of factors.

Introduction

Very important examples in ergodic theory have been constructed in the class of rank-one transformations, which is closely connected to the notion of transformations with fast cyclic approximation [START_REF] Katok | Approximation of ergodic dynamical systems by periodic transformations[END_REF]: If the rate of approximation is sufficiently fast, then the transformation will be inside the rank-one class. The notion of rankone transformations has been defined in [START_REF] Ornstein | On the root problem in ergodic theory[END_REF], where mixing examples have appeared. Later, Daniel Rudolph used them for a machinery of counterexamples [START_REF] Daniel | An example of a measure preserving map with minimal self-joinings, and applications[END_REF].

Jonathan King contributed to the theory of rank-one transformations by several deep and interesting facts. His Weak-Closure-Theorem (WCT) [START_REF] King | The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam[END_REF] is now a classical result with applications even out of the range of Z-actions (see for example [START_REF] Tikhonov | Embeddings of lattice actions in flows with multidimensional time[END_REF]). He also proved the minimal-self-joining (MSJ) property for rank-one mixing automorphisms (see [START_REF]Joining-rank and the structure of finite rank mixing transformations[END_REF]), the rigidity of non-trivial factors [START_REF] King | The commutant is the weak closure of the powers, for rank-1 transformations, Ergodic Theory Dynam[END_REF], and the weak closure property for all joinings for flat-roof rank-one transformations [START_REF]Flat stacks, joining-closure and genericity[END_REF].

A natural question is whether the corresponding assertions remain true for flows (R-actions) and for Z n -actions. We show that for flows the situation is quite similar: The joining proof of the Weak-Closure Theorem given in [START_REF] Ryzhikov | Mixing, rank and minimal self-joining of actions with invariant measure[END_REF] (see also [START_REF]Self-joinings of rank-one actions and applications[END_REF]) can be adapted to the situation of a rank-one R-action (Theorem 5.2). We also give in the same spirit a proof of the rigidity of non-trivial factors of rank-one flows (Theorem 6.2) which, with some simplification, provides a new proof of King's result in the case of Z-actions. We prove a flat-roof flow version as well (Theorem 7.1). Note that a proof of the Weak-Closure Theorem for rank-one flows had already been published in [START_REF] Zeitz | The centralizer of a rank-one flow[END_REF]. Unfortunately it relies on the erroneous assumption that if (T t ) t∈R is a rank-one flow, then there exists a real number t 0 such that T t0 is a rank-one transformation (see beginning of Section 3.2 in [START_REF] Zeitz | The centralizer of a rank-one flow[END_REF]).

Concerning multidimensional rank-one actions, the situation is quite different. The Weak-Closure Theorem is no more true [START_REF] Downarowicz | Weak closure theorem fails for Z 2 -actions[END_REF], and factors may be non-rigid [START_REF] Downarowicz | Phenomena in rank-one Z 2 -actions[END_REF]. Rank-one partially mixing Z-actions have MSJ [START_REF] King | A canonical structure theorem for finite joiningrank maps[END_REF], however it is proved in [START_REF] Downarowicz | Phenomena in rank-one Z 2 -actions[END_REF] that for Z 2 -actions this is generally not true. We remark that it was an answer for Z 2 -action to Jean-Paul Thouvenot's question: Whether a mildly mixing rank-one action possesses MSJ, though this interesting problem remains open for Z-actions. Regardless these surprising results, there are some partial versions of WCT: Commuting automorphisms can be partially approximated by elements of the action (Corollary 8.4), and non-trivial factors must be partially rigid (Corollary 8.5). We present these results as consequences of A. Pavlova's theorem (Theorem 8.3, see also [START_REF]Rank, rigidity of factors, and weak closure of measure-preserving Z n -actions[END_REF]) .

Preliminaries and notations

Weak convergence of probability measures. We are interested in groups of automorphisms of a Lebesgue space (X, A , µ), where µ is a continuous probability measure. The properties of these group actions are independent of the choice of the underlying space X, and for practical reasons we will assume that X = {0, 1} Z , equipped with the product topology and the Borel σ-algebra. This σ-algebra is generated by the cylinder sets, that is sets obtained by fixing a finite number of coordinates. On the set M 1 (X) of Borel probability measures on X, we will consider the topology of weak convergence, which is characterized by

ν n w ----→ n→∞ ν ⇐⇒ for all cylinder set C, ν n (C) ----→ n→∞ ν(C),
and turns M 1 (X) into a compact metrizable space.

We will often consider probability measures on X ×X, with the same topology of weak convergence. We will use the following observation: If ν n and ν in M 1 (X ×X) have their marginals absolutely continuous with respect to our reference measure µ, with bounded density, then the weak convergence of ν n to ν ensures that for all measurable sets A and

B in A , ν n (A × B) ----→ n→∞ ν(A × B).
Self-joinings. Let T = (T g ) g∈G be an action of the Abelian group G by automorphism of the Lebesgue space (X, A , µ). A self-joining of T is any probability measure on X×X with both marginals equal to µ and invariant by T ×T = (T g ×T g ) g∈G .

For any automorphism S commuting with T , we will denote by ∆ S the self-joining concentrated on the graph of S -1 , defined by

∀A, B ∈ A , ∆ S (A × B) := µ(A ∩ SB).
In particular, for any g ∈ G we will denote by ∆ g the self-joining ∆ Tg . In the special case where S = T 0 = Id, we will note simply ∆ instead of ∆ 0 or ∆ Id .

If F is a factor (a sub-σ-algebra invariant under the action (T g )), we denote by µ ⊗ F µ the relatively independent joining above F , defined by

µ ⊗ F µ(A × B) := X E µ [½ A |F ] E µ [½ B |F ] dµ.
Recall that µ ⊗ F µ coincides with ∆ on the σ-algebra F ⊗ F .

Flows.

A flow is a continuous family (T t ) t∈R of automorphisms of the Lebesgue space (X, A , µ), with T t • T s = T t+s for all t, s ∈ R, and such that (t, x) → T t (x) is measurable. We recall that the measurability condition implies that for all measurable set A, µ(A △ T t A) ---→ t→0 0.

Lemma 2.1. Let (T t ) t∈R be an ergodic flow on (X, A , µ). Let Q be a dense subgroup of R, and λ be an invariant probability measure for the action of (T t ) t∈Q . Assume further that λ ≪ µ, with dλ dµ bounded by some constant C. Then λ = µ. Proof. Let t ∈ R, and let (t n ) be a sequence in Q converging to t. For any measurable set A, we have

λ T t A △ T tn A ≤ Cµ T t A △ T tn A ----→ n→∞ 0.
Hence λ(T t A) = lim n λ(T tn A) = λ(A). This proves that λ is T t -invariant for each t ∈ R. Since µ is ergodic under the action of (T t ) t∈R , we get λ = µ.

3. Rank-one flows Definition 3.1. A flow (T t ) t∈R is of rank one if there exists a sequence (ξ j ) of partitions of the form

ξ j =    E j , T sj E j , T 2 sj E j , . . . , T hj-1 sj E j , X \ hj -1 i=0 T i sj E j   
such that ξ j converges to the partition into points (that is, for every measurable set A and every j, we can find a ξ j -measurable set A j in such a way that µ(A △ A j ) ---→ j→∞ 0), s j /s j+1 are integers, s j → 0 and s j h j → ∞.

Several authors have generalized the notion of a rank-one transformation to an R-action using continuous Rokhlin towers (see e.g. [START_REF] Prikhod | Stochastic constructions of mixing systems of positive local rank[END_REF]). One can show that the above definition includes all earlier definitions of rank-one flows with continuous Rokhlin towers. The above definition without the requirement that s j /s j+1 be integers was given by the third author in [START_REF] Ryzhikov | Mixing, rank and minimal self-joining of actions with invariant measure[END_REF].

Lemma 3.2. Let (T t ) t∈R be a rank-one flow. Then the sequences (s j ) and (h j ) in the definition can be chosen so that

s 2 j h j ---→ j→∞ ∞.
Proof. Let (s j ) and (h j ) be given as in the definition. Recall that h j s j → ∞.

For each j, let n j > j be a large enough integer such that s j s nj h nj > j. Define ℓ j := s j /s nj ∈ Z + . We consider the new partition

ξj :=    Ẽj , T sj Ẽj , • • • , T hj -1 sj Ẽj , X \ hj -1 i=0 T i sj Ẽj    where Ẽj := ℓj-1 i=0 T i sn j E nj
and hj := [h nj /ℓ j ]. One can easily check that ξj still converges to the partition into points. Moreover we have s 2 j hj = s 2 j [h nj s nj /s j ] → ∞.

Lemma 3.3 (Choice Lemma for flows, abstract setting). Let (T t ) t∈R be an arbitrary flow, and let ν be an ergodic invariant measure under the action of (T t ) t∈R . Let a family of measures (ν k j ) satisfy the conditions:

• There exist sequences (d j ) and (s j ) of positive numbers with d j ---→ j→∞ 0, s j /s j+1 is an integer for all j, and s j ---→ j→∞ 0, such that for all measurable set A and all k, j

(1) ν k j (T sj A) -ν k j (A) < s j d j ; • There exists a family of positive numbers (a k j ) with k a k j = 1 for all j, such that

(2) k a k j ν k j w ---→ j→∞ ν.
Then there is a sequence (k j ) such that ν

kj j w ---→ j→∞ ν.
Proof. Given a cylinder set B, an integer j ≥ 1 and ε > 0, we consider the sets K j of all integers k such that ν(B) -ν k j (B) > ε. Suppose that the (sub)sequence K j satisfies the condition k∈Kj a k j ≥ a > 0.

Let λ be a limit point for the sequence of measures ( k∈Kj a k j ) -1 k∈Kj a k j ν k j . Then λ = ν since λ(B) ≤ ν(B) -ε, but by (2), we have λ ≪ ν, and dλ/dν ≤ 1/a. Moreover, the measure λ is invariant by T sp for all p. Indeed, for j ≥ p, since s p /s j is an integer, we get from (1) that

ν k j (T sp A) -ν k j (A) < s p d j ---→ j→∞ 0.
By Lemma 2.1, it follows that λ = ν. The contradiction shows that k∈Kj a k j → 0.

Thus, for all large enough j, most of the measures ν k j satisfy

|ν k j (B) -ν(B)| < ε. Let {B 1 , B 2 , .
. . } be the countable family of all cylinder sets. Using the diagonal method we find a sequence k j such that for each n

|ν kj j (B n ) -ν(B n )| ---→ j→∞ 0, i.e. ν kj j w ---→ j→∞ ν.
Columns and fat diagonals in X × X. Assume that (T t ) t∈R is a rank-one flow defined on X, with a sequence (ξ j ) of partitions as in Definition 3.1. For all j and |k| < h j -1, we define the sets C k j ∈ X × X, called columns:

C k j := 0≤r,ℓ≤hj -1 r-ℓ=k T r sj E j × T ℓ sj E j .
Given 0 < δ < 1, we consider the set

D δ j := [δhj ] k=-[δhj ]
C k j .

(See Figure 1.)

T sj E j T

hj-1 sj

E j C 3 j E j T hj-1 sj E j E j T sj E j δh j D δ j Figure 1.
Columns and fat diagonals in X × X

Approximation theorem

Recall from Section 2 that, given a flow (T t ) t∈R , ∆ t stands for the self-joining supported by the graph of T -t . Lemma 4.1. Let ν be an ergodic joining of the rank-one flow (T t ) t∈R . Let 0 < δ < 1 be such that

(3) ℓ δ := lim j ν(D δ j ) > 0.
Then there exists a sequence (k j ) with -δh j ≤ k j ≤ δh j such that

∆ kj sj ( • |C kj j ) w ---→ j→∞ ν.
Proof. Our strategy is the following: First we prove that the joining ν can be approximated by sums of parts of off-diagonal measures, then applying the Choice Lemma we find a sequence of parts tending to ν. By definition of D δ j , we have

ν D δ j △ (T sj × T sj )D δ j ≤ C h j .
It follows that for any fixed p, the sets D δ j are asymptotically T sp × T sp -invariant: Indeed, since T sp = T sp/sj sj where s p /s j is an integer when j ≥ p, we get 3), λ is absolutely continuous with respect to ν, and dλ dν ≤ 1 ℓ δ < ∞. By Lemma 2.1, it follows that λ = ν. Hence we have ( 4)

ν D δ j △ (T sp × T sp )D δ j ≤ s p s j C h j ---→ j→∞ 0 (recall that s j h j → ∞). Let λ be a limit measure of ν( • | D δ j ). Then λ is T sp × T sp -invariant for each p, by (
ν( • | D δ j ) w ---→ j→∞ ν.
We now prove that ( 5)

[δhj] k=-[δhj ] ν(C k j |D δ j )∆ ksj ( • |C k j ) w ---→ j→∞ ν.
For arbitrary measurable sets A, B we can find ξ j -measurable sets A j , B j such that

ε j := µ(A △ A j ) + µ(B △ B j ) → 0.
We have

k ν(C k j |D δ j )∆ ksj (A × B|C k j ) -ν(A × B) = M 1 + M 2 + M 3 + M 4 ,
where

M 1 := k ν(C k j |D δ j ) ∆ ksj (A × B|C k j ) -∆ ksj (A j × B j |C k j ) , M 2 := k ν(C k j |D δ j )∆ ksj (A j × B j |C k j ) -ν(A j × B j |D δ j ), M 3 := ν(A j × B j |D δ j ) -ν(A × B|D δ j ), M 4 := ν(A × B|D δ j ) -ν(A × B). The density of the projections of the measure ∆ ksj ( • |C k j ) with respect to µ is bounded by (1 -δ) -1 . Hence M 1 ≤ ε j /(1 -δ).
Since A j , B j are ξ j -measurable,

ν(A j × B j |C k j ) = ∆ ksj (A j × B j |C k j
), and we get M 2 = 0.

The absolute value of the third term M 3 can be bounded above as follows

|M 3 | ≤ ν(D δ j ) -1 ν (A j × B j ) △ (A × B) ≤ ε j ν(D δ j ) → 0.
The last term M 4 goes to zero as j → ∞ by ( 4), and this ends the proof of (5).

To apply the Choice Lemma for the measures ν k j = ∆ ksj ( • |C k j ) and a k j = ν(C k j |D δ j ), it remains to check the first hypothesis of the lemma. By construction of the columns C k j , we have for any measurable subset A ∈ X × X and all

k ∈ {-[δh j ], . . . , [δh j ]}, (6) ∆ ksj (T sj × T sj A|C k j ) -∆ ksj (A|C k j ) < C h j
where C is a constant. We get the desired result by setting d j := C s j h j .

The Choice Lemma then gives a sequence (k j ) with -δh j ≤ k j ≤ δh j such that

∆ kj sj ( • |C kj j ) w ---→ j→∞ ν.
Theorem 4.2. Let a flow T = (T t ) t∈R be of rank-one and ν be an ergodic selfjoining of (T t ) t∈R . Then there is a sequence

(k j ) such that ∆ kj sj w ---→ j→∞ 1 2 ν + 1 2 ν ′ for some self-joining ν ′ : For all measurable sets A, B µ(A ∩ T kj sj B) → 1 2 ν(A × B) + 1 2 ν ′ (A × B).
Proof. For any 1/2 < δ < 1, we have

lim j→∞ ν(D δ j ) > 1 -2(1 -δ) = 2δ -1 > 0.
Hence we can apply Lemma 4.1 for any 1/2 < δ < 1. By a diagonal argument, we get the existence of (k j ) and (δ j ) ց 1 2 with -δ j h j ≤ k j ≤ δ j h j such that ∆ kj sj • |C

kj j w ---→ j→∞ ν.
Let us decompose ∆ kj sj as

∆ kj sj = ∆ kj sj • |C kj j ∆ kj sj (C kj j ) + ∆ kj sj • |X × X \ C kj j 1 -∆ kj sj (C k j ) .
Since lim inf j→∞ ∆ kj sj (C kj j ) ≥ 1/2, we get the existence of some self-joining ν ′ such that ∆ kj sj w ---→

j→∞ 1 2 ν + 1 2 ν ′ .
Corollary 4.3. A mixing rank-one flow has minimal self-joinings of order two.

Proof. Let ν be an ergodic self-joining of a mixing rank-one flow (T t ) t∈R . Let (k j ) be the sequence given by Theorem 4.2. If |k j s j | → ∞, since T is mixing we have

∆ kj sj w ---→ j→∞ µ × µ, hence µ× µ = 1 2 ν + 1 2 ν ′ for some self-joining ν ′ .
The ergodicity of µ× µ then implies that µ × µ = ν. Otherwise, along some subsequence we have k j s j → s for some real number s. Then ∆ s = 1 2 ν + 1 2 ν ′ for some self-joining ν ′ , and again the ergodicity of ∆ s yields ν = ∆ s . Thus T has minimal self-joinings of order two..

Weak Closure Theorem for rank-one flows

Lemma 5.1 (Weak Closure Lemma). If the automorphism S commutes with the rank-one flow (T t ) t∈R , then there exist 1/2 ≤ d ≤ 1, a sequence (k j ) of integers and a sequence of measurable sets (Y j ) such that, for all measurable sets A, B

µ(A ∩ T kj sj B ∩ Y j ) → d µ(A ∩ SB)
, where Y j has the form

Y d,- j := 0≤i<dhj T i sj E j or Y d,+ j := (1-d)hj<i≤hj T i sj E j .
Proof. This lemma is a consequence of the proof of Theorem 4.2, when the joining ν is equal to ∆ S . Given a sequence (δ j ) ց 1 2 , the proof provides a sequence (k j ) where -δ j h j ≤ k j ≤ δ j h j , such that ∆ kj sj ( • |C So we start from the following statement: We have a sequence of sets {Y j }, of the form given in Lemma 5.1, such that for all measurable A, B µ(A ∩ T kj sj B ∩ Y j ) → dµ(A ∩ SB). Then a similar statement holds when Y j is replaced by SY j : Indeed, since S commutes with T and µ is invariant by S, we have

Y kj j = hj i=kj T i sj E j if k j ≥ 0 hj+kj i=0 T i sj E j if k j < 0. We then have ∆ kj sj ( • |C kj j ) = ∆ kj sj ( • |Y
µ(A ∩ T kj sj B ∩ SY j ) = µ(S -1 A ∩ T kj sj S -1 B ∩ Y j ) ---→ j→∞ d µ(S -1 A ∩ SS -1 B) = d µ(A ∩ SB).
Let λ be a limit point for the sequence of probability measures {ν j } defined on X × X by

ν j (A × B) := 1 µ(Y j ∪ SY j ) µ A ∩ T kj sj B ∩ (Y j ∪ SY j ) .
Then λ ≤ 2 ∆ S . Moreover, the measure λ is invariant by T sp × T sp for all p. Indeed, for j ≥ p, we have

µ(T sp Y j △ Y j ) = µ(T sp/sj sj Y j △ Y j )
which is of order sp sj hj , hence vanishes as j → ∞. Since ∆ S is an ergodic measure for the flow {T t × T t }, we can apply Lemma 2.1, which gives λ = ∆ S . We obtain

µ A ∩ T kj sj B ∩ (Y j ∪ SY j ) → u µ(A ∩ SB)
, where u := lim j µ(Y j ∪ SY j ) (if the limit does not exist, then we consider some subsequence of {j}).

Our aim is to show that u = 1, which will end the proof of the theorem. Let us introduce

W j :=   0≤i≤hj T i sj E j   \ Y j .
Assume that u < 1, then (denoting by Y c the complementary of Y ⊂ X)

lim j ∆ S (W j × W j ) = lim j µ(W j ∩ SW j ) = lim j µ(Y c j ∩ SY c j ) = 1 -u > 0.
Let us consider the case where Y j has the form Y d,- j = 0≤i<dhj T i sj E j . Then W j = dhj≤i≤hj T i sj E j , and we define for any

δ ′ < 1 -d W j (δ ′ ) := (1-δ ′ )hj <i≤hj T i sj E j ⊂ W j .
In the same way, if Y j has the form Y d,+ j = (1-d)hj<i≤hj T i sj E j , we set for δ ′ < 1-d

W j (δ ′ ) := 0<i<δ ′ hj T i sj E j ⊂ W j .
In both cases, note that

∆ S (W j × W j ) \ (W j (δ ′ ) × W j (δ ′ )) ≤ 2(1 -d -δ ′ ).
Thus, for δ ′ close enough to 1 -d, we get lim sup

j ∆ S W j (δ ′ ) × W j (δ ′ ) ≥ 1 -u -2(1 -d -δ ′ ) > 0. Since W j (δ ′ ) × W j (δ ′ ) ⊂ D δ ′ j , this ensures that lim sup ∆ S (D δ ′ j ) > 0. Lemma 4.1 then provides a sequence (k ′ j ) with -δ ′ h j ≤ k ′ j ≤ δ ′ h j , such that ∆ k ′ j sj ( • |C k ′ j j ) w ---→ j→∞ ∆ S ,
and the projections Y

k ′ j j of C k ′ j j on the first coordinate satisfy lim j µ(Y k ′ j j ) ≥ 1 -δ ′ > d,
which contradicts the maximality of d. Hence u = 1.

6. Rigidity of factors of rank-one flows Lemma 6.1. Let F be a non-trivial factor of a rank-one flow (T t ) t∈R . Then there exist 1/2 ≤ d ≤ 1, a sequence of integers (k j ) with |k j s j | 0 and a sequence of measurable sets (Y j ) such that, for all measurable sets A, B ∈ F

µ(A ∩ T kj sj B ∩ Y j ) → d µ(A ∩ B)
, where Y j has the form

Y d,- j := 0≤i<dhj T i sj E j or Y d,+ j := (1-d)hj<i≤hj T i sj E j .
Proof. We start with the relatively independent joining above the factor F (see Section 2). Since F is a non-trivial factor, µ ⊗ F µ = ∆, hence we can consider an ergodic component ν such that ν({(x, x), x ∈ X}) = 0. Observe however that for any sets A, B ∈ F , we have ν(A × B) = µ(A ∩ B).

We repeat the proof of Lemma 5.1 with ν in place of ∆ S . This provides sequences (k j ) and (Y j ) and a real number 1/2 ≤ d ≤ 1, such that for all measurable sets A, B µ(A ∩ T kj sj B ∩ Y j ) → d ν(A × B). If we had k j s j → 0, then the left-hand side would converge to d µ(A ∩ B), which would give ν(A × B) = µ(A ∩ B) for all A, B ∈ A , and this would contradict the hypothesis that ν gives measure 0 to the diagonal. Theorem 6.2. Let F be a non-trivial factor of a rank-one flow (T t ) t∈R . Then there exists a sequence of integers (k j ) with |k j s j | → ∞ such that, for all measurable sets

A, B ∈ F µ(A ∩ T kj sj B) → µ(A ∩ B).
Proof. Again we fix some ergodic component ν such that ν({(x, x), x ∈ X}) = 0. We consider the maximal number d for which the statement of Lemma 6.1 is true. We thus have a sequence of sets {Y j }, of the form given in Lemma 6.1, such that

(7) ∀A, B ∈ F , 1 µ(Y j ) E µ ½ A ½ T k j s j B ½ Yj → µ(A ∩ B). In the above equation, one can replace ½ Yj by φ j (x) := E ν [½ Yj (x ′ )|x]: Indeed, since ν coincides with ∆ on F ⊗ F , we have ½ A (x ′ ) = ½ A (x) and ½ T k j s j B (x ′ ) = ½ T k j s j B (x) ν-a.s. Hence, E µ ½ A ½ T k j s j B ½ Yj = E ν ½ A (x)½ T k j s j B (x)½ Yj (x ′ ) = E µ ½ A (x)½ T k j s j B (x)φ j (x) .
We note that ( 8)

E µ |φ j -φ j • T sj | ≤ µ(Y j △ T sj Y j ) = O 1 h j .
For any ε > 0, let U ε j := {x : φ j (x) > ε} .

We would like to prove that (7) remains valid with ½ Yj replaced by ½ U ε j for ε small enough. To this end, we need almost-invariance of U ε j under T sj , which does not seem to be guaranteed for arbitrary ε. Therefore, we use the following technical argument to find a sequence (ε j ) for which the desired result holds. Fix ε > 0 small enough so that µ(U ε j ) > µ(Y j )/2 for all large j. By Lemma 3.2, we can assume that s 2 j h j → ∞. Let δ j = o(s j ) such that (δ j h j ) -1 = o(s j ). We divide the interval [ε/2, ε] into ε/(4δ j ) disjoint subintervals of length 2δ j . One of these subintervals, called I j , satisfy [START_REF] Pavlova | Self-joinings and weak closure of rank one actions[END_REF] µ ({x :

φ j (x) ∈ I j }) ≤ 4δ j ε .
Let us call ε j the center of the interval I j . Observe that

µ U εj j △ T sj U εj j ≤ µ ({x : |φ j (x) -ε j | < δ j })+µ {x : |φ j (x) -φ j (T sj (x))| ≥ δ j } .
By ( 9) and ( 8), we get that

(10) µ U εj j △ T sj U εj j = O δ j + 1 δ j h j = o(s j ).
Taking a subsequence if necessary, we can assume that the sequence of probability measures λ j , defined by

∀A, B ∈ A , λ j (A × B) := 1 µ(U εj j ) E µ ½ A ½ T k j s j B ½ U ε j j ,
converges to some probability measure λ, which is invariant by T sp × T sp for all p by [START_REF] Prikhod | Stochastic constructions of mixing systems of positive local rank[END_REF]. Recall that µ(U εj j ) > µ(Y j )/2 and that ½ U ε j j ≤ φ j /ε j . Then, since

ε j > ε/2, we have λ| F ⊗F ≤ 4 ε ∆| F ⊗F .
Since ∆| F ⊗F is an ergodic measure for the flow {T t × T t }| F ⊗F , we can apply Lemma 2.1, which gives λ| F ⊗F = ∆| F ⊗F . This means that (7) remains valid with ½ Yj replaced by ½ U ε j j .

The analogue of ( 7) is also valid when we replace ½ Yj by ½ Yj ∪U ε j j : Indeed, we also have the almost-invariance property

µ (Y j ∪ U εj j ) △ T sj (Y j ∪ U εj j ) = o(s j ) and ½ Yj ∪U ε j j ≤ ½ Yj + ½ U ε j j
. We conclude by a similar argument. Since ε can be taken arbitrarily small, we can now use a diagonal argument to show that (7) remains valid with ½ Yj replaced by ½ Yj ∪U ε j j where the sequence (ε j ) now satisfies ε j → 0. Hence, taking a subsequence if necessary to ensure that µ(Y j ∪ U εj j ) converges to some number u, we get

∀A, B ∈ F , E µ ½ A ½ T k j s j B ½ Yj ∪U ε j j → uµ(A ∩ B).
It now remains to prove that u = 1, which we do by repeating the end of the proof of Theorem 5.2. Assume that u < 1. Let us introduce

W j :=   0≤i≤hj T i sj E j   \ Y j .
We have

lim j ν(W j × W j ) = lim j ν(Y c j × Y c j ) = lim j E µ ½ Y c j (1 -φ j ) . Observe that (1 -φ j ) ≥ ½ (U ε j j ) c -ε j . Hence lim j ν(W j × W j ) ≥ lim j E µ ½ Y c j ½ (U ε j j ) c = 1 -u > 0.
Let us consider the case where Y j has the form Y d,- j = 0≤i<dhj T i sj E j . Then W j = dhj≤i≤hj T i sj E j , and we define for any

δ ′ < 1 -d W j (δ ′ ) := (1-δ ′ )hj <i≤hj T i sj E j ⊂ W j .
In the same way, if Y j has the form

Y d,+ j = (1-d)hj<i≤hj T i sj E j , we set for δ ′ < 1-d W j (δ ′ ) := 0<i<δ ′ hj T i sj E j ⊂ W j .
In both cases, note that

ν (W j × W j ) \ (W j (δ ′ ) × W j (δ ′ )) ≤ 2(1 -d -δ ′ ).
thus, for δ ′ close enough to 1 -d, we get lim sup

j ν W j (δ ′ ) × W j (δ ′ ) ≥ 1 -u -2(1 -d -δ ′ ) > 0. Since W j (δ ′ ) × W j (δ ′ ) ⊂ D δ ′ j , this ensures that lim sup ν(D δ ′ j ) > 0. Lemma 4.1 then provides a sequence (k ′ j ) with -δ ′ h j ≤ k ′ j ≤ δ ′ h j , such that ∆ k ′ j sj ( • |C k ′ j j ) w ---→ j→∞ ν.
In particular, ∆ k ′ j sj ( • |C

k ′ j j )| F ⊗F w ---→ j→∞ ∆| F ⊗F . Since the projections Y k ′ j j of C k ′ j j
on the first coordinate satisfy

lim j µ(Y k ′ j j ) ≥ 1 -δ ′ > d,
this contradicts the maximality of d. Hence u = 1.

King's theorem for flat-roof rank-one flow

We consider a rank-one flow (T t ) t∈R . We say that (T t ) t∈R has flat roof if we can choose the sequence ξ j = {E j , T sj E j , . . . , T

hj-1 sj

E j , X \ hj -1 k=0 T k sj E j } in the definition such that µ T hj sj E j △ E j µ(E j ) ---→ j→∞ 0.
Theorem 7.1. Let (T t ) t∈R be a flat-roof rank-one flow, and ν be an ergodic selfjoining of (T t ) t∈R . Then there exists a sequence (k j ) such that ∆ kj sj w ---→ j→∞ ν.

Proof. Let us defined, for 0 ≤ k ≤ h j -1

a j k := ν T k sj E j × E j and b j k := ν E j × T hj-k sj E j .
We claim that the flat-roof property implies (11)

h j hj -1 k=1 |a j k -b j k | ---→ j→∞ 0.
Indeed, by invariance a j k = ν T

hj sj E j × T

hj-k sj E j . Hence

|a j k -b j k | ≤ ν (T hj sj E j △ E j ) × T hj-k sj E j , and
hj -1

k=1 |a j k -b j k | ≤ ν (T hj sj E j △ E j ) × X = µ (T hj sj E j △ E j ) .
The claim follows, since µ(

E j ) ∼ 1/h j . C k-hj j C k j T k sj E j a j k T hj-k sj E j b j k E j E j Figure 2. The union of C k j and C k-hj j is denoted by G k j .
We gather the columns C k j in pairs, defining for 1

≤ k ≤ h j -1, G k j := C k j ⊔C k-hj j . (See Figure 2.) We also set G 0 j := C 0 j . Note that ν(G k j ) = (h j -k)a j k + kb j k . Observe also that ν   hj -1 k=0 G k j   = ν   hj -1 k=0 T k sj E j × hj -1 k=0 T k sj E j   ---→ j→∞ 1.
Hence, [START_REF] Daniel | An example of a measure preserving map with minimal self-joinings, and applications[END_REF] hj -1

k=0 ν(G k j ) ν( • |G k j ) w ---→ j→∞ ν.
We claim that, using the flat-roof property, we can in the above equation replace ν( • |G k j ) by ∆ ksj . Let A and B be ξ j -measurable sets, which are unions of T i sj E j (0 ≤ i ≤ h j -1). We denote by r k (respectively ℓ k ) the number of elementary cells of the form T i1 sj E j × T i2 sj E j which are contained in A × B and which belong to the column C k j (respectively C k-hj j

). We have

(13) ν(A × B|G k j )ν(G k j ) = ℓ k b j k + r k a j k .
Moreover, we will show that the flat-roof property ensures the existence of a sequence (ε j ) with ε j ---→ j→∞ 0 such that

(14) ∆ ksj (A × B) - ℓ k + r k h j ≤ ε j .

Indeed, let us cut

A into A 1 := A∩ 0≤i≤k-1 T i sj E j and A 2 := A∩ k≤i≤hj -1 T i sj E j . We have ∆ ksj (A 2 × B) = r k µ(E j ),
and

∆ ksj (A 1 × B) = ℓ k ∆ ksj (E j × T hj-k sj E j ) + ∆ ksj (A 1 × B) \ C k-hj j . Recalling that ∆ ksj (E j × T hj-k sj E j ) = µ(E j ∩ T hj sj E j ), we get (15) ∆ ksj (A×B) = (r k +ℓ k )µ(E j )-ℓ k µ(E j \T hj sj E j )+∆ ksj (A 1 × B) \ C k-hj j
.

The second term of the right-hand side is bounded by h j µ(E j ∆T

hj sj E j ), which goes to 0 by the flat-roof property. To treat the last term, we consider the particular case A = B = 0≤i≤hj -1 T i sj E j , for which this last term is maximized. We have then

1 -∆ ksj (A × B) ≤ 2µ   X \ 0≤i≤hj -1 T i sj E j   ---→ j→∞ 0.
On the other hand, [START_REF]Self-joinings of rank-one actions and applications[END_REF] gives

∆ ksj (A 1 × B) \ C k-hj j = ∆ ksj (A × B) -h j µ(E j ) + kµ(E j \ T hj sj E j ).
Since h j µ(E j ) → 1, and kµ(E j \ T

hj sj E j ) ≤ h j µ(E j ∆T

hj sj E j ) → 0, we get that the last term of (15) goes to 0 uniformly with respect to k, A and B. It follows that

∆ ksj (A × B) -(ℓ k + r k )µ(E j ) ---→ j→∞ 0,
uniformly with respect to k, A and B. This concludes the proof of [START_REF]Rank, rigidity of factors, and weak closure of measure-preserving Z n -actions[END_REF].

Equations ( 14) and [START_REF] Ryzhikov | Mixing, rank and minimal self-joining of actions with invariant measure[END_REF] give

hj -1 k=0 ν(A × B|G k j ) -∆ ksj (A × B)) ν(G k j ) ≤ hj -1 k=0 |a j k -b j k | ℓ k - k h j (ℓ k + r k ) + ε j ≤ h j hj -1 k=0 |a j k -b j k | + ε j
which goes to 0 as j → ∞ by [START_REF] Arthur Robinson | Rank-one Z d actions and directional entropy[END_REF].

Recalling [START_REF] Daniel | An example of a measure preserving map with minimal self-joinings, and applications[END_REF], we obtain

hj -1 k=0 ν(G k j )∆ ksj w ---→ j→∞ ν.
It remains to apply the Choice Lemma to conclude the proof of the theorem.

Z n -Rank-one action

We consider now an action of Z n (n ≥ 1). For k ∈ Z n , we denote by k(1), . . . , k(n) its coordinates. Definition 8.1. A Z n -action {T k } k∈Z n is of rank one if there exists a sequence (ξ j ) of partitions converging to the partition into points, where ξ j is of the form

ξ j = (T k E j ) k∈Rj , X \ k T k E j ,
and R j is a rectangular set of indices:

R j = {0, . . . , h j (1) -1} × • • • × {0, . . . , h j (n) -1}.
Note that the above definition corresponds to so-called R-rank one actions defined in [START_REF] Arthur Robinson | Rank-one Z d actions and directional entropy[END_REF] with the additional condition that the shapes in the sequence R be rectangles. The sequence (ξ j ) in the above definition being fixed, we define as for the rank-one flows the notions of columns and fat diagonals: For any k ∈ Z n , we set

C k j := r,ℓ∈Rj r-ℓ=k T r E j × T ℓ E j ,
and given 0 < δ < 1,

D δ j := k: i (hj (i)-|k(i)|)≥(1-δ) i hj (i) C k j .
Lemma 8.2. For any self-joining ν of the rank-one action {T k } k∈Z n , for any δ > 1 -1/2 n , we have

lim inf j→∞ ν(D δ j ) > 0.
Proof. We can find ε > 0, small enough such that

1 2 -ε n > 1 -δ. Let r ∈ Z n be such that ∀i, 1 2 -ε h j (i) < r(i) < 1 2 + ε h j (i).
Then, for any ℓ ∈ R j , we have for all i:

|r(i) -ℓ(i)| < 1 2 + ε h j (i). Hence i h j (i) -|r(i) -ℓ(i)| > (1 -δ) i h j (i), which means that for any ℓ ∈ R j , the column C r-ℓ j is contained in D δ j . It follows that   r: ∀i, |r(i)-hj(i)/2|<εhj (i) T r E j   ×   ℓ∈Rj T ℓ E j   ⊂ D δ j .
We then get

lim inf j→∞ ν(D δ j ) ≥ lim inf j→∞ µ   r: ∀i, |r(i)-hj(i)/2|<εhj (i) T r E j   = (2ε) n .
We can now state the analogue of Theorem 4.2 for Z n -rank-one action, which was first proved by A.A. Pavlova in [START_REF] Pavlova | Self-joinings and weak closure of rank one actions[END_REF]. Theorem 8.3. Let ν be an ergodic self-joining of the Z n -rank-one action {T k } k∈Z n . Then we can find a sequence (k j ) in Z n and some self-joining ν ′ such that ∆ kj w ---→ j→∞ 1

2 n ν + 1 -1 2 n ν ′ : For all measurable sets A, B µ(A ∩ T kj B) → 1 2 n ν(A × B) + 1 - 1 2 n ν ′ (A × B).
Proof. The proof follows the same lines as for Theorem 4.2. First note that Lemma 4.1 can be easily adapted to the Z n -situation. Hence, by Lemma 8.2, using a diagonal argument, we get the existence of (k j ) and (δ

j ) ց 1 -1 2 n with C kj j ⊂ D δj j such that ∆ kj • |C kj j w ---→ j→∞ ν.
To conclude, it remains to prove that lim inf ∆ kj (C kj j ) ≥ 1/2 n . To this aim, we count the number of pairs (r, ℓ) such that T r E j × T ℓ E j ⊂ C kj j . We can easily check that these are exactly the pairs (r, ℓ) such that, for all 1 ≤ i ≤ n, there exists m(i) ∈ {0, . . . , h j (i) -1 -|k j (i)|} with r(i), ℓ(i) = k j (i) + m(i), m(i) if k j (i) ≥ 0 m(i), -k j (i) + m(i) otherwise.

Hence ∆ kj (C kj j ) = i h j (i) -1 -|k j (i)| µ(E j ). Using the fact that C kj j ⊂ D δj j , we get the desired result.

When n ≥ 2, it is known that the Weak Closure Theorem fails (counterexamples have been given in [START_REF] Downarowicz | Weak closure theorem fails for Z 2 -actions[END_REF][START_REF] Downarowicz | Phenomena in rank-one Z 2 -actions[END_REF]). However, as a consequence of Theorem 8.3, we get the following: Corollary 8.4 (Partial Weak Closure Theorem for Z n -rank-one action). Let S be an automorphism commuting with the Z n -rank-one action {T k } k∈Z n . Then we can find a sequence (k j ) in Z n and some self-joining ν ′ such that

∆ kj ---→ j→∞ 1 2 n ∆ S + 1 - 1 2 n ν ′ .

Moreover, if S /

∈ {T k k ∈ Z n }, then {T k } k∈Z n is partially rigid: There exists a sequence (k ′ ℓ ) in Z n with |k ′ ℓ | → ∞ such that for all measurable sets A and B lim inf

ℓ→∞ µ A ∩ T k ′ ℓ B ≥ 1 2 2n µ(A ∩ B).
Proof. The first part is a direct application of Theorem 8.3 with ν = ∆ S . If moreover S / ∈ {T k k ∈ Z n }, then the sequence (k j ) of the theorem must satisfy |k j | → ∞. Let us enumerate the cylinder sets as {A 0 , A 1 , . . . , A ℓ , . . .}. Let (ε ℓ ) be a sequence of positive numbers decreasing to zero. For any ℓ, we can find a large enough integer j 1 (ℓ) such that for all cylinder sets A, B ∈ {A 0 , A 1 , . . . , A ℓ },

µ T k j 1 (ℓ) A ∩ SB ≥ 1 2 n -ε ℓ µ(SA ∩ SB) = 1 2 n -ε ℓ µ(A ∩ B).
Then, we can find a large enough integer j 2 (ℓ) with |j 2 (ℓ)| > 2|j 1 (ℓ)| such that for all cylinder sets A, B ∈ {A 0 , A 1 , . . . , A ℓ }, µ T k j 1 (ℓ) A ∩ T k j 2 (ℓ) B ≥ 1 2 n -ε ℓ µ(T k j 1 (ℓ) A ∩ SB).

It follows that for all ℓ ≥ 0 and all cylinder sets A, B ∈ {A 0 , A 1 , . . . , A ℓ },

µ A ∩ T k j 2 (ℓ) -k j 1 (ℓ) B ≥ 1 2 n -ε ℓ 2 µ(A ∩ B).
This proves the result announced in the corollary when A and B are cylinder sets with k ′ ℓ := k j2(ℓ) -k j1(ℓ) , and this extends in a standard way to all measurable sets.

The counterexample given in [START_REF] Downarowicz | Phenomena in rank-one Z 2 -actions[END_REF] also shows that the rigidity of factors is no more valid when n ≥ 2. Theorem 8.3 only ensures the partial rigidity of factors of Z n -rank-one actions. Proof. This is a direct application of Theorem 8.3 where ν is an ergodic component of the relatively independent joining above the factor F .

∆

  S , and ∆ kj sj (C kj j ) converges to some number d ≥ 1/2. Let Y kj j be the projection on the first coordinate of C kj j , that is

.Theorem 5 . 2 (

 52 kj j × X), and µ(Y kj j ) = ∆ kj sj (C kj j ) → d.This yields, for all measurable sets A, B,µ(A ∩ T kj sj B ∩ Y kj j ) → d µ(A ∩ SB).If there exist infinitely many j's such that k j ≥ 0, then along this subsequence, we have µ j -k j )/h j → d. A similar result holds along the subsequence of j's such that k j < 0, with Y d,+ j replaced by Y d,- j Weak Closure Theorem for rank-one flows). If the automorphism S commutes with the rank-one flow (T t ) t∈R , then there exists a sequence of integers (k j ) such that ∆ kj sj → ∆ S : For all measurable sets A, B, µ(A ∩ T kj sj B) → µ(A ∩ SB). Proof. We fix T and consider the set of real numbers d for which the conclusion in the statement of Lemma 5.1 holds. It is easy to show by a diagonal argument that this set is closed. Hence we consider its maximal element, which we still denote by d. (If d = 1, the theorem is proved.)

Corollary 8 . 5 (

 85 Partial rigidity of factors of Z n -rank-one action). Let F be a nontrivial factor of the Z n -rank-one action {T k } k∈Z n . Then there exists a sequence(k j ) in Z n with |k j | → ∞ such that, for all measurable sets A, B ∈ F lim inf µ(A ∩ T kj B) ≥ 1 2 n µ(A ∩ B).

This work is partially supported by the grant NSh 8508.2010.1.