ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC

arXiv:1406.2608v1 [math.DS] 10 Jun 2014

ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES E. H. EL ABDALAOUI Abstract. It is shown that the multiple ergodic averages of commuting invertible measure preserving transformations of a Lebesgue probability space converge almost everywhere provided that the maps are weakly mixing with an ergodic extra condition.

1. Introduction The purpose of this note is to give a partial affirmative answer to the well-known open problem of the pointwise convergence of the Furstenberg ergodic averages. This problem can be formulated as follows: let k ≥ 2, (X, B, µ, Ti )ki=1 be a finite family of dynamical systems where µ is a probability measure, Ti are commuting invertible measure preserving transformations and f1 , f2 , · · · , fk a finite family of bounded functions. Does the following averages N k 1 XY fi (Tin x) N n=1 i=1

convergence almost everywhere? The classical Birkhoff theorem correspond to the case k = 1. The case k = 2 with Ti = T pi , and pi ∈ N∗ for each i, is covered by Bourgain double ergodic theorem [11]. The L2 version of this problem has been intensively studied and the topics is nowadays very rich. These studies were originated in the seminal work of Furstenberg on Szemeredi’s theorem in [15]. Furstenberg-Kaztnelson-Ornstein in [16] proved that the L2 -norm convergence holds for Ti = T i and T weakly mixing. Twenty three years later, Host and Kra [19] and independently T. Ziegler [31] extended Furstenberg-Katznelson-Ornstein result by proving that for any transformation preserving measure, the L2 -norm convergence for Ti = T i holds . In 1984, J. Conze and E. Lesigne in [12] gives a positive answer for the case k = 2. Under some extra ergodicity assumptions, Conze-Lesigne result was extended to the case k = 3 by Zhang in [30] and for any k ≥ 2 by Frantzikinakis and Kra in [14]. Without these assumptions, this result was proved by T. Tao in [29]. Subsequently, T. Austin in [6] gives a joining alternative proof of Tao result, and recently, M. Walsh Date: June 11, 2014. 2010 Mathematics Subject Classification. Primary 37A05, 37A30, 37A40; Secondary 42A05, 42A55. Key words and phrases. ergodic theorems, non-singular maps, Furstenberg ergodic averages, Bourgain double recurrence theorem, Birkhoff theorem, Frobenius-Perron operator, Koopman operator, Lebesgue probability space, strictly ergodic topological model, joining. 1

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extended Tao result by proving that the L2 -norm convergence holds for the maps (Ti )ki=1 generate a nilpotent group [33]. This solve a Bergelson-Leibman conjecture stated in [10]. Therein, the authors produced a counter-examples of maps generated a solvable group for which the L2 -norm convergence doesn’t holds. For the pointwise convergence, partial results were obtained in [13] and [3] under some ergodic and spectral assumptions. In a very recent preprint [5], I. Assani obN serve that the action of the maps Ti induced a dynamical system (X k , ki=1 B, ν, φ) where φ(x1 , · · · , xk ) = (Ti xi )ki=1 is a non-singular map with respect to the probability measure ν given by 1X 1 ν(A1 × A2 × · · · Ak ) = µ (φ−n (A1 × A2 × · · · Ak )), |n| ∆ 3 2 n∈Z with µ∆ is a diagonal probability measure on X k define on the rectangle A1 × A2 × · · · × Ak by µ∆ (A1 × A2 × · · · Ak ) = µ(A1 ∩ A2 ∩ · · · Ak ). It is easy to see that the Radon-Nikodym derivative of the pushforward measure of ν under φ verify dν ◦ φ 1 ≤ ≤ 2. 2 dν We associate to φ the Koopman operator Uφ defined by Uφ (f ) = f ◦ φ, where f is a measurable function on X k . Since Uφ maps L∞ on L∞ and φ is non-singular, the adjoint operator Uφ∗ acting on L1 can be defined by the relation Z Z Uφ (f ).gdν = f .Uφ∗ (g)dν, for any f ∈ L∞ and g ∈ L1 . For simplicity of notation, we write φ∗ instead of Uφ∗ and φ instead of Uφ when no confusion can arise. φ∗ is often referred to as the Perron-Frobenius operator or transfer operator associated with φ. The basic properties of φ∗ can be found in [21] and [1]. It is well known that the pointwise ergodic theorem for φ can be characterized by φ∗ . Indeed, Y. Ito established that the validity of the L1 -mean theorem for φ∗ implies the validity of the pointwise ergodic theorem for φ from L1 to L1 [20]. Moreover, the subject has been intensively studied by many authors ( Ryll-Nardzewski [26], Assani [2], Assani-Wo´s [4], Ortega Salvador [23], R. Sato [28], [27]). Here, we will use and adapt the Ryll-Nardzewski approach [26]. Notice that the L2 -norm convergence implies that for any k-uplet of Borel set (Ai )ki=1 , we have k k N  Y  Y  1 X Ai , = µF Ai µ∆ φ−n N −→+∞ N n=1 i=1 i=1

lim

where µF is an invariant probability measure under the actions of (Ti ). Following T. Austin, µF is called a Furstenberg self-joining of (Ti ) [6],[7], [8]. Therein, T. Austin stated the multiple recurrence theorem in the following form ∀(Ai )ki=1 ∈ B k , µF (A1 × A2 × · · · × Ak ) = 0 =⇒ µ∆ (A1 × A2 × · · · × Ak ) = 0,

ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES

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We further point out that if the maps Ti = T i and T is a weakly mixing map, then µF = ⊗ki=1 µ, that is, µF is singular with respect to µ∆ and this is not incompatible with the multiple recurrence theorem, since the absolutely continuity holds only one the sub-algebra generated by the rectangle Borel sets. The proof of T. Austin is based on the description of µF in terms of the measure µ and various partially-invariant factors of B. This is done using a suitable extension of the originally-given system (X, B, µ, Ti ). Here, we will further need the method used by Hansel-Raoult in [18] and its generalization to Zd action obtain by B. Weiss in [32]. Therein, the authors gives a generalization of the Jewett theorem using the Stone representation theorem combined with some combinatorial arguments. The proof given by Weiss yields a ”uniform” extension of Rohklin towers lemma. This allows us, under a suitable assumption, to produce a strictly ergodic topological model for which we are able to show under some condition that the Furstenberg averages converge almost everywhere. In summary, our proof is essentially based on two kind of arguments. On one hand, the Assani observation [5] combined with Ryll-Nardzewski theorem [26] and some basic fact from the non-singular ergodic theory, and on the other hand on the C-method introduced by Austin [6] combined with Hansel-Raoult-Weiss procedure [18]. Let us mention that it is obvious that our result is not yet a generalization of classical ergodic Birkhoff theorem and cannot be considered as a generalization of the Bourgain double ergodic theorem. The paper is organized as follows In section 2, we state our main result and we recall the main ingredients need it for the proof. In section 3, we establish under our assumptions the L2 -norm convergence. In section 4, we recall the Stone representation theorem and Hansel-RaoultWeiss procedure used to produce a strictly ergodic topological model. Finally, In section 5, we gives a proof of our main result.

2. Main result Let (X, B, µ) be a Lebesgue probability space, that is, X is a Polish space (i.e. metrizable separable and complete), whose Borel σ-algebra B is complete with respect to the probability measure µ on X. The notion of Lebesgue space is due to Rokhlin [25], and it is well known [22] that a Lebesgue probability space is isomorphic (mod 0) to ordinary Lebesgue space ([0, 1), C, λ) possibly together with countably S many atoms, that is, there are x0 , x1 , · · · , ∈ X, X0 ⊂ X, Y0 ⊂ [0, 1), and φ : X0 {xi } −→ Y0 , which S is invertible such that the pushforward measure of µ under φ is λ with µ(X0 {xi }) = 1 = λ(Y0 ) = 1 . Here, we will deal only with non-atomic Lebesgue space. A dynamical system is given by (X, B, µ, T ) where (X, B, µ) is a Lebesgue space and T is an invertible bi-measurable transformation which preserves the probability measure µ. In this context, we state our main result as follows

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Theorem 2.1. Let k ∈ N∗ and (X, B, µ, Ti )ki=1 be a finite family of dynamical systems where (X, B, µ) is Lebesgue probability space, and assume that T1 , T2 , · · · , Tk are commuting weakly mixing transformations on X such that for any i 6= j, the map Ti ◦ Tj−1 is ergodic. Then, for every fi ∈ L∞ (µ), i = 1, · · · , k, the averages N k 1 XY fi (Tin x) N n=1 i=1

converge almost everywhere to

k Z Y

fj dµ.

j=1

The proof will follows from the Ryll-Nardzewski theorem [26] combined with the machinery of C-systems introduced by T. Austin. We remind that this machinery allows T. Austin to obtain a joining proof of the Tao theorem on the L2 -norm convergence of the Furstenberg ergodic averages. We will use the C-systems machinery in the section 3. Here, we recall the Ryll-Nardzewski theorem [26] Theorem 2.2 (Ryll-Nardzewski [26]). Let (X, B, ν) be a probability measure and φ an invertible map on X such that the pushforward measure of ν under φ is absolutely continuous with respect to ν. Then the following conditions are equivalent. (a) The operator T f = f ◦ φ satisfies the pointwise ergodic theorem from L1 (ν) to L1 (ν), that is, for any function f ∈ L1 (ν) there is a function g ∈ L1 (ν) such that N 1 X n lim (T f )(x) = g(x) ν.a.e. N −→+∞ N n=1 (b) There is a constant K such that N 1 X ν(φ−n A) ≤ Kν(A), N −→+∞ N n=1

lim sup

for each Borel set A. The condition (b) is called Hartman condition, and the proof of Theorem 2.2 is essentially based on the notion of Mazur-Banach limit. Precisely, using the MazurBanach limit, Ryll-Nardzewski proved that there is a finite measure ρ such that, for any Borel set A, we have (i) 0 ≤ ρ(A) ≤ Kν(A); (ii) if A = φ−1 A then ρ(A) = ν(A); (iii) ρ(φ−1 A) = ρ(A). This insure that ρ is φ-invariant and we can apply the Birkhoff ergodic theorem to conclude. For more details and the rest of the proof, we refer the reader to the Ryll-Nardzewski paper [26]. Let us further point out that therein, Ryll-Nardzewski produce a counter-example for which the pointwise ergodic theorem in L1 (ν) doesn’t imply the ergodic theorem in L1 (ν). We remind that the ergodic theorem in L1 (ν) holds, if for any f ∈ L1 (ν), there is a function g ∈ L1 (ν) such that N

1 X

f (φn (x)) − g(x) −−−−−→ 0.

N n=1 1 N →+∞

ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES

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We notice that under our assumption, and following Ryll-Nardzewski ideas, one may choose a non-decreasing sequence of Borel set (Am ) such that lim Am = X. Hence, for any Borel set, the sequence νm (A) defined by νm (A) =

N −1 1 X ν(φn (A) ∩ Am ) N n=0

is a bounded sequence. But, it is an easy exercise to see that we can extend the operator lim on the space of real bounded sequences to obtain a bounded operator on ℓ∞ by Hahn-Banach theorem. We denote such operator by MBlim. This allows us to define a sequence of probability measure νm on X given by νm (A) = M Blim

−1   1 NX ν(φn (A) ∩ Am ) . N n=0

It follows that if A = φ(A) then, for any m ∈ N, we have νm (A) = ν(A ∩ Am ). We further have, for any Borel set A, νm (A) ≤ Kν(A), and νm (φ(A))

−1   1 NX ν(φn+1 (A) ∩ Am ) = MBlim N n=0 −1  1 NX ν(A ∩ Am ) ν(φN (A) ∩ Am )  = MBlim + ν(φn (A) ∩ Am ) − N n=0 N N −1  1 NX  = MBlim ν(φn (A) ∩ Am ) N n=0

= νm (A), since

 ν(A ∩ A ) ν(φN (A) ∩ A )  m m = 0. + MBlim − N N That is, νm is invariant under φ. Now, the sequence (νm (A)) is a bounded nondecreasing sequence. Therefore, we can put ρ(A) =

lim

m−→+∞

νm (A),

and it is easy to check that (i),(ii) and (iii) holds. Remark 2.3. Ryll-Nardzewski proved Theorem 2.2 in the more general setting where ν is a σ-finite measure. 3. L2 -norm convergence of Furstenberg averages We start by proving the following Theorem 3.1. Let k ∈ N∗ and (X, B, µ, Ti )ki=1 be a finite family of dynamical systems where µ is a probability measure space, and T1 , T2 , · · · , Tk are commuting

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weakly mixing transformations on X such that for any i 6= j, the map Ti ◦ Tj−1 is ergodic. Then, for every fi ∈ L∞ (µ), i = 1, · · · , k, the averages k N 1 XY fi (Tin x) N n=1 i=1

converge in L2 (X, µ) to

k Z Y

fj dµ.

j=1

The proof is based on van der Corput trick and the C-sated systems method introduced by T. Austin. In our case the C-sated systems are trivial, and the van der Corput lemma can be stated in the following form Lemma 3.2 (van der Corput [9]). Let (un ) be a bounded sequence in a Hilbert space. Then, −1 H−1 N

2 X

1 NX 1 X



un ≤ lim sup lim sup hun+h , un i . lim sup N n=0 H n=0 h=0

As a simple consequence it follows that if

H−1 N 1 XX hun+h , un i = 0, H−→+∞ N −→+∞ H n=0

lim

lim

h=0

then

−1

1 NX

un −−−−−→ 0.

N →+∞ N n=0

We remind that if (Ti )ki=1 are a commuting maps on X then the associated Csystems are the dynamical systems (X, C, µ, Ti ), i = 1, · · · , k, for which C = IT1 ∨ IT2 T −1 ∨ IT2 T −1 · · · ∨ ITk T −1 , 1

1

1

where, for any transformation S, IS is the factor σ-algebra of S-invariant Borel sets, that is, n o IS = A : µ(A∆S −1 A) = 0 . Notice that under our assumption the σ-algebra C is trivial. The key notion in the Austin proof is the notion of C-sated system defined as follows [7]

Definition 3.3. Let k be a integer such that k ≥ 2. The system (X, B, (Ti )ki=1 ), k ≥ 2 is C-sated if any joining λ of X with any C-system Y is relatively independent over the largest C-factor XC of X, that is, for any bounded measurable function f on X, we have Eλ (f (x)|Y ) = Eλ (EX (f (x)|XC )|Y ), Where Eλ (.|•) is a conditional expectation operator. We denote the expectation operator by Z E(f ) = f dµ, f ∈ L2 (X). Following this setting, the Tao L2 -norm convergence theorem can be stated as follows

ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES

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Theorem 3.4 (Austin-Tao [6],[29]). Let (X, B, Ti ), i = 1, · · · , k, be a C-sated system. Then, for any f1 , f2 , · · · , fk functions in L∞ , N

1 X

f1 ◦ T1 · · · fk ◦ Tk −−−−−→ 0. EX (f1 |XC ) = 0 =⇒ N n=1 2 N →+∞

Now, we are able to give the proof of Theorem 3.1.

Proof of Theorem 3.1. We use induction on k to prove the theorem. The statement is obvious for k = 1. For the case k = 2, by our assumption the C-system is trivial and we can write N

1 X

f1 (T1n x)f2 (T2n x) − E(f1 )E(f2 )

N n=1 2 N N

1 X

1 X

≤ (f1 − E(f1 ))(T1n x)f2 (T2n x) + f2 (T2n x)E(f1 ) − E(f1 )E(f2 ) N n=1 N n=1 2

N N

1 X

1 X



(f1 − E(f1 ))(T1n x)f2 (T2n x) + f2 (T2n x)E(f1 ) − E(f1 )E(f2 ) ≤ N n=1 N n=1 2 2

Hence, by the L2 -norm convergence and the von Neumann ergodic theorem, we have N

1 X

f1 (T1n x)f2 (T2n x) − E(f1 )E(f2 ) −−−−−→ 0.

N n=1 2 N →+∞

In the general case, applying the van der Corput Lemma 3.2, the L2 convergence of Furstenberg average of order k, can be reduced in the class of C-systems to the case of k − 1 commuting maps. In this case, the L2 convergence gives N

1 X

E(f1 ) = 0 =⇒ f1 (T1n x)f2 (T2n x) · · · fk (Tkn x) −−−−−→ 0. N n=1 2 N →+∞

Therefore, suppose that the result holds for some integer k ≥ 1, and assume that (X, µ, T1 , · · · , Tl ) is a system of order k + 1. Then, N

1 X

n f1 (T1n x)f2 (T2n x) · · · fk+1 (Tk+1 x) − E(f1 )E(f2 ) · · · E(fk+1 )

N n=1 2

N

1 X

n = (f1 − E(f1 ))(T1n x)f2 (T2n x) · · · fk+1 (Tk+1 x) + N n=1

E(f1 )

N

1 X

n f2 (T2n x) · · · fk+1 (Tk+1 x) − E(f1 )E(f2 ) · · · E(fk+1 ) N n=1 2

N

1 X

n ≤ (f1 − E(f1 ))(T1n x)f2 (T2n x) · · · fk+1 (Tk+1 x) N n=1 2

N

1 X

n + |E(f1 )| f2 (T2n x) · · · fk+1 (Tk+1 x) − E(f2 ) · · · E(fk+1 ) −−−−−→ 0. N n=1 2 N →+∞

This complete the proof of the theorem.



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As a consequence we deduce the following result Corollary 3.5. Let k ∈ N∗ and (X, B, µ, Ti )ki=1 be a finite family of dynamical systems where µ is a probability measure space, assume that Ti , i = 1 · · · k are commuting weakly mixing transformations on X. Then, for every Ai ∈ B, i = 1, · · · , k, the averages N 1 X µ∆ (T1−n A1 × T2−n A2 × · · · × Tk−n Ak ) N n=1

converge to µ(A1 )µ(A2 ) · · · µ(Ak ). Form this we deduce the following crucial lemma Lemma 3.6. Let k ∈ N∗ and (X, B, µ, Ti )ki=1 be a finite family of dynamical systems where µ is a probability measure space, and assume that T1 , T2 , · · · , Tk are commuting weakly mixing transformations on X. Then, for every fi ∈ L∞ (µ), i = 1, · · · , k, the averages N Z k 1 X Y fi (Tin x)dν N n=1 i=1 converge to

k Y

µ(fi ).

i=1

Proof. It is suffice to prove the lemma for any finite family of Borel set (Ai )ki=1 . Indeed, by Corollary 3.5, we have N   1 X µ∆ φ−n (A) N n=1

converge to

k O

µ(A), where A = A1 × A2 · · · × Ak . This gives also that the

i=1

convergence holds when replacing µ∆ by µφj ∆ , for any j ∈ Z, where µφj ∆ is the pushforward measure of µ∆ under φj . Therefore, for any M ∈ N, the convergence holds for νM where 1 X 1 νM (A) = µφn ∆ (A). 3 2|n| |n|≤M

Now, let ε > 0 then there is a positive integer M0 such that for any M ≥ M0 , we have |νM (A) − ν(A)| ≤ ε, ∀A ∈ Ak . Hence, for any N ∈ N, N N 1 X
1 X −n ν (φ (A)) ν φ−n (A) < ε. − M N n=1 N n=1

By letting N and M goes to infinity we obtain k O i=1

N N k O 1 X 1 X ν(φ−n (A)) ≤ lim sup ν(φ−n (A)) ≤ µ(A)+ε. N −→+∞ N N −→+∞ N n=1 n=1 i=1

µ(A)−ε ≤ lim inf

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Since ε was chosen arbitrarily, we conclude that k N O

1 X −n µ A . ν φ (A) −−−−−→ N →+∞ N n=1 i=1

This proves the lemma.



4. Stone representation theorem and Furstenberg averages Let us consider a dynamical system (X, B, µ, T ). Then, there exists a countable algebra A dense in B for the pseudo-metric d(A, B) = µ(A∆B), A and B in A. We further have that A separates the points of X, that is, for each x, y ∈ X with x 6= y, there is A ∈ A such that either x ∈ A, y 6∈ A or else y ∈ A, x 6∈ A. Hence, by the Stone representation theorem, we associate to A a Stone algebra Ab on the b of all ultrafilter on X such that for any A ∈ A, A b = {UA ∈ X/A b set X ∈ UA }. b b b Consequently A = {A, A ∈ A} is algebra of subsets of X, which is isomorphic to A. Ab is called the Stone algebra. Assuming that T is ergodic, Hansel and Raoult proved in [18] that there is a dense and invariant countable algebra AT such that for any A of AT we have N

1 X

1A (T n x) − µ(A) −−−−−→ 0,

N n=1 ∞ N →+∞

that is, A is a uniform ergodic set.

Their result was extended to the ergodic Zd -action by B. Weiss in [32]. Indeed, B. Weiss proved that if T = (Ti )di=1 is a generator of ergodic Zd -action then there is a dense and T -invariant countable algebra AT such that all elements A in AT are uniform ergodic sets, that is,

1 X i

1A (T x) − µ(A) −−−−−→ 0,

|Rn | ∞ n→+∞ i∈Rn

where Rn is the square {i ∈ Zd : |i|∞ ≤ n}.

Applying a Stone representation theorem to Hansel-Raoult-Weiss algebra, and b be equipped with the topology which has Ab as a base of clopen sets. It letting X b is metrizable space (since Ab is countable), compact (by the standard follows that X b is called the Stone space of A. ultrafilter lemma) and totaly disconnected. X Furthermore, the probability µ induces a mapping µ′ from Ab into [0,1], such that (1) for any A ∈ Ab \ {∅}, µ′ (A) > 0, b and µ′ (X) b = 1. (2) µ′ is σ-additive on A,

Hence, by Carath´eodory’s extension theorem, µ′ has an unique extension as a probb We further have that ability measure µ b on the Borel σ-algebra Bb generate by A. b for every non-empty open subset O ⊂ X, µ b(O) > 0. This gives in particular that b ∈ B, b if µ b = 1 then A b is a dense in X. b for any A b(A) The Zd -action with generators T induces a Zd -action with generators Tb = (Tbi )di=1 such that, by construction and due to intrinsic properties of Lebesgue space, the

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E. H. EL ABDALAOUI

b We further two action are isomorphic and for each i, Tbi is a homeomorphism on X. b the sequence of continuous function have that for any A, 1 X 1Ab ◦ Tbn |Rn | n∈RN

b Then, it follows by Birkhoff’s ergodic theorem that converges uniformly to µ b(A). d the Z -action with generators Tb = (Tbi )di=1 is strictly ergodic, that is, µ b is a unique b probability measure Tb-invariant with µ b(O) > 0 for every nonempty open set O ⊂ X. b B, b Tb) is called the Stone-Jewett-Weiss topological model. The topological model (X, For the case d = 1, we call it the Stone-Jewett-Hansel-Raoult topological model. Finally, let us denote by B(A) the Banach space of all scalar-valued functions that are uniform limits of sequences of A-measurable step functions, equipped with b the Banach the supremum norm. Then, B(A) is isometrically isomorphic to C(X), b space of all continuous functions on the Stone space X. 5. Proof of the main result

b B, bµ Let (X, b, Tb = (Tbi )ki=1 ) be the Stone-Weiss topological model associated to (X, B, µ, T ), and for any A1 × A2 × · · · Ak ∈ B k , put 1X 1 νb(A1 × A2 × · · · Ak ) = µ b (Tbn (A1 × A2 × · · · Ak )), |n| ∆ 3 2 n∈Z

n b k associated to µ where Tbn = (Tbi )ki=1 and µ b∆ is the diagonal measure on X b.

b Tb), where b B, b λ, From this, let us consider the non-singular dynamical system (X, Nk b b = νb + i=1 µ . λ 2 Hence, by Lemma 3.6, for any A1 × A2 × · · · Ak ∈ Abk , we have Z

N −1   1 X b −−−−−→ ⊗k µ . 1A1 ×A2 ×···Ak ◦ Tbn dλ b A × A × · · · A 1 2 k i=1 N →+∞ N n=0

b k , we get Therefore, for any f1 ⊗ f2 ⊗ · · · fk ∈ C(X) Z Z N −1 1 X b −−−−−→ f1 ⊗ f2 ⊗ · · · ⊗ fk d(⊗k µ f1 ⊗ f2 ⊗ · · · ⊗ fk ◦ Tbn dλ i=1 b). N →+∞ N n=0

bk This gives, by the standard argument, that for any continuous functions f on X Z Z N −1 d(⊗ki=1 µ b) b 1 X bn b T f dλ −−−−−→ f dλ. b N →+∞ N n=0 dλ

b λ), b Applying again a standard density argument, it follows that for any f ∈ L∞ (X, we have Z Z N −1 d(⊗ki=1 µ b) b 1 X bn b dλ. T f dλ −−−−−→ f b N →+∞ N n=0 dλ

ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES

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b k , we get Hence, for any Borel A in X

N −1 1 X b bn b b(A) ≤ 2λ(A). λ(T A) −−−−−→ ⊗ki=1 µ N →+∞ N n=0

b Tb). b B, b λ, Consequently, the Hartman condition holds for our dynamical system (X, b Therefore, by Ryll-Nardzewski theorem (Theorem 2.2), T satisfy the pointwise ergodic theorem. We thus get that the pointwise ergodic theorem holds for νb, which yields that the pointwise ergodic theorem holds for µ b∆ since µ b∆ is absolutely continuous with respect to νb.

Remark 5.1. Notice that our proof yields that Tb satisfy the pointwise ergodic theorem with respect to the product measure, which follows also by the Birkhoff ergodic theorem. Moreover, our last density argument is based essentially on the classical Lusin theorem [24, p.53]. We further notice that our proof gives more, namely if b Hence, fi ∈ Lk (µ), then by generalized H¨ older inequality, f1 ⊗ f2 ⊗ · · ·⊗ fk ∈ L1 (λ). the pointwise ergodic theorem holds. In view of this last observation, it is natural to ask Question. If (X, A, µ, Ti )ki=1 , k ≥ 2 is a finite family of Lebesgue dynamical systems where µ is a probability measure space, and if (Ti ) are commuting weakly mixing transformations on X. Does, for any p ≥ 1 and for every functions (fi )ki=1 ∈ (Lp (X))k , the following averages N k 1 XY fi (Tin x) N n=1 i=1

converge a.e.? Notice that the case k = 1 follows from Dunford-Schwartz theorem [17, p.27] and the case k = 2 with Ti = T pi , pi ∈ N∗ follows from the Bourgain double ergodic theorem and the following observation Fact. Assume that the pointwise convergence of Furstenberg average holds for L∞ functions then the pointwise convergence holds for Lp -functions by L∞ approximations combined with the Marcinkiewicz interpolation theorem [17, p.25] and the maximal ergodic inequality. Acknowledgment 1. I would like to express my warmest thanks to Professor XiangDong Ye for his invitation to Hefei University of Technology and for bringing to my attention the problem of the pointwise convergence of Furstenberg average and the arXiv-paper of Professor Assani [5]. My thanks go also to Professor Idris Assani for his invitation to the Ergodic Workshop in Chapel Hill and for the conversations on Bourgain double ergodic theorem and his recent work with Ryo Moore∗. Finally, my posthumous thanks to Professor Antoine Brunel who introduce me to the work of R. Sato. ∗

A joint work on the extension of their result is in preparation.

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References [1] J. Aaronson, An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, 50. American Mathematical Society, Providence, RI, 1997. [2] I. Assani, Quelques r´ esultats sur les op´ erateurs positifs ` a moyennes born´ ees dans Lp , (French) [Some results on Lp mean bounded positive operators] Ann. Sci. Univ. Clermont-Ferrand II Probab. Appl., No. 3 (1985), 65-72. [3] I. Assani, Multiple recurrence and almost sure convergence for weakly mixing dynamical systems, Israel J. Math., 103 (1998), 111-124. [4] I. Assani & J. Wo´s, J. An equivalent measure for some nonsingular transformations and application, Studia Math., 97 (1990), no. 1, 1-12. [5] I. Assani, A.e. Multiple recurrence for weakly mixing commuting actions, available on arXiv:1312.5270. [6] T. Austin, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems, 30 (2010), 321-338. [7] T. Austin, Pleasant extensions subject to some algebraic constraints, and applications, Preliminary notes available on arXiv:0905.0518, 2009. [8] T. Austin, Topics in Ergodic Theory, Notes 8: Multiple Recurrence III: structure of the Furstenberg self-joining, notes available on http://www.math.brown.edu/∼timaustin. [9] V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349. [10] V. Bergelson and A. Leibman, A nilpotent Roth theorem, Invent. Math., 147 (2002), 429-470. [11] J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math., 404 (1990), 140-161. [12] J.-P. Conze and E. Lesigne, Th´ eor` emes ergodiques pour des mesures diagonales, Bull. Soc. Math. France, 112 (1984), 143-175. [13] J-M. Derrien and E. Lesigne, Un th´ eor` eme ergodique polynomial ponctuel pour les endomorphismes exacts et les K-systemes, Ann. Inst. H. Poincare Probab. Statist., 32 (1996), no. 6, 765-778. [14] N. Frantzikinakis and B. Kra, Convergence of multiple ergodic averages for some commuting transformations, Ergodic Theory Dynam. Systems, 25 (2005), 799-809. [15] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemer´ edi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256. [16] H. Furstenberg, Y. Katznelson, D. Ornstein, The ergodic theoretical proof of Szemer´ edi’s theorem, Bull. Amer. Math. Soc., (N.S.) 7 (1982), no. 3, 527-552. [17] A. Garcia, Topics in almost everywhere convergence, Markham Publishing Company, Chicago, 1970. [18] G. Hansel, J. P. Raoult, Ergodicity, uniformity and unique ergodicity, Indiana Univ. Math. J., 23 (1973/74), 221-237. [19] B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math., 161 (2005), 397-488. [20] Y. Ito, Uniform integrability and the pointwise ergodic theorem, Proc. Amer. Math. Soc., 16 (1965), 222-227. [21] U. Krengel, Ergodic theorems, With a supplement by Antoine Brunel, de Gruyter Studies in Mathematics, 6. Walter de Gruyter & Co., Berlin, 1985. [22] K. Petersen, Ergodic theory. Cambridge Studies in Advanced Mathematics 2., Cambridge University Press, Cambridge, 1983. [23] P. Ortega Salvador, Weights for the ergodic maximal operator and a.e. convergence of the ergodic averages for functions in Lorentz spaces, Tohoku Math. J., (2) 45 (1993), no. 3, 437-446. [24] W. Rudin, Real and complex analysis, Third edition. McGraw-Hill Book Co., New York, 1987. [25] V. A. Rohklin, On the fundamental ideas of measure theory, Amer. Math. Soc. Translation, (1952), 55 pp. [26] C. Ryll-Nardzewski, On the ergodic theorems, I, Generalized ergodic theorems, Studia Math., 12, (1951), 65-73. [27] R. Sato, Pointwise ergodic theorems in Lorentz spaces L(p, q) for null preserving transformations, Studia Math., 114 (3) (1995), 227-236.

ON THE POINTWISE CONVERGENCE OF MULTIPLE ERGODIC AVERAGES

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[28] Sato, Ryotaro Pointwise ergodic theorems for functions in Lorentz spaces Lpq with p 6= ∞, Studia Math., 109 (1994), no. 2, 209-216. [29] T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. 1 PN ) n=1 f1 (Rn x)f2 (S n x)f3 (T n x), Monatsh. [30] Q. Zhang, On convergence of the averages ( N Math., 122 (1996), 275-300. [31] T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97. [32] B. Weiss, Strictly ergodic models for dynamical systems, Bull. Amer. Math. Soc., (N.S.) 13 (1985), no. 2, 143-146. [33] M. Walsh, Norm convergence of nilpotent ergodic averages, Annals of Math, Volume 175 (2012), Issue 3, 1667-1688. Normandie Universit´ e, Universit´ e de Rouen-Math´ ematiques, Labo. de Maths Raphael e, BP.12 76801 Saint Etienne du Rouvray SALEM UMR 60 85 CNRS Avenue de l’Universit´ - France E-mail address: [email protected]