JOINING PRIMENESS AND DISJOINTNESS FROM INFINITELY DIVISIBLE SYSTEMS

MARIUSZ LEMAŃCZYK, FRANÇOIS PARREAU, AND EMMANUEL ROY

Abstract. We show that ergodic dynamical systems generated by infinitely divisible stationary processes are disjoint in the sense of Furstenberg with distally simple systems and systems whose maximal spectral type is singular with respect to the convolution of any two continuous measures.

Introduction

The notion of disjointness of dynamical systems introduced by Furstenberg [9] in 1967 leads to a natural program of identifying disjoint classes of dynamical systems. We refer to the recent monograph [11] and also to [34] for the joining theory of dynamical systems. In this note we introduce the notion of joining prime (JP) system. Such a system T has the property that every ergodic joining λ of T with the Cartesian product S_{1} × … × S_{k} of weakly mixing systems is the direct product of a joining of T and some S_{i} with the remaining factors S_{j}. In particular, a weakly mixing JP system does not admit a Cartesian product representation, and it is natural to expect that it will be disjoint from classes of systems which are sufficiently divisible in some sense. Indeed, we will show that the JP class is disjoint from a class of systems which includes systems of probabilistic origin, namely those arising from infinitely divisible (ID) stationary processes, and also from infinitely divisible systems in the ergodic theoretical sense (to avoid confusion, we will refer to the latter as ergodically ID systems).

The JP class includes two recently considered classes of automorphisms: (A) some generalizations of simple systems (2-quasi-simple systems), considered by Ryzhikov and Thouvenot in [33], and later in [16] (distally simple systems); (B) systems whose maximal spectral type is singular with respect to the convolution of any two continuous measures on the circle.

Simple systems were introduced by Veech [36] and del Junco and Rudolph [18]. We recall here that this class includes some horocycle flows [27], and in fact all horocycle flows are factors of simple systems [34]. Another subclass of simple systems comes from the finite rank theory (see [17], [19]). In [15], simple systems were shown to be disjoint from Gaussian systems; earlier, in [34], Thouvenot showed disjointness of Gaussian systems from systems with the minimal self-joining property. This disjointness result was generalized in two directions. In [34], Thouvenot introduced ergodically ID systems, extending some properties of Gaussian systems: an ergodically ID system possesses an infinite tree of splitting factors, in which for every path the intersection of the decreasing family of factors is trivial.

2000 Mathematics Subject Classification. Primary 37A05; secondary 37A10, 37A30, 37A50.

Key words and phrases. Joinings, disjointness, infinite divisibility, distal simplicity, spectral singularity and convolutions.

The research of the first author is partially supported by Polish MNiSzW grant N N201 384834, Marie Curie “Transfer of Knowledge” EU program – project MTKD-CT-2005-030042 (TODEQ), the research of the first and third authors are partially supported by MSRI (Berkeley), program “Ergodic Theory and Additive Combinatorics”.

Besides a larger class of systems was recently considered in [33], namely those systems for which all ergodic self-joining different from the product measure are relatively isometric over the marginal factors. Disjointness of such systems from some subclasses of ergodically ID systems including Gaussian systems was then proved in [33] and [35]. If we replace relatively isometric by relatively distal and we assume additionally the so called PID property (each pairwise independent selfjoining of such a system is independent), then we get the class of so-called distally simple systems which are shown to be disjoint with all ergodically ID systems in [16]. All flows with the Ratner property [27] are distally simple. Also, partially mixing finite rank transformations lie in the class of distally simple systems [22], [34]. Moreover, recently in [8], it is proved that some smooth flows on surfaces satisfy the Ratner property and hence are distally simple.

On the other hand, the JP class includes those systems whose maximal spectral type is singular with respect to the convolution of any two continuous measures, property which we refer to as the CS property. A stronger property is that the Gaussian automorphism constructed from the (reduced) maximal spectral type has a simple spectrum (simple convolutions property). The famous Chacon automorphism has both the minimal self-joinings property [17] and the simple convolution property [1]. Ageev [2] and Ryzhikov [32] recently showed that many other rank one transformations enjoy the simple convolution property, including typical automorphisms and mixing examples. These properties are also related to the property that the convolution powers of the (reduced) maximal spectral type are mutually singular (see [20], and also [14] and [34] for usefulness of such a property in ergodic theory).

The extension of the definitions and results to flows and actions of other locally compact abelian groups is straightforward. In Section 6, we will show that when T = (T_{t})_{t∈ℝ} is a measurable flow and in the weak closure of the unitary group coming from time t automorphisms we find an operator of the form f(T ), where f : ℝ → ℂ is analytic and different from a multiple of a character then the flow has the CS property, and hence it has also the JP property. The assumption turns out to be satisfied for many natural examples of flows (see Section 6). In [23], even the simple convolutions property is shown to hold for a large class of smooth flows on the torus.

We show that systems with the JP property are disjoint with ergodically ID systems, which extends the result of [16] for distally simple systems. In particular a consequence is that a typical dynamical system is disjoint from every ergodically ID system. In fact we give a general criterion (Proposition 5) for such a disjointness which is apparently satisfied for a larger class than ergodically ID systems. The criterion applies to systems which can be embedded in symmetric factors of infinitely many Cartesian products, and hence is especially adapted to systems given by stationary ID processes (and in particular to Poisson suspension systems). However, we also give simple arguments showing that systems generated by stationary ID processes, like Poisson suspensions, are factors of ergodically ID systems.

Most results of this paper have been obtained during the visit of the first author at the University Paris 13 in Fall 2006.

1. Basic definitions and notation

We will firstly consider automorphisms acting on standard probability Borel spaces and their ergodicity will be tacitly assumed. Joinings between automorphisms T and S are meant as measures on the corresponding product space having the coordinate projections as the original measures and invariant under T ×S. The set of joinings between these automorphisms is denoted by J(T, S) (with an obvious change of notation when more automorphisms are involved). The subset of ergodic joinings will be denoted by J^{e}(T, S). Following [9], systems T and S between which the only possible joining is the product measure are called disjoint; we write then T ⊥ S. If T and S act on (X, B, μ) and (Y, C, ν) respectively and if λ ∈ J(T, S) then it determines a Markov operator Φ_{λ} : L^{2}(X, B, μ) → L^{2}(Y, C, ν),

(1)  ∫_{Y} Φ_{λ}(f)g dν = ∫_{X×Y} f ⊗ g dλ, Φ_{λ} ∘ T = S ∘ Φ_{λ}

(we still denote by T and S the corresponding unitary operators on the L^{2} spaces). Φ_{λ} is doubly stochastic: Φ_{λ}(1) = Φ_{λ}^{∗}(1) = 1, and the image via Φ_{λ} of a non-negative function is non-negative. Notice that Φ_{λ}(f) = E^{λ}(f ⊗ 1|Y), where by E^{λ}(·|Y) we denote the conditional expectation with respect to the σ-algebra {∅, X} ⊗ C. In fact (1) establishes a 1-1 correspondence between joinings and Markov operators, see e.g. [11], Chapter 6 for more details.

Let R be an ergodic automorphism of a standard probability Borel space (Z, D, ρ). Following [18], R is said to have the pairwise independence property (PID) if every (finite or infinite) pairwise independent self-joining of R is equal to the product measure. Ryzhikov in [31] proved the following important lemma.

Lemma 1. Assume that R has the PID property and let T, S be automorphisms. Assume that λ ∈ J(T, S, R) is pairwise independent. Then it is the product measure.

For the definition of distality and relative distality we refer the reader to [10]. An automorphism R is called 2-fold distally simple (see [16]) if each λ ∈ J^{e}(R, R) is either the product measure, or the extension (R × R, λ) is relatively distal over the marginal factors R. A 2-fold distally system which has the PID property is called distally simple.

As proved in [16], each 2-fold distally simple automorphism either has purely discrete spectrum or is weakly mixing. Another important property of a 2-fold distally simple system R is that (see [16]):

(2) if S is an arbitrary automorphism and λ ∈ J^{e}(R, S) then either λ is the product measure or the extension (R × S, λ) is relatively distal over the factor S.

For sake of completeness, since (2) is used in the sequel, we briefly recall its justification from [16]. Denote by Ĉ the maximal factor of (R × S, λ) containing {∅, Z} ⊗ C for which R × S|Ĉ → S is relatively distal. Then R × S|_{D⊗C} → R × S|Ĉ is relatively weakly mixing and the relatively independent square of (R × S, λ) over Ĉ is ergodic. We can view this latter system as an ergodic joining (R × R × S, κ) of two copies of R and S. Moreover, by basic properties of relative products (see [10]) the extension (R × R × S, κ) → (R × S, λ) is still relatively weakly mixing.

If λ is not the product measure, the two copies of R cannot be independent. Then (R × R, κ|_{D⊗D}) → R_{1} is relatively distal and (R × R × S, κ) → (R × S, λ) is also relatively distal. It follows that this extension is trivial and Ĉ = D ⊗ C.

We also recall (see [9]) that

(3) if T is ergodic, the extension T → T_{1} is relatively distal and S is weakly mixing, then the only extension of T_{1} × S to an ergodic joining of T and S is the direct product.

A dynamical system T will be said to have the convolution singularity (CS) property if its reduced maximal spectral type σ_{T} (i.e. the maximal spectral type of T on L_{0}^{2}(X, B, μ), the orthocomplement of the constant functions) is singular with respect to the convolution product λ_{1} ∗ λ_{2} of two arbitrary continuous measures λ_{1}, λ_{2} on the circle. CS systems have purely singular spectra, thus they automatically have the PID property by the result of Host [12]. It can be seen ([25]) that the CS property holds if the symmetric tensor product operator V_{σ} ⊙ V_{σ} has simple spectrum, where

V_{σ} : L^{2}(𝕋, σ) → L^{2}(𝕋, σ), V_{σ}(f)(z) = zf(z).

This relates CS property of T with spectral properties of the Gaussian automorphism determined by σ_{T}. In particular if this Gaussian automorphism has a simple spectrum then T has CS property (see [3]).

In what follows, if no confusion arises, we will identify automorphisms with their actions on σ-algebras, in particular factor automorphisms will be identified with the underlying invariant sub-σ-algebras.

2. Systems with joining primeness property and disjointness

An ergodic system R acting on (Z, D, ρ) is said to have the joining primeness (JP) property if for every pair of weakly mixing systems S_{1} and S_{2} and every λ ∈ J^{e} (R, S_{1} × S_{2}) we have,

λ = λ_{Z,Y_{1}} ⊗ ν_{2} or λ = λ_{Z,Y_{2}} ⊗ ν_{1},

where by λ_{Z,Y_{i}} we denote the projection of λ to the corresponding two coordinates.

Of course this definition really makes sense only for a system which is not disjoint from a weakly mixing system.

Notice that assuming S_{1} and S_{2} isomorphic would give an equivalent definition. Indeed, let λ ∈ J^{e} (R, S_{1} × S_{2}) where S_{1} and S_{2} are two weakly mixing systems, not necessarily isomorphic. If S′_{1} and S′_{2} are isomorphic copies of S_{1} and S_{2} respectively, then, by forming the direct product of the previous joining with the direct product S′_{1} × S′_{2}, we obtain an ergodic joining of R with the direct product (S_{1} × S′_{2}) × (S′_{1} × S_{2}) of two isomorphic weakly mixing transformations and the equivalence of the definitions follows. Remark also that we could have required R to satisfy a seemingly stronger property: for any n ≥ 2, any familly S_{1}, …, S_{n} of weakly mixing transformations and any λ ∈ J^{e}(R, S_{1} × … × S_{n}), there exists j(λ) ∈ {1, …, n} such that

(4)  λ = λ_{Z,Y_{j(λ})} ⊗ (⊗_{j≠j(λ)}ν_{j}) .

Once again, it is not difficult to check that the resulting definition would be equivalent to the previous one and that it is still so if S_{1}, …, S_{n} are assumed isomorphic.

Given λ ∈ J^{e}(R, S_{1} × … × S_{n}), we identify D and the C_{i} to sub-σ-algebras of the joining system. It is easy to see that λ satisfies (4) if and only if

(5)  Φ_{λ}(f) ∈ L^{2}(C_{j(λ)})

for each f ∈ L_{0}^{2}(Z, D, ρ).

Proposition 1. The class of weakly mixing JP systems is closed under factors, inverse limits and distal extensions which are weakly mixing.

Proof. The first two properties are obvious. The fact that the JP property is closed under weakly mixing distal extensions follows from (3). □

Let us notice that whenever a JP automorphism R has a common factor with a direct (weakly mixing) product S_{1} ×S_{2} then in fact this common factor is contained in either the first or the second coordinate σ-algebra of C_{1} ⊗ C_{2}. Hence using del Junco-Rudolph’s criterion from [18] of non-disjointness of a system from a simple system we obtain the following.

Proposition 2. A JP system is disjoint with a weakly mixing simple system if and only if they do not have a common non-trivial factor. □

The result below is proved in [33] and [16] in a slightly different form and we give a proof only for completeness.

Proposition 3. If R is weakly mixing and distally simple then it has the JP property.

Proof. Assume that λ ∈ J^{e}(R, S_{1} ×S_{2}). By (2), the extension (R×S_{1}, λ_{Z,Y_{1}}) → S_{1} is either relatively distal or we have independence. If it is relatively distal, since S_{2} is weak mixing, we have λ = λ_{Z,Y_{1}} ⊗ ν_{2} by (3). Similarly, if (R × S_{2}, λ_{Z,Y_{2}}) → S_{1} is relatively distal, we get λ = λ_{Z,Y_{2}} ⊗ ν_{1}. Furthermore, if D is independent from C_{1} and C_{2}, then by the PID property of R (and Lemma 1), λ is just the product measure. □

In order to prove that CS implies JP we need a lemma.

Lemma 2. Assume that λ ∈ J^{e}(R, S_{1} × S_{2}) and it satisfies

Φ_{λ}(L_{0}^{2}(Z, D, ρ)) ⊂ L_{0}^{2}(Y_{1}, C_{1}, ν_{1}) ⊕ L_{0}^{2}(Y_{2}, C_{2}, ν_{2}).

Then λ = λ_{Z,Y_{1}} ⊗ ν_{2} or λ = λ_{Z,Y_{2}} ⊗ ν_{1}.

Proof. Denoting by p_{i} the orthogonal projection from L^{2}(Y_{1} × Y_{2}, ν_{1} ⊗ ν_{2}) on L^{2}(Y_{i}, ν_{i}) and E the orthogonal projection on constant functions, by the assumption

(6)  Φ_{λ} + E = p_{1} ∘ Φ_{λ} + p_{2} ∘ Φ_{λ}.

But p_{i} ∘ Φ_{λ} is still a Markov operator, so p_{i} ∘ Φ_{λ} = Φ_{λ_{i}} for some joining λ_{i}, i = 1, 2. Now the condition in (6) simply means that

λ + μ ⊗ ν_{1} ⊗ ν_{2} = λ_{Z,Y_{1}} ⊗ ν_{2} + λ_{Z,Y_{2}} ⊗ ν_{1}

and since the four joinings under consideration are ergodic (the factor joinings λ_{Z,Y_{i}} are ergodic and the S_{i} are weakly mixing), the assertion follows. □

Proposition 4. If R is weakly mixing and has the CS property then it has the JP property.

Proof. For each f ∈ L^{2}(Z, D, ρ), it is a general fact that the spectral measure of Φ_{λ}(f) is absolutely continuous with respect to the spectral measure of f and therefore with respect to σ_{R}. Since

L_{0}^{2}(Y_{1} × Y_{2}, ν_{1} ⊗ ν_{2}) = L_{0}^{2}(Y_{1}, ν_{1}) ⊕ L_{0}^{2}(Y_{2}, ν_{2}) ⊕ (L_{0}^{2}(Y_{1}, ν_{1}) ⊗ L_{0}^{2}(Y_{2}, ν_{2}))

and on L_{0}^{2}(Y_{1}, ν_{1}) ⊗ L_{0}^{2}(Y_{2}, ν_{2}) the maximal spectral type is the convolution of the reduced maximal spectral types of S_{1} and S_{2}, we obtain the assumption of Lemma 2 satisfied. Therefore, the result follows. □

We will give examples of automorphisms with the CS property in Section 6.

3. Joining primeness property and symmetric factors

The aim of this section is to prove that JP systems are disjoint from dynamical systems that possess the probabilistic property of infinite divisibility (not to be mistaken with the ergodic theoretical notion sharing the same name, see next section).

Let us recall the definition of an infinitely divisible (ID) stationary process: A stationary process {X_{n}}_{n∈Z} of distribution P on (R^{Z}, B^{⊗Z}) is ID if, for any

integer k, there exists a probability distribution P_{k} on (R^{Z}, B^{⊗Z}) such that P = P_{k} ∗ ⋯ ∗ P_{k} (k terms) where ∗ denotes the convolution operation with respect to the usual coordinatewise addition on R^{Z}. In other words, the stationary process {X_{n}}_{n∈Z} is ID if, for any integer k, it can be realized as an independent sum

X_{n} = X_{n}^{(1,k)} + ⋯ + X_{n}^{(k,k)} where the processes {X_{n}^{(i,k)}}_{n∈ℤ} are stationary with the same distribution P_{k}. If P is infinitely divisible, then the distributions P_{k} are uniquely determined and are also infinitely divisible. Moreover if (R^{Z}, B^{⊗Z}, P, S) (where S stands for the shift) is ergodic, then it is weakly mixing (see [28]) and so is (R^{Z}, B^{⊗Z}, P_{k}, S) for any k. Any Gaussian stationary process is infinitely divisible as well as any stable stationary process among many others.

The important ergodic property here that will play the key role for our disjointness result comes from the following observation: the system determined by a stationary ID process is, for any integer k ≥ 1, a factor of the symmetric factor of the Cartesian k-th power of a dynamical system (by symmetric factor we mean the sub-σ-algebra of subsets which are invariant under all permutations of coordinates). Indeed, (R^{Z}, B^{⊗Z}, P, S) is a factor of (R^{Z}, B^{⊗Z}, P_{k}, S)^{×k} via the factor

map ({x_{n}^{1}}_{n∈Z} , …, {x_{n}^{k}}_{n∈Z}) ↦ {x_{n}^{1} + ⋯ + x_{n}^{k}}_{n∈Z}.

Closely related to ID stationary processes are Poisson suspensions. Let us recall the definition.

Consider a dynamical system (X, A, μ, T) where μ is σ-finite and form the Poisson measure (X^{∗}, A^{∗}, μ^{∗}) where X^{∗} is the space of counting measures on (X, A), A^{∗} the smallest σ-algebra such that for any A ∈ A^{∗} the map N_{A} : ν ↦ ν (A) is measurable and μ^{∗} is the unique probability measure such that for any family A_{1}, …, A_{k} of disjoint sets in A of finite μ-measure, the random variables N_{A_{i}} are independent and Poisson distributed with parameters μ(A_{i}). Let furthermore T_{∗} : X^{∗} → X^{∗} be the map which assigns to each ν ∈ X^{∗} its direct image under T. Then T_{∗} is μ^{∗}-preserving and (X^{∗}, A^{∗}, μ^{∗}, T_{∗}) is called the Poisson suspension over the base (X, A, μ, T) (see [3], and also recent [5], [29], [30] for the joining theory of Poisson suspensions). Convolution in this space is well defined, arising from addition of measures. It is well known that μ^{∗} = (\frac{1}{k} μ)^{∗} ∗ ⋯ ∗ (\frac{1}{k} μ)^{∗} (k terms) for every k ≥ 1, and this shows the ID character of a Poisson suspension. We therefore derive the same conclusion as above: for any integer k a Poisson suspension is a factor of the symmetric factor of a direct product of k copies of a dynamical system.

The disjointness of JP systems from systems given by ID stationary processes and Poisson suspensions is based on the relationship between JP systems and symmetric factors. In fact, it will be a consequence of a bit more general criterion given in Proposition 5 below. We first need a lemma.

Let S be a weakly mixing automorphism of a standard probability space (Y, C, ν). Given n ≥ 1, by F_{n} = F_{n}(C) = F_{n}(S) we denote the symmetric factor of S^{×n}.

Proposition 5. Let T be an automorphism of a standard probability space (X, B, μ). Assume that, for each g ∈ L^{2}(X, B, μ), there exist a sequence (k_{j})_{j≥1} of integers going to infinity and a sequence (S_{j})_{j≥1} of weakly mixing automorphisms such that T is a factor of S_{j}^{×k_{j}} and moreover

dist(g, L^{2}(F_{k_{j}}(S_{j})) = o (\frac{1}{\sqrt{k_{j}}} ) ·

Then T is disjoint from every weakly mixing JP system.

Proof. Let λ_{0} ∈ J(R, T), where R : (Z, D, ρ) → (Z, D, ρ) is a JP automorphism and let f be any function in L_{0}^{2}(Z, D, ρ). We have to prove that g := Φ_{λ_{0}}(f) = 0.

By the assumption, given an arbitrary ε > 0, there exist an arbitrarily large integer k > 0 and a weakly mixing automorphism S on a standard space (Y, C, ν) such that T is a factor of S^{×k} and moreover

(7)  dist(g, L^{2}(F_{k}(S)) ≤ \frac{ε}{\sqrt{k}} ·

Let λ̃_{0} be the relatively independent extension of λ_{0} to a joining of R with S^{×k}, so that Φλ̃_{0} is the composition of Φ_{λ_{0}} and the embedding of L^{2}(X, B, μ) into L^{2}(Y^{×n}, C^{⊗n}, ν^{⊗n}), and thus Φλ̃_{0}(f) = Φ_{λ_{0}}(f) = g.

Consider now an ergodic decomposition of λ̃_{0}:

λ̃_{0} = ∫_{J^{e}(R,S^{×k})} λdP (λ).

For λ ∈ J^{e}(R, S^{×k}), we write the underlying σ-algebra as D∨C_{1}∨…∨C_{k}, where the C_{i}’s algebras remain independent. When Φ_{λ}(f) ≠ 0, since R has the JP property, there exists exactly one 1 ≤ j = j(λ) ≤ k such that Φ_{λ}(f) ∈ L^{2}(C_{j(λ)}).

For 1 ≤ j ≤ k, denote by M_{j} the set of those λ ∈ J^{e}(R, S^{×k}) for which Φ_{λ}(f) ≠ 0 and j(λ) = j. The sets M_{j} are disjoint and clearly measurable. Let

f_{j} := ∫_{M_{j}} Φ_{λ}(f) dP ∈ L^{2}(C_{j}).

We obtain the orthogonal decomposition

g = ∫_{J^{e}(R,S^{×k})} Φ_{λ}(f) dP = ∑_{j=1}^{k} f_{j} ∈ ⨁_{j=1}^{k} L^{2}(C_{j})

and moreover there exists some j_{0} ∈ \{1, …, k\} such that P (M_{j_{0}}) ≤ \frac{1}{k}, whence

(8)  ∥f_{j_{0}}∥ ≤ \frac{1}{k} ∥f∥.

Now we claim that for all 1 ≤ j ≤ k

(9)  |∥f_{j}∥ − ∥f_{j_{0}}∥| ≤ \frac{2ε}{\sqrt{k}} ·

Indeed, consider a permutation π of {1, …, k} sending j_{0} to j, and the corresponding unitary operator U_{π} on L^{2}(Y^{×n}, C^{⊗n}, ν^{⊗n}). Since all functions in L^{2}(F_{k}(S)) are fixed by U_{π},

∥f_{j} − f_{j_{0}}∥ ≤ (∑_{ℓ=1}^{k} ∥f_{π(ℓ)} − f_{ℓ}∥_{L^{2}(C)}^{2})^{1/2} = ∥U_{π}(g) − g∥ ≤ 2 dist(g, L^{2}(F_{k}(S))

and (9) follows from (7).

So, by (8), ∥f_{j}∥ ≤ \frac{1}{k} ∥f∥ + \frac{2ε}{\sqrt{k}} for all 1 ≤ j ≤ k and

∥g∥ = (∑_{j=1}^{k} ∥f_{j}∥^{2})^{1/2} ≤ \frac{1}{\sqrt{k}} ∥f∥ + 2ε.

Since k was arbitrarily large and ε was arbitrarily small, this proves that g = 0. □

Remark 1. We notice that T satisfying the assumption of Proposition 5 is still disjoint from all roots of JP maps. Indeed, if T satisfies the assumptions of Proposition 5 then each (non-zero) power of it does so; moreover if two automorphisms are non-disjoint then also their powers are non-disjoint and the disjointness result with roots of JP maps follows.

4. JP property and infinite divisibility in the ergodic theoretical sense

Let us recall that an ergodic automorphism T is said to be infinitely divisible if there exists a sequence of factors {B_{ω} : ω ∈ {0, 1}^{∗}} of B where B_{ε} = B, B_{ω} = B_{ω0} ⊗ B_{ω1} and for each f ∈ L_{0}^{2}(X, B, μ), η ∈ {0, 1}^{ℕ}

(10)  lim_{n→∞} E(f|B_{η[0,n)}) = 0.

As in the introduction, to avoid confusion, this ergodic theoretical notion of infinite divisibility will be referred to as ergodically ID in the sequel.

Proposition 6. Ergodically ID automorphisms are disjoint from JP systems.

Proof. Assume that R is a JP automorphism, and let λ ∈ J^{e}(R, T), where T is ergodically ID. For each f ∈ L_{0}^{2}(Z, D, ρ) and n ≥ 1, by the JP property of R, there is an ω ∈ {0, 1}^{n} such that Φ_{λ}(f) belongs to L^{2}(B_{ω}) (indeed, B = ⨂_{|ω|=n} B_{ω}). Then the result follows immediately from (10). □

In order to complete the picture we will now give a general argument which in particular shows that dynamical systems induced by stationary ID processes are factors of ergodically ID dynamical systems.

Proposition 7. Let T be an automorphism of a standard probability space (X, B, μ). Assume that there exists a sequence (S_{n})_{n≥0} of automorphisms of standard probability spaces (Y_{n}, C_{n}, ν_{n}) such that S_{0} = T and, for every n ≥ 0, S_{n} is isomorphic to a factor of the symmetric factor of S_{n+1} × S_{n+1}. Then T is a factor of an ergodically ID system.

Proof. Let, at each stage n ≥ 0, T_{n} := Π_{ω∈{0,1}^{n}}S_{ω} be the direct product of 2^{n} copies S_{ω} of S_{n} acting each on a space (Y_{ω}, C_{ω}, ν_{ω}), and T_{n} acting on the product space (X_{n}, B_{n}, μ_{n}). From the assumption we represent each S_{ω} as a factor of the symmetric factor of S_{ω0} × S_{ω1}. So each T_{n} is a factor of T_{n+1} and we obtain an inverse system of dynamical systems. Let T̃ stand for the inverse limit, acting on (X̃, B̃, μ̃). We will show that T̃ is ergodically ID.

We consider each B_{n} and C_{ω} as sub-σ-algebras of B̃: then B̃ is generated by the union of the increasing sequence of factors B_{n} and each C_{ω} is contained in the symmetric factor F_{2}(B_{ω0}, B_{ω1}). Let moreover, for ω ∈ {0, 1}^{∗}

B_{ω} = ⋁_{η∈{0,1}^{∗}} C_{ωη}.

We obtain a splitting sequence of independent factors as required in the definition of an ergodically ID system. We need to show that (10) holds true and it is enough here to consider f ∈ L_{0}^{2}(B_{n}) for some n ≥ 0. Then, for ω ∈ {0, 1}^{n},

(11)  E(f|B_{ω}) = E(f|C_{ω}).

Indeed, it is again enough here to consider f of the form f = ⊗_{ω′∈{0,1}^{n}}f_{ω′} with f′_{ω} measurable with respect to C_{ω′}, and then f_{ω′} is still independent from B_{ω} whenever ω′ ≠ ω.

Now, represent E(f|C_{ω}) as an orthogonal sum

E(f|C_{ω}) = E(f|C_{ω0}) + E(f|C_{ω1}) + g,

Since C_{ω} is contained in the symmetric factor of S_{ω0} × S_{ω1}, we have that E(f|C_{ω1}) corresponds to E(f|C_{ω0}) by the given isomorphism between S_{ω0} and S_{ω1}, and in particular it has the same norm. Therefore

∥E(f|C_{ωi})∥ ≤ \frac{1}{\sqrt{2}}∥E(f|C_{ω})∥ ≤ \frac{1}{\sqrt{2}}∥f∥

for i = 0, 1.

We may repeat the same argument and thus we obtain for every η ∈ {0, 1}^{∗}

∥E(f|C_{ωη})∥ ≤ (\frac{1}{\sqrt{2}})^{|η|} ∥f∥.

Moreover, since B_{n} ⊂ B_{p} for p ≥ n, we still have E(f|B_{ωη}) = E(f|C_{ωη}) by (11) and the result follows. □

Applying this result to a Poisson suspension (X^{∗}, A^{∗}, μ^{∗}, T_{∗}) and following the lines of the construction, it can be checked that the inverse limit is indeed isomorphic to the Poisson suspension of

(X × [0, 1), A ⊗ B([0, 1)), μ ⊗ λ, T × Id)

where λ is Lebesgue measure on [0, 1). Indeed, we recall that (X^{∗}, A^{∗}, μ^{∗}, T_{∗}) is a

factor of the symmetric factor of (X^{∗}, A^{∗}, (\frac{1}{2}μ)^{∗} , T_{∗})^{×2} but this latter system is isomorphic to the Poisson suspension of

(X × \{0, 1\} , A ⊗ P (\{0, 1\}) , μ ⊗ (\frac{1}{2}δ_{0} + \frac{1}{2}δ_{1}) , T × Id)

and the result follows, proceeding inductively.

5. Remarks concerning JP property of higher order

We can extend the definition of JP, considering a weaker property for multiple joinings: an ergodic system R is said to have the joining primeness property of order n ≥ 1 (JP(n)) if, for every k ≥ n+1, every direct product of k weakly mixing automorphisms S_{1}, …, S_{k}, for every λ ∈ J^{e} (R, S_{1} × … × S_{k}), there exist i_{1}, …, i_{n} in {1, …, k} such that

λ = λ_{Z,Y_{i_{1}} ,...,Y_{in}} ⊗ ⨂_{j≠i_{1},...,i_{n}} ν_{j}.

Clearly JP(m)⊆JP(n) whenever m ≤ n and JP(1) is just the class JP considereded in this paper. Note that, as in this latter case, we can replace in the definition possibly non isomorphic S_{i}’s by isomorphic ones.

Proposition 8. JP(n) systems are disjoint from ergodically ID systems.

Proof. Let R belong to the JP(n) class, T be ergodically ID and consider an ergodic joining λ of R and T and the corresponding Markov operator Φ_{λ} : L^{2}(Z, D, ρ) → L^{2}(X, B, μ).

With the notation of previous section, let Ω_{k} be the set of all ω ∈ {0, 1}^{∗} with |ω| = k, for k ≥ 0, so that B = ⨂_{ω∈Ω_{k}} B_{ω}. By the JP(n) property of R, we can find subsets Δ of Ω_{k} of cardinality ≤ n such that λ is the product of its restriction to D ∨ ⨂_{ω∈Δ} B_{ω} and the other factors, or equivalently

(12)  Φ_{λ}(L^{2}(D)) ⊂ L^{2} (⨂_{ω∈Δ} B_{ω}) .

Since, for Δ, Δ′ ⊂ Ω_{k}, we have by independence

L^{2} (⨂_{ω∈Δ} B_{ω}) ∩ L^{2} ( ⨂_{ω∈Δ′} B_{ω}) = L^{2} ( ⨂_{ω∈Δ∩Δ′} B_{ω}) ,

it follows that there is a smallest set Δ ⊂ Ω_{k} satisfying (12), which we denote by Ω_{k}^{λ}. Let also n_{k} be its cardinality (n_{k} ≤ n for all k).

Now, from B_{ω} = B_{ω0} ⊗ B_{ω1} for all ω, it follows that Ω_{k}^{λ} is contained in the set of all ω[0, k) for ω ∈ Ω_{k+1}^{λ}, and conversely Ω_{k+1}^{λ} is contained in the set of all ω0 and all ω1 for ω ∈ Ω_{k}^{λ}. In particular, n_{k} is non decreasing and, being bounded by n, it is eventually constant. This implies that there exists K such that for k ≥ K n_{k} = n_{K} and for each ω ∈ Ω_{k}^{λ} there is exactly one ω′ ∈ {ω0, ω1} in Ω_{k+1}^{λ}. Hence there exist η_{1}, … η_{n_{K}} in {0, 1}^{ℕ} such that

Φ_{λ}(L^{2}(D)) ⊂ ⋂_{k≥K} L^{2} (⨂_{i=1}^{n_{K}} B_{η_{i}[0,k)}) .

Finally, again by independence, for each k ≥ K, the orthogonal projection from L^{2} (⨂_{i=1}^{n_{K}} B_{η_{i}[0,K)}) onto L^{2} (⨂_{i=1}^{n_{K}} B_{η_{i}[0,k)}) is the tensor product of the orthogonal projections of the factors. By (10) for each i ∈ {1, …, n_{K}} the projection of every f ∈ L_{0}^{2}(B_{η_{i}[0,K)}) in L^{2}(B_{η_{i}[0,k)}) goes to zero as k tends to infinity and it follows that the same holds for the tensor product. Therefore Φ_{λ} is the orthogonal projection onto the constant functions and λ is the product measure. □

5.1. Examples. We present here a family of systems which are JP(2) but not JP(1). Recall that a direct product of two weakly mixing automorphisms cannot have the JP property, so we get such examples from the proposition below if we assume moreover the automorphisms to be weakly mixing.

Proposition 9. If R_{1} has the JP property and R_{2} is distally simple then R_{1} × R_{2} is JP(2).

Proof. Let S_{1}, S_{2} and S_{3} be weakly mixing automorphisms and consider an ergodic joining λ between R_{1} × R_{2} and S_{1} × S_{2} × S_{3}. In this joining we denote D_{i} and C_{j} the factor σ-algebra corresponding to each transformation R_{i} and S_{j}.

By the JP property of R_{1} we can assume, up to exchange of coordinates, that D_{1} ∨ C_{1} is independent from C_{2} ∨ C_{3}, i.e. that D_{1} ∨ C_{1}, C_{2} and C_{3} are independent. Since R_{2} also has the JP property, D_{2} is independent from either C_{2} or C_{3} and we may assume D_{2} ⊥ C_{3}.

Now, D_{1} ∨ C_{1} ∨ C_{2} is independent from C_{3} and, since R_{2} is distally simple, by (2), either D_{1} ∨ D_{2} ∨ C_{1} ∨ C_{2} is relatively distal over D_{1} ∨ C_{1} ∨ C_{2}, or D_{2} is independent from D_{1} ∨ C_{1} ∨ C_{2}. In the first case, since S_{3} is weakly mixing, we get by (3) that D_{1} ∨ D_{2} ∨ C_{1} ∨ C_{2} remains independent from C_{3}. In the latter case, D_{1} ∨ C_{1} ∨ C_{2}, D_{2} and C_{3} are pairwise independent, whence independent by the PID property of R_{2} and Lemma 1, and we get again D_{1} ∨ D_{2} ∨ C_{1} ∨ C_{2} ⊥ C_{3}.

□

6. Flows with the convolution singularity property

As noticed in the introduction, all definitions and results of this paper extend to the case of flows. In this section we will deal with flows since special flows over irrational rotations can often be shown to satisfy the assumptions of Proposition 10 below. Besides, it is not hard to see that a flow (T_{t})_{t∈ℝ} has the CS property if and only if any given non-zero time t automorphism of it has the CS property.

Assume that (T_{t})_{t∈ℝ} is a (measurable) measure-preserving flow of (X, B, μ) and let σ be its reduced maximal spectral type. We also denote by (T_{t}) the associated unitary one-parameter group on L^{2}(X, B, μ). Then, given t ∈ ℝ, T_{t} on L_{0}^{2} corresponds spectrally to the multiplication by s ↦ e^{2πits} on L^{2}(ℝ, σ) and, if we define the operator V on L^{2}(X, B, μ) as the weak limit of a sequence (T_{t_{n}})_{n∈ℕ}, V corresponds to the multiplication by a function ϕ_{V} (being the weak-∗ limit of the sequence (e^{2πit_{n}·}) in L^{∞}(ℝ, σ)).

Proposition 10. Assume that there exists a sequence (T_{t_{n}})_{n∈ℕ} converging weakly to V which is not of the form cT_{s} and such that ϕ_{V} is analytic. Then σ is singular with respect to the convolution of any two continuous measures.

Proof. Let us notice first that the result is purely spectral: If σ ⊥̸ λ_{1} ∗ λ_{2}, where λ_{i} are positive continuous Borel measures on ℝ, and if e^{2πit_{n}·} → ϕ weakly-∗ in L^{∞}(ℝ, σ), where ϕ is analytic, then ϕ must be a multiple of a character.

Since e^{2πit_{n}·} → ϕ_{V} in L^{∞}(ℝ, σ) implies the same weak-∗ convergence in L^{∞}(ℝ, η) for each η ≪ σ, w.l.o.g. we can assume that 0 < σ ≪ λ_{1} ∗ λ_{2} and moreover that ϕ_{V} is not identically zero. By passing to a subsequence of (t_{n}) if necessary and thanks to the weak-∗ compactness, e^{2πit_{n}·} converges weakly-∗ to ψ_{1}, ψ_{2} and ψ in L^{∞}(ℝ, λ_{1}), L^{∞}(ℝ, λ_{2}) and L^{∞}(ℝ, λ_{1} ∗ λ_{2}) respectively, where ψ_{i} and ψ are Borel functions which we assume to be defined everywhere. We thus obtain the “generalized character” property

(13)  ψ (s_{1} + s_{2}) = ψ_{1} (s_{1}) ψ_{2} (s_{2})

for λ_{1} ⊗ λ_{2}-almost all (s_{1}, s_{2}) ∈ ℝ × ℝ. This can be found in [13], but we give a proof for completeness. For f_{i} ∈ L^{∞}(ℝ, λ_{i}) we put dλ′_{i} = f_{i} dλ_{i}; we have

∫_{ℝ} e^{2πit_{n}s} dλ′_{1} ∗ λ′_{2}(s) = ∫_{ℝ×ℝ} e^{2πit_{n}(s_{1}+s_{2})}f_{1}(s_{1})f_{2}(s_{2}) dλ_{1}(s_{1})dλ_{2}(s_{2})

= ∫_{ℝ} e^{2πit_{n}s_{1}}f_{1}(s_{1}) dλ_{1}(s_{1}) ∫_{ℝ} e^{2πit_{n}s_{2}}f_{2}(s_{2}) dλ_{2}(s_{2}).

Thus, by passing to the weak-∗ limits

∫_{ℝ} ψ(s) dλ′_{1} ∗ λ′_{2}(s) = ∫_{ℝ} ψ_{1}(s_{1}) dλ′_{1}(s_{1}) ∫_{ℝ} ψ_{2}(s_{2}) dλ′_{2}(s_{2}).

It follows that

∫_{ℝ×ℝ} ψ(s_{1} + s_{2}) dλ′_{1}(s_{1})dλ′_{2}(s_{2}) = ∫_{ℝ} ψ_{1}(s_{1}) dλ′_{1}(s_{1}) ∫_{ℝ} ψ_{2}(s_{2}) dλ′_{2}(s_{2}),

whence (13) holds since functions of the form f_{1} ⊗ f_{2} generate a linearly dense subset of L^{1}(ℝ × ℝ, λ_{1} ⊗ λ_{2}).

Of course, we also have ψ = ϕ_{V} σ-almost everywhere. Put h = \frac{dσ}{dλ_{1}∗λ_{2}} and define the measure σ̃ on ℝ × ℝ by dσ̃ = h̃ dλ_{1} ⊗ λ_{2}, where h̃ (s_{1}, s_{2}) = h (s_{1} + s_{2}). Then by (13) and the definition of σ̃

(14)  ϕ_{V} (s_{1} + s_{2}) = ψ_{1} (s_{1}) ψ_{2} (s_{2})

for σ̃-almost all (s_{1}, s_{2}) ∈ ℝ^{2}.

Put C = {(s_{1}, s_{2}) ∈ ℝ^{2} : ϕ_{V} (s_{1} + s_{2}) = ψ_{1}(s_{1})ψ_{2}(s_{2})}. Then C is a Borel subset of ℝ^{2}. Since h̃ > 0 on a set of positive λ_{1} ∗ λ_{2}-measure, λ_{1} ⊗ λ_{2}(C) > 0. There exist Borel subsets A and B of ℝ such that

(15)  λ_{1} ⊗ λ_{2} ((A × B) ∩ C) ≥ \frac{3}{4}λ_{1} (A) λ_{2} (B) > 0.

Given s ∈ B define

A_{s} = {s_{1} ∈ A : (s_{1}, s) ∈ C}

and let

B′ = \{s ∈ B : λ_{1} (A_{s}) > \frac{1}{2}λ_{1} (A)\} .

In view of (15), λ_{2} (B′) > 0. If we fix s_{2} ∈ B′, the equality ϕ_{V} (s_{1} + s_{2}) = ψ_{1} (s_{1}) ψ_{2} (s_{2}) is satisfied for all s_{1} ∈ A_{s_{2}}. But λ_{1}(A_{s_{2}}) > 0, so A_{s_{2}} is uncountable and since ϕ_{V} is analytic and not identically zero, we must have ψ_{2} (s_{2}) ≠ 0. We have shown that ψ_{2}(s_{2}) ≠ 0 for s_{2} ∈ B′.

Now fix s_{2} = s′ − s′′ with s′ and s′′ in B′. For s_{1} ∈ A_{s′} ∩ A_{s′′}, we obtain

(16)  \frac{ϕ_{V} (s_{1} + s′)}{ψ_{2} (s′)} = \frac{ϕ_{V} (s_{1} + s′′)}{ψ_{2} (s′′)}.

But λ_{1} (A_{s′} ∩ A_{s′′}) > 0 from the definition of B′, and in particular A_{s′} ∩ A_{s′′} is uncountable. Both functions in (16) being analytic functions of s_{1}, the equality holds for all s_{1} ∈ ℝ. Replacing now s_{1} by s_{1} − s′′, we obtain that

(17)  ϕ_{V} (s_{1} + s_{2}) = ϕ_{V} (s_{1}) \frac{ψ_{2} (s′)}{ψ_{2} (s′′)}

for every s_{1} ∈ ℝ, and in particular by taking s_{1} = 0, ϕ_{V} (s_{2}) = ϕ_{V} (0) \frac{ψ_{2}(s′)}{ψ_{2}(s′′)}. But the set B′ − B′ is still uncountable, so we must have ϕ_{V} (0) ≠ 0. It now follows from (17) that

ϕ_{V} (s_{1} + s_{2}) = \frac{ϕ_{V} (s_{1}) ϕ_{V} (s_{2})}{ϕ_{V} (0)}

which holds for s_{2} ∈ B′ − B′ and all s_{1} ∈ ℝ. Once s_{1} is fixed the number of s_{2} being uncountable, we hence obtain that this equality holds for all (s_{1}, s_{2}) ∈ ℝ^{2}. We conclude that \frac{ϕ_{V}}{ϕ_{V} (0)} is a continuous character of ℝ and the proof is complete. □

In order to obtain natural examples of flows satisfying the assumptions of Proposition 10 let us first notice that if P is a probability Borel measure on ℝ and we have the weak convergence

(18)  T_{t_{n}} → ∫_{ℝ} T_{t} dP (t)

of Markov operators in L^{2}(X, B, μ) then

e^{2πit_{n}·} → P̂(·) weakly-∗ in L^{∞}(ℝ, σ)

(see e.g. the proof of Cor. 5.2 in [7]). Now, for example if P has a compact support and is not a Dirac measure, or if P is a Gaussian distribution, then P̂ is an analytic function different from a multiple of a character.

This is satisfied for many examples of special flows over irrational rotations: see e.g. [4], [6], [7], [20], [23], [24], [37].

Added in October 2009: It is proved in [21] that there are CS systems which are not DS. In fact, in [21] it is proved that a typical automorphism T has a non-trivial relatively weakly mixing extension T̂ such that T̂ has CS property.

References

[1] O.N. Ageev, On the spectrum of Cartesian powers of classical automorphisms, (Russian) Mat. Zametki 68 (2000), 643–647; translation in Math. Notes 68 (2000), 547–551.

[2] O.N. Ageev, Mixing with staircase multiplicity function, Ergodic Theory Dynam. Systems 28 (2008), 1687–1700.

[3] I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, Ergodic Theory, Springer-Verlag, New York, 1982.

[4] B. Fayad, A. Windsor, A dichotomy between discrete and continuous spectrum for a class of special flows over rotations, J. Modern Dynam. 1 (2007), 107-122.

[5] Y. Derriennic, K. Fra̧czek, M. Lemańczyk, F. Parreau, Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings, Coll. Math. 110 (2008), 81–115.

[6] K. Fra̧czek, M. Lemańczyk, On symmetric logarithm and some old examples in smooth ergodic theory, Fund. Math. 180 (2003), 241–255.

[7] K. Fra̧czek, M. Lemańczyk, A class of special flows over irrational rotations which is disjoint from mixing flows, Ergodic Theory Dynam. Systems 24 (2004), 1083–1095.

[8] K. Fra̧czek, M. Lemańczyk, On mild mixing of special flows over irrational rotations under piecewise smooth maps, Ergodic Theory Dynam. Syst. 26 (2006), 719-738.

[9] H. Furstenberg, Disjointness in ergodic theory, minimal sets and diophantine approximation, Math. Syst. Theory 1 (1967), 1-49.

[10] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, 1981.

[11] E. Glasner, Ergodic Theory via Joinings, Math. Surveys and Monographs vol. 101, 2003.

[12] B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum, Israel J. Math. 76 (1991), 289–298.

[13] B. Host, F. Méla, F. Parreau, Analyse Harmoniques des Mesures, Astérisque 135-136, 1986.

[14] A. del Junco, M. Lemańczyk, Generic spectral properties of measure-preserving maps, and applications, Proc. Amer. Math. Soc. 115 (1992), 725-736.

[15] A. del Junco, M. Lemańczyk, Simple systems are disjoint from Gaussian systems, Studia Math. 133 (1999), 249-256.

[16] A. del Junco, M. Lemańczyk, Joinings of distally simple systems, preprint (2005).

[17] A. del Junco, M. Rahe, L. Swanson, Chacon’s automorphism has minimal self-joinings, J. Analyse Math. 37 (1980), 276–284.

[18] A. del Junco, D. Rudolph, On ergodic actions whose self-joinings are graphs, Ergodic Theory Dynam. Systems 7 (1987), 531-551.

[19] A. del Junco, D. Rudolph, A rank-one, rigid, simple, prime map, Ergodic Theory Dynam. Systems 7 (1987), 229–247.

[20] A. Katok, Combinatorial constructions in ergodic theory and dynamics, Amer. Math. Soc., Providence, RI, 2003.

[21] A. Katok, M. Lemańczyk, Some new cases of realization of spectral multiplicity function for ergodic transformations, to appear in Fundamenta Math.

[22] J. King, J.-P. Thouvenot, A canonical structure theorem for finite joining-rank maps, J. Analyse Math. 56 (1991), 211–230.

[23] M. Lemańczyk, F. Parreau, Special flows over irrational rotations with the simple convolution property, preprint (2007).

[24] M. Lemańczyk, M. Wysokińska, On analytic flows on the torus which are disjoint from systems of probabilistic origin, Fund. Math. 195 (2007), 97–127.

[25] F. Parreau, E. Roy, Joinings of Poisson suspensions, preprint (2007).

[26] A.A. Prikhodko, V.V. Ryzhikov, Disjointness of the convolutions for Chacon’s automorphism. Dedicated to the memory of Anzelm Iwanik, Colloq. Math. 84/85 (2000), 67–74.

[27] M. Ratner, Horocycle flows, joinings and rigidity of products, Annals Math. 118 (1983), 277-313.

[28] J. Rosiński, T. Żak, The equivalence of ergodicity and weak mixing for infinitely divisible processes, J. Theoret. Probab. 10 (1997), 73-86.

[29] E. Roy, Ergodic properties of Poissonian ID processes, Ann. Probab. 23 (2007), 1163-1187.

[30] E. Roy, Mesures de Poisson, infinie divisibilité et propriétés ergodiques, thesis 2005, Paris VI. http://www.math.univ-paris13.fr/∼ roy/TheseRoy.pdf

[31] V.V. Ryzhikov, Joinings, intertwinings operators, factors, and mixing properties of dynamical systems, Russian Acad. Sci. Izv. Math. 42 (1994), 91-114.

[32] V.V. Ryzhikov, Weak limits of powers, the simple spectrum of symmetric products, and mixing constructions of rank 1, Mat. Sb. 198 (2007), 137-159; translation in Sb. Math. 198 (2007), 733–754.

[33] V.V. Ryzhikov, J.-P. Thouvenot, Disjointness, divisibility, and quasi-simplicity of measurepreserving actions, Functional Analysis and its Appl. 40 - 3, (2006), 237-240.

[34] J.-P. Thouvenot, Some properties and applications of joinings in ergodic theory, in: Ergodic Theory and its Connections with Harmonic Analysis, London Math. Soc., Lecture Notes Ser. 205, Cambridge Univ. Press, 1995, 207-235.

[35] J.-P. Thouvenot, Les systèmes simples sont disjoints de ceux qui sont infiniment divisibles et plongeables dans un flot. Dedicated to the memory of Anzelm Iwanik, Coll. Math. 84-85 (2000), 481-483.

[36] W.A. Veech, A criterion for a process to be prime, Monatsh. Math. 94 (1982), 335-341.

[37] D. Volný, Constructions of smooth and analytic cocycles over irrational circle rotations, Comment. Math. Univ. Carolinae 36, 4 (1995), 745 - 764.

(Mariusz Lemańczyk) Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87–100 Toruń, Poland

E-mail address: mlem@mat.uni.torun.pl

(François Parreau & Emmanuel Roy) Laboratoire d’Analyse, Géométrie et Applications – UMR 7539 Université Paris 13 et CNRS, 99 av. J.-B. Clément, 93430 Villetaneuse, France

E-mail address: parreau@math.univ-paris13.fr, roy@math.univ-paris13.fr