Alexandre I Danilenko
email: alexandre.danilenko@gmail.com

Zemer Kosloff
email: zemer.kosloff@mail.huji.ac.il

Emmanuel Roy
email: roy@math.univ-paris13.fr

B Verkin

GENERIC NONSINGULAR POISSON SUSPENSION IS OF TYPE III 1

est destinée au dépôt

Introduction

In this paper we continue to study nonsingular Poisson suspensions for nonsingular transformations of infinite Lebesgue spaces (X, B, µ) that we initiated in our previous work [DaKoRo]. Let X * denote the space of σ-finite integer-valued nonnegative measures on X. We endow X * with the smallest σ-algebra B * of subsets such that the map N B : X * ω → ω(B) ∈ Z + is B * -measurable for each B ∈ B. There exists a unique probability measure µ * on B * such that -for each finite family B 1 , . . . , B n of mutually disjoint subsets from B of finite µ-measure, the random variables N B 1 , . . . , N B n are independent, -for each B ∈ B with 0 < µ(B) < ∞, the measure µ * • N -1 B is the Poisson distribution on Z + with parameter µ(B).

The triplet (X * , B * , µ * ) is called the Poisson suspension of (X, B, µ). Completing B * with respect to µ, we obtain that the Poisson suspension is a Lebesgue space. We showed in [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Corollary 4.1] that there is a maximal subset Aut 2 (X, µ) of the group Aut(X, µ) of all nonsingular transformations T of (X, B, µ) for which the Poisson suspension T * is well defined as a nonsingular invertible transformation of (X * , B * , µ * ). As in the classical measure preserving case, by T * ω we mean the measure ω • T -1 ∈ X * for each ω ∈ X * . Moreover, Aut 2 (X, µ) is a Polish group in an appropriate topology d 2 which is stronger than the usual weak topology [DaKoRo, Theorem D]. The suspension T * admits an equivalent invariant probability measure if and only if T admits an equivalent invariant measure ν such that dν dµ -1 ∈ L 2 (X, µ). In this paper we consider the problem:

Let T * do not admit an equivalent probability measure. Can T * be ergodic?

If yes, what is the Krieger type of T * ?

We answer affirmatively the first question and contribute partly to the second one. Though we are unable so far to construct a concrete example of an ergodic conservative T of type III whose Poisson suspension is ergodic and of type III, we instead utilize the Baire category tools to prove a stronger "existence" result.

The research of Z.K. was partially supported by ISF grant No. 1570/17.

Typeset by A M S-T E X

Theorem A (main result). The subset of all T ∈ Aut 2 (X, µ) such that T is ergodic and of type III 1 and T * is ergodic and of type III 1 is a dense G δ in (Aut 2 (X, µ), d 2 ).

To prove this theorem we first construct a concrete example of a totally dissipative T with T * being ergodic and of type III 1 . The construction is motivated by a recent progress in the theory of nonsingular Bernoulli shifts achieved in [START_REF] Kosloff | On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel[END_REF] and [DaLe], because if T is totally dissipative then T * is always a nonsingular Bernoulli shift. However the aforementioned papers deal only with the shifts on {0, 1} Z while we encounter indeed with the product spaces A Z with A uncountable. This situation is considerably more difficult and we are not sure that the techniques developed in [START_REF] Kosloff | On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel[END_REF] and [DaLe] extends to it in the full generality. However, we need only a very particular case which, in turn, can be reduced further to the shift on (Z N + , n∈Z κ n ), where (κ n ) n∈Z is a specially selected sequence of Poisson distributions. Then we prove that T * is of type III 1 by showing that the Maharam extension of T * is conservative and has nonsingular property K. For that we use essentially properties of Skellam distributions (see Appendix A) and Lévy's continuity theorem in addition to the theory of nonsingular endomorphisms and measurable orbit theory that were utilized in [START_REF] Kosloff | On the K property for Maharam extensions of Bernoulli shifts and a question of Krengel[END_REF] and [DaLe].

Secondly, we prove that the conjugacy class of T is dense in Aut 2 (X, µ). Main Theorem follows from that and an additional fact that the subset of ergodic type III 1 transformations in Aut 2 (X, µ) is a dense G δ (in d 2 ) [DaKoRo, Theorem E].

The group Aut 1 (X, µ) and a homomorphism χ : Aut 1 (X, µ) → R were introduced in [START_REF] Yu | Categories of symmetries and infinite-dimensional groups[END_REF]Chapter X,§4]. We showed in [DaKoRo, Theorem D] that Aut 1 (X, µ) ⊂ Aut 2 (X, µ), Aut 1 (X, µ) is a Polish group in a topology d 1 which is stronger than d 2 , and χ is continuous. If χ(T ) = 0 then T is dissipative [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Proposition 4.13]. Thus, every ergodic transformation from Aut 1 (X, µ) is contained in the proper closed subgroup Ker χ. The following statement is proved in the same way as Theorem A.

Theorem B. The subset of all T ∈ Ker χ such that T is ergodic and of type III 1 and T * is ergodic and of type III 1 is a dense G δ in (Ker χ, d 1 ).

It is interesting to note that dynamical properties of T * are determined not only by the dynamical properties of T but also by the choice of µ inside its equivalence class. Indeed, if (X, µ, T ) is totally dissipative then we can find three different measures µ 1 ∼ µ 2 ∼ µ 3 in the equivalence class of µ such that (X * , µ * 1 , T * ) is a probability preserving Bernoulli shift, (X * , µ * 2 , T * ) is an ergodic system of type III 1 and (X * , µ * 3 , T * ) is a totally dissipative system. We "refine" further this phenomenon in a rather surprising way: for each T ∈ Aut 2 (X, µ) and t > 0 we consider a dynamical system (X * , µ * t , T * ), where µ t is the scaling of µ by t, i.e. µ t (A) := tµ(A) for each Borel subset A ⊂ X. Of course, dµ t •T dµ t = dµ•T dµ for each t > 0. We then encounter with the following phase transition phenomenon.

Theorem C. Let T ∈ Aut 1 (X, µ). If there is α > 1 such that α -1 < dµ•T n dµ (x) < α for each n > 0 at a.e. x ∈ X then there is t 0 ∈ [0, +∞] such that the Poisson suspension (X * , µ * t , T * ) is conservative for each t ∈ (0, t 0 ) and the Poisson suspension (X * , µ * t , T * ) is totally dissipative for each t ∈ (t 0 , +∞). The most interesting case is when the bifurcation point t 0 is proper, i.e. 0 < t 0 <

+∞. 2

Example D. There is a totally dissipative T ∈ Aut 1 (X, µ) and t 0 ∈ ( 1 6 , 4) such that the Poisson suspension (X * , (tµ) * , T * ) is weakly mixing of stable type III 1 if 0 < t < t 0 and the Poisson suspension (X * , (tµ) * , T * ) is totally dissipative for t > t 0 .

We recall that a nonsingular transformation T has conservative index n ∈ N if the n-th Cartesian power of T is conservative and the (n + 1)-th is dissipative.

Proposition E. Let the condition of Theorem C hold and let the bifurcation point t 0 be proper. Then for each n ∈ N, if t 0 n+1 < t < t 0 n then (X * , (tµ) * , T * ) has conservative index n.

As a byproduct, we apply the techniques developed in this paper to study conservativeness of the Poisson suspensions of general dissipative transformations. In particular, we show the following.

Theorem F. If T ∈ Aut 1 (X, µ) and χ(T ) = 0 then T * is totally dissipative.
The outline of the paper is as follows. The first two sections are of preliminary nature. We present there some concepts and facts from the theory of measured equivalence relations and their cocycles ( §1) and nonsingular endomorphisms and their extensions ( §2) to be used below in the paper. In § 3 we study Poisson suspensions of transformations defined on purely atomic measure spaces. Since the purely atomic case was not considered in our previous paper on the Poisson suspensions, we first establish some basic results related to the nonsingularity and conservativeness of such suspensions independently of [DaKoRo] (see Propositions 3.1 and 3.2). Then we prove some necessary (Proposition 3.3) and sufficient (Proposition 3.4) conditions for conservativeness of the Poisson suspensions in terms of the underlying dynamical system. Theorem 3.5 plus Corollary 3.7 provide some conditions on a measure on Z under which the Poisson suspension of the underlying unit translation on Z possesses the nonsingular property K. This is technically the most involved result of the paper. Example 3.8 gives a concrete example of a measure on Z satisfying those conditions. In §4 we show how to pass from the purely atomic case studied in § 3 to the continuous case. In particular, we construct a totally dissipative transformation on a nonatomic Lebesgue space whose Poisson suspension is weakly mixing and of type III 1 (see Theorem 4.2 and a remark just below it). In § 5 we prove Theorems A and B (see Theorem 5.3). § 6 is devoted to Poisson extensions of general dissipative transformations. We prove there Theorem F (see Theorem 6.1). Some extension of Theorem F to the more general case where T ∈ Aut 2 (X, µ) is also discussed in that section (see Remark 6.2). In § 7 we study the phase transitions for the conservativeness of Poisson suspensions when scaling the underlying intensity. Theorem C and Proposition E are proved there. Example D is also provided in § 7. The paper has Appendix A which is devoted completely to the Skellam distributions and their properties that we utilize in the proof of Theorem 3.5.

countable. If R is countable and A

∈ B then the R-saturation R(A) := x∈A R(x) of A belongs to B. An R-saturated subset is also called R-invariant. If for each Borel subset A of zero measure, R(A) is also of zero measure then R is called µ- nonsingular. If the sub-σ-algebra {R(A) | A ∈ B} of R-invariant Borel subsets is trivial (mod 0) then R is called µ-ergodic.
From now on R is countable and µ-nonsingular. The full group [R] of R consists of all nonsingular transformations R of (X, µ) such that Rx ∈ R(x) at a.e. x ∈ X. Given a locally compact Polish group G, a Borel map α : R → G is called a cocycle of R if there is a Borel subset N of zero measure such that α(x, y)α(y, z) = α(x, z) for all x, y ∈ X \ N such that (x, y), (y, z) ∈ R.

In the later sections of this paper we deal only with the cases where G is either R or the multiplicative group R * + . A cocycle β : R → G is cohomologous to α if there is a Borel function φ : X → G and a Borel subset N of zero measure such that

β(x, y) = φ(x)α(x, y)φ(y) -1 for all x, y ∈ X \ N . Fix a left Haar measure λ G on G. The α-skew product equivalence relation R(α) on the product space (X × G, µ × λ G ) is defined by: (x, g) ∼ (y, h) if (x, y) ∈ R and h = α(x, y)g.

This equivalence relation is countable and (

µ × λ G )-nonsingular. If R(α) is ergodic then α is called ergodic. Of course, if α is ergodic then R is ergodic. If a cocycle β is cohomologous to α and α is ergodic then β is also ergodic.
We now isolate an important cocycle of R with values in the multiplicative group R * + . It is called the Radon-Nikodym cocycle of R and denoted by ∆ R . To define it, we first fix a countable subgroup Γ of Borel bijections of X that generates R. Such a group exists according to [START_REF] Feldman | Ergodic equivalence relations, cohomology and von Neumann algebras, I[END_REF]Theorem 1

]. Now we set ∆ R (x, γx) := dµ • γ dµ (x), x ∈ X, γ ∈ Γ.
It follows from the chain rule for the Radon-Nikodym derivatives that ∆ R is a cocycle of R. This cocycle is well defined, i.e. it does not depend on the choice of Γ generating R1 .

Example 1.1. Let A be a countable set and let λ n be a non-degenerated distribution on A, i.e. λ n (a) > 0 if a ∈ A and a∈A λ n (a) = 1, for each n ≥ 1. We set (X, λ) := (A N , ∞ n=1 λ n ). Denote by S the tail equivalence relation on X, i.e. two points x = (x n ) ∞ n=1 and y = (y n ) ∞ n=1 from X are S-equivalent if there is N > 0 such that x n = y n for all n > N . Then S is an ergodic λ-nonsingular countable equivalence relation on X and

∆ S (x, y) = ∞ n=1 λ n (y n ) λ n (x n ) = N n=1 λ n (y n ) λ n (x n )
for all (x, y) ∈ S.

If we change µ with an equivalent σ-finite measure then the Radon-Nikodym cocycle of R related to the new measure is cohomologous to the original ∆ R . The ∆ R -skew product equivalence relation R(∆ R ) is called the Maharam extension of R. If it is ergodic, i.e. ∆ R is ergodic, then R is said to be of Krieger's type III 1 . It follows from the aforementioned cohomology property of ∆ R that the property of R to be of type III 1 does not change if we replace µ with an equivalent σ-finite measure.

Suppose now that R is µ-ergodic. Given a cocycle α of R with values in an Abelian locally compact Polish group G, we say that an element g ∈ G is an essential value of α if for each neighborhood U of g in G and each Borel subset A ⊂ X of positive measure there are a Borel subset B ⊂ A of positive measure and a Borel one-to-one map γ : B → A such that (x, γx) ∈ R and α(x, γx) ∈ U for each x ∈ B. The set r(α) of all essential values of α is a closed subgroup of G ( [START_REF] Schmidt | Cocycles on ergodic transformation groups[END_REF]Lemma 3.3], [START_REF] Feldman | Ergodic equivalence relations, cohomology and von Neumann algebras, I[END_REF]Proposition 8.5]). The cocycle α is ergodic if and only if r(α) = G [START_REF] Schmidt | Cocycles on ergodic transformation groups[END_REF]Corollary 5.4] (see also [START_REF] Feldman | Ergodic equivalence relations, cohomology and von Neumann algebras, I[END_REF]Theorem 8]).

The following standard approximation lemma (see [START_REF] Choksi | Abelian cocycles for nonsingular ergodic transformations and the genericity of type III 1 transformations[END_REF]Lemma 2.1]) is useful for computation of r(α).

Lemma 1.2. Let A ⊂ B be a semiring such the corresponding ring is dense in B. Let 0 < δ < 1 and let g ∈ G. If for each A ∈ A of positive measure and a neighborhood U of g there are a subset B ⊂ A and a one-to-one Borel map γ : B → A such that µ(B) > δµ(A), (x, γx) ∈ R, α(x, γx) ∈ U and δ < ∆ R (x, γx) < δ -1 for all x ∈ B then g is an essential value of α.

An ergodic invertible transformation T of (X, B, µ) is called of Krieger's type III 1 if the T -orbit equivalence relation is of type III 1 . If, moreover, T × S is of type III 1 for each ergodic probability preserving transformation S then T is called of stable type III 1 .

For more information on the measurable orbit theory and detailed proofs of the aforementioned facts we refer the reader to [FeMo], [Sc], [DaSi].

Nonsingular endomorphisms, their extensions and associated equivalence relations

Let (X, B, µ) be a standard σ-finite measure space. A Borel map T :

X → X is called a µ-nonsingular endomorphism if µ • T -1 ∼ µ. Consider the follow- ing decreasing sequence B ⊃ T -1 B ⊃ T -2 B ⊃ • • • of sub-σ-algebras in B. If +∞ n=1 T -n B = {∅,
X} (mod 0) then T is called exact. We will consider only aperiodic endomorphisms, i.e. we assume that

µ n>0 {x ∈ X | T n x = x} = 0.
Let T be µ-nonsingular endomorphism. We recall that a measurable function ω :

X → R * + is called markovian for (X, µ, T ) if f • T ω dµ = f dµ for each f ∈ L 1 (X, µ) [SiTh1].
Such a function may not be unique (see [START_REF] Silva | The subadditive ergodic theorem and recurrence properties of markovian transformations[END_REF]Example 1.3(c)]). However if we assume that the measure µ • T -1 is σ-finite then a standard verification shows that there exists a unique T -1 B-measurable markovian function for (X, µ, T ). We call it the Radon-Nikodym derivative of T and denote by ρ T . One can check that

ρ T = dµ dµ•T -1 • T . Of course, if T is invertible then ρ T = dµ•T dµ .
Let κ denote a measure on R * + equivalent to the Lebesgue measure and such that κ(aB) = a -1 κ(B). Given T such that µ • T -1 is σ-finite, we can define a σfinite measure preserving endomorphism T ρ T of the product space (X × R * + , µ × κ) by setting [START_REF] Silva | A skew product entropy for nonsingular transformations[END_REF]Definition 4.1]. The existence and uniqueness (up to a natural isomorphism) of the natural extension was proved in [START_REF] Silva | A skew product entropy for nonsingular transformations[END_REF]Theorem 4.2]. We will denote it by T . The natural extension of the Maharam extension of T is canonically isomorphic to the Maharam extension of the natural extension of T . If S ∈ Aut(Y, ν) is the natural extension of an exact endomorphism then S is called a nonsingular K-automorphism. We recall that a nonsingular invertible transformation is called weakly mixing if the Cartesian product of it with every ergodic probability preserving transformation is ergodic. A nonsingular K-automorphism is weakly mixing whenever it is conservative [START_REF] Silva | A skew product entropy for nonsingular transformations[END_REF]Proposition 4.8].

T ρ T (x, t) := (T x, ρ T (x)t). It is called the Maharam extension of T . If (Y, Y, ν) is a standard σ-finite measure space, S is a transformation from Aut(Y, ν), π : Y → X is a Borel map such that µ • π -1 = ν, πS = T π, ρ S = ρ T • π and Y = n>0 S n π -1 B then S is called the natural extension of T [
If for a.e. x ∈ X, the set T -1 {x} is at most countable then T is called countableto-one. From now on we will consider only endomorphisms which are countableto-one. Given such a T , we can associate an equivalence relation S T on X by setting:

(2-1) (x, y) ∈ S T if there is n ≥ 0 such that T n x = T n y.

Then S T is countable and µ-nonsingular. It is ergodic if and only if T is exact [Ha].

If (2-1) holds, we set

α ρ T (x, y) := ρ T (x) • • • ρ T (T n-1 x)ρ T (T n-1 y) -1 • • • ρ T (y) -1 .
Then α ρ T is a well defined cocycle of S T with values in R * + .2 Remark 2.1. We note that α ρ T depends also on µ, i.e., in fact, α ρ T = α ρ T ,µ . If we replace µ with an equivalent measure λ and the Radon-Nikodym derivative dλ dµ is measurable with respect to the σ-algebra T -1 B then the cocycle α ρ T ,λ is cohomologous to α ρ T ,µ .

Denote by S T (α ρ T ) the α ρ T -skew product extension of S. Then S T ρ T = S T (α ρ T ). From the aforementioned facts we deduce the following proposition (see the beginning of the proof of [START_REF] Danilenko | K-property for Maharam extensions of nonsingular Bernoulli and Markov shifts[END_REF]Theorem 4.1,Claim B] for details).

Proposition 2.2. Let T be countable-to-one nonsingular endomorphism of a probability space. The Maharam extension of the natural extension of T is a Kautomorphism if and only if the cocycle α ρ T of S T is ergodic.

We illustrate the aforementioned concepts with the following example.

Example 2.3. Let (X, µ) := (A N , ∞ n=1 µ n ), where A and (µ n ) ∞
n=1 stand for the same objects as in Example 1.1. Denote by T : X → X the one-sided Bernoulli shift on X. Of course, it is countable-to-one. By the Kakutani criterion [Ka], T is µ-nonsingular if and only if

∞ n=1 a∈A µ n (a) -µ n+1 (a) 2 < ∞.
Moreover, if T is nonsingular then we can compute the Radon-Nikodym derivative of T :

(2-2) ρ T (x) = ∞ n=1 µ n (x n+1 ) µ n+1 (x n+1 ) at µ-a.e. x = (x n ) n∈N ∈ X.
It is easy to see that S T is the tail equivalence relation on X. Of course, T is exact. The natural extension of T is the two-sided Bernoulli shift on the space (A Z , n∈Z µ n ), where

µ n := µ n if n > 0 and µ n := µ 1 if n ≤ 0. The corresponding projection π : A Z → A N is given by the formula π((x n ) n∈Z ) := (x n ) n∈N .
Given a standard probability space (Y, Y, ν) and a transformation S ∈ Aut(Y, ν), we denote by U S the associated (with

S) Koopman unitary operator in L 2 (Y, ν), i.e. U S f := f • S -1 • √ S for all f ∈ L 2 (Y, ν) 3 . We recall that S is said to be -conservative if for each subset B ∈ Y of positive measure, there is n > 0 such that ν(S -n B ∩ B) > 0, -dissipative if it is not conservative, -totally dissipative if there is B ∈ Y such that n∈Z T n B = Y (mod 0).
The following lemma from [Ko1, Lemma 2.2] will be used in the subsequent sections.

Lemma 2.4. If ∞ n=1 U n S 1, 1 < ∞ then S is totally dissipative.
We remind Hopf's criterion of conservativeness for nonsingular maps.

Lemma 2.5 (see, e.g., [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF]Proposition 1.3

.1]). A nonsingular transformation S of a standard probability space (Y, Y, ν) is conservative if and only if ∞ n=1 dν•S n dν (y) = +∞ at a.e. y ∈ Y .
We will also need the following sufficient condition for conservativeness of nonsingular transformations.

Proposition 2.6. Let S be a nonsingular transformation on a standard probability space (Y, ν). Assume that

dν dν•S -n ∈ L 2 (Y, ν) for each n > 0 and write b(n) := dν dν•S -n 2 2 . If there is a sequence (a(n)) ∞ n=1 of positive reals such that ∞ n=1 a(n) = ∞ but ∞ n=1 a(n) 2 b(n) < ∞ then S is conservative.
3 For the sake of simplicity, we use here and below the notation S for the Radon-Nikodym derivative of S, i.e. S :

= dν•S -1 dν . Proof. We set A n := {y ∈ Y | dν•S -n dν (y) < a(n)}. By Markov's inequality, ν(A n ) = ν y ∈ Y dν dν • S -n (y) 2 > 1 a(n) 2 ≤ a(n) 2 b(n). Since ∞ n=1 a(n) 2 b(n) < ∞,
it follows from the Borel-Cantelli lemma that for ν-a.e. y, there is N (y) > 0 such that for each n > N (y),

dν • S -n dν (y) ≥ a(n). Now the condition ∞ n=1 a(n) = ∞ yields that ∞ n=1 dν•S -n dν (y) = ∞. Hence S is conservative by Lemma 2.5.
We also recall definition of an r.f.m.p. extension for nonsingular maps. A nonsingular endomorphism T of (X, B, µ) is called a relatively finite measure preserving (r.f.m.p.) extension of a nonsingular endomorphism

S of (Y, Y, ν) if there is a Borel map π : X → Y such that ν = µ • π -1 , πT = Sπ and ρ T = ρ S • π.
For instance, the natural extension of a nonsingular endomorphism is r.f.m.p.

For more information on nonsingular endomorphisms we refer to [Aa], [Ha], [START_REF] Silva | A skew product entropy for nonsingular transformations[END_REF] and [DaLe].

Poisson suspensions of type III 1 over a discrete base

Let (X, B, µ) be a σ-finite measure space and µ(X) = ∞. We first recall some definitions, notation and facts from [DaKoRo]. Aut(X, µ) stands for the group of all µ-nonsingular invertible transformations. The weak topology on Aut(X, µ) is induced from the weak (equivalently, strong) operator topology on the unitary group U of the Hilbert space L 2 (X, µ) via the embeding Aut(X, µ) S → U S ∈ U. We also let

Aut 2 (X, µ) := {T ∈ Aut(X, µ) | √ T -1 ∈ L 2 (X, µ)} and Aut 1 (X, µ) := {T ∈ Aut(X, µ) | T -1 ∈ L 1 (X, µ)}.
The two sets are Borel subgroups of Aut(X, µ) endowed with the weak topology and Aut 1 (X, µ) ⊂ Aut 2 (X, µ) [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Proposition 4.14]. For j = 1, 2, we define a topology d j on Aut j (X, µ) by saying that a sequence (T n ) ∞ n=1 of transformations from Aut j (X, µ) converges to a transformation T from Aut j (X, µ) if T n → T weakly and (T n ) 1/j -(T ) 1/j j → 0 as n → ∞. Then Aut j (X, µ) endowed with d j is a Polish group [DaKoRo, Theorem D]. There exists a continuous homomorphism χ : Aut 1 (X, µ) → R defined by the formula [START_REF] Yu | Categories of symmetries and infinite-dimensional groups[END_REF]Chapter X,§4] and [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Theorem 4.11]).

χ(T ) := X (T -1) dµ (see
From now on and till the end of the section let X = Z and let T denote the unit translation, i.e. T n := n + 1 for all n ∈ Z. Given a measure µ on X, we set

a n := µ(n) for each n ∈ Z. Then T is µ-nonsingular if and only if µ is non- degenerated, i.e. a n > 0 for each n ∈ Z. Moreover, T (n) = a n-1 a n for all n ∈ Z.
Of course, Aut(X, µ) is the group of all permutations of Z. We see that T ∈ Aut 2 (X, µ) if and only if

(3-1) n∈Z a n-1 a n -1 2 a n = n∈Z ( √ a n-1 - √ a n ) 2 < ∞.
In a similar way, T ∈ Aut 1 (X, µ) if and only if

n∈Z |a n-1 -a n | < ∞.
If the latter inequality holds then there exist the two limits a +∞ := lim n→+∞ a n and a -∞ := lim n→-∞ a n . Moreover, it is easy to verify that

(3-2) χ(T ) = a +∞ -a -∞ .
Let (X * , µ * , T * ) denote the Poisson suspension of the dynamical system (X, µ, T ). The mapping ω → (ω(n)) n∈Z is an isomorphism of (X * , µ * ) onto the infinite product space (Z Z + , n∈Z κ n ), where κ n is the Poisson distribution with parameter a n for each n ∈ Z. Moreover, this mapping conjugates T * with the shift on Z Z + . For this reason, from now on we view T * as the shift Z Z + defined by

(T * y) n := y n+1 for all y = (y n ) n∈Z ∈ Z Z + .
It was shown in [DaKoRo, Theorem A] that in the case where µ is non-atomic, T * is µ * -nonsingular if and only if T ∈ Aut 2 (X, µ). We now verify that the same holds also in our (purely atomic) case.

Proposition 3.1. T * is µ * -nonsingular if and only if (3-1) is satisfied. In this case for a.e. y = (y n ) n∈Z ∈ X * , (T * ) (y) = n∈Z κ n-1 (y n ) κ n (y n ) .
Proof. We first recall definition of the Hellinger distance H(λ, ξ) between two probability measures λ, ξ on a countable set C:

H 2 (λ, ξ) := 1 - C dλ dξ dξ = 1 2 c∈C λ(c) -ξ(c) 2 .
Given two Poisson distributions ν a and ν b on Z + with parameters a and b respectively, then the Hellinger distance H(ν a , ν b ) between ν a and ν b satisfies

H 2 (ν a , ν b ) = 1 -e -1 2 ( √ a- √ b) 2 .
Therefore it follows from the Kakutani criterion [Ka] that T * is µ * -nonsingular if and only if

(3-3) ∞ > 2 n∈Z H 2 (κ n , κ n+1 ) = 2 n∈Z (1 -e -1 2 ( √ a n - √ a n+1 ) 2
).

The series in the righthand side of (3-3) converges if and only if (3-1) holds.

In a similar way, the Kakutani criterion [Ka] can be applied to prove the following claim.

Proposition 3.2. Let λ be another non-degenerated measure on Z.

Then λ * is equivalent to µ * if and only if n∈Z λ(n) -µ(n) 2 < ∞.
Our next purpose is to investigate when T * is conservative or dissipative in terms of the original system (X, µ, T ). Let L -(µ) denote the set of limit points of the sequence (a n ) n<0 and let L + (µ) denote the set of limit points of the sequence (a n ) n>0 . If T ∈ Aut 2 (X, µ) then (3-1) implies that there exist reals

β + ≥ α + ≥ 0 and β -≥ α -≥ 0 such that L -(µ) = [α -, β -] and L + (µ) = [α + , β + ]. Proposition 3.3. (i) If T ∈ Aut 2 (X, µ) and L -(µ) ∩ L + (µ) = ∅ then T * is totally dissipative. (ii) If T ∈ Aut 1 (X, µ) and χ(T ) = 0 then T * is totally dissipative.
Proof. Since (ii) follows from (i) and (3-2), it suffices to prove (i). Let H(., .) denote the Hellinger distance on the set of probability measures on Z + . It follows from the condition of (i) that there is δ > 0 and N > 0 such that H(κ i , κ j ) > δ whenever i < -N and j > N . This inequality and Proposition 3.1 yield that if n > 3N then

U n T * 1, 1 = n∈Z Z + dκ k-n dκ k (y k ) dκ k (y k ) = n∈Z (1 -H 2 (κ k-n , κ k )) ≤ n 3 <k< 2n 3 (1 -H 2 (κ k-n , κ k )) < (1 -δ 2 ) n/3 . Hence ∞ n=1 U n T * 1, 1 < ∞ n=1 (1 -δ 2 ) n/3 < ∞.
It remains to use Lemma 2.4. We will also need a sufficient condition for conservativeness of T * . Proposition 3.4. Let T ∈ Aut 1 (X, µ), χ(T ) = 0 and

dµ dµ•T -n 2 -1 ∈ L 1 (X, µ) for each n ∈ N. If there is a sequence (b n ) ∞ n=1 of positive reals such that ∞ n=1 b n = ∞ but ∞ n=1 b 2 n e X dµ dµ•T -n 2 -1 dµ < ∞ then T * is conservative. Proof. Since χ(T ) = 0, it follows that k∈Z (a k -a k-n ) = 0 for each n > 0. Therefore log dµ * dµ * • T -n * (y) 2 = - k∈Z 2 log κ k-n (y k ) κ k (y k ) = -2 k∈Z y k log a k-n a k + a k -a k-n = -2 k∈Z y k log a k-n a k = -y, 2 log dµ • T -n dµ
at a.e. y ∈ X * . Here we consider y as a measure on X and use the notation y, f for the integral of a function f ∈ L 1 (X, y) with respect to y. Utilizing the above formula and the Laplace transform we obtain that

dµ * dµ * • T -n * 2 2 = X * e -y,2 log dµ•T -n dµ dµ * (y) = e X e -2 log dµ•T -n dµ -1 dµ = e X dµ dµ•T -n 2 -1 dµ .
It remains to apply Proposition 2.6.

From now on we will assume that a n := ae n for some a > 0 and a sequence ( n ) n∈Z of reals such that

(3-4) n = 0 if n ≤ 1, lim n→+∞ n = 0, ∞ n=1 2 n = ∞ but ∞ n=1 4 n < +∞.
Let R denote the one-sided Bernoulli shift on the space ((Z + ) N , ∞ n=1 κ n ), and let π denote the projection

(Z + ) Z (x n ) n∈Z → (x n ) n>0 ∈ (Z + ) N . Then Rπ = πT * . In view of Example 2.3, T * is the natural extension of R. Theorem 3.5. If (3-4) holds then the cocycle log α ρ R of the equivalence relation S R associated with R is ergodic. Proof. According to Example 2.3, S R is the tail equivalence relation on (Z + ) N . By (2-2), ρ R (v) = ∞ j=2 κ j-1 (v j ) κ j (v j ) for each v = (v j ) ∞ j=1 ∈ (Z + ) N . Hence for each n > 0, ρ R (v) • • • ρ R (R n-1 v) = n j=2 κ 1 (v j ) κ j (v j ) j>n κ j-n (v j ) κ j (v j )
.

Suppose now that R n v = R n w. Then v j = w j for all j > n. Therefore

α ρ R (v, w) := ρ R (v) • • • ρ R (R n-1 v) ρ R (w) • • • ρ R (R n-1 w) = n j=2 κ j (w j ) κ j (v j ) n j=2 κ 1 (v j ) κ 1 (w j ) = n j=2 a j a 1 w j -v j .
Thus we obtain that for each (v, w)

∈ S R , log α ρ R (v, w) = j>1 (w j -v j ) j .
Let ∆ S R denote the Radon-Nikodym cocycle of S R . Utilizing Example 1.1 and the definition of κ j we obtain that

log ∆ S R (v, w) = log n j=1 κ j (w j ) κ j (v j ) = j>0 (w j -v j ) log a j -log w j ! v j ! .
Now we fix an infinite subset J ⊂ {2, 3, 4, . . . } with j∈J 2 j < ∞. Then it follows from (3-4) that j ∈J 2 j = ∞. Hence we can change ( n ) n∈Z with another sequence coinciding with the original one if n ∈ J and such that n = 0 for each n ∈ J. Of course, the new sequence satisfies (3-4) and determines a new measure µ on X. Proposition 3.2 yields that µ * ∼ µ * . Moreover, the Radon-Nikodym derivative v → d µ * dµ * (v) does not depend on the first coordinate of v = (v j ) ∞ j=1 because 1 ∈ J. Therefore according to Remark 2.1, the cocycle α ρ R , µ * is cohomologous to α ρ R ,µ * . Hence α ρ R , µ * is ergodic if and only if α ρ R ,µ * is ergodic. In the sequel, by α ρ R we mean α ρ R , µ * . Thus, we may assume without loss of generality that the triplet ((

Z + ) N , ∞ n=1 κ n , S R ) is isomorphic to (Z, η, T ), where Z = (Z + × Z + ) N , η = ∞ j=1 (κ j ⊗ κ 0 )
and T is the tail equivalence relation on Z. The corresponding isomorphism is given by the map

Z N + v → ((v j ) j ∈J , (v j ) j∈J ) ∈ (Z + × Z + ) N .
Indeed, we use here the fact that the sets J and N \ J are both infinite and hence admit a bijection onto N and, furthermore, κ j = κ 0 if j ∈ J.

Let z = (v j , w j ) j>0 and z = (v j , w j ) j>0 are two T -equivalent points with v j , w j , v j , w j ∈ Z + . Computing log α ρ R and log ∆ S R in the "new coordinates" we obtain that

log α ρ R (z, z ) = j>0 (v j -v j ) j and log ∆ S R (z, z ) = j>0 (v j -v j ) log a j + (w j -w j ) log a 0 -log v j ! v j ! -log w j ! w j ! .
Denote by τ the flip on

Z + × Z + , i.e. τ (v, w) = (w, v). Define a transformation τ n of Z by setting τ n := τ × • • • × τ n times × Id × Id • • • . Of course, τ n ∈ [T ] and (3-5) log α ρ R (z, τ n z) = n j=1 (w j -v j ) j = log ∆ S R (z, τ n z)
at a.e. z ∈ Z for each n > 0.

Claim 1. For each p > 0,

lim n→∞ η({z ∈ Z | log α ρ R (z, τ n z) > -p}) = 0.
To prove this claim we first define mappings X j : Z → R by setting

X j (z) := (w j -v j ) j
for each z = (v m , w m ) m>0 and j ∈ N. Then the following are satisfied: -X 1 , X 2 , . . . is a sequence of independent random variables, -for each j > 1, the measure η • ( 1 j X j ) -1 is the Skellam distribution χ a 0 ,a j with parameters (a 0 , a j ) (see Appendix A) and -for each n > 0 and a.e. z ∈ Z,

(3-6) log α ρ R (z, τ n z) = n j=1 X j (z).
We now have (see Appendix A):

E(X j ) = j (a 0 -a j ) = a j (1 -e j ), σ 2 (X j ) = 2 j (a 0 + a j ) = a 2 j (1 + e j )
and for each t ∈ R,

(3-7) φ X j -E(X j ) (t) := E(e it(X j -E(X j )) )
= e -(a+ae j )+ae it j +ae j e -it j -itE(X j )

= e a((e it j -1+e -it j -1)+(e j -1)(e -it j -1+it j ))

= e -4a sin 2 t j 2 +a(e j -1)(e -it j -1+it j ) .

Let

β n := n j=1 2 j -1
2 . Utilizing (3-7), we now compute the characteristic function of the random variable

Y n := β n n j=1 (X j -E(X j )) at a point t ∈ R: (3-8) φ Y n (t) = n j=1 φ X j -E(X j ) (β n t) = e -4a n j=1 sin 2 t j β n 2 +a n j=1 (e j -1)(e -itβ n j -1+it j β n )) .
Since the sequence ( n ) ∞ n=1 is bounded and

∞ j=1 2 j = ∞, it follows that β n max 1≤j≤n | j | → 0 as n → ∞.
Therefore there is

C = C(t) > 0 such that n j=1 sin 2 t j β n 2 - n j=1 t j β n 2 2 ≤ Cβ 4 n n j=1 4 j and (3-9) n j=1 (e j -1)(e -itβ n j -1 + it j β n )) ≤ Cβ 2 n n j=1 | j | 3 . (3-10)
For each > 0, there is N > 0 such that | n | < whenever n > N . Therefore

β 4 n n j=1 4 j ≤ β 4 n N j=1 4 j + 2 n j=N +1 2 j ( n j=1 2 j ) 2 < 2 and β 2 n n j=1 | j | 3 ≤ β 2 n N j=1 | j | 3 + n j=N +1 2 j n j=1 2 j < 2
if n is large enough. It follows that the lefthand sides in (3-9) and (3-10) go to 0 as n → ∞. Hence (3-9) yields that lim n→∞ n j=1 sin 2 t j β n 2 = t 2 4 for each t ∈ R. From this, (3-8) and (3-10), we deduce that

lim n→∞ φ β n n j=1 (X j -E(X j )) (t) = e -at 2 for each t ∈ R.
However, the map R t → e -at 2 is the characteristic function of a normal distribution. Hence by Lévy's continuity theorem, (β n n j=1 (X j -E(X j ))) ∞ n=1 converges in distribution to a Gaussian random variable. Since

n j=1 2 j → ∞, it follows that lim n→∞ β n n j=1 E(X j ) = lim n→∞ a n j=1 j (1 -e j ) n j=1 2 j = -∞. Therefore β n n j=1 X j → -∞ in distribution. This, in turn, yields that n j=1 X j → -∞ in distribution. Thus Claim 1 is proved.
Claim 2. For η-a.e. z ∈ Z and each > 0, there is N such that |X n (z)| < for each n > N . To prove this claim we let

B n := {z = (v j , w j ) j∈N ∈ Z | |X n (z)| > | n |} = {z ∈ Z | |w n -v n | > | n | -1/2 }. Then η(B n ) = χ a 0 ,a n ({k ∈ Z | |k| > | n | -1/2 }). According to (A-1), there is L > 0 such that if | n | -1/2 > L then |k|>| n | -1/2 χ a 0 ,a n (k) < 4 n . Hence η(B n ) < 4 n . Therefore ∞ n=1 η(B n ) < ∞ n=1
4 n < ∞ according to (3-4). Claim 2 follows from this via the Borel-Cantelli lemma.

Claim 3. Each r < -1 is an essential value of log α ρ R . We first recall that given d > 0 and elements c 1 , . . . , -

c d ∈ Z + × Z + , the cylinder [c 1 , . . . , c d ] d 1 is the set {z = (z n ) n∈Z ∈ Z | z j = c j , 1 ≤ j ≤ d}. To prove the claim, take > 0, a positive integer k and a cylinder C = [(a 1 , b 1 ), . . . , (a k , b k )] k 1 ⊂ Z, where a 1 , b 1 , . . . , a k , b k ∈ Z + . It
η(A) > 0.5η(C), -max z∈A |X j (z)| < whenever N ≥ j > M and -max z∈A N j=M +1 X j (z) < -r. For z ∈ A, let l(z) be the smallest number l > M such that l j=M +1 X j (z) < -r. Then l(z) ≤ N and (3-11) l(z) j=M +1 X j (z) -r ≤ |X l(z) (z)| < .
We now set

ψ(z) := ((a 1 , b 1 ), . . . , (a k , b k ), z k+1 , . . . , z M , τ (z M +1 ), . . . , τ (z l(z) ), z l(z)+1 , . . . ) for each z ∈ A. Then ψ(z) ∈ C, (z, ψ(z)) ∈ T and (3-12) | log α ρ R (z, ψ(z)) -r| <
in view of (3-11) and (3-6). We now show that the map ψ : A z → ψ(z) ∈ C is oneto-one. Suppose that ψ(z) = ψ(z ) for some z = (z j ) j∈Z ∈ A and z = (z j ) j∈Z ∈ A.

If l(z) = l(z ) then obviously z = z . Therefore suppose that l(z) > l(z ). Then the equality ψ(z) = ψ(z ) implies that z j = z j if 1 ≤ j ≤ M and τ (z j ) = τ (z j ) if

M < j ≤ l(z ). Hence z j = z j if 1 ≤ j ≤ l(z ). Therefore -r > l(z ) j=M +1 X j (z ) = l(z )
j=M +1 X j (z). This yields that l(z) ≤ l(z ), a contradiction. Thus, ψ is one-toone. It follows from and (3-5) and(3-12) that -r -< log ∆ S R (z, τ n z) < -r + for all z ∈ A. Lemma 1.2 implies now that r is an essential value of log α ρ R . Thus, Claim 3 is proved.

Since the essential range of log α ρ R is a closed subgroup of R containing an arbitrary real less than -1 (in view of Claim 3), it follows that this subgroup is R. Hence log α ρ R is ergodic.

Remark 3.6. While proving Theorem 3.5 we also showed as a byproduct that the cocycle ∆ S R is also ergodic, i.e. S R is of type III 1 . In fact, we proved a stronger result. Let E denote the orbit equivalence relation on (Z + ) N generated by the group of finite permutations of the coordinates. Then E is a proper subrelation of S R . It follows from the proof of Theorem 3.5 that E is of type III 1 and the restriction of the cocycle log α ρ R to E is ergodic.

The next corollary is the main result of this section.

Corollary 3.7. If (3-4) holds and T * is conservative then the Maharam extension of T * is a weakly mixing K-automorphism. In particular, T * is weakly mixing and of stable type III 1 .

Proof. It follows from Theorem 3.5 that α ρ R is ergodic. Since T * is the natural extension of R, the Maharam extension (T * ) ρ T * of T * is a nonsingular Kautomorphism by Proposition 2.2. Since T * is conservative, (T * ) ρ T * is also conservative according to the Maharam theorem (see [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF]Theorem 3.4.1], [Sc, Theorem 5.5]). It follows that (T * ) ρ T * is weakly mixing. This implies that T * is weakly mixing and of type III 1 .

We now provide concrete examples of conservative T * such that (3-4) holds.

Example 3.8. Let n := 0 if n ≤ 1 and n := -n -1/2 if n > 1. Then (3-4) holds. We claim that if 0 < a < 1 6 then T * is conservative. For that we will utilize Proposition 3.4. We first note that

X dµ dµ • T -n 2 -1 dµ = a k∈Z e 2( k -k-n ) -1 e k = a k∈Z e 3 k -2 k-n -e k and |3 k -2 k-n | ≤ 3| k |.
Therefore it follows from the Taylor expansion of the exponential function that

e 3 k -2 k-n -e k = 2 k -2 k-n + 1 2 (3 k -2 k-n ) 2 - 1 2 2 k + O( 3 k ) as k → ∞. Since k∈Z | k | 3 < ∞ and k∈Z ( k -k-n ) = 0, k∈Z (e 3 k -2 k-n -e k ) = 1 2 k∈Z (3 k -2 k-n ) 2 -2 k + O(1) = 2 k∈Z 2 2 k + 2 k-n -3 k k-n + O(1) = 4 n+1 k=2 1 k + 2 k>n+1 1 k -n - 1 k + 3 k - 3 k(k -n) + O(1) = 6 n+1 k=2 1 k + 6 k>n+1 1 k - 1 k(k -n) + O(1) = 6 log n + 6 +∞ n+1 1 t - 1 t(t -n) dt + O(1) = 6 log n + 6 log t -log t - n 2 + t(t -n) +∞ n+1 + O(1) = 6 log n + O(1)
as n → +∞. Thus, we have shown that

X dµ dµ • T -n 2 -1 dµ = 6a log n + O(1).
It follows that there is a constant C > 0 such that for each n > 0, (3-13)

X dµ dµ • T -n 2 -1 dµ < 6a log n + C. Choose a real β such that 1 ≥ β > 1 2 + 3a. It exists because a < 1 6 . Now let b n := n -β . Then ∞ n=1 b n = ∞ but in view of (3-13), ∞ n=1 b 2 n e X dµ dµ•T -n 2 -1 dµ ≤ e C ∞ n=1 1 n 2β-6a < ∞.
It follows from Proposition 3.4 that T * is conservative. It is weakly mixing and of type III 1 by Corollary 3.7.

We now show that the restriction a < 1 6 in Example 3.8 can not be dropped. For that we'll need a criterion for dissipativity from [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Corollary 6.8].

Lemma 3.9.

If T ∈ Aut 2 (X, µ) and n≥0 e -1 2 dµ•T n dµ -1 2 2 < ∞ then T * is totally dissipative.
Proposition 3.10. Let (X, µ, T ) be as in Example 3.8 but a > 4. Then T * is totally dissipative.

Proof. For each n > 0,

dµ • T n dµ -1 2 2 = a k>-n e n+k -k 2 -1 2 e k = a 1 k=-n+1 e n+k 2 -1 2 + a ∞ k=2 e n+k -k 2 -1 2 e k = a n+1 k=1 e k 2 -1 2 + a ∞ k=2 e n+k 2 -e k 2 2 = a 4 n+1 k=1 1 √ k + O 1 k 2 + a 4 +∞ k=2 1 √ n + k - 1 √ k + O 1 k 2 = a 4 n+1 k=1 1 k + a 4 +∞ k=2 1 n + k + 1 k - 2 k(n + k) + O(1)
as n → ∞. Since

n+1 k=1 1 k = log n + O(1) and +∞ k=2 1 n + k + 1 k - 2 k(n + k) = +∞ 2 1 n + t + 1 t - 2 t(n + t) dt + o(1) = log(t(n + t)) -2 log t + n 2 + t(n + t) +∞ 2 + o(1) = log n + O(1),
we obtain that

dµ•T n dµ -1 2 2 = a 2 log n + O(1) as n → ∞. It follows that there is a real D such that for each n > 0, (3-14) dµ • T n dµ -1 2 2 ≥ D + a log n 2 .
Since a > 4, we deduce from (3-14) that

n≥0 e -1 2 dµ•T n dµ -1 2 2 ≤ n≥0 e -1 2 (D+ a log n 2 ) = e -D 2 n≥0 n -a 4 < ∞.
It follows now from Lemma 3.9 that T * is totally dissipative.

Poisson suspensions of type III 1 over a non-atomic base

In this subsection we construct concrete examples of weakly mixing Poisson suspensions of type III 1 which are Poisson suspensions of nonsingular transformations defined on non-atomic σ-finite spaces.

Let (X, µ, T ) be as in § 3, i.e. X = Z, µ a non-degenerated measure on X and T the unit translation on X. Let K = [0, 1]. Denote by λ the Lebesgue measure on K. Let Y := K × Z and let Q : Y → Y be the direct product of the identity on K and T on X, i.e. Q(k, n) := (k, n + 1) for all k ∈ K and n ∈ Z. Then the product measure

ν := λ ⊗ µ is a non-atomic measure on Y . Of course, Q is ν-nonsingular. Moreover, Q (k, n) = T (n) = a n-1 a n ,
where a n := µ({n}). The dynamical system (X, µ, T ) is a factor of (Y, ν, Q). Moreover, the corresponding projection

(4-1) ϑ : Y (k, n) → n ∈ X
is relatively finite measure preserving (r.f.m.p.

(Q) = χ(T ) = lim n→+∞ a n -lim n→-∞ a n .
To describe the Poisson suspension of (Y, ν, Q) we first write Y as the "disjoint" union Y = n∈Z K × {n}. Under the natural identification, we may assume that

Y * = n∈Z (K × {n}) * and ν * = n∈Z ν * n ,
where ν n := (ν (K × {n})). If we identify naturally K × {n} with K then ν n corresponds to a n • λ for each n ∈ Z. Hence the Poisson suspension Q * of Q corresponds to the 2-sided Bernoulli shift on the infinite product space

(Y * , ν * ) = (K * ) Z , n∈Z (a n • λ) * .
We also recall that we view (X * , µ * ) as the infinite product ((Z + ) Z , n∈Z κ n ), where κ n := (µ {n}) * (see § 3). The factor map ϑ from (4-1) generates a factor map ϑ * : Y * → X * . Thus we have that ϑ

* Q * = T * ϑ * and µ * = ν * • (ϑ * ) -1 .
Since ϑ is r.f.m.p., ϑ * is r.f.m.p. whenever Q * is ν * -nonsingular. It is important to note that ϑ * has a "coordinate-wise" structure, i.e. ϑ * y = (ϑ * n y n ) n∈Z for each y = (y n ) n∈Z ∈ Y * , where ϑ * n is a measure preserving mapping from the probability space (K * , ν * n ) onto the probability space (Z + , κ n ) given by K * y n → y n (K) ∈ Z + . Corollary 4.1. The Bernoulli shift Q * is ν * -nonsingular if and only if T * is µ *nonsingular, i.e. if and only if (3-1) is satisfied. In this case, the mapping

ϑ * : Y * y = (y n ) n∈Z → (ϑ * n y n ) n∈Z := (y n (K)) n∈Z ∈ X * intertwines Q * with T * , maps ν * onto µ * and dν * • Q * dν * (y) = dµ * • T * dµ * (ϑ * y) = n∈Z κ n-1 (ϑ * n y n ) κ n (ϑ * n y n ) for a.e. y = (y n ) n∈Z ∈ Y * . Hence (Y * , ν * ) is isomorphic to the product space (K Z × X * , λ Z × µ *
) in such a way that (i) ϑ * corresponds to the projection to the second coordinate and (ii) Q * corresponds the direct product B × T * , where B :

K Z → K Z is the Bernoulli shift preserving λ Z . Proof. For each n ∈ Z, we disintegrate ν * n relative to κ n (via ϑ * n ): (4-2) ν * n = k∈Z + κ n (k)ξ n,k ,
where (ξ n,k ) k∈Z + is the corresponding sequence of conditional probabilities on K * . Using (4-2), we obtain the disintegration of ν * relative to µ * (via ϑ * ):

ν * = X * n∈Z ξ n,k n dµ * ((k n ) n∈Z ).
Since ϑ * intertwines the Bernoulli shifts on Y * and X * respectively, the r.f.m.p. property of ϑ * means that n∈Z ξ n,k n+1 = n∈Z ξ n+1,k n+1 and hence ξ n,k n+1 = ξ n+1,k n+1 for all n ∈ Z and µ * -a.e. (k n ) n∈Z ∈ X * . Hence there is a sequence (ξ k ) k∈Z + of probability measures on K * such that ξ n,k = ξ k and therefore

ν * = X * n∈Z ξ k n dµ * ((k n ) n∈Z ).
The measure ξ k is non-atomic for each k > 0 and the measure ξ 0 is a delta-measure. For a.e. k = (k n ) n∈Z ∈ X * , there exist infinitely many positive n with k n > 0 and infinitely many negative n with k n > 0. Hence for such a k and every n ∈ Z, there are (uniquely defined) two integers l n (k) and r n (k) such that the following are satisfied:

-

l n (k) ≤ n ≤ r n (k), -k l n (k) = 0 and k r n (k)+1 = 0 and -if l n (k) < n ≤ r n (k) then k n = 0.
Of course, the mappings X * k → l n (k) ∈ Z and X * k → r n (k) ∈ Z are measurable for every n ∈ Z. For each n ∈ Z \ {0} and l ∈ N, there is a measure theoretical isomorphism

τ k,l : ((K * ) l , ξ k ⊗ ξ 0 ⊗ • • • ⊗ ξ 0 ) → (K l , λ l ).
Next, for each point k = (k n ) n∈Z ∈ X * and y = (y n ) n∈Z ∈ (K * ) Z , we define a point k, y = (z n ) n∈Z of K Z in the following way: z n is the (n -l n (k) + 1)-th symbol in the block

τ l n (k),r n (k)-l n (k)+1 [y l n • • • y r n ] ∈ K r n (k)-l n (k)+1 .
We now define an isomorphism τ of (Y * , ν * ) onto (K Z × X * , λ Z ⊗ µ * ) by setting τ (y) := ( ϑ * y, y , ϑ * y).

It is straightforward to verify that (i) and (ii) hold.

We now state the main result of this section. . This result will be improved in § 7 below.

Generic Poisson suspension is of type III 1

Throughout this section we assume that µ is an infinite σ-finite non-atomic measure on standard Borel space (X, B). Given two subsets A, B ⊂ X of finite measure, let τ A,B denote a µ-nonsingular bijection from A onto B such that

dµ•τ A,B dµ (x) = µ(B) µ(A) for all x ∈ A.
We now recall a definition and some facts from [DaKoRo].

Definition 5.1. A nonsingular transformation T of a σ-finite standard measure space (X, µ) is locally aperiodic if there is a subset A ⊂ X of positive finite measure such that T x = x if x ∈ A and T n x = x for each x ∈ A and n > 0.

Of course, each locally aperiodic transformation T belongs to Aut 1 (X, µ). It is easy to verify that χ(T ) = 0. It was also shown in [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Proposition 5.4 and Theorem 5.8(2)] that:

(F1) the conjugacy class of T in Aut 2 (X, µ) is d 2 -dense in Aut 2 (X, µ) and (F2) the conjugacy class of T in Aut 1 (X, µ) is d 1 -dense in Ker χ.
Proposition 5.2. Let T ∈ Aut(X, µ) and there are a partition X = n∈Z W n of X into subsets W n of finite measure and a sequence (a n ) n∈Z of reals such that T W n = W n+1 and T (x) = a n at a.e. x ∈ W n and for each n ∈ Z. Suppose that there exists a limit lim |n|→∞ µ(W n ) ∈ (0, +∞).

(i) If T ∈ Aut 2 (X, µ) then the conjugacy class of T is dense in Aut 2 (X, µ).

(ii) If T ∈ Aut 1 (X, µ) then the conjugacy class of T is dense in Ker χ.

Proof. (i) Since T ∈ Aut 2 (X, µ), ∞ > √ T -1 2 2 = n∈Z W n ( T (x) -1) 2 dµ(x) = n∈Z ( √ a n -1) 2 µ(W n ). Since a n = µ(W n+1 )/µ(W n ), we obtain that (5-1) n∈Z ( µ(W n+1 ) -µ(W n )) 2 < ∞.
If we show that the d 2 -closure of the conjugacy class of T contains a locally aperiodic transformation then (F1) yields that the conjugacy class of T is dense in Aut 2 (X, µ).

We now construct such a transformation. For each n > 0, we set h n := 2 n . Let B 0 be a subset of X with µ(B 0 ) = 1 and let S be a µ-preserving transformation 20 of X such that Sx = x if x ∈ B 0 and the restriction of S to B 0 is isomorphic to the 2-adic odometer. Then S is locally aperiodic and there exists a decreasing infinite sequence of subsets

B 0 ⊃ B 1 ⊃ • • • in X such that S i B n ∩ S j B n = ∅ if 0 ≤ i < j < h n and h n -1 i=0 S i B n = B 0 .
We see that S has a "cyclic structure of period h n on B n ". The idea of the following argument is to define for each n > 0, a transformation S n extending the cyclic structure of S from B n to a larger subset in such a way S n → S and S n is conjugate to a cyclic permutation of the finite sequence W -h n-1 , . . . , W h n-1 -1 that approaches to T as n → ∞: Since the convergence is considered in d 2 , we should choose carefully the transformations in these constructions to control their Radon-Nikodym derivatives.

More precisely, we now construct a sequence of µ-preserving transformations (S n ) ∞ n=1 and a sequence (B n ) ∞ n=1 of subsets in X such that -S n B 0 = S for each n and S n → S weakly (and hence in

d 2 ) as n → ∞, -B n ⊃ B n , µ(B n ) = µ(W -h n-1 ), S i n B n ∩ S j n B n = ∅ for all 0 ≤ i < j < h n and each n. -If we set Y n := h n -1 i=0 S i n B n and Y • n := Y n \ B n then for each subset A ⊂ X of finite measure, lim n→∞ µ(A ∩ Y • n ) = µ(A). To see that such a construction is possible, it is convenient to think that X = [0, +∞), µ is a Lebesgue measure on X, B 0 = [0, 1), B n := [0, h -1 n ).
To define B n and S n , we first select a real δ n > 0 and an integer

D n > 0 such that (a) δ n h n < n -1 and (b) µ(B n ) + δ n D n = µ(W -h n-1 ). Now we set B n := B n D n -1 j=0 [1 + jh n δ n , 1 + (jh n + 1)δ n ) and S n x :=          Sx, if x ∈ B 0 x + δ n , if 1 ≤ x < 1 + (D n h n -1)δ n x -(D n h n -1)δ n , if 1 + (D n h n -1)δ n ≤ x < 1 + D n h n δ n x, if x ≥ 1 + D n h n δ n .
It follows from (a) that S n → S weakly as n → ∞. (b) implies that µ(B n ) = µ(W -h n-1 ). The sets S i n B n , i = 0, . . . , h n -1, are mutually disjoint and their union

Y n equals [0, 1 + h n D n δ n ). In particular, Y n ⊃ [0, 1 + h n b/2) eventually in n, where b := lim |m|→∞ µ(W m ). Hence n→∞ Y n = X. Since the set B n \ B n is "uniformly distributed" along Y n \ B 0 , we have that µ(Y • n ∩ [0, E)) → µ([0, E)) for each E > 1. Thus, the sequences (S n ) ∞ n=1 and (B n ) ∞ n=1 are as desired. We now set X n := h n -1 i=0 T i W -h n-1 and X • n := X n \ W -h n-1 . Then X • 1 ⊂ X • 2 ⊂ • • • and ∞ n=0 X • n = X.
For each n > 0, we select a measure preserving Borel bijection τ n of X \ X n onto X \ Y n and define a transformation R n of X by setting

R n x := S i n τ W -h n-1 ,B n T -i if x ∈ W i-h n-1 , 0 ≤ i < h n . τ n x if x ∈ X n . Then R n ∈ Aut(X, µ), R n X n = Y n and hence R n ∈ Aut 1 (X, µ). A straightforward verification shows that R n T -1 R -1 n = S -1 n on Y • n . Therefore S n R n T -1 R -1 n → Id weakly as n → ∞.
Next, we note that the Radon-Nikodym derivatives of T -1 and R n are constant on each level W j , j ∈ Z. Since S n preserves µ and T -1 W j = W j-1 for each j ∈ Z, it follows that the Radon-Nikodym derivative of the transformation

S n R n T -1 R -1 n is constant on the subset R n W j for each j ∈ Z. Hence we compute easily that (S n R n T -1 R -1 n ) (x) equals 1, if x ∈ Y • n µ(S n R n W i-h n-1 -1 ) µ(R n W i-h n-1 ) , if x ∈ R n W i-h n-1 and i ∈ {1, . . . , h n -1}. If i ∈ {1, . . . , h n } then µ(S n R n W i-h n-1 -1 ) = µ(W i-h n-1 -1 ). If i = h n then µ(S n R n W i-h n-1 -1 ) = µ(S n S h n -1 n τ W -h n-1 ,B n W -h n-1 ) = µ(W -h n-1 ). Therefore (S n R n T -1 R -1 n ) (x) =          1, if x ∈ Y • n µ(W i-h n-1 -1 ) µ(W i-h n-1 ) , if x ∈ R n W i-h n-1 and i ∈ {1, . . . , h n } µ(W -h n-1 ) µ(W h n -h n-1 ) , if x ∈ R n W h n -h n-1 . Therefore (S n R n T -1 R -1 n ) -1 2 2 = i ∈{1,...,h n } µ(W i-h n-1 -1 ) -µ(W i-h n-1 ) 2 + µ(W -h n-1 ) -µ(W h n -h n-1 ) 2 . Since b < ∞, it follows that µ(W -h n-1 ) -µ(W h n-1 ) → 0 as n → ∞.
Utilizing this fact and (5-1) we obtain that

lim n→∞ (S n R n T -1 R -1 n ) -1 2 = 0. Therefore S n R n T -1 R -1 n → Id in d 2 as n → ∞. Since S n → S in d 2 , we obtain that SR n T -1 R -1 n → Id in d 2 as n → ∞.
Thus S belongs to the d 2 -closure of the conjugacy class of T in Aut 2 (X, µ), as desired.

(ii) is proved in a similar way. First, we note that the conditions T ∈ Aut 1 (X, µ) and lim |n|→∞ µ(W n ) < ∞ imply that T ∈ Ker χ. Indeed,

χ(T ) = X (T -1) dµ = n∈Z W n (T -1) dµ = n∈Z (µ(W n-1 ) -µ(W n )) = 0.
Secondly, we will use (F2) in place of (F1). Instead of (5-1) we now have that n∈Z |µ(W n+1 ) -µ(W n )| < ∞. Define R n in the same way as in the proof of (i). As was noted there, R n ∈ Aut 1 (X, µ). Moreover, it is easy to see that R n ∈ Ker χ. Slightly modifying the above argument and considering the L 1 -norm instead of the square of the L 2 -norm, we obtain that (

S n R n T -1 R -1 n ) -1 1 → 0. The latter yields that SR n T -1 R -1 n → Id in d 1 as n → ∞.
We recall two more facts from [DaKoRo]. Let E denote the subset of all ergodic transformations of type III 1 in Aut(X, µ). [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Theorem 5.8(3)]. We now state and prove the main result of the section.

(F3) E ∩ Aut 2 (X, µ) is a dense G δ in (Aut 2 (X, µ), d 2 ) [DaKoRo, Theorem 5.5]. (F4) E ∩ Aut 1 (X, µ) = E ∩ Ker χ is a dense G δ in (Ker χ, d 1 )
Theorem 5.3.

(i) The subset

E * 2 := {T ∈ E ∩ Aut 2 (X, µ) | T * is ergodic and of type III 1 } is a dense G δ in (Aut 2 (X, µ), d 2 ). (ii) The subset E * 1 := {T ∈ E ∩ Ker χ | T * is ergodic and of type III 1 } is a dense G δ in (Ker χ, d 1 ).
Proof. (i) Since the set of ergodic transformations of type III 1 is G δ in Aut(X * , µ * ) endowed with the weak topology [ChHaPr] and the map Aut 2 (X, µ) T → T * ∈ Aut(X * , µ * ) is continuous in view of [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Theorem 4.15 and Definition 4.5], it follows that the subset

E 2 := {T ∈ Aut 2 (X, µ) | T * is ergodic of type III 1 } is a G δ in (Aut 2 (X,
= E 2 ∩ E ∩ Aut 2 (X, µ).
We finally deduce (i) from these facts and (F3).

(ii) is proved in a similar way with usage of (F4) instead of (F3).

Poisson suspensions over dissipative bases

Let (X, B, µ) be a σ-finite infinite standard nonatomic measure space and let T ∈ Aut 2 (X, µ). Let T be totally dissipative. This means that there is a subset B ∈ B such that X = n∈Z T n B. Suppose that µ(B) < ∞. Since T ∈ Aut 2 (X, µ), it follows from [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Lemma 3.2] that µ(T n B) < ∞ for each n ∈ Z. Then without loss of generality we can assume that X = [0, 1]×Z and T (y, n) = (y, n+1) for all y ∈ [0, 1] and n ∈ Z. Let λ denote the Lebesgue measure on [0, 1]. We can also assume that there exist a sequence (a n ) n∈Z of functions a n ∈ L 1 ([0, 1], λ) such that a n > 0 a.e. and for each

F ∈ L 1 ([0, 1] × Z, µ), (6-1) X F dµ = n∈Z [0,1] F (y, n)a n (y) dλ(y).
It is straightforward to verify that T ∈ Aut 2 (X, µ) if and only if

n∈Z [0,1] a n+1 (y) a n (y) -1 2 a n (y) dλ(y) = n∈Z √ a n+1 - √ a n 2 2 < ∞.
In a similar way, T ∈ Aut 1 (X, µ) if and only if n∈Z a n+1 -a n 1 < ∞. The latter inequality implies that there exist two nonnegative functions a, b

∈ L 1 ([0, 1], λ) such that (6-2) lim n→∞ a -a n 1 = 0 and lim n→-∞ b -a n 1 = 0. Since χ(T ) = lim m→∞ (µ(T ([0, 1] × {-m, . . . , m})) -µ([0, 1] × {-m, . . . , m})), it follows that (6-3) χ(T ) = lim m→∞   m+1 j=-m+1 a j dλ - m j=-m a j dλ   = a 1 -b 1 .
The following theorem is the main result of this section.

Theorem 6.1. If T ∈ Aut 1 (X, µ) and χ(T ) = 0 then T * is totally dissipative.

Proof. Consider the Hopf decomposition of X: there are two invariant subsets X c and X d of X such that the dynamical system (X c , µ X c , T X c ) is conservative and the dynamical system (X d , µ X d , T X d ) is totally dissipative [START_REF] Aaronson | An introduction to infinite ergodic theory[END_REF]§1.1].

Since χ(T ) = 0, it follows that T is dissipative [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Proposition 4.13] and hence µ(

X d ) > 0. Of course, T X d ∈ Aut 1 (X d , µ d ), T X c ∈ Aut 1 (X c , µ c ) and χ(T ) = χ(T X d ) + χ(T X c ) = χ(T X d ).
Hence χ(T X d ) = 0. Moreover, T * is isomorphic to the Cartesian product of (T X d ) * and (T X c ) * . Therefore if (T X d ) * is totally dissipative then so is T * . In view of this, we may assume without loss of generality that T is totally dissipative itself. Hence there is a subset B ⊂ X such that X = n∈Z T n B. We may also assume that µ(B) < ∞. Indeed, if µ(B) = ∞ then we partition B into countably many subsets B m , m ∈ N of finite measure. For each m > 0, let

B m := n∈Z T n B m . Then B m is invariant under T and ∞ m=1 B m = X. Of course, T B m ∈ Aut 1 (B m , µ B m ) for each m > 0 and χ(T ) = ∞ m=1 χ(T B m
). Therefore there exists m 0 > 0 such that χ(T B m 0 ) = 0. Since T * is isomorphic to the Cartesian product of (T B m 0 ) * and (T (X \ B m 0 )) * , it follows that if (T B m 0 ) * is totally dissipative then so is T * . Thus, it suffices to consider only the case where µ(B) < ∞.

In this case we can consider X as [0, 1] × Z and T as the unit rotation along the second coordinate. We will also use the notation (a n ) n∈Z and λ introduced in the beginning of this section to describe µ via (6-1). Then T * is the Bernoulli shift on the infinite product space X * := n∈Z ([0, 1] * , κ * n ), where κ n is a measure equivalent to λ on [0, 1] with dκ n dλ = a n . We are going to apply Lemma 3.9 to prove that T * is totally dissipative. We will use the following equality:

(6-4) dµ • T n dµ -1 2 2 = k∈Z [0,1] a k+n a k -1 2 a k dλ = k∈Z √ a k+n - √ a k 2 2 .
Since for two nonnegative functions e, f ∈ L

1 ([0, 1], λ), e -f 1 = √ e -f , √ e + f ≤ √ e -f 2 • √ e + f 2 , and √ e + √ f 2 ≤ √ e 2 + √ f 2 = e 1 + f 1 , we obtain that (6-5) √ a k+n - √ a k 2 ≥ 1 2D a k+n -a k 1
for all k and n, where D := sup k∈Z a k 1 < ∞. It follows from (6-2) and (6-3) that there is N > 0 such that (6-6) a i -a j 1 > 1 3 |χ(T )| whenever i > N and j < -N .

Taking n > 3N and utilizing (6-4), (6-5), (6-6) we obtain that

dµ • T n dµ -1 2 2 ≥ k∈Z a k+n -a k 2 1 4D 2 ≥ -n 3 >k>-2n 3 a k+n -a k 2 1 4D 2 ≥ n |χ(T )| 2 108D 2 . Therefore n>0 e -1 2 dµ•T n dµ -1 2 2 < n>0 e -|χ(T )| 2 216D 2 n < ∞.
Hence T * is totally dissipative by Lemma 3.9.

Remark 6.2. This theorem can be extended to the case where T ∈ Aut 2 (X, µ) in the following way. Suppose that T is dissipative. Select a subset B ⊂ X of finite positive measure such that T n B ∩ T m B = ∅ if n, m ∈ Z and n = m. Then we represent the restriction of T to n∈Z T n B in the same way as in the beginning of this section such that (6-1) holds. If there exist δ > 0 and N > 0 such that for all n, m > N , we have that √ a n -√ a -m 2 > δ then T * is totally dissipative. This fact is proved in the very same way as Theorem 6.1. We leave details to the reader.

Section 7. Phase transition for conservativeness when scaling the intensity of Poisson suspensions Let (X, B, µ) be a σ-finite standard measure space with µ nonatomic and infinite. For t > 0, let µ t denote the measure on (X, B) given by µ t (B) := tµ(B).

Definition 7.1. Let T ∈ Aut 2 (X, µ). We say that T * is conservatively concrete for intensity scaling (CCIS) if for each t > 0, the Poisson suspension (X * , B * , µ * t , T * ) is either conservative or totally dissipative.

If T ∈ Aut 1 (X, µ) and χ(T ) > 0 then T is CCIS. On the other hand, if T ∈ Aut 2 (X, µ) and there is a T -invariant subset A ⊂ X of finite positive measure such that T A is totally dissipative and T (X \ A) preserves µ (X \ A) then T * is not CCIS. Indeed, (X * , µ * , T * ) is isomorphic to the Cartesian product

(A * , (µ A) * , (T A) * ) × ((X \ A) * , (µ (X \ A)) * , (T (X \ A)) * ).
We now set [A] j := {ω ∈ X * | ω(A) = j} for j = 0, 1. Then [A] j is invariant under T * and µ * ([A] j ) > 0 for j = 0, 1. Since [A] 0 is a singleton (modulo (µ A) * ), it follows that T * is conservative when restricted to this subset. One can check that T

* [A] 1 is totally dissipative. Since (T (X \ A)) * is conservative (because it is probability preserving), it follows that (T A) * × (T (X \ A)) * is conservative on [A] 0 × (X \ A) * and dissipative on [A] 1 × (X \ A) * . Hence T * is not CCIS.
Problem. What are necessary and sufficient conditions on (X, µ, T ) under which T * is CCIS?

In this paper we prove CCIS for a certain family of Poisson suspensions. In the proof of the following theorem we use an idea similar to what was used in [Ko2, Theorem 3] (see also [START_REF] Danilenko | Weak mixing for nonsingular Bernoulli actions of countable amenable groups[END_REF]Theorem 3.5]).

Theorem 7.2. Let T ∈ Aut 1 (X, µ). Suppose also that there is α > 1 such that α -1 < (T n ) (x) < α for each n > 0 at a.e. x ∈ X. Then T * is either conservative or totally dissipative. Hence T * is CCIS.

Proof. In view of the aforementioned remark, it suffices to consider only the case where χ(T ) = 0.

Let S be the subgroup of all µ-preserving transformations on (X, µ) such that for each S ∈ S, there is a subset A S ⊂ X of finite measure with Sx = x whenever

x ∈ A S . Let D := ω ∈ X * ∞ n=1 (T n * ) (ω) < ∞ .
We first show that D is invariant under S * for each S ∈ S. Indeed, for each n > 0, ω ∈ D and S ∈ S, we have (utilizing [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Corollary 4.1(3)

]) that (T n * ) (S * ω) = ω({x})>0 (T n ) (Sx) = x ∈A S ω({x})>0 (T n ) (x) x∈A S ω({x})>0 (T n ) (Sx) = (T n * ) (ω) x∈A S ω({x})>0 (T n ) (Sx) x∈A S ω({x})>0 (T n ) (x) < (T n * ) (ω) • α 2ω(A S ) . Hence ∞ n=1 (T n * ) (ω) ≤ α 2ω(A S ) ∞ n=1 (T n * ) (ω)
< +∞, i.e. S * ω ∈ D, as desired. We now prove that the group S * := {S * | S ∈ S} of µ * -preserving transformations of X * is ergodic4 . For that, we select a µ-preserving totally dissipative transformation Q of X and a sequence (S n ) ∞ n=1 of transformations from S that converges weakly to Q. Then the sequence (

(S n ) * ) ∞ n=1 weakly converges to Q * in Aut(X * , µ * ). If A is an S * -invariant subset of X * then A is invariant Q * because µ * (Q * A A) = lim n→∞ µ * ((S n ) * A A) = 0. Since Q * is isomorphic to a probabil- ity preserving Bernoulli shift, it is ergodic and hence µ * (A)(1 -µ * (A)) = 0. Hence S * is ergodic.
Since D is S * -invariant, µ * (D)(1 -µ * (D)) = 0 and the assertion of the theorem follows.

We also prove a general result related to CCIS.

Proposition 7.3. Let T ∈ Aut 1 (X, µ). (i) If (X * , µ * , T * ) is totally dissipative then (X * , µ * t , T * ) is totally dissipative for each t > 1. (ii) If (X * , µ * , T * ) is conservative then (X * , µ * t , T * ) is conservative for each t ∈ (0, 1).
Proof. (i) If χ(T ) = 0 then the assertion of the proposition is trivial. Hence from now on we assume that χ(T ) = 0. It follows from the condition of the proposition that the Cartesian product (X * × X * , µ * ⊗ µ * t-1 , T * × T * ) is totally dissipative. Therefore, by Lemma 2.5, (7-1)

+∞ n=1 dµ * • T n * dµ * (ω) dµ * t-1 • T n * dµ * t-1
(τ ) < +∞ at (µ * ⊗ µ * t-1 )-a.e. (ω, τ ) ∈ X * × X * . Since T n ∈ Aut 1 (X, µ), it follows from [START_REF] Danilenko | Nonsingular Poisson suspensions[END_REF]Corollary 4.1(3)] that for each n > 0, i.e. ω + τ is contained in the dissipative part of (X * , µ * , T * ). Since µ * t * µ * 1-t = µ * , the dissipative part of (X * , µ * , T * ) is of positive measure, a contradiction From Proposition 7.3 and Theorem 7.2 we deduce the main result of this section on a phase transition for conservativeness of Poisson suspensions while scaling the underlying intensity.

dµ * • T n * dµ * (ω) dµ * t-1 • T n * dµ * t-1 (τ ) = e X log
Corollary 7.4. Let T ∈ Aut 1 (X, µ). Suppose that there is α > 1 such that α -1 < (T n ) (x) < α for each n > 0 at a.e. x ∈ X. Then there is t 0 ∈ [0, +∞] such that the Poisson suspension (X * , µ * t , T * ) is conservative for each t ∈ (0, t 0 ) and the Poisson suspension (X * , µ * t , T * ) is totally dissipative for each t ∈ (t 0 , +∞). We call t 0 the bifurcation point. Of course, it is interesting when the bifurcation point is proper, i.e. 0 < t 0 < +∞. A concrete example of such a Poisson suspension was constructed in § 4 (see Remark 4.3). Combining it with Corollary 7.4 we obtain the following theorem.

Theorem 7.5. Let X = R, T x = x + 1 for all x ∈ X and µ be an absolutely continuous measure on X such that dµ(x) = f (x)dx with f (x) = 1, if x < 2 e -n -1/2 , if n ≤ x < n + 1 for an integer n > 1.

Then there exists t 0 ∈ [ 1 6 , 4] such that (X * , µ * t , T * ) is weakly mixing of stable type III 1 for each t ∈ (0, t 0 ) and (X * , µ * t , T * ) is totally dissipative for each t ∈ (t 0 , +∞). We note that Corollary 7.4 and Theorem 7.5 are the "nonsingular Poisson" analogues of a phase transition phenomenon discovered recently for the nonsingular Gaussian actions introduced by Y. Arano, Y. Isono and A. Marrakchi in [ArIsMa] (see Proposition 5.1 there).

We conclude this section with computing the conservativeness index of the Poisson suspension (X * , µ * t , T * ) in the case where the bifurcation point is proper.

Proposition 7.6. Let the condition of Corollary 7.4 hold. Suppose that the bifurcation point t 0 is proper. Then for each n ∈ N, if t 0 n+1 < t < t 0 n then (X * , µ * t , T * ) has conservative index n.

Proof. We have to prove that the n-th Cartesian power of (X * , µ * t , T * ) is conservative and the (n + 1)-th Cartesian power of (X * , µ * t , T * ) is dissipative. We first note that the n-th convolution of µ t is µ * nt . Arguing as in the proof of Proposition 7.3(i), we obtain that for each m > 0,

dµ * t • T m * dµ * t (ω 1 ) • • • dµ * t • T m * dµ * t (ω n ) = dµ * nt • T m * dµ * nt (ω 1 + • • • + ω n )
for (µ * t ⊗ • • • ⊗ µ * t )-a.e. (ω 1 , . . . , ω n ) ∈ X * × • • • × X * . Since nt < t 0 , the system (X * , µ * nt , T * ) is conservative and hence the above equality and Lemma 2.5 yield that the n-the Cartesian power of (X * , µ * t , T * ) is conservative. On the other hand, the (n + 1)-th convolution power of µ * t equals µ * (n+1)t with (n + 1)t > t 0 . Now, for each m > 0, we have that

dµ * t • T m * dµ * t (ω 1 ) • • • dµ * t • T m * dµ * t (ω n+1 ) = dµ * θ • T m * dµ * θ (ω 1 + • • • + ω n+1 )
for (µ * t ⊗ • • • ⊗ µ * t )-a.e. (ω 1 , . . . , ω n+1 ) ∈ X * × • • • × X * . Since (X * , µ * (n+1)t , T * ) is dissipative, the above equality and Lemma 2.5 yield that the direct product of the (n + 1)-th Cartesian power of (X * , µ * t , T * ) is dissipative, as desired. This yields the following estimation: for each A > 0, there is an integer L > 0 such that for each l > L (A-1) sup 0<a,b<A |k|≥l χ a,b (k) ≤ l -8 .

  follows from Claim 1 and Claim 2 that there are N > M > k and a subset I ⊂ (Z + × Z + ) N -k such that the subsetA := (z k+1 ,...,z N )∈I [(a 1 , b 1 ), . . . , (a k , b k ), z k+1 , . . . , z N ] N 1 ⊂ Csatisfies the following three conditions:

=

  Appendix A. Skellam distributions The Skellam distribution with parameters (a, b) is the distribution χ a,b of the difference X -Y of two independent random variables X and Y , each Poissondistributed with respective parameter (expected value) a ≥ 0 and b≥ 0. It is known that E(X -Y ) = a -b and σ 2 (X -Y ) = a + b. The characteristic function φ X-Y (t) := E(e it(X-Y ) ) of X -Y is e -(a+b)+ae it +be -it at each t ∈ R. Of course, χ a,b (Z) = 1. For each k ∈ Z,χ a,b (k) = e a-b a b where I k is the modified Bessel function of the first kind, i.e. I k (z) = I |k| (zj + |k|)! . From the two above formulas we deduce that for each L > 0, |k|≥L χ a,b (k) ≤ e a-b e a-b+ab a L e a + b L e b L! .

  ). It is straightforward to verify that Q ∈ Aut 2 (Y, ν) if and only if T ∈ Aut 2 (X, µ), i.e. if and only if (3-1) is satisfied. In a similar way, Q ∈ Aut 1 (Y, ν) if and only if T ∈ Aut 1 (X, µ). Hence if Q ∈ Aut 1 (Y, ν) then the limits lim n→+∞ a n and lim n→-∞ a n exist and χ

  If, moreover, T * is conservative then Q * is weakly mixing and of stable Krieger's type III 1 .Proof. Consider Q * as the direct product B × T * described in Corollary 4.1. Then the Maharam extension of Q * is isomorphic to the product of B with the Maharam extension of T * . It remains to apply Theorem 3.5 and a simple fact that the direct product of a (conservative) K-automorphism with a probability preserving Bernoulli shift is a (conservative) K-automorphism.Remark 4.3. Utilizing Example 3.8 we obtain concrete examples of ergodic conservative suspensions Q * of type III 1 for Q defined on a nonatomic infinite measure space. Moreover, it follows from Example 3.8 and Proposition 3.10 that Q * is totally dissipative if a > 4 and conservative (and hence weakly mixing and of stable type III 1 ) if 0 < a < 1 6

	Theorem 4.2. If (3-1) and (3-4) are satisfied then the Maharam extension of
	(Y

* , ν * , Q * ) is a K-automorphism.

  µ), d 2 ). By Theorem 4.2 and Example 3.8, E 2 = ∅. Moreover, there exists a transformation T ∈ E 2 satisfying the conditions of Proposition 5.2. Hence, by Proposition 5.2, the conjugacy class of T is dense in (Aut 2 (X, µ), d 2 ). Since E 2 is conjugacy invariant, it follows that E 2 is a dense G δ . Of course, E * 2

  It remains to apply Lemma 2.5.(ii) Let D t := {ω ∈ X * | < ∞} stand for the dissipative part of (X * , µ * t , T * ). Assume in the contrapositive that µ * t (D t ) > 0. Then the set D t × X * is contained in the dissipative part of the product (X * × X * , µ * t ⊗ µ * 1-t , T * × T * ). Hence for (µ * t ⊗ µ * 1-t )-a.e. (ω, τ ) ∈ D t × X * ,

			dµ•T n dµ	dω+ X log dµ•T n dµ	dτ
	= e X log dµ t •T n dµ t	d(ω+τ )
	=	dµ * t • T n * t dµ *	(ω + τ ).
	Since µ * * µ * t-1 = µ * t , we deduce from (7-1) that		+∞ n=1	dµ * t •T n * dµ * t	(ω) < +∞ at µ * t -a.e.
	ω. +∞ n=1 (ω) ∞ > dµ * t •T n * dµ * t +∞ n=1 dµ * t • T n * dµ * t (ω) dµ * 1-t • T n * 1-t dµ *	(τ ) =	n=1 +∞	dµ

* • T n * dµ * (ω + τ ),

In this paper we do not distinguish between objects (such as subsets, maps, cocycles, etc.) which agree almost everywhere.

The reader should not confuse α ρ T with ∆ S T . The former is the cocycle generated by the Radon-Nikodym derivative of T . The later is the Radon-Nikodym cocycle of S T . If T is invertible then α ρ T = ∆ S T . However, in the general case, the two cocycles are different-see the proof of Theorem

3.5 below.

This follows also from[START_REF] Shimomura | Poisson measures on the configuration space and unitary representations of the group of diffeomorphisms[END_REF] Theorem 2.3]. However we present here an alternative very short proof.

Acknowledgements. We thank the anonymous referee for careful reading the paper and making useful remarks.

Measured equivalence relations and their cocycles

Let (X, B, µ) be a standard σ-finite measure space. A Borel equivalence relation R ⊂ X × X is countable if for each x ∈ X, the R-equivalence class R(x) is