Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings

(1)

VOL.110 2008 NO.1

ERGODIC AUTOMORPHISMS WHOSE WEAK CLOSURE OF OFF-DIAGONAL MEASURES CONSISTS OF ERGODIC

SELF-JOININGS BY

Y.DERRIENNIC(Brest),K.FRCZEK(Toru«andWarszawa), M.LEMACZYK(Toru«andWarszawa)andF.PARREAU(Paris)

Abstra t. Basi ergodi properties of the ELF lass of automorphisms, i.e.of the lassofergodi automorphismswhoseweak losureofmeasuressupportedonthegraphsof iteratesof

T

onsistsofergodi self-joiningsareinvestigated.DisjointnessoftheELF lass with:2-foldsimpleautomorphisms,intervalex hangetransformationsgivenbyaspe ial typepermutationsandtime-onemapsofmeasurableowsisdis ussed.Allergodi Poisson suspensionautomorphismsaswellasdynami alsystemsdeterminedbystationaryergodi symmetri

α

-stablepro essesareshowntobelongtotheELF lass.

Introdu tion. The notion of disjointness between measure-preserving automorphisms of standard probability Borel spa es was introdu ed by Furstenberg [9 ℄ in 1967. Sin e then many results showing disjointness of some lasses have been proved (see e.g. [9 ℄, [12 ℄, [14 ℄, [19 ℄, [21 ℄, [26 ℄, [28 ℄, [29 ℄,[46 ℄, [47 ℄).

In[6 ℄ these ond and the third named authors of this paper introdu ed the notion of ELF (

1

) ow. An ELF ow is, by denition, an ergodi ow su hthatwhenwepasstotheweak losureofitstime-

t

maps onsideredas Markov operators of the underlying

L

2

-spa e, then all the weak limits are inde omposable Markov operators. The ELFpropertyis interesting only in thenon-mixing ase,andindeedin ontrastwiththisproperty,some lassi al weaklymixingbutnon-mixingspe ialowsoverirrationalrotationsor,more generally, over interval ex hange transformations turn out to have in the weak losure of Markov operators given by their time-

t

maps su iently de omposableMarkovoperators.Su howsareoftenspe ialrepresentations of some smooth ows on surfa es and a motivation to introdu e the ELF propertywastoprovedisjointness(inthesenseofFurstenberg)ofsu hows

2000Mathemati sSubje tClassi ation:37A05,37A50. Keywordsandphrases:joinings,ELFproperty,disjointness.

Resear hpartiallysupportedbyKBNgrant1P03A03826andMarieCurieTransfer ofKnowledge program,proje tMTKD-CT-2005-030042(TODEQ).

(

1

) Thea ronymELF omesfromtheFren habbreviationofergodi itédeslimites faibles.

[81℄

(2)

from the ELF lass (see [6 ℄, [7 ℄, [8 ℄). In parti ular, some lassi al smooth weakly mixing ows on surfa es (e.g. onsidered in [25 ℄) turn out to be disjoint fromtheELF lass.

Ontheotherhand,theELFpropertywasalso introdu edinthehopeof expressingthefa tthatagivenowisofprobabilisti origin.Indeed,arst attempt to dene a system to be of probabilisti origin might be via the Kolmogorovgroup propertyof the spe trum. However, ea h weakly mixing systemhas anergodi extension whi h has theKolmogorovgroup property, simply by taking the innite dire t produ t of the system. Therefore this spe tralproperty istoo weak to singleout systemsof probabilisti origin. As noti ed in [6 ℄ Gaussian ows enjoy the ELF property (this result also follows from some earlier results of [29 ℄). The present paper and, indepen-dently,the PhD thesis of E.Roy[36 ℄ are a further onrmation of the fa t that dynami al systems whose origin are well-known lasses of stationary pro esses(see below)are insidetheELF lass. We also mentionthat inthe general ase,in luding mixing,another joiningproperty(satised for exam-plebyowswithRatner'sproperty[35 ℄)hasbeenintrodu edin[43 ℄toshow disjointness fromGaussian systems.

In this paper, instead of ows, we onsider the ELF property for au-tomorphisms. One of the main results of the paper states that all ergodi Poisson suspension automorphismsenjoytheELFproperty.Thisresultisa onsequen e of Theorem 1 belowsaying thatPoissonian joiningsof ergodi Poisson automorphisms remain ergodi ; the same result is also proved in there ent,independent paper[36 ℄.Moreover, we onsiderso alled

α

-stable automorphisms, i.e. ergodi automorphisms a ting on a spa e whose mea-surable stru ture is determined by an invariant real subspa e in whi h all variablesaresymmetri

α

-stable (

0 < α < 2

,for

α = 2

we omeba kto the Gaussian ase). We prove (Theorem 3 below) that

α

-stable self-joinings of su hautomorphismsmustne essarily beergodi , fromwhi htheELF prop-ertydire tlyfollows.Intheaforementionedthesis[36℄,afurtherstepforward isevenmade:itisprovedthatgivenanergodi stationaryinnitelydivisible pro ess, ea h innitely divisible self-joining of the orresponding measure-preserving automorphismremains ergodi , andin parti ular we also obtain theELFpropertyinthismost general ase.

Furthermore, we show (Proposition 12 below) that weakly mixing but non-mixing2-foldsimpleautomorphismsaredisjointfromtheELF lass.Itis alsoshownthatthetime-onemapsofows onsideredin[8℄aredisjointfrom any ELF automorphism, and therefore the time-one maps of Ko hergin's smoothows from[25 ℄ aredisjoint fromanyELFautomorphism.

Re ently, some attention has been devoted to joining properties of in-terval ex hange transformations(see e.g. [4℄,[5℄).Here we areableto prove

(3)

ingathree-intervalex hange transformationweobtaindisjointnessfromthe ELF lass. In fa t, this resultis a onsequen e of a more general statement provedinthepaper.Namelygiven

k ≥ 3

we onsiderspe ialpermutationsof

{1, . . . , k}

andweprovethatfora.a. hoi esoflengthsofpartitionintervalsof

[0, 1)

theresultingautomorphismsaredisjointfromallELFautomorphisms. Some results in this paper have been obtained during the visit of the fourth-named author at Ni olaus Coperni us University inSeptember 2003 and during the visit of the third-named author at Université de Bretagne O identale intheSpring 2004.

1.Preliminaries

1.1. Fa tors, joinings and Markov operators. Assume that

T

is an er-godi automorphism of a standard probability Borel spa e

(X, B, µ)

. The asso iated unitarya tion of

T

on

L

2 _{(X, B, µ)}

isgivenby

U

T

(f ) = f ◦ T

(but wewilloftenwrite

T (f )

insteadof

f ◦ T

).Wedenoteby

C(T )

the entralizer of

T

, thatis, the set of all automorphisms of

(X, B, µ)

ommuting with

T

. Endowedwiththestrongoperatortopologyof

U (L

2 _{(X, B, µ))}

the entralizer be omes a Polish group. Any

T

-invariant sub-

σ

-algebra

A ⊂ B

is alled a fa tor of

T

. The quotient a tion of

T

on the quotient spa e

(X/A, A, µ|

A

)

will be denoted by

T |

A

or even by

A

if no onfusion arises. We say that

T

is rigid if the set

{T

n

_{: n ∈ Z}}

has an a umulation point in

C(T )

. It follows thatinthe rigidity ase the entralizer is un ountable and for some in reasing sequen e

(q

n

)

,

T

q

n

_{→ Id}

.Automorphismswhi h have no rigidity at all are alledmildly mixing (see [11 ℄). More pre isely,

T

is alled mildly mixing if its only rigidfa tor isthe one-point fa tor.

Assumenowthat

S

isanotherergodi automorphismofastandard prob-ability Borel spa e

(Y, C, ν)

.By a joining of

T

and

S

we mean any

T × S

-invariant measure

̺

on

(X × Y, B ⊗ C)

whose marginals

̺

X

and

̺

Y

satisfy

̺

X

= ̺|

X

= µ

and

̺

Y

= ̺|

Y

= ν

respe tively.Thesetofjoiningsbetween

T

and

S

isdenoted by

J(T, S)

.Whenevertheautomorphism

T × S

a ting on

(X ×Y, B ⊗C, ̺)

(forshortwewillalsowrite

(T ×S, ̺)

)isergodi ,thejoining

̺

is alledergodi andthesetofergodi joiningsisdenotedby

J

e

_{(T, S)}

.The formula \

X×Y

f ⊗ g d̺ =

\

Y

Φ

̺

(f ) · g dν

establishesaone-to-one orresponden ebetweentheset

J(T, S)

and theset

J (T, S)

ofallMarkovoperatorsfrom

L

2 _{(X, B, µ)}

to

L

2 _{(Y, C, ν)}

intertwining

U

T

and

U

S

(seee.g. [42 ℄,[29℄for moredetails). Re all thatapositive linear operator

Φ : L

2 _{(X, B, µ) → L}

2 _{(Y, C, ν)}

is alled Markov if

Φ(1

X

) = 1

Y

and

Φ

∗

₍₁

Y

) = 1

X

,and then

Φ = Φ

̺

where

̺(A × B) =

T

B

Φ(1

A

) dν

for measur-able sets

A ∈ B

and

B ∈ C

. The set of Markov operators is losed in the weak operator topology of

B(L

2 _{(X, B, µ), L}

2 _{(Y, C, ν))}

(4)

and

J(T, S)

are ompa t (on the latter set we transport the topology of

J (T, S)

). Ergodi joinings orrespond to so alled inde omposable Markov operators,i.e.to theextremalpointsintheset

J (T, S)

,whi hhas anatural stru tureofaChoquetsimplex.NotethattheMarkovoperator orrespond-ingtothe produ tmeasure

µ ⊗ ν

equals

Π

X,Y

(f ) =

T

X

f dµ

.Ifonemore er-godi automorphism

R

on

(Z, D, η)

isgivenand

Φ

̺

∈ J (T, S)

,

Φ

κ

∈ J (S, R)

then

Φ

κ

◦ Φ

̺

∈ J (T, R)

and the orresponding joining of

T

and

R

will be denotedby

κ ◦ ̺

.

Whenever

S = T

we will write

J

2 (T )

and

J

e

2 (T )

instead of

J(T, T )

and

J

e

(T, T )

respe tively.Note thatif

W ∈ C(T )

thenthe formula

µ

W

(A × B)

= µ(A ∩ W

−1

_B)

determines a self-joining, alled a graph joining, of

T

, and moreover

µ

W

∈ J

e

2 (T )

(for

W = T

n

we speak about o-diagonal self-joinings). We saythat

T

is 2-fold simple (see [49 ℄, [21 ℄)if the only ergodi self-joinings of

T

are graph joinings or the produ t measure

µ ⊗ µ

. The measure

µ

Id

will alsobe denotedby

∆

X

or

∆

µ

.

Wesaythat

T

isrelativelyweaklymixingwithrespe ttoafa tor

A ⊂ B

if theself-joining

λ

( alledtherelativelyindependent extensionof thediagonal measure on

A

) given by

λ(A × B) =

\

X/A

E(1

A

| A) · E(1

B

| A) d(µ|

A

)

is ergodi . If

A

1 ⊂ A

is another fa tor and

T |

A

is relatively weakly mixing over

A

1

then

T

is still relatively weakly mixing over

A

1

(forthis hainrule seee.g. [20 ℄).

Following[9℄wesaythattwoergodi automorphisms

T

and

S

aredisjoint if

J(T, S) = {µ ⊗ ν}

. Re all that

J

e

_{(T, S) = {µ ⊗ ν}}

implies disjointness of

T

and

S

. Given a lass

R

of ergodi automorphisms, we denote by

R

⊥

the lassofallergodi automorphismsdisjointfromanymemberof

R

.Then bya multiplier (see [12 ℄) of

R

⊥

we mean an ergodi automorphism ea h of whose ergodi joinings with an automorphism belonging to

R

⊥

gives rise to another member of

R

⊥

. The lass of multipliers of

R

⊥

is then denoted by

M(R

⊥

₎

.

Inwhat follows, we willneed thefollowing.

Proposition1([1 ℄). Let

T

be anergodi automorphismof

(X, B, µ)

.If

̺ ∈ J

e

2 (T )

andalso

̺ ◦ ̺ ∈ J

e

2 (T )

then

(T × T, ̺)

isrelatively weakly mixing over the two marginal fa tors

B ⊗ {∅, X}

and

{∅, X} ⊗ B

.

Assumethat

T

isweaklymixing and

̺ ∈ J

e

2 (T )

.Then dire tlyfrom the hainrulefor therelative weakmixing propertywe obtain thefollowing. (1) If

(T × T, ̺)

isrelativelyweaklymixingoverthemarginalfa tors,then

(5)

We will alsoneed the following simple lemma.

Lemma 2. Assume that

T

is a weakly mixing automorphism of a stan-dard probability Borelspa e

(X, B, µ)

. Assume that

N

0 ⊂ N

andthe density of

N \ N

0

equals zero. Assume moreover that for ea h

f, g ∈ L

2 _{(X, B, µ)}

,

hf ◦ T

n

, gi → hf, 1ih1, gi

as

n → ∞

,

n ∈ N

0

.Assumethat

̺ ∈ J(T )

.Thenforall

f, g, h ∈ L

∞

_{(X, B, µ)}

, \

X×X

f (T

n

x)g(x)h(T

n

y) d̺(x, y) →

\

X

f (x)h(y) d̺(x, y)

\

X

g(x) dµ(x).

Proof. Wehave \

X×X

f (T

n

x)g(x)h(T

n

y) d̺(x, y) =

\

X

f (T

n

x)g(x)Φ

∗

_̺

(h ◦ T

n

)(x) dµ(x)

=

\

X

(f · Φ

∗

_̺

(h)) ◦ T

n

· g dµ

→

\

X

f · Φ

∗

_̺

(h) dµ

\

X

g dµ =

\

X×X

f ⊗ h d̺

\

X

g dµ.

Formoreinformationonjoiningswereferthereadertothemonographby E.Glasner[13 ℄.Forthespe traltheoryofdynami alsystemsseee.g.[3℄,[33 ℄. 1.2. Sub-joiningsand sub-Markovoperators in innite measure-preserv-ing ase. Given two automorphisms

T

and

S

a ting on

σ

-nite standard Borelspa es

(X, B, µ)

and

(Y, C, ν)

respe tively,byasub-joining of

T

and

S

we meanea hpositive

σ

-nite

T × S

-invariant measure

̺

on

(X × Y, B ⊗ C)

whose marginals

̺

X

and

̺

Y

satisfy

̺

X

≤ µ

and

̺

Y

≤ ν

.By theformula

\

X×Y

f (x)g(y) d̺(x, y) =

\

Y

V (f ) · g dν,

there is a one-to-one orresponden e between the set of sub-joinings and theset of sub-Markov operators

V : L

2 _{(X, B, µ) → L}

2 _{(Y, C, ν)}

intertwining

U

T

and

U

S

, where by a sub-Markov operator we mean a positive operator

V : L

2 (X, B, µ) → L

2 (Y, C, ν)

su h that

V f ≤ 1

for all

f ∈ L

2 _{(X, B, µ)}

satisfying

0 ≤ f ≤ 1

,and

V

∗

_{g ≤ 1}

forall

g ∈ L

2 _{(Y, C, ν)}

satisfying

0 ≤ g ≤ 1

. Remark1. Notethateveninthe ase

T = S

,althoughtheo-diagonal measures

µ

T

n

havethepropertythattheirmarginalsareequalto

µ

(equiva-lently,

T

X

T

n

(1

A

) dµ = µ(A)

forea h

A ⊂ X

ofnitemeasure),thefa tthat the onstant fun tion

1 X

is not integrable may ause thatthe marginalsof aweaklimit

̺

ofasequen eofo-diagonal measuresneed notbe equalto

µ

(nevertheless,wewill have

̺

X

≤ µ

).

(6)

1.3.Co y lesand ompa tgroupextensions. Assumethat

T

isanergodi automorphism of a standard probability Borel spa e

(X, B, µ)

. Let

G

be a ompa t metri group with the

σ

-algebra

B(G)

of Borel sets and Haar measure

m

G

. Let

ϕ : X → G

be a measurable map. It generates a o y le

ϕ

( · )

( · ) : Z × X → G

bythe formula

ϕ

(n)

(x) =











ϕ(T

n−1

x) · ϕ(T

n−2

x) · . . . · ϕ(x)

if

n > 0

,

1

if

n = 0

,

(ϕ(T

−1

_{x) · . . . · ϕ(T}

n

_x))

−1

if

n < 0

.

We denote by

T

ϕ

the skew produ t automorphism dened on

(X × G,

B ⊗ B(G), µ ⊗ m

G

)

bythe formula

T

ϕ

(x, g) = (T x, ϕ(x) · g).

We all

T

ϕ

a ompa t group extension of

T

.

Denote by

τ

g

the map on

X × G

given by

τ

g

(x, g

1 ) = (x, g

1 g

−1

₎

. Note that

τ

g

∈ C(T

ϕ

)

for ea h

g ∈ G

.

Compa t group extensions have the so alled relative unique ergodi ity (RUE) property: whenever the produ t measure

µ ⊗ m

G

is ergodi , it is theonly

T

ϕ

-invariant measureof

(X × G, B ⊗ B(G))

whose proje tion on

X

equals

µ

(seee.g. [10℄).

We say that a o y le

ϕ : X → G

is ergodi if

T

ϕ

onsidered with

µ ⊗ m

G

isergodi . Inthis aseergodi self-joinings of

T

ϕ

whose proje tions on

X × X

are

∆

X

arene essarily graphjoinings orrespondingto

τ

g

,

g ∈ G

(see [21 ℄).

1.4. Gaussian automorphisms. An ergodi automorphism

T

of a stan-dard probability Borel spa e

(X, B, µ)

is alled Gaussian if there exists a

U

T

-invariant subspa e

H ⊂ L

2 _{(X, B, µ)}

of real-valued fun tions generating

B

andsu h thatea h non-zerovariablefrom

H

has aGaussiandistribution. For a joining theory of Gaussian automorphisms we refer the reader to [29 ℄ (seealso [3 ℄forageneral theory ofGaussian automorphisms).Inparti ular, it is proved in [29 ℄ that there is a spe ial subset

J

g

2 (T ) ⊂ J

2 e

(T )

alled the set ofGaussian self-joinings (for

̺ ∈ J

g

2 (T )

,

(T × T, ̺)

remains a Gaussian automorphism).Roughlyspeaking,thisset orrespondstoall ontra tionsof therst haos

H

intertwiningthe unitarya tion of

T

on

H

(all o-diagonal self-joinings

µ

T

n

are in

J

g

2 (T )

). It follows that

J

g

2 (T )

is losed in the weak topology ofjoinings.

AGaussianautomorphism

T

isentirely determinedbythespe tral mea-sure

σ

of

U

T

on

H

(c)

_{= H + iH}

.Moreover,

T

isergodi i

σ

is ontinuous. Themaximal spe tral type of

T

isthesumof onse utive onvolutions

σ

(n)

(7)

automor-Ea hvariable

f ∈ H

,viewed as a map

f : X → R

,is alleda Gaussian o y le.Itis alleda Gaussian oboundary if

f = g − g ◦ T

for some

g ∈ H

. The subspa e

H

onsists entirely of Gaussian oboundaries i 1 is not in the topologi al support of

σ

([27℄). We refer the reader to [27 ℄ for more information about ergodi ity of ir le group extensions of the form

T

e

2πif

, where

f

is aGaussian o y le.

1.5. Integral automorphisms andspe ial ows. Let

T

be an ergodi au-tomorphism of a standard probability Borel spa e

(X, B, µ)

. Assume that

f : X → N

is a measurable fun tion with nite integral. Let

X

f

⊂ X × N

be given by

S

n∈N

X

n

× {n}

, where

X

n

= {x ∈ X : f (x) ≤ n}

. Let

B

f

denote the restri tion of the produ t

σ

-algebra of

B

and the

σ

-algebra of all subsets of

N

to the set

X

f

. Let

µ

f

denote the restri tion of the prod-u t measure

µ ⊗

P

n∈N

δ

{n}

to

X

f

.Bytheintegral transformation built over theautomorphism

T

andunderthefun tion

f

we meanthe transformation

T

f

: (X

f

, B

f

, µ

f

) → (X

f

, B

f

, µ

f

)

dened by

T

f

(x, k) =

_{(x, k + 1)}

if

f (x) < k

,

(T x, 1)

if

f (x) = k

.

Supposethat

A ∈ B

haspositivemeasure.Itiseasyto he kthat

(T

A

)

τ

A

and

T

aremetri allyisomorphi ,where

T

A

: A → A

istheindu edautomorphism and

τ : A → N

stands fortherst returntimefun tion (see[3, Chapter1℄). Denote by

m

R

the Lebesgue measure on

R

. Assume that

f : X → R

is a measurable positive fun tion su h that

T

X

f dµ = 1

. The spe ial ow

T

f

_{= {(T}

f

₎

t

}

t∈R

built from

T

and

f

is dened on the spa e

X

f

₌

{(x, t) ∈ X × R : 0 ≤ t < f (x)}

( onsidered with

B

f

, the restri tion of the produ t

σ

-algebra, and

µ

f

, the restri tion of the produ t measure

µ ⊗ m

R

of

X × R

). Under the a tion of the spe ial ow ea h point in

X

f

moves verti allyatunitspeed,andweidentifythepoint

(x, f (x))

with

(T x, 0)

(see e.g. [3 , Chapter 11℄). In the spe ial ase where

f ≡ 1

the spe ial ow

T

f

a tson

X × [0, 1)

andis alledthe suspensionow forthe automorphism

T

. Then we write

T

b

instead of

T

f

and

( b

X, b

B, b

µ)

instead of

(X

f

_{, B}

f

_{, µ}

f

₎

. Let

π : b

X = X × [0, 1) → X

denote thenatural proje tion. Then the

σ

-algebra

π

−1

_{(B) ⊂ b}

_B

is

( b

T )

1

-invariant and

π : ( b

X, π

−1

_{(B), b}

_{µ) → (X, B, µ)}

establishes an isomorphism between automorphisms

( b

T )

1

of

( b

X, π

−1

_{(B), b}

_µ)

and

T

of

(X, B, µ)

.Finally,noti e thattheows

T

b

f

and

T

f

areisomorphi whenever

f : X → N

.

Lemma 3. Let

T

be an ergodi automorphism of

(X, B, µ)

and let

f :

X → N

be a measurable fun tion with nite integral. Suppose that

(a

n

)

is a sequen e of integers su h that

(T

f

₎

a

n

→ p((T

f

₎

1 )

weakly, where

p

is a trigonometri polynomial. Then

T

a

n

(8)

Proof. Sin e the operators

(T

f

₎

1

a ting on

L

2 _(X

f

_{, B}

f

_{, µ}

f

₎

and

( b

T

f

)

1

a ting on

L

2 _{( b}

_X

f

, b

B

f

, b

µ

f

)

areunitarily isomorphi ,

( b

T

f

)

a

n

→ p(( b

T

f

)

1 )

intheweakoperatortopologyon

L

2 _{( b}

_X

f

, b

B

f

, b

µ

f

)

.Let

π : b

X

f

= X

f

×[0, 1) →

X

f

be the natural proje tion. Sin e

L

2 _{( b}

_X

f

, π

−1

(B

f

), b

µ

f

) ⊂ L

2 ( b

X

f

, b

B

f

, b

µ

f

)

is an invariant subspa e with respe t to the operators

( b

T

f

)

a

n

(

n ∈ N

),

( b

T

f

)

a

n

→ p(( b

T

f

)

1 )

in the weak operator topology on

L

2 _{( b}

_X

f

, π

−1

(B

f

), b

µ

f

)

. Sin ethe operators

T

f

on

L

2 _(X

f

, B

f

, µ

f

)

and

( b

T

f

)

1

on

L

2 _{( b}

_X

f

, π

−1

(B

f

), b

µ

f

)

are unitarily isomorphi ,

T

a

n

f

→ p(T

f

)

in the weak operator topology on

L

2 (X

f

, B

f

, µ

f

)

.

2. Basi properties of ELF automorphisms. An ergodi automor-phism

T

of a standard Borel spa e

(X, B, µ)

is said to have theELF prop-erty if

{µ

T

n

: n ∈ Z} ⊂ J

e

2 (T )

,or equivalently,the weak losureofthesetof Markov operators

{T

n

_{: n ∈ Z}}

onsists of inde omposable Markov opera-tors.For short,wewill speakabout ELF automorphisms.

It is lear that ergodi dis rete spe trum automorphisms and mixing automorphisms are examplesof ELF automorphisms. By what was said in Se tion 1.4, Gaussian automorphisms also enjoy the ELF property (see [6℄ for adire tproof of thatfa t).

The following two onsequen es of Proposition 1 have already been no-ti ed in[6 ℄.

Proposition 4 ([6 ℄). If

T

is an ELF automorphism and if

̺ ∈

{µ

T

n

: n ∈ Z}

then

(T × T, ̺)

is relatively weakly mixing with respe t tothe two natural marginal

σ

-algebras.

Proposition 5 ([6 ℄). Assume that

T

is an ELF automorphism andlet

̺ ∈ {µ

T

n

: n ∈ Z}

. Let

S

be an ergodi automorphism on

(Y, C, ν)

.Assume that

̺

1

isan ergodi joining of

T

and

S

. Then

̺

1 ◦ ̺

isstill ergodi .

2.1. Disjointness of ELF automorphisms from time-one maps of some measurableows. Proposition 5,similarlyto [6 ℄,allowsus toprove disjoint-nessofthe lassofELFautomorphismsfromautomorphismshavingapie e of integral Markov operator in the weak losure of its powers. Indeed, as-sumethat

S

isanautomorphismof

(Y, C, ν)

.Let

P

beaprobabilitymeasure dened on the Borel

σ

-algebra of

C(S)

. We dene a Markov operator

M

P

on

L

2 _{(Y, C, ν)}

byputting

M

P

(f ) =

\

C(S)

f ◦ R dP (R).

(9)

Theintegral ontherighthand side ismeant intheweaksense,i.e.for ea h

f, g ∈ L

2 (Y, C, ν)

,

D

\

C(S)

f ◦ R dP (R), g

E

=

\

C(S)

hf ◦ R, gi dP (R).

Inorder to seethatthis denitionis orre twe dene

hhf, gii =

\

C(S)

hf ◦ R, gi dP (R)

and he kthatwehaveobtainedabilinearformon

L

2 _{(Y, C, ν)}

whi h,bythe S hwarzinequality,is bounded. Clearly,

M

P

∈ J

2 (S)

.

Proposition 6. Let

S : (Y, C, ν) → (Y, C, ν)

be an ergodi automor-phism. Assume that there exist an in reasing sequen e

(t

n

)

of natural num-bersand a probability Borelmeasure

P

on

C(S)

su h that

S

t

n

_{→ a}

\

C(S)

R dP (R) + (1 − a)Φ

intheweakoperatortopologyon

B(L

2 _{(Y, C, ν))}

,where

a > 0

and

Φ ∈ J

2 (S)

. Assume that

P ({R ∈ C(S) : R

isweakly mixing

}) > 0

. Then

S

is weakly mixing. If moreover

P

isnot Dira andeither

(i)

P

is on entrated on

{S

i

_{: i ∈ Z}}

,or

(ii)

P

is on entrated on

{S

t

: t ∈ R}

, where

S

1 = S

(i.e. we assume in parti ular that

S

isembeddable in a measurable ow),

then

S

isdisjoint fromall ELF automorphisms.

Proof. First, let us show that

S

is weakly mixing. Indeed, if

f

is its eigenfun tion then

kf k

2 _L

2 = |hS

t

n

f, f i| →

a

\

C(S)

hf ◦ R, f i dP (R) + (1 − a)hΦ(f ), f i

_.

Sin e

|hf ◦ R, f i| ≤ kf k

2

and

|hΦ(f ), f i| ≤ kf k

2

, a onvexity argument shows that we must have

hf ◦ R, f i = kf k

2

for

P

-a.e.

R ∈ C(S)

(and also

hΦ(f ), f i = kf k

2

provided

a < 1

).So for su han

R

,wehave

f ◦ R = c(R)f

(

c(R) ∈ C

), andsin e

R

maybe taken weakly mixing,

f

is onstant.

Let

T

be an ELF automorphism on

(X, B, µ)

. Let

Ψ : L

2 _{(Y, C, ν) →}

L

2 _{(X, B, µ)}

be an inde omposable Markov operator intertwining

S

and

T

. Then

Ψ ◦ S

t

n

_{= T}

t

n

_{◦ Ψ}

andbypassing toasubsequen e of

(t

n

)

ifne essary, we nd

Ψ ◦ (aM

P

+ (1 − a)Φ) = Φ

̺

◦ Ψ,

(10)

inde om-posable. Ontheother hand,

Ψ ◦ (aM

P

+ (1 − a)Φ) = a

\

C(S)

Ψ ◦ R dP (R) + (1 − a)Ψ ◦ Φ,

and hen e we must have

Ψ ◦ R = Φ

̺

◦ Ψ

for

P

-a.e.

R ∈ C(S)

. This means that for a set of full

P ⊗ P

-measure of

(R

1 , R

2 ) ∈ C(S) × C(S)

, we have

R

2 ◦ R

−1

₁

◦ Ψ

∗

= Ψ

∗

.Noti ehoweverthat both assumptions (i)and (ii)and the fa t that

P

is not Dira imply that for some weakly mixing element

R ∈ C(S)

we have

R ◦ Ψ

∗

_{= Ψ}

∗

and therefore

Ψ = Π

Y,X

.

Supposenowthat

(S

t

)

t∈R

isa measurable,weaklymixing owa tingon

(Y, C, ν)

.Supposethatfora sequen e

(r

n

)

ofreal numberswith

r

n

→ ∞

we have (2)

S

r

n

→ a

\

R

S

t

dQ(t) + (1 − a)Φ,

where

Q

isnotDira .Bypassingtoasubsequen eifne essarywe anassume that the sequen e

({r

n

})

of fra tional parts of

r

n

onverges to

0 ≤ b ≤ 1

. Sin e the ow is measurable,

S

{r

n

}

→ S

b

in the strong operator topology. It follows that the sequen e

(S

1 )

[r

n

]

_{= S}

r

n

◦ S

−{r

n

}

onverges in the weak operatortopology and we have

(S

1 )

[r

n

]

→ a

\

R

S

t−b

dQ(t) + (1 − a)Φ ◦ S

−b

.

We have provedthe following.

Corollary7. Assumethat

(S

t

)

t∈R

isameasurable,weaklymixingow for whi h

(2)

holds with

Q

whi h is not Dira . Then the time-one map

S

1

isdisjoint fromallELF automorphisms.

Remark 2. The assumptions of Corollary 7 are satised for time-one maps of some lassi al examples of spe ial ows over irrational rotations and overinterval ex hange transformations(see [6 ℄[8 ℄)and inparti ular it issatised for some smooth owson surfa es(see [8 ℄).

2.2. Fa tors and dire t produ ts of ELF automorphisms. The following propositionshowsthatthe lassofELFautomorphismsis losedundersome basi operations.

Proposition 8. The lass of ELF automorphisms is losed under fa -tors and inverse limits. The dire t produ t of weakly mixing ELF automor-phisms remains an ELFautomorphism.

Proof. Closedness undertakingfa tors and inverse limitsisobvious. Assume that

T

i

is a weakly mixing ELF automorphism of

(X

i

, B

i

, µ

i

)

,

(11)

Suppose that

T

n

i

_{→ Φ}

̺

for some

̺ ∈ J

2 (T )

.By applying thediagonalizing pro edure if ne essary, we an assume that for ea h

j ≥ 1

,

T

n

i

j

→ Φ

̺

j

for some

̺

j

∈ J

e

2 (T

j

)

.Iteasilyfollowsthat

̺ = ̺

1 ⊗ ̺

2 ⊗ · · ·

andbe auseof(1),

̺

isergodi , whi h ompletestheproof.

Remark 3. Notehoweverthat an ergodi self-joining of an ELF auto-morphism need not be an ELF automorphism. Indeed, even if

T

is mixing thenanergodi self-joiningneednot give risetoanELFautomorphism.For example,bySmorodinskyThouvenot's resultfrom[45 ℄itfollowsthatgiven an ergodi zero entropy automorphism

S

and a Bernoulli automorphism

T

we an nd an ergodi self-joining

̺

of

T

su h that

(T × T, ̺)

has

S

as its fa tor.

2.3.Lifting theELF propertyto ompa t groupextensions. Wewill now dis uss the possibility of lifting theELF property by a ompa t group ex-tension.Soassumethat

T

isanELFautomorphismandlet

ϕ : X → G

bea o y le, where

G

isa ompa tmetri group.Re all rst thatif

T

is mixing and the extension

T

ϕ

is weakly mixing then

T

ϕ

is infa t mixing (see [37 ℄). A lookat ashort joining proof (dueto A. delJun o) of thatfa t gives rise to a riterion oflifting the ELFproperty.

Proposition9. Assumethat

T

hastheELFpropertyand

ϕ : X → G

is ergodi . Assumemoreoverthat for ea h

̺ ∈ {µ

T

n

: n ∈ Z}

the o y le

ϕ × ϕ

over

(T × T, ̺)

isergodi .Then

T

ϕ

has the ELF property.

Proof. Assume that

(T

ϕ

)

m

i

_{→ Φ}

e

̺

. We must show that

̺

e

is ergodi . We an assume that

m

i

→ ∞

, otherwise the result is lear. We then have

T

m

i

_{→ Φ}

̺

, where

̺

is theproje tion of

̺

e

on

X × X

.Now,

̺

e

is a

T

ϕ

× T

ϕ

-invariant measure whose proje tion is

̺

.However, byour standing assump-tionthemeasure

̺ ⊗ m

G

⊗ m

G

has thesame propertyanditisergodi .The resultnowfollows fromthe relative uniqueergodi ityproperty for ompa t groupextensions.

The above proof suggests that in general we have no han e to lift the ELF property and in fa t we will loose this property when the base has dis rete spe trum.

Proposition 10. An ergodi isometri extension

T

b

of a dis rete spe -trumautomorphism

T

hastheELFpropertyitheextensionalsohasdis rete spe trum.

Proof. We an assumethat

T

isan ergodi rotation(

T x = x + x

0

) of a ompa t metri monotheti group

X

.Moreoverassume that

ϕ : X → G

is anergodi o y leforwhi h

T

istheKrone kerfa torand

T

b

isthequotient a tion of

T

ϕ

on

X × G/H

. All we need to show is that under all these assumptions

T

b

doesnot have theELFproperty.

(12)

To thisend rst hoosea sequen e

(n

i

)

of density1 su hthat (3)

U

n

i

b

T

→ 0

weaklyon

L

2 _{(X × G/H, m}

X

⊗ m

G/H

) ⊖ L

2 (X, m

X

),

whi h is possible be ause

T

is the Krone ker fa tor of

T

b

and therefore the spe traltypeof

U

b

T

on

L

2 _{(X ×G/H, m}

X

⊗m

G/H

)⊖L

2 (X, m

X

)

is ontinuous. Sin e the density of

(n

i

)

is 1, there exists a subsequen e

(m

i

)

of

(n

i

)

su h that

T

m

i

_{→ Id}

.Indeed,givenaneighbourhood

W ∋ 0

in

X

,bythepointwise ergodi theoremfor stri tly ergodi systems the average timeof visiting

W

bytheorbit of an arbitrary point of

X

is equal to

m

X

(W )

,hen e positive. Therefore we an nd

n

j

= n

j

(W )

so that

n

j

x

0 ∈ W

. Letting

W → {0}

provesthe laim.

It follows from (3) that

T

b

m

i

onverges weakly to the operator

E( · | X)

whi h orresponds to the joining

∆

X

⊗ m

G/H

⊗ m

G/H

. However, this last joiningisnotergodi :thefun tion

F (x, gH, x, g

′

_{H) = g}

−1

_g

′

_H

isnot onstant but it is

T × b

b

T

-invariant

∆

X

⊗ m

G/H

⊗ m

G/H

-a.e. Therefore,

T

b

does not have theELFpropertyand theresultfollows.

Thefollowing orollaryfollows dire tlyfromProposition 10.

Corollary 11. If an extension of a rotation

T

has the ELF property, then the extension isrelatively weaklymixing over

T

.

Remark 4. In [52 ℄ there are expli it onstru tions of ELF automor-phisms whi hare relatively weakly mixing extensionsof some irrational ro-tations.

Let us now show however that the riterion of Proposition 9 may work insome ases ofmildlymixing ELFautomorphismswhi harenot mixing.

We onsidersymmetri probabilitymeasures

σ

on

T

su hthat (4) all weak losurepointsofthesequen e

{z

n

_{: n ∈ Z}}

in

L

2 _{(T, σ)}

arein the set

{az

n

_{: |a| < 1, n ∈ Z}}

.

Sin e

σ

isasymmetri measure,thenumbers

a

in(4)havetobereal.Under the above assumption, the Gaussian automorphism asso iated to

σ

has to bemildlymixing.Re allthat lassi alRieszprodu tsyieldexamplesofsu h measures,in ludingexampleswhi harenotRaj hmanmeasures sothatthe setof weak losurepointsisnot trivial (see[16 , Ch. II, Se t. 7℄).

Proposition 12. Assumethat

T

isamildly mixingGaussian automor-phism determined bya measure satisfying

(4)

forwhi ha ertain

a 6= 0

isin theweak losureof hara ters(su h a

T

isnotmixing).Take

f

fromtherst real haos. Assume that

f

is not a Gaussian oboundary. Then

T := T

(13)

Proof. Assume that

(5)

(T

e

2πif

)

n

t

→ Φ

_̺

_e

forsomesequen e

(n

t

)

with

n

t

→ ∞

.Then

T

n

t

_{→ Φ}

̺

,where

̺

isthe proje -tionof

̺

e

on

X ×X

.Withoutlossofgeneralitywe anassumethat

z

n

t

_{→ a}

in theweaktopologyof

L

2 _{(T, σ)}

forsomereal

a

with

|a| < 1

.Wehavetoprove that

̺ ∈ J

e

2 (T

e

2πif

)

isergodi . If now

̺

isthe produ t measurethen so is

̺

e

, sin e

T

e

2πif

isweaklymixing([27℄)andwemayapplytherelativeunique er-godi itypropertyfor ompa tgroupextensionsto on lude.Notethat

̺

an-not be agraphmeasure, sin e

T

isassumed tobemildlymixing. Moreover, sin e

z

n

t

_{→ a}

intheweaktopologyof

L

2 _{(T, σ)}

,

T

n

t

restri tedtotherstreal haostendstomultipli ationby

a

,andhen e

Φ

̺

ismultipli ationby

a

onthe rst real haos. ByProposition 9 allwe need to showisthat

T

e

2πif

× T

_e

2πif

isergodi as a

T × T

-extensionof

(T × T, ̺)

.Following Proposition6in[27 ℄ itis su ient to show thatthe o y le

lf (x) + mf (y)

(with

(l, m) 6= (0, 0)

) isnot aGaussian oboundary (fortheGaussian automorphism

(T × T, ̺)

). If for ea h

r ∈ N

we put

f

(r)

_{(x) = f (x) + f (T x) + · · · + f (T}

r−1

_x)

then we have

klf

(r)

(x) + mf

(r)

(y)k

2 _L

2 _(̺)

= (l

2 + m

2 )kf

(r)

k

2 + 2lmhf

(r)

(x), f

(r)

(y)i

_L

2 _(̺)

= (l

2 + m

2 )kf

(r)

k

2 + 2lm

\

X

(Φ

̺

f

(r)

)(y)f

(r)

(y) dµ(y)

= (l

2 + m

2 + 2ml · a)kf

(r)

k

2 .

Now,

f

is not aGaussian oboundary,so

kf

(r

t

)

_{k → ∞}

alonga subsequen e

(r

t

)

(see [27 ℄)andsin e

|a| < 1

,

(l

2 + m

2 + 2ml · a)kf

(r

t

)

_{k → ∞}

or equivalently

klf

(r

t

)

_{(x) + mf}

(r

t

)

_(y)k

L

2 _(̺)

→ ∞,

whi hmeans(see[27 ℄)thatindeed

lf (x)+mf (y)

isnotaGaussian obound-ary.

3.PoissonautomorphismshavetheELFproperty. Inthisse tion we will dene and study a spe ial lass of self-joinings for the lass of au-tomorphisms obtained byPoisson suspension of innite measure-preserving maps.

3.1.Poisson suspension automorphisms. Assumethat

T

isan automor-phismofastandard Borelspa e

(X, B, µ)

,where

µ

is

σ

-nite.Wedenoteby

e

T

thePoisson suspension automorphism a ting on

( e

X, e

B, e

µ)

.The points of

e

(14)

nite measurewe dene

N

A

: e

X → N

byputting

N

A

(e

x) = #{n ∈ N : x

n

∈ A}.

Then we dene

B

e

as the smallest

σ

-algebra of subsets of

X

e

for whi h all variables

N

A

,

µ(A) < ∞

, are measurable. The measure

µ

e

is dened by the requirement that the variables

N

A

satisfy the Poisson law with parameter

µ(A)

and moreover that for ea h family of pairwise disjoint subsets of

X

of nite measure the orresponding variables are independent (see [24 ℄ for details).Finally,welet

T

e

a tbytheformula

T ({x

e

n

}) = ({T x

n

})

toobtainan automorphism of

( e

X, e

B, e

µ)

.The spa e

L

2 _{( e}

_{X, e}

_{B, e}

_µ)

admits a de omposition into invariant haos

L

n≥0

H

(n)

, where

H

(0)

is the subspa e of onstants,

H = H

X

= H

(1)

is the subspa e generated by the entred variables

N

0 A

=

N

A

− µ(A)

and

H

(n)

is the ortho omplement of the sum of haos

H

(i)

,

0 ≤ i ≤ n − 1

,in the subspa e generated by the produ ts of

n

variables of theform

N

A

(see[32 ℄).Themap

1 A

7→ N

0 A

anbeextendedtoanisometry

I

of

L

2 _{(X, B, µ)}

onto

H

andit onjugates

U

T

with

U

e

T

|

H

.Moreoverweobtain a natural isometry between

H

(n)

and the

n

th symmetri tensor produ t

H

⊙n

of

H

underwhi h

⊙

n

i=1

N

A

0 i

orresponds to theproje tion of

Q

n

i=1

N

A

i

in

H

(n)

. Theoperator

U

e

T

preserves the haosand,for ea h

n ≥ 0

,its restri tion to

H

(n)

orresponds to

(U

e

T

|

H

)

⊙n

by this natural isometry. In su h a ase, we will saythatan operator a ts well on the haos.

If

0 6= f ∈ L

2 _{(X, B, µ)}

is an eigenfun tion of

U

T

orresponding to

c

(with

|c| = 1

), then

f

is an eigenfun tion of

U

T

orresponding to

c

. Then

I(f ) ⊙ I(f ) ∈ H

(2)

and it is a

U

e

T

-invariant fun tion. Furthermore, if

σ

denotesthemaximalspe traltypeof

U

e

T

on

H

(whi hisequaltothemaximal spe tral type of

U

T

on

L

2 _{(X, B, µ)}

) then the maximal spe tral type of

U

e

T

on the

n

th haos is equal to the

n

th onvolution

σ

(n)

_{= σ ∗ · · · ∗ σ}

. Re all that

σ

(n)

is ontinuous i

σ

is ontinuous.

ThereforethePoissonsuspensionautomorphism

T

e

on

( e

X, e

B, e

µ)

isergodi ifandonlyifthespe traltypeof

T

on

L

2 _{(X, B, µ)}

is ontinuous;equivalently, ithere areno

T

-invariant subsetsof

X

ofnite positive measureor elsei

L

2 (X, B, µ)

doesnot ontainnon-zero

T

-invariant fun tions. In this ase

T

e

is weakly mixing. Finally, note that, in parti ular, if a Poisson suspension automorphism isergodi then ne essarily themeasure

µ

is innite.

3.2. Fa tors and Poisson joinings. If

X

1

is a

T

-invariant subset of

X

, then

T

e

isthedire tprodu toftwoPoissonsuspensionsof

T

a tingon

X

1

and on

X \ X

1

,inparti ular,

]

T |

X

1

isafa torof

T

e

.Assumenowthat

S

a tingon another

σ

-nite standardBorelspa e

(Y, C, ν)

isafa tor of

(X

1 , µ|

X

1 , T |

X

1 )

inthesense thatthereis ameasurable map

F : X

1 → Y

su h

F

∗

(µ|

X

(15)

and

F ◦ T = S ◦ F

on

X

1

.Then

S

e

a ting on

( e

Y , e

C, eν)

isafa tor of

]

T |

X

1

via themap

F : e

e

X

1 → e

Y

given by

F ({x

e

n

}) = {F (x

n

)}

.

Thenthe asso iated operator

V

e

F

: L

2 ( e

Y , e

ν) → L

2 ( e

X

1 , g

µ|

X

1 )

,

g 7→ g ◦ e

F

, a ts wellon the haos.

By Poisson fa tors of

T

e

we will mean fa tors

S

e

obtained as above. We willalso saythatthe map

F

isapartialmapof

X

to

Y

semi- onjugating

T

and

S

.Note thatif

F : X

1 → Y

establishes a semi- onjugation of

T

and

S

thenthe asso iated isometry

V

F

isa sub-Markovoperator from

L

2 _{(Y, ν)}

to

L

2 (X

1 , µ|

X

1 )

.

Assume that

T

and

S

are automorphisms of

σ

-nite spa es

(X, B, µ)

and

(Y, C, ν)

respe tively. A joining

η

of

T

e

and

S

e

is alled a Poisson join-ing if the asso iated Markov operator

V = Φ

e

η

a ts well on the haos and, via the natural isomorphisms of the rst haos

H

X

of

( e

X, e

B, e

µ)

and

H

Y

of

( e

Y , e

C, eν)

with

L

2 _{(X, B, µ)}

and

L

2 _{(Y, C, ν)}

, the operator

V |

e

H

X

or-responds to the sub-Markov operator

V

asso iated to a sub-joining of

T

and

S

.

Proposition 13. The lass of Poisson joinings between

T

e

and

S

e

is losed in the weak topology of joinings, in parti ular, the lass of Poisson self-joiningsof

T

e

ontains the weak losure of

{ e

T

n

_{: n ∈ Z}}

.

Moreover, the relative produ t of

T

e

over a Poisson fa tor is a Poisson self-joining.

Proof. The rst part follows dire tly from the fa t that the set of sub-Markovoperatorsis losed inthe weakoperator topology.

Toprove these ondparttake aPoissonfa tor whi hisdetermined bya partialfun tion

F : X

1 ⊂ X → Y

.ThentheMarkovoperator orresponding totherelativeprodu toverthisfa torisgivenby

V

e

F

◦V

F

e

∗

◦P

L

2 _{( e}

_X

₁

₎

.Sin e

V

e

F

,

V

∗

e

F

,

P

L

2 _{( e}

_X

₁

₎

a twellonthe haosandtheirrestri tionsto therst haos an naturally be identied with

V

F

,

V

∗

F

,

L

2 _{(X) ∋ f 7→ f |}

X

1 ∈ L

2 (X

1 )

resp., so aresub-Markovoperators,the omposition

V

e

F

◦V

F

e

∗

◦P

L

2 ( e

X

1 )

istheasso iated Markovoperator ofa Poisson self-joining.

3.3. Ergodi ity of Poisson joinings. Assume that

̺

is a sub-joining of

T

and

S

and denote by

V

the orresponding sub-Markov operator from

L

2 _{(X, B, µ)}

to

L

2 _{(Y, C, ν)}

.We will nowpass to a onstru tion of a Poisson joining

η

of

T

e

and

S

e

orrespondingto

̺

,i.e.ifweput

V = Φ

e

η

then

V |

e

H

≡ V

. This Poisson joining turns out to be unique, so the stru ture of Poisson joiningswill be understood.

Set

µ

′

_{= µ − ̺}

X

and

ν

′

_{= ν − ̺}

Y

.Letusdene a

σ

-nite standardspa e

(Z, ̺

′

₎

as a formaldisjoint union of

(X, µ

′

₎

,

(Y, ν

′

₎

(16)

we dene

R

on

(Z, ̺

′

₎

byputting

R|

X

= T

,

R|

Y

= S

and

R|

X×Y

= T × S

. Sin e

̺

X

and

̺

Y

are

T

- and

S

-invariant respe tively,

̺

′

is

R

-invariant. We nowhave the partial mapping

F : X ∪ (X × Y ) ⊂ Z → X

whi h to

x ∈ X

orto

(x, y) ∈ X ×Y

asso iates

x

,andforea h

A ⊂ X

ofnitemeasure we have

̺

′

_(F

−1

_{(A)) = µ}

′

_{(A) + ̺}

X

(A) = µ(A)

.Clearly,

F ◦ R = T ◦ F

,so

F

establishesasemi- onjugationof

R

and

T

.Themapping

G : Y ∪ (X × Y ) ⊂

Z → Y

whi h to

y ∈ Y

or to

(x, y) ∈ X × Y

asso iates

y

has similar properties. Hen e, the two maps

F : e

e

Z → e

X

and

G : e

e

Z → e

Y

are fa tor mappingsof

R

e

to

T

e

and

S

e

respe tively.It followsthat

( e

F , e

G) : e

Z → e

X × e

Y

denesajoining

η = ( e

F , e

G)

∗

( e

̺

′

₎

of

T

e

and

S

e

,thatis,forea h

f ∈ L

2 _{( e}

_{X, e}

_{B, e}

_µ)

and

g ∈ L

2 _{( e}

_{Y , e}

_{C, e}

_ν)

we have \

f (e

x)g(e

y) dη(e

x, e

y) =

\

f ◦ e

F · g ◦ e

G d e

̺

′

_.

It follows that theMarkov operator asso iated to

η

is equal to

V = V

e

∗

e

G

V

F

e

. Hen e,

V

e

a ts well on the haos and learly its restri tion to

H

X

an be naturally identied with

V

∗

G

V

F

. Let us now show that

V

∗

G

V

F

= V

. Indeed, take

A ⊂ X

and

B ⊂ Y

of nite measure. Noti e that

1 A

◦ F · 1

B

◦ G

equalszerooutside of

X × Y

,andon

X × Y

thisfun tion isequalto

1 A×B

. Therefore \

V

_G

∗

V

F

1 A

· 1

B

dν =

\

1 A

◦ F · 1

B

◦ G d̺

′

= ̺(A × B) =

\

V 1

A

· 1

B

dν,

when e

V

∗

G

V

F

= V

.

Theorem 14. Ea h Poisson joining of two ergodi Poisson suspension automorphisms remainsergodi .In parti ular,ea hergodi Poisson suspen-sion automorphism has the ELF property.

Proof. Noti e that the se ond assertion follows from the rst one and Proposition 13.

Assume that

T

e

and

S

e

are ergodi . It follows that

X

and

Y

have no invariant sets of nitepositive measure. Letus showthatin

Z

there areno

R

-invariantsetsofnitepositive

̺

′

-measure.Indeed,supposethat

h = 1

C

∈

L

2 _{(Z, ̺}

′

₎

is

R

-invariant. Then

V

∗

F

h

is

T

-invariant, so equal to zero

µ

-a.e.In parti ular for ea h subset

A ⊂ X

ofnite measurewehave

0 = hV

_F

∗

h, 1

A

i = hh, V

F

(1

A

)i =

\

h · (1

A

◦ F ) d̺

′

that is,

̺

′

_{(C ∩ F}

−1

_{(A)) = 0}

. Sin e

F

−1

_(A)

is a formal disjoint union of

A

and

A × Y

,

̺

′

_{(C ∩ X) = 0}

and by a similar argument

̺

′

_{(C ∩ Y ) = 0}

togetherwith

̺

′

_{(C ∩ (A × B)) = 0}

for ea h

A ⊂ X

,

B ⊂ Y

ofnitemeasure. Therefore

̺

′

_{(C) = 0}

.ItfollowsthatthePoisson suspension

R

e

of

R

isergodi and thereforeits fa tor

( e

T × e

S, η)

remains ergodi .

(17)

Remark 5. Independently, using dierent arguments, the resulton er-godi ityofPoissonian joiningshas also been proved byE.Royin[36 ℄.

4. Self-joinings of symmetri

α

-stable automorphisms. In this se tionwewill dene and study

α

-stableself-joinings for

α

-stable automor-phisms, i.e. automorphismsgiven by stationaryergodi symmetri

α

-stable pro esses (see [17 ℄, [30 ℄, [44 ℄). We will show that ea h ergodi symmetri

α

-stableautomorphism has the ELFproperty.

4.1. Auxiliary lemmas. The proofs of the following two elementary in-equalities areslight adaptations ofthe proofs from [44 ,pp.9192℄.

Lemma 15. If

0 < α < 1

then for all

x, y ∈ R

we have

(6)

|x|

α

_{+ |y|}

α

_{− |x + y|}

α

_{≥ (2 − 2}

α

_{) min(|x|}

α

_{, |y|}

α

_).

If

1 ≤ α < 2

then for all

x, y ∈ R

we have (7)

2(|x|

α

_{+ |y|}

α

_{) − (|x + y|}

α

_{+ |x − y|}

α

_{) ≥ 2(2 − 2}

α/2

_{) min(|x|}

α

_{, |y|}

α

_).

Inparti ular, (6)implies (8)

| |x + y|

α

_{− |y|}

α

_{| ≤ |x|}

α

for

x, y ∈ R

and

0 < α ≤ 1,

and bytheHölder inequality

(9)

|x + y|

α

_{≤ max(1, 2}

α−1

_)(|x|

α

_{+ |y|}

α

₎

for

x, y ∈ R

and

0 < α ≤ 2.

Thefollowing resultis a onsequen e of (8)and the Hölderinequality.

Lemma 16. Assume that

0 < α ≤ 2

. Let

(Ω, F , m)

be a nite measure spa e. Let

(A

n

)

n≥1

⊂ F

. Assume that

(f

n

), (g

n

) ⊂ L

α

_{(Ω, m)}

satisfy \

A

n

|f

n

|

α

dm → 0

and \

Ω

|g

n

|

α

dm =

O

(1)

as

n → ∞.

Then \

A

n

(|f

n

+ g

n

|

α

− |g

n

|

α

) dm → 0

as

n → ∞.

4.2. Symmetri

α

-stable pro esses. Re all that a real random variable

X

has a stable distribution if for any

a, b > 0

we an nd

c > 0

and a real number

d

su h that the distributions of

aX

1 + bX

2

and of

cX + d

are the same,where

X

1 , X

2

areindependent opies of

X

(oneprovesthenthatthere exists

α = α(X)

,

0 < α ≤ 2

,su hthat

c = (a

α

_{+ b}

α

₎

1/α

₎

.Inwhatfollowswe will onsideronly the symmetri ase(i.e. thedistribution of

X

andof

−X

arethesame, f.theGaussian ase).Inthis ase, the hara teristi fun tion of

X 6= 0

isof theform

E(e

itX

_{) = e}

−|t|

α

_σ

for some positive

σ

(

t ∈ R

). Let

0 < α ≤ 2

. Let

S

be an arbitrary ountable set. Let

X = (X

s

)

s∈S

be a pro ess dened on a probability spa e

(Ω, F , P )

. We say that

X

is (symmetri )

α

-stable if ea h nite linear ombination

Y =

P

m

i=1

a

i

X

s

i

(18)

(

a

i

∈ R

,

i = 1, . . . , m

) is a symmetri

α

-stable variable, i.e. there exists

σ ≥ 0

su h that

Ee

itY

_{= e}

−|t|

α

_σ

for all

t ∈ R

(and

σ > 0

whenever

Y 6= 0

). We thenwrite

((Y ))

α

= σ

1/α

.

Remark 6. For

1 ≤ α < 2

(

α = 2

leads to the Gaussian ase)

((Y ))

α

turns outto bea normintheBana hspa e

T

0<r<α

L

r

(Ω, P )

,andfor ea h

0 < r < α

thereexists

c = c

α,r

su hthat

((Y ))

α

= c

r,α

kY k

r

forea h

Y

whi h is

α

-stable. For

0 < α < 1

ina similarmanner we obtaina Fré het spa e.

Thefollowing theoremhas beenprovedin[30 , pp.127128℄.

Theorem 17. Assume that

0 < α ≤ 2

. Assume moreover that

X =

(X

s

)

s∈S

isan

α

-stablepro ess.Then there existsanite positive Borel mea-sure ( alled a spe tralmeasure of

X

)

m

on

R

S

su h that

E exp

i

n

X

j=1

a

j

X

s

j

= exp

−

1

2

\

R

S

n

X

j=1

a

j

x

s

j

α

dm(x)

for arbitrary

a

1 , . . . , a

n

∈ R

and

s

1 , . . . , s

n

∈ S

, where

x = (x

s

)

s∈S

. Remark7. Itfollowsthat

((

P

n

j=1

a

j

X

s

j

))

α

=

1

2

T

R

S

|

P

n

j=1

a

j

x

s

j

|

α

dm(x)

. Remark 8. When

0 < α < 2

,the measure

m

is notunique.

4.3.

α

-stableautomorphisms. Wesaythatanautomorphism

T

ofa stan-dard probability Borelspa e is

α

-stable if there exists a linear spa e

B

0

of real fun tions on

X

su hthat

1.

B(B

0 ) = B

,

2. for ea h

0 6= f ∈ B

0

,

f

is an

α

-stablevariable, 3.

B

0

is

T

-invariant.

The following riterion as well as the method of proof arevery lose to the ergodi ity riteria in[17 ℄and [15 ℄.