liftingrokkhlincocycles-2B(2).pdf

LiftingmixingpropertiesbyRokhlincocycles

M. Lemanczyk

F. Parreau

Abstract

We study the problem of lifting various mixing properties from a

base automorphism T 2 Aut(X;B;) to skew products of the form

T ';S , where ' : X ! G is a cocycle with values in a locally compact

Abelian group G, S = (S g ) g2G is a measurable representation of G in

Aut(Y;C;) and T ';S acts on the product space (X Y;BC;)

T ';S (x;y) = (Tx;S ' ( x ) (y)):

It is also shown that whenever T is ergodic (mildly mixing, mixing)

but T ';S is not ergodic (is not mildly mixing, not mixing), then on

a non-trivial factor A C of S the corresponding Rokhlin cocycle

x 7! S ' ( x ) j A is a coboundary (a quasi-coboundary).

Introduction

Given an ergodic automorphism T of a standard Borel space (X;B;) we

can study various extensions T of it. Among such extensions a special role is

played by so called compact group extensions or, more generally, isometric

extensions (see [8], [12] and [30]). In particular, one can ask which ergodic

properties of T are lifted by isometric extensions. The two papers 1 by Dan

Rudolph [25] and [26] are beautiful examples of the mechanism that once the

extension enjoys some \minimal" ergodic property then it shares some strong

ergodic properties assumed to hold for its base. By iterating the procedure

of taking isometric extensions we can hence lift ergodic properties of T to

weakly mixing distal extensions of it.

Research partly supported by Polish MNiSzW grant N N201 384834

1 In [25] it is proved that Bernoullicity is lifted whenever the extension is weakly mixing,

while in [26] it is shown that mixing (multiple mixing) lifts whenever the extension is

weakly mixing.

The notion complementary to distality is relative weak mixing [8], [12],

[30] and a natural question arises what happens with lifting ergodic proper-

ties from T to T when T is relatively weakly mixing over the factor T. This,

by Abramov-Rokhlin's theorem [2], leads to the study of so called Rokhlin

cocycle extensions which are automorphisms of the form T = T acting on

(X Y;BC;) by the formula

T (x;y) = (Tx; x (y));

where : X ! Aut(Y;C;) is measurable 2 . Since the above formula de-

scribes all possible (ergodic) extensions of T, it is hard to expect interesting

theorems on such a level of generality { one has to specify subclasses of

Rokhlin cocycles for which one can obtain some results. We will focus on

the following class.

Let G be a second countable locally compact Abelian (LCA) group.

Assume that we have a measurable action S of this group given by g 7!

S g 2 Aut(Y;C;). Let ' : X ! G be a cocycle. The automorphism T ';S

acting on (X Y;BC;) given by

T ';S (x;y) = (Tx;S '(x) (y))

will be called the Rokhlin (';S)-extension 3 of T.

A systematic study of the problem of lifting ergodic properties from T to

T ';S was originated by D. Rudolph in [27]. Since then, extensions T ';S ! T

have been studied in numerous papers, see e.g. [5], [11], [12], [13], [21], [22],

[24] and [28].

The present paper is a continuation of investigations from [21] and [22],

and, due to a new approach presented here, makes them complete. This new

approach is based on a harmonic analysis result from [17], and it consists

in showing that given an action S = (S g ) g2G of a second countable LCA

group G on a probability standard Borel space (Y;C;) and a saturated Borel

subgroup G, the spectral space of functions in L 2 (Y;C;) whose spectral

measures are concentrated on is the L 2 -space of an S-invariant sub--

algebra A C (a measure-theoretic factor of S). This will systematically

be used in our study because the group of L 1 -eigenvalues of the Mackey

G-action associated to T and ' is saturated and hence yields an S-factor.

2 The map is often called a Rokhlin cocycle.

3 We would like to emphasize that, as noticed in [5], if we admit G to be non-Abelian

locally compact, then each ergodic extension e T = T is of the form T ';S ; more specically,

a general Rokhlin cocycle x 7! (x) is cohomologous to a cocycle x 7! S '(x) for some G;'

and S.

Using that we will prove natural necessary and sucient conditions for

weak mixing of T ';S and relative weak mixing of T ';S over T. We also com-

pute possible eigenvalues of T ';S and determine the relative Kronecker factor

whenever T ';S is ergodic. The idea of a factor determined by a saturated

group allows us to prove that if T is ergodic but T ';S is not, then the Rokhlin

cocycle x 7! S '(x) j A is a coboundary as a cocycle taking values in Aut(A),

where A is the non-trivial factor of S corresponding to the above-mentioned

eigenvalue group. Finally, by replacing coboundary by quasi-coboundary, a

similar conclusion is achieved when T is mildly mixing but T ';S is not, and

when T is mixing but T ';S is not.

Another tool explored here is a use of mixing sequences of weighted

unitary operators, that is, of operators on L 2 (X;B;) given by the formula

f 7! f T for each f 2 L 2 (X;B;)

determined by a measurable : X !Tand an automorphism T. This,

in particular, will solve the problem of lifting mild mixing property, and

complete the picture from [22] of lifting mixing and multiple mixing.

1Preliminaries

We briey recall basic denitions, some known results and x notation for

the rest of the paper.

1.1Self-joiningsofanautomorphism,relativeconcepts

Assume that T is an automorphism of a standard probability Borel space

(X;B;), which we denote T 2 Aut(X;B;) 4 . Denote by J(T) the set of

self-joinings of T, that means the set of TT-invariant probability measures

on (XX;BB) whose both marginals are equal to . To each self-joining

2 J(T) one associates a Markov operator

of L 2 (X;B;) given by

f(y)g(y) d(y) =

f(x)g(y) d(x;y)

for each f;g 2 L 2 (X;B;). Moreover, the T T{invariance of means that

T = T :

(1)

4 We shall also denote by T the unitary operator f 7! f T on L 2 (X;B;).

5 A linear bounded operator of L 2 (X;B;) is calledMarkovif (1) = 1 = (1)

and f 0 whenever f 0.

Notice also that we always have k fk kfk and thus

k k = 1.

On the other hand each Markov operator on L 2 (X;B;) for which (1)

holds determines a self-joining by the formula

(AB) =

(1 A ) d

for each A;B 2B. Then

and = :

(2)

Therefore the set J(T) can naturally be identied with the set J(T) of

Markov operators on L 2 (X;B;) satisfying (1). The set J(T) is a closed

subset in the weak operator topology and hence it is compact. Thus

n ! i h n f;gi!hf;gi for each f;g 2 L 2 (X;B;):

By transferring the weak operator topology via (2) we obtain the weak

topology on J(T) and

n ! i n (AB) ! (AB) for each A;B 2B:

Since the composition of two Markov operators is Markov, J(T) is a compact

semitopological semigroup. By the same token, J(T) is also a compact

semitopological semigroup ( 1 2 := 1 2 ).

Given a factor

, i.e. a T-invariant sub--algebra AB, let

x x d(x)

X=A

be the disintegration of over the factor A. By setting

A =

x x x d(x):

X=A

we obtain a self-joining A which is often called the relative product over

A. Note that A j AA = A , where A (A 1 A 2 ) = (A 1 \A 2 ) for each

A 1 ;A 2 2A. Moreover, we have A = E(jA).

Assume additionally that T is ergodic. Then we can speak about ergodic

self-joinings of T and the set of such joinings will be denoted by J e (T). By

J e (T) we denote the subset of J(T) corresponding to J e (T). The elements

6 Up to a little abuse of not a tion, we dene thefactorsystemTj A : (X=A;A;j A ) !

(X=A;A;j A ) in which cosets x 2 X= A are giv en by those points which cannot be distin-

guished by the sets from A; then Tj A (x) = Tx.

of J e (T) are exactly the extremal points in the natural simplex structure of

J(T). Recall that T is said to be relatively weakly mixing over a factor A if

E(jA) 2J e (T).

The notion which is complementary to relative weak mixing is the con-

cept of relative Kronecker factor [8], [30]. More precisely, if A is a factor then

the relative Kronecker factor K(A) (of T over Tj A ) is the smallest -algebra

making all relative eigenfunctions 7 measurable (AK(A)).

For more about joinings or relative concepts in ergodic theory, see e.g.

[8], [12], [19], [28] and [30].

1.2 G-actions

Assume that G is a second countable LCA group. By a G-action S =

(S g ) g2G we mean a measurable representation of G on a probability stan-

dard Borel space (Y;C;), that is a group homomorphism g 7! S g , G !

Aut(Y;C;). Then we also denote by S = (S g ) g2G the associated uni-

tary representation of G on L 2 (Y;C;), which is continuous. For each

f 2 L 2 (Y;C;), by f;S (or f is S is understood) we denote the spec-

tral measure of f, i.e. the measure on the character group G 8

determined

by the Fourier transform 9

b f;S (g) :=

(g) d f;S () =

f S g f d:

b G

We denote G(f) = spanfS g f : g 2 Gg. Then the correspondence f ! 1 b G

yields the canonical isomorphism of Sj G(f) with the representation V f =

(V g ) g2G of L 2 ( G;B( G); f ), where V g j() = (g)j(). The maximal

spectral type of S on L 0 (Y;C;) (the subspace of zero mean function in

L 2 (Y;C;)) will be denoted by S 10 .

7 By arelative(with respect to A)eigenvalueof T one means an A-measurable map

c : (X=A;A;j A ) ! U(n) for which there is M : (X;B;) !C n satisfying the following:

A =

A for a.e. x 2 X;

M 1 (x)

M n (x)

M 1 (Tx)

M n (Tx)

c(x)

(3)

M i ? A M j for i 6= j and E(jM i j 2 jA) = 1; i;j = 1;:::;n: (4)

The map M satisfying (3) and (4) is called arelativeeigenfunctioncorrespondingtoc.

8 Since G is second countable LCA, also b G is second countable LCA.

9 By Pontryagin Duality Theorem, the character group of b G has a natural identication

with G.

10 Formally speaking, it is the class of equivalence of measures which are maximal spec-

tral measures but in what follows we abuse the vocabulary and often speak about a given

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