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STUDIA MATHEMATICA 156 (1) (2003)

Semisimple extensions of irrational rotations

Mariusz Lema«czyk (Toru«), Mieczysªaw K. Mentzen (Toru«)

and Hitoshi Nakada (Yokohama)

Abstract. We show that semisimple actions of l.c.s.c. Abelian groups and cocycles

with values in such groups can be used to build new examples of semisimple automorphisms

(Z-actions) which are relatively weakly mixing extensions of irrational rotations.

Introduction. It is an important problem in ergodic theory to study

classes of automorphisms with a \given" set of self-joinings (see [24]). His-

torically, such an approach was rst presented in [20] by D. Rudolph, where

automorphisms with a minimal structure of self-joinings (called MSJ) were

shown to exist. A generalization of this notion appeared in [25] and then in

[9]. In these two articles the notion of 2-fold simplicity was introduced and

studied. An ergodic automorphism is called 2-fold simple if its only ergodic

self-joinings are either graphs or the product measure. A further generaliza-

tion was proposed in [8], where the notion of semisimplicity was introduced.

An ergodic automorphism is said to be semisimple if for each of its ergodic

self-joinings the automorphism corresponding to the self-joining is relatively

weakly mixing with respect to both marginal -algebras. As proved in [8],

such automorphisms have still strong ergodic properties, and in particular

the structure of their factors can be easily described. Based on some earlier

results of J.-P. Thouvenot, it was already remarked in [8] that some Gaussian

automorphisms are semisimple (Gaussian automorphisms are never 2-fold

simple). In [16], F. Parreau, J.-P. Thouvenot and the rst author developed

a far reaching study of Gaussian automorphisms with a minimal (in the

category of Gaussian automorphisms) set of self-joinings. All such Gaussian

systems turn out to be semisimple.

Almost all historical examples of automorphisms presented above are

weakly mixing. In fact, the only exception are ergodic rotations which are

2-fold simple but not weakly mixing. More precisely, the MSJ property im-

plies weak mixing, while in the class of 2-fold simple automorphisms we have:

2000 Mathematics Subject Classication : Primary 37A05; Secondary 37A25.

Research partly supported by KBN grant 5 P03A 027 21.

[31]

M. Lema«czyk et al.

either such an automorphism is weakly mixing or it is a rotation. In the class

of semisimple automorphisms, there has been the question of whether the

existence of a discrete part in the spectrum forces a decomposition into a

direct product of the form \discrete spectrum automorphism weakly mix-

ing automorphism". The question is natural because, as noticed in [8], an

ergodic distal automorphism is semisimple if and only if it is a rotation.

Actually more is true: if an ergodic automorphism is semisimple then it is

relatively weakly mixing over its Kronecker factor (see Section 1.2).

In this article we will construct semisimple weakly mixing extensions of

irrational rotations answering the above question. The main idea of the con-

struction comes from papers by D. Rudolph and E. Glasner and B. Weiss [7].

Roughly, we x a simple (or even semisimple) action of an Abelian l.c.s.c.

group which will serve as bre automorphisms of a skew product whose

base is an irrational rotation. Under some assumptions on the relevant bre

cocycle, the skew product turns out to be semisimple (it cannot be 2-fold

simple). In order to see that we have constructed a completely new class

(in particular, no aforementioned direct product decomposition exists) of

semisimple automorphisms we use some recent results from [14]: the class

we will consider is disjoint in the sense of Furstenberg from all weakly mix-

ing automorphisms, on the other hand the automorphisms from this class

are relatively weakly mixing extensions of the base irrational rotation.

A slightly technical part of our paper is to show the existence of some

cocycles over irrational rotations, taking values in Abelian l.c.s.c. groups and

having strong ergodic properties (see Section 3). Here, we consider two well

known examples of cocycles (one real-valued and the other integer-valued)

over the rotation by an irrational , where has bounded partial quotients.

A part of this work was done during the visit of the third author at

Nicolaus Copernicus University in Torun in April 2000.

The authors would like to thank Francois Parreau for fruitful discussions

on the subject.

1. Preliminaries

1.1. Z -action cocycles taking values in Abelian l.c.s.c. groups. Assume

that ( X; B ; ) is a standard probability space and T : ( X; B ; ) ! ( X; B ; )

is an ergodic automorphism. Let G be an Abelian locally compact second

countable (l.c.s.c.) group. Assume moreover that ' : X ! G is a cocycle.

More precisely, this means that ' is measurable and the formula

: ' ( x ) + ' ( Tx ) + ::: + ' ( T n 1 x )

if n > 0,

' ( n ) ( x ) =

if n = 0,

( ' ( T n x ) + ::: + ' ( T

1 x ))

if n < 0,

Semisimple extensions of irrational rotations

denes a cocycle for the Z-action given by n 7! T n ( n 2 Z). We will say

that ' is ergodic if the corresponding cylinder ow

T ' : ( X G; BB ( G ) ; m G ) ! ( X G; BB ( G ) ; m G ) ;

T ' ( x;g ) = ( Tx;' ( x ) + g ) ;

is ergodic. Here B ( G ) denotes the -algebra of Borel subsets of G and m G

stands for an (innite whenever G is not compact) Haar measure on B ( G ). As

follows from [22], ergodicity of ' is \ c ontrolled" by the group E ( ' ) of essen-

tial values of ' . More precisely, let G = G [f1g be the one- p oint compact-

ication of G (if G is compact th en G = G ). We dene g 2 E ( ' ) if for each

open neighbourhood U 3 g in G , for each A 2B of positive measure, there

ex ists N 2 Z such that ( A \ T

N A \ [ ' ( N ) 2 U ]) > 0. Then we put E ( ' ) :=

E ( ' ) \ G . It turns out ([22]) that E ( ' ) is a closed subgroup of G and:

(i) ' is ergodic i E ( ' ) = G ,

(ii) ' is a c ob oundary (i.e. ' ( x ) = f ( x ) f ( Tx ) for a measurable map

f : X ! G ) i E ( ' ) = f 0 g .

We say that two cocycl es '; : X ! G are cohomologous if ' is

a coboundary. In this case E ( ' ) = E ( ). Given a cocycle ' : X ! G , let

: X ! G=E ( ' ) be the corresponding quotient cocycle.

Lemma 1 ([22]). E ( '

) = f 0 g .

Following [22], we say that the cocycle ' is regular if E ( '

) = f 0 g . Then

' is cohomologous to a cocycle : X ! E ( ' ) and the latter is ergodic as a

cocycle with values in the closed subgroup E ( ' ) (see [22]).

In particular, if E ( ' ) is cocompact then ' is regular and as a direct

consequence we nd that all cocycles taking values in compact groups are

regular.

The following proposition appeared in [17].

Proposition 1. Let T be ergodic. Assume that G;H are Abelian l.c.s.c.

groups and let : G ! H be a continuous group homomorphism. If ' :

X ! G is a cocycle , then

( E ( ' )) E ( ' ) :

Moreover , ( E ( ' )) = E ( ' ) whenever ' is regular.

Given T : ( X; B ; ) ! ( X; B ; ) and ' : X ! G we denote by '

the image of on G via ' . Recall also that an increasing sequence ( q n ) of

integers is called a rigidity time for T if T q n ! Id weakly. We will make use

of the following essential value criterion.

Proposition 2 ([17]). Assume that T is ergodic and let ' : X ! G be a

cocycle with values in an Abelian l.c.s.c. group G . Let ( q n ) be a rigi di ty time

for T. Suppose that ( ' ( q n ) ) ! weakly on G. Then supp( ) E ( ' ) .

M. Lema«czyk et al.

1.2. Self-joinings of ergodic automorphisms. Let ( X; B ; ) be a standard

probability space. We denote by Aut( X; B ; ) the group of all measure-

preserving automorphisms of ( X; B ; ). Let T 2 Aut( X; B ; ) be ergodic.

By the centralizer C ( T ) of T we mean the subgroup f S 2 Aut( X; B ; ) :

ST = TS g . Any T -invariant sub- -algebra A of B is called a factor of

T (more precisely the corresponding factor is the quotient action of T

on ( X= A ; A ; )). If there is no ambiguity on T , we shall also say that B

(or T ) has A as its factor or that B is an extension of A , which we denote

by B ! A . The maximal factor of T with discrete spectrum (which does

exist) is called the Kronecker factor of T .

Let T i : ( X i ; B i ; i ) ! ( X i ; B i ; i ), i = 1 ; 2, be two automorphisms (of

probability standard spaces). We denote by J ( T 1 ;T 2 ) the set of all joinings

of them. More precisely, 2 J ( T 1 ;T 2 ) if is a T 1 T 2 -invariant probability

measure on ( X 1 X 2 ; B 1 B 2 ) whose marginals are equal to i . We should

emphasize that by a joining we will also mean the corresponding automor-

phism T 1 T 2 on ( X 1 X 2 ; B 1 B 2 ; ) and the notation ( T 1 T 2 ; ) will

often appear. The subset J e ( T 1 ;T 2 ) of ergodic joinings consists of those

for which ( T 1 T 2 ; ) is ergodic. If each T i is ergodic then J e ( T 1 ;T 2 ) is

the set of extremal points of J ( T 1 ;T 2 ) and the ergodic decomposition of

each joining consists of elements of J e ( T 1 ;T 2 ). In case T 1 = T 2 = T we

speak about self-joinings and use the notation J ( T ), J e ( T ). In this case,

if A B is a factor of T then we denote by J ( T; A ) (resp. J e ( T; A )) the

set of self-joinings (resp. ergodic self-joinings) of the factor action of T on

( X= A ; A ; ).

If T : ( X; B ; ) ! ( X; B ; ) is an ergodic automorphism, then to each

S 2 C ( T ) we associate the graph self-joining S given by

S ( A B ) = ( A \ S

1 B )

(1)

Clearly, S is ergodic. If S = Id then instead of Id we will also write X .

Following [9], we say that T is 2- fold simple if each ergodic self-joining of T is

either a graph or equals . If T is 2-fold simple and C ( T ) = f T i : i 2 Z g

then T has the 2- fold minimal self-joining (MSJ) property. If A is a factor

of T then the relative product over A is the self-joining A in J ( T ) given

for each A;B 2B :

(2)

A ( A B ) =

X= A E ( A jA ) E ( B jA ) d for each A;B 2B :

This self-joining need not be ergodic. We say that an automorphism T is

relatively weakly mixing over A , or that B!A is relatively weakly mixing ,

if A 2 J e ( T ). The relative product is a particular case of the following

construction. Assume that 2 J ( T; A ), that is, is a self-joining of a factor

A of T . Then the self-joining

of T given by

Semisimple extensions of irrational rotations

( A B ) =

X= A X= A E ( A jA )( x ) E ( B jA )( y ) d ( x;y )

( A;B 2B )

is called the relatively independent extension of .

In the case of two automorphisms T i : ( X i ; B i ; i ) ! ( X i ; B i ; i ), i = 1 ; 2,

easy extensions of formulas (1) and (2) allow us to dene joinings between T 1

and T 2 when an isomorphism S : ( X 1 ; B 1 ; 1 ;T 1 ) ! ( X 2 ; B 2 ; 2 ;T 2 ) is given,

or, more generally, when there is an isomorphism between a non-trivial factor

of T 1 and a factor of T 2 (in the latter case we say that T 1 and T 2 have a

common factor ).

A notion complementary to weak mixing is distality (see [26] for the

denition). Given a factor AB there exists exactly one factor

A such that

A ! A is distal (see

[26] or [4, Th. 6.17 and the nal remark on page 139]). The decomposition

B !

A B , B !

A is relatively weakly mixing and

A ! A is called the Furstenberg{Zimmer decomposition of B ! A .

It follows that, given a factor A , there exists a smallest factor

AA such

that T is relatively weakly mixing over

A . If A is trivial, then

A = D is the

maximal distal factor of T .

Following [8], we say that an ergodic automorphism T is semisimple if for

each ergodic 2 J e ( T ) the extension ( BB ; ) ! ( B X; ) is relatively

weakly mixing (clearly, ( B X; ) can be identied with ( B ; )). It has

been noticed in [8] that an ergodic distal automorphism is semisimple i

it is isomorphic to a rotation. Moreover, if T is semisimple and B ! A is

relatively weakly mixing then A is also semisimple ([8]). It follows that if T is

semisimple and D stands for its maximal distal factor then D is semisimple

because B!D is relatively weakly mixing. We have shown the following.

Proposition 3. If T is semisimple then it is a relatively weakly mixing

extension of its Kronecker factor.

Two automorphisms T i : ( X i ; B i ; i ) ! ( X i ; B i ; i ), i = 1 ; 2, are said to

be disjoint if J ( T 1 ;T 2 ) = f 1 2 g ([3]). We will then write T 1 ? T 2 .

1.3. Actions of Abelian locally compact second countable groups. Assume

that G is an Abelian l.c.s.c. group and let G = f R g g g 2 G be a Borel action of

this group on a Borel space ( Y; C ) (we always suppose that such a space is

standard, that is, up to isomorphism, Y is a Polish space, while C stands for

the -algebra of Borel sets), meaning that the map G Y 3 ( g;y ) 7! R g y 2

Y is measurable. If now is a probability measure invariant under the action

of G then the notions dened in the previous section for Z-actions can be

extended to corresponding notions for actions of G on ( Y; C ; ) (see also [9]).

We say that G is mildly mixing if it has no non-trivial rigid factors, that is,

whenever for A 2C there exists ( g n ) G , g n !1 and ( R g n A 4 A ) ! 0

( n !1 ), then ( A ) = 0 or 1 (see [23], also [5], [14], [15]).

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