exactnessRokhlin(2).pdf

EXACTNESS OF ROKHLIN ENDOMORPHISMS

AND WEAK MIXING OF POISSON BOUNDARIES.

& MARIUSZ LEMANCZYK

JON AARONSON

ABSTRACT. We give conditions for the exactness of Rokhlin endomorphisms, apply

these to random walks on locally compact, second countable topological groups and

obtain that the action on the Poisson boundary of an adapted random walk on such

a group is weakly mixing.

§ 0 INTRODUCTION

Rokhlin endomorphisms.

By a non-singular endomorphism we mean a quadruple ( X,B,m,T ) where

( X,B,m ) is a standard probability space and T : X 0 →X 0 is a measurable trans-

formation of X 0 ∈B, m ( X 0 ) = 1 satisfying m ( T −1 A ) = 0 ⇔m ( A ) = 0 ( A∈B ).

As in [Ro2], the endomorphism T is called exact if T ( T ) :=

∞ n=0 T −n B = {∅,X} .

Let G be a locally compact, Polish topological (LCP) group.

By a non-singular G -action on a probability space ( Y,C,ν ) we mean a measurable

homomorphism S : G→ Aut ( Y ) where Aut ( Y ) denotes the group of invertible,

non-singular transformations of Y equipped with its usual Polish topology.

The

action S is called probability preserving if each S g preserves ν .

Given a non-singular endomorphism ( X,B,m,T ), a non-singular G -action S :

G→ Aut ( Y ) of a LCP group G and a measurable function f : X→G , we consider

the Rokhlin endomorphism

T =

T f,S : X×Y→X×Y dened by

T ( x,y ) := ( Tx,S f (x) y ) .

Rokhlin endomorphisms rst appeared in [Ro1] (see also [AR]). We give conditions

for their exactness (theorem 2.3).

These conditions are applied to random walk endomorphisms. Meilijson (in

[Me]) gave sucient conditions for exactness for random walk endomorphisms over

G = Z . We clarify Meilijson’s theorem proving a converse (proposition 4.2), extend

it to LCP Abelian groups (theorem 4.1), characterize the exactness of the Rokhlin

endomorphism for a steady random walk (theorem 4.5) and obtain that the group

action on the Poisson boundary (see § 4) of an adapted (i.e.

globally supported)

random walk is weakly mixing (proposition 4.4).

1991 Mathematics Subject Classication . 37A20, 37A15 (37A40, 60B15, 60J50).

c 2004. Aaronson was partially supported by the EC FP5, IMPAN-BC Centre of Excellence

when this work was begun. Lemanczyk’s research was partially supported by KBN grant 1 P03A

03826. Both authors thank MPIM, Bonn for hospitality when this work was nished.

Typeset by A M S -T E X

& MARIUSZ LEMA NCZYK

JON AARONSON

Tools employed include the ergodic theory of “associated actions” (see § 1), and

the boundary theory of random walks (see § 4).

The authors would like to thank the referee for some useful comments including

a simplication in the proof of proposition 3.4.

§ 1 ASSOCIATED ACTIONS

For a non-singular endomorphism ( Z,D,ν,R ) set

•I ( R ) := {A∈D : R −1 A = A} – the invariant σ -algebra, and

•T ( R ) :=

∞

n=0 R −n D – the tail σ -algebra.

Let ( X,B,m,T ) be a non-singular endomorphism. Let G be a locally compact,

Polish topological (LCP) group, let f : X→G be measurable.

There are two associated (right) G -actions arising from the invariant and tail

σ -algebras of T f , which are dened as follows:

• dene the (left) skew product endomorphism T f : X×G→X×G by T f ( x,g ) :=

( Tx,f ( x ) g ) and x P∈P ( X×G ) , P∼m×m G ;

• for t∈G , dene Q t : X×G→X×G by Q t ( x,g ) := ( x,gt −1 ), then Q t ◦T f =

T f ◦Q t .

The associated invariant action.

The invariant factor of ( X×G,B ( X×G ) , P,T f ) is a standard probability space

( ,F,P ) = ( I ,F I ,P I ) equipped with a measurable map π : X×G→ such

that P◦π −1 = P, π◦T f = π and π −1 F = I ( T f ).

Since Q t I ( T f ) = I ( T f ) (because T f ◦Q t = Q t ◦T f ),

•∃ a P -non-singular G -action P : → so that π◦Q = P◦π .

Proposition 1.1.

( ,F,P, P ) is ergodic i ( X,B,m,T ) is ergodic.

Proof. If ( X,B,m,T ) is ergodic, then so is ( X×G,B ( X×G ) , P,T f ,Q ) where

T f ,Q denotes the Z + ×G action dened by ( n,t ) →T f ◦Q t ∈ Aut ( X×G ). Any

P -invariant, measurable function on lifts by π to a T f ,Q -invariant, measurable

function on X×G , which is P -a.e. constant.

Conversely, any T -invariant function on X lifts to a T f -invariant function on

X×G which is also Q -invariant and thus the lift of a P -invariant, measurable

function on . If ( ,F,P,g ) is ergodic this function is constant (a.e.).

€

The non-singular G -action ( ,F,P, P ) is called the invariant- or Poisson

G -

action associated to ( T,f ) and denoted P = P ( T,f ).

This action is related to the Mackey range of a cocycle (see [Zi2] and § 3), and

the Poisson boundary of a random walk (see § 4).

The associated tail action.

The tail factor of ( X×G,B ( X×G ) , P,T f ) is a standard probability space

( ,F,P ) = ( T ,F T ,P T ) equipped with a measurable map π : X×G→ such

that P◦π −1 = P, π −1 F = T ( T f ).

Since Q t T ( T f ) = T ( T f ) (because T f ◦Q t = Q t ◦T f ),

•∃ a P -non-singular G -action τ : → so that π◦Q = τ◦π .

Proposition 1.2.

( ,F,P,τ ) is ergodic i ( X,B,m,T ) is exact.

Proof. Suppose rst that ( X,B,m,T ) is exact. Any τ -invariant, measurable func-

tion F : →R lifts by π to a Q -invariant, T ( T f )-measurable function F : X×G→

R . In particular ∃ ! F n : X×G→R

( n≥ 0), measurable so that F = F n ◦T n . It

EXACTNESS

follows from Q t ◦T f = T f ◦Q t that each F n is Q -invariant, whence ∃F n : X→R

measurable so that F n ( x,y ) =

F n ( x ) for P -a.e. ( x,y ) ∈X×G . Thus F = F 0 =

F 0

is T ( T )-measurable, whence constant P -a.e. by exactness of T .

Conversely, any T ( T )-measurable function on X lifts to a T ( T f )-measurable

function on X×G which is also Q -invariant and thus the lift of a τ -invariant, mea-

surable function on . If ( ,F,P,τ ) is ergodic this function is constant (a.e.).

€

The non-singular G -action ( ,F,P,τ ) is called the associated tail G -action (of

( T,f )) and denoted τ = τ ( T,f ). As with the Poisson action, the tail action is

related to the Mackey range of a cocycle, and also to the tail boundary of a random

walk (see § 4).

§ 2 CONDITIONS FOR EXACTNESS AND A CONSTRUCTION OF ZIMMER

We begin with a proposition generalising Zimmer’s construction (in [Zi1]) of

a G -valued cocycle over an ergodic, probability preserving transformation with a

prescribed ergodic, non-singular G -action as Mackey range.

Proposition 2.1.

Suppose that G is a LCP group,

that ( X,B,m,T ) is a non-singular endomorphism, and suppose that f : X→G

measurable.

Let S be a non-singular G -action on a probability space ( Y,C,ν ) and dene

T =

T f,S : X×Y→X×Y by

T ( x,y ) := ( Tx,S f (x) y );

then

T,f ) ∼ = P ( T,f ) ×S, & τ (

T,f ) ∼ = τ ( T,f ) ×S.

P (

Proof.

Dene π : X×Y×G→X×G×Y by π ( x,y,g ) := ( x,g,S g −1 y ) . Evidently π

is a bimeasurable bijection.

Fix p∈P ( G ) , p∼m G . A calculation shows that

( m×ν×p ) ◦π −1 ∼m×p×ν,

(1)

indeed

d ( m×ν×p ) ◦π −1

d ( m×p×ν )

( x,g,y ) = dν◦S g

dν

( y ) =: D ( g,y ) .

Next, we claim that

(2)

π◦

T f = ( T f × Id | Y ) ◦π.

To see this,

π◦

T f ( x,y,g ) = π ( Tx,S f (x) y,f ( x ) g ) = ( Tx,f ( x ) g,S (f (x)g) −1 S f (x) y )

= ( Tx,f ( x ) g,S −1 y ) = T f × Id ◦π ( x,y,g ) .

& MARIUSZ LEMA NCZYK

JON AARONSON

It follows from (2) that

T f ) = π −1 ( I ( T f × Id)) ,T (

T f ) = π −1 ( T ( T f × Id)) .

I (

Now, in general

I ( T f × Id) = I ( T f ) ⊗B ( Y ) ,T ( T f × Id) = T ( T f ) ⊗B ( Y )

mod m×p×ν

and so

T f ) = π −1 ( I ( T f ) ⊗B ( Y )) ,T (

T f ) = π −1 ( T ( T f ) ⊗B ( Y ))

I (

mod m×p×ν.

P i ) ( i = P, τ ) be the invariant or tail factors

of ( X×G,m×p,T f ) and ( X×Y×G,m×ν×p,

Now let ( i ,F i ,P i ) and let (

i ,

F i ,

T f ) respectively, according to

the value of i = P, τ . By (1) and (2), π induces a measure space isomorphism of

(

i ,

P i ) with ( i ×Y,P i ×ν ).

Denoting the associated G -actions by

Q t ( x,y,g ) := ( x,y,gt −1 ) and Q t ( x,g ) :=

( x,gt −1 ), we note that

Q t = ( Q t ×S t ) ◦π.

The proposition now follows from this.

π◦

€

Corollary 2.2.

T is ergodic i P ( T,f ) ×S is ergodic.

T is exact i τ ( T,f ) ×S is ergodic.

3) If both T f and S are ergodic, then

T,f ) ∼ = S.

T is ergodic and P (

T,f ) ∼ = S.

Proof. Parts 1) and 2) follow from propositions 1.1 and 1.2. Parts 3) and 4) follow

from these and form essentially Zimmer’s construction.

4) If T f is exact and S is ergodic, then

T is exact and τ (

€

§ 3 LOCALLY INVERTIBLE ENDOMORPHISMS

In this section, we obtain additional results for a non-singular, exact endomor-

phism ( X,B,m,T ) of a standard measure space which is locally invertible in the

sense that ∃ an at most countable partition α⊂B so that T : a→Ta is invertible,

non-singular ∀a∈α . Under the assumption of local invertibility, the associated

actions of § 1 are Mackey ranges of cocycles (as in [Zi2]). See proposition 3.2 (below).

As in [S-W], we call an ergodic, non-singular G -action ( X,B,m,U ) properly er-

godic if m ( U G ( x )) = 0 ∀x∈X and call a properly ergodic, non-singular G -action

S : G→ Aut ( Y ) mildly mixing if U×S is ergodic for any properly ergodic

non-singular G -action ( X,B,m,U ). As shown in [S-W]:

• there are no mildly mixing actions of compact groups,

• a mildly mixing action of non-compact LCP group has an equivalent, invariant

probability. Moreover,

• a probability preserving G -action ( G a non-compact LCP group) ( Y,C,ν,S ) is

mildly mixing i

L 2 (ν)

−→f⇒f is constant .

G −→∞, f◦S g N

f∈L 2 ( ν ) , g n ∈G, g n

We prove

EXACTNESS

Theorem 3.1.

Suppose that ( X,B,m,T ) is a non-singular, locally invertible, exact endomor-

phism of a standard measure space, that

is a LCP, non-compact, Abelian group

and that f : X→G is measurable.

Either

T f,S is exact for every mildly mixing probability preserving G -action S :

G→ Aut ( Y ) , or ∃ a compact subgroup

K≤G, t∈G

and f : X→K, g : X→G

measurable so that

f = g−g◦T + t + f.

Note that the invertible version of this generalizes corollary 6 of [Ru]. The rest

of this section is the proof of theorem 3.1.

Tail relations.

Let ( X,B,m,T ) be a non-singular, locally invertible endomorphism of a standard

probability space. Consider the tail relations

T ( T ) := { ( x,y ) ∈X×X : ∃k≥ 0 , T k x = T k y} ;

G ( T ) := { ( x,y ) ∈X 0 ×X 0 : ∃k, ℓ≥ 0 , T k x = T ℓ y}

where X 0 := {x∈X : T n+k x = T k x∀n, k≥ 1 } . We assume that m ( X\X 0 ) = 0

(which is the case if T is ergodic and m is non-atomic) and so T ( T ) ⊂G ( T )

mod m . Both T ( T ) and G ( T ) are standard, countable, m -non-singular equivalence

relations in sense of [F-M] whose invariant sets are given by

I ( G ( T )) = I ( T ) , I ( T ( T )) = T ( T )

respectively.

Given a LCP group G and f : X→G measurable, dene f n : X→G

( n≥ 1)

f n ( x ) := f ( T n−1 x ) f ( T n−2 x ) ...f ( Tx ) f ( x )

and dene f : G ( T ) →G by

f ( x,x ′ ) := f ℓ ( x ′ ) −1 f k ( x ) for k, ℓ≥ 0 such that T k x = T ℓ x ′

(this does not depend on the k, ℓ≥ 0 such that T k x = T ℓ x ′ for x,x ′ ∈X 0 ).

It follows that f : G ( T ) →G is a (left) G ( T ) -orbit cocycle in the sense that

f ( y,z ) f ( x,y ) = f ( x,z ) ∀ ( x,y ) , ( y,z ) ∈G ( T ) .

Note that since T ( T ) ⊂G ( T )

mod m , the restriction f : T ( T ) →G

is a (left)

T ( T )-orbit cocycle.

Mackey ranges of cocycles. Let R be a countable, standard, non-singular equiv-

alence relation on the standard measure space ( X,B,m ) and let : R→G be a

left R -orbit cocycle. It follows from theorem 1 in [F-M], there is a countable group

and a non-singular -action ( X,B,m,V ) so that

R = R V := { ( x,V γ x ) : γ∈ , x∈X}.

Let f ( γ,x ) := ( x,V γ x ) ( f = f ,V : ×X→G ) be the associated left V -cocycle

(satisfying f ( γγ ′ ,z ) = f ( γ,V γ ′ z ) f ( γ ′ ,z )).

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