endomorphism

8/22/2019 endomorphism

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.JOU RNAL OF MATHE MATICAL ANALYSIS AND APPLIC ATIONS 204, 854867 1996

ARTICLE NO. 0472

A Construction of a Non-Measure-Preserving

E ndomorphism U sing Quotient R elations and

Automorphic FactorsU

Karma Dajani

Department of Mathematics, Uni ersiteit Utrecht, The Netherlands

and

Jane Hawkins

Department of Mathematics, Uni ersity of North Carolina at Chapel Hill, Chapel Hill,North Carolina 27599

Submitted by Dorothy Maharam Stone

Received November 10, 1995

1. INTRODUCTION

The main purpose of this paper is to construct ergodic, nonsingular,

conservative n-to-one endomorphisms which preserve no equivalent -w xfinite measure. While such examples a re known to exist 6, 9, 12, 19 , our

examples are fundamentally different from the existing ones. It was no-w xticed independently in 6, 19 that the t wo-sided type I II B ernoulli shift of

w xH ama chi 9 is the nat ural extension of an exact two-to-one endomorphismw xwith no equivalent -finite invariant measure. Also in 12 it is shown tha t

Cartesian products of finite measure-preserving exact with type III auto-

morphisms yield examples of n-to-one endomorphisms with no equivalent

invariant measure. In Section 5 we give an example of a two-to-one

endomorphism whose ergodic nonsingular measure is neither exact nor is

it equivalent to a product measure of an automorphism with an exact

endomorphism.

* The a uthors were pa rtially supported by NSF G rant D MS-910356.

854

0022-247Xr96 $18.00

Copyright 1996 by Academic Press, Inc.

All rights of reproduction in any form reserved.


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C O NSTR U C TI O N O F AN E N D O M O R P H I SM 855

O ur construction is based on the connections between the R adonNiko-

dym derivative of the original endomorphism and that of its maximal

automorphic factor. U sing a chain rule for factor maps, we prove a

theorem giving a condition for a maximal automorphic factor which does

not preserve any equivalent measure to force the endomorphism to have .that same property Theorem 4.5 of this paper . As a n important step in

proving that the hypotheses of Theorem 4.5 are satisfied, we characterize

the maximal automorphic factor of an endomorphism as the action of a

quotient relation of two natural relations present in every endomorphism.

In 1989, Feldman, Sutherland, and Z immer studied ergodic relations,w xsubrelations, and quotient relations 8 . They are also discussed in an

w xearlier paper by K. Schmidt 17 . In this paper, we compute the quotientrelation of the orbit relation for a noninvertible ergodic countable-to-one

endomorphism by a natural subrelation which is trivial precisely when the

map is invertible. We show that we obtain the group of integers acting by

the maximal automorphic factor of the endomorphism as the quotient. We

use duality properties of the relations to characterize exactness of anw xendomorphism in P roposition 3.8; this extends a n earlier result from 1 .

Some results about measure theoretic properties of ergodic endomor-phisms are proved in Section 2; the characterization of the maximal

automorphic factor as an action of a quotient relation is presented in

Section 3. The construction is done in Sections 4 and 5. This paper is the

third in a study conducted by the authors on nonsingular endomorphisms, w x.their fa ctors, and their natural extensions cf. 5, 6 .

2. PRELIMINARY MEASURE THEORETIC RESULTS

ABOUT NONSINGULAR ENDOMORPHISMS

.Throughout this paper we will assume that X, BB, is a Lebesgueprobability space and T : XX is a nonsingular conservat ive ergodicendomorphism which is surjective and countable-to-one almost every-

.where. The a ssumption that X - results in no loss of generality. w xwhen T is n-to-one; this is shown in Lemma 2.4. B y a result of R ohlin 16

w x.see also 20 , we can assume by replacing X by a measurable T-invariantsubset of full measure if necessary that T is forward nonsingular as well,

. y1 .so that T satisfies that for all A gBB, A s 0m T A s 0m . w x TA s 0. We apply a well-known result of R ohlin 16 to obtain a

4measurable partition s A , A , A , . . . o f X into at most countably1 2 3many pieces satisfying:

. .i A ) 0 for each i;i .ii the restriction of T to each A , which we will write as T, isi i

one-to-one;


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D AJANI AND HAWKINS856

.iii ea ch A is of maximal measure in X_D A with respect toi j- i j .property ii ;

.iv T is one-to-one and onto X.1

When we say that the endomorphism T is n-to-one, we mean that every 4 . .partition s A , A , A , . . . satisfying i iv conta ins precisely n atoms1 2 3and that T is one-to-one and onto X for each i s 1, . . . , n. Equivalently,ifor -a.e. x gX, the set Ty1x contains exactly n points.

. . . .For ea ch x gA , let J x s d Trd x , and for x gX, let J xi T i Ti .s J x . This is the Jacobian function for T, defined by W. Parryi T Ai i

w x15 , and is independent of the choice of . Our nonsingularity assump-

tions imply that J ) 0 a.e. In order to define the RadonNikodymTderivative of T, we consider the following identities holding a.e. see

w x w x.10 o r 18 :

d Ty1 1 x ' x s ; 1 . . .T d J y .

y1 TygT x

d 1 x ' Tx s . 2 . . .T y1 Txd T .

T

1 .The function satisfies for every fgL X, BB, ,T

f Tx x d x s f x d x . 3 . . . . . .

H HTX X

.For any function satisfying 3 in place of , w e s a y t h a t isT

w xMarkovian for T and , and it was shown in 19 that is the onlyT

Ty1BB measurable function which is Markovian for T and . Thus, if is

a BB measurable function which is Ma rkovian for T and then g1 . y1 . y1 .L X, BB, and E T BB s . H e r e E h T BB denotes the

T

1 .conditional expecta tion of h gL X, BB, with respect to the sub--alge-bra Ty1BB. We call the function the RadonNikodym deri ati e of T.

T

. yk . k . ykkSimilarly, x s drd T T x is the unique T BB measurableT

k .function which is Markovian for T and . It is easily seen that k, x

. . ky1 .' x Tx T x is a BB measurable function which isT T T

k . yk . kMarkovian for T and so that E k, T BB s . T

From these observations we show that even though we do not have achain rule we have a related identity which we call the Pseudo-Chain rule .Proposition 2.3 below . Before we can prove that, we generalize a result

w xmentioned in 12 .


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LEMMA 2.1. Assume that T is a nonsingular endomorphism which is .n-to-one on X, BB, , with a -finite measure on BB. Then for eery

k g, is -finite on TykBB.

Proof. We fix any k G 1. The endomorphism Tk is m snk-to-one so 4we can f ind a measurable partition s A , A , A , . . . , A such that the1 2 3 m

restriction of Tk to each atom is one-to-one. G iven a ny set B g TykBB :

BB, since is -finite on BB, we consider the set B lA sB and we1 1 .write it as B s D B with B - . Now the B s determine,1 j s1 1, j 1, j 1, j1 1 1 1

via symmetric points, a unique countable partition of B lA sB in fact2 2. of B lA for every p s 1, . . . , m . We write B s D B , wherep 2 j s1 2, j1 1

yk

k

. .B s T T B ; if B s for any j , we subdivide further if2, j 2 1, j 2, j 11 1 1necessary since is -finite. We write B s D D B , with2 j s1 j s1 2, j j1 2 1 2B s D B and each B of finite measure; we use this parti-2, j j s1 2, j j 2, j j1 2 1 2 1 2tion to refine the partition of B obtained previously. We proceed induc-1tively, refining all previous partitions at each step, and this process stops

after a f inite number of steps, when we reach A . Finally we writemB s D D D C , with C s D B gj s1 j s1 j s1 j j . . . j j j . . . j ps1 p, j j . . . j1 2 m 1 2 m 1 2 m 1 2 m

yk .T BB and C - .j j . . . j1 2 m

Remark 2.2. Lemma 2.1 is false when T is countable-to-one. A coun-terexample is Xs with s Lebesgue measure, and Tx s tan x.Lebesgue measure is clearly -finite on with respect to the -algebra BB

of Borel sets, but it is not -finite with respect to Ty1BB.w xIn 10 a generalization of the conditional expectation operator onto

Ty1BB, denoted E1, is defined as a linear operator on the space of

measurable functions on X when T is finite-to-one and is -finite. It is1 . . . ..y1defined by E h x s h y rJ y . Similarly, we define y g T T x . T T

k . . . ..yk k k kfor each k g, E h x s h y rJ y . Clearly for y g T T x . T T

k . ykeach BB measurable h, the function E h is T BB measurable, and if

1 . 1 . y1 .h gL X, BB, , by Lemma 2.1 we have E h sE h T BB .

.P ROPOSITION 2.3 P seudo-C hain R ule . Assume that T is a nonsingular

.endomorphism which is n-to-one on X, BB, , with a -finite measure onBB. For each k g, for eery i s 1, . . . , k y 1, and a.e. x gX we hae

k sEk i ky i (Ti . .T T T

.If X - , and T is countable to one, then

k sE i Tyk

BB ky i (Ti

. .T T T

Proof. We fix any k g, and choose any i s 1, . . . , k y 1. It is easilyshown that i ky i (Ti is a BB measurable Markovian function for

T T


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k w x ykkT and . B y 19, E xample 1.3 , is the unique T BB measurableT

Markovian. Furthermore, since ky i (Ti is TykBB measurable we haveT

k i . k . ii ky i i ky iE (T sE (T . The result follows if we show T T T T

k k .that if is Markovian for T and , then E is also Markovian. Since

k . . . .. .yk k k kE x s y rJ y x , we ha ve for a ny fg y g T T x . T T1 .L X, BB, ,

f x d x s f(Tk x x d x . . . . .H HX X

since is Markovian;

nk 1k yk yks f(T T x T x d x . . . H j j yk

kJ T xX .js1 T j

yk 4nk

by a change of variables and since T X forms a disjoint partition ofj js1X;

nk yk T x .js f x d x . . H yk

kJ T xX .js1 T j

y .s f x d x . .H /kJ y .X yk T .ygT x

and since k is Markovian,T

y .k

kf(T x x d x . . . .H T /kJ y .X yk k T .ygT T xThe following lemma shows that the assumption that is a probabilitymeasure does not result in any loss of generality for n-to-one maps.

X .LEMMA 2.4. If T is a nonsingular n-to-one endomorphism on X, BB, with

Xa -finite measure, then there exists a finite measure ; X such that . y1X s h(Trh for some T BB measurable function h.

T T

w x X XProof. B y definition 11 , a measure sh d is cohomologous to . XXif s h(Trh . Then is cohomologous to if and only if h is

T Ty1 w xT BB-measurable 11 , a nd is finite if a nd only if h is integrable. There-1 y1 X .fore it suffices to find a positive h gL X, T BB, ; such an h exists

X y1since is -finite on T BB by Lemma 2.1.


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3. TH E QU OTIE NT R E LATIO N R r S AND TH ET TMAXIMAL AUTOMORPHIC FACTOR FOR

C OU NTAB LE -TO -ONE MAP S T

For each countable-to-one nonsingular endomorphism T, we define two w x.amenable equivalence relations R and S :R cf. 4, 10 . The relationT T T

.R :X=X is defined as follows: x, y gR if and only if there existT Tm, n G 1 such that Tnx s Tmy. We also associate a subrelation S :RT T

. n n:X=X by x, y g S if and only if T x s T y for some n G 1. WhenTthe endomorphism T is clearly understood, we write R and S for R andT

. . 4 . .S . For x gX, let R x ' y gX: x, y gR and S x ' y gX: x, yT4 . yn n w xg S . One can verify that S x s D T T x 10 . Similarly we definen G 0

. . 4for each set A gBB, R A ' y: x, y gR for some x gA , and we say . ..that R is nonsingular if A s 0m R A s 0 for a ll A gBB. We say

. . . .R is ergodic with respect to if R A sA A s 0 or X_ A s 0.We have identical definitions for the subrelation S.

If T is one-to-one, then S is trivia l each equivalence class consists of.exactly one point and R is the usual equivalence relation associated to

w xorbits. I n 10 the f ollowing connections were proved to exist between thema p T and its associated relations R and S.

.1 T is nonsingular mR is nonsingular.

.2 T is ergodic mR is ergodic.

. .3 T is nonsingular S is nonsingular and the converse is false .

.4 T is exactm

S is ergodic.

Also, it is easily checked that:

. n . n .5 For all n g , S T x s T Sx for a .e. x gX.

.6 S is a subrelation of R of infinite index if and only if T is notinvertible.

We apply a technique defined by Feldman, Sutherland, and Zimmer tow xdefine the quotient relation R rS in a measurable wa y 8 . In general theT T

quotient of a countable ergodic relation by a subrelation can only be

described as a groupoid, but in our case we obtain a genuine group, ,

acting naturally by T on a factor space of X. B y convention T0x sx.We first define the choice maps of a relation and subrelation, following

w x 48 . Let s A , A , A , . . . denote a partition chosen to satisfy condi-1 2 3 . . k k k 4t ions i iv of Section 1. For each k g, let s A , A , . . . denote1 2the partition of X defined by k s Eky 1Tyi; it satisfies conditionsis0 . . k k k ki iv for T , so that the restriction of T to A , written as T , is1 1


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one-to-one and onto X. For each n g we define , : XX byn

Tnx if n G 0 x s y1n k T x if n s yk , k g. .1

The maps are called choice maps, a nd are d efined in t his wa y to satisfynthe following easily verified properties.

LEMMA 3.1.

.1 For eery n g , each is measurable.n . . . ..2 For each n g , for eery x, y g S, x , y g S.n n . . ..3 For each x gX, R x s D S x .nsy n . . ..4 ( x , x g S for all n, m g n m nqm.

We now consider the ergodic decomposition of with respect to the .relation S. We obtain a Lebesgue space Y, FF, and a canonical system

4 .of measures on X such tha t for a ny A gBB, A sy y g Y . . w x . .H A d y 16 . Let : X, BB, Y, FF, be the canonical projec-Y y

tion; then y1FF:BB is the -algebra of S-invariant subsets. We show that 4the choice maps on X induce automorphisms on Y, by definingn n

. .for each n g and y g Y, x ' x .n n

P ROPOSITION 3.2. For each n g , the map is an automorphism of Y.n

Proof. We assume first that n g so that s Tn; then by definitionn . n . yn y1 y1 . x s T x . It suffices to show that T FFs FF mod 0 . Letn

A g Tyny1FF; then A s TynB for some B g y1FFand SB sB. We writeyn yn yn y1 .A s T B s T SB s ST B s SA g FF by applying 5 above. The

reverse containment is shown by taking B g y1FF; then B s Tyn TnB sTyn TnSB s Tyn STnB g Tyny1FF.

The proposition is trivially true for n s 0 since s Id . For n g0, Xwe now prove the result for . We have shown that is an automor-yn n

y1 y1 .phism so exists and is an automorphism. We claim that x sn n . y1 . . x a.e., hence s . By Lemma 3.1 4 , x, ( x g S soyn yn n n yn

. . ..that x s ( x s x for a .e. x.n yn n yn

The following result is immediate from Lemma 3.1 and the proof of

Proposition 3.2.

. .C OROLLARY 3.3. 1 For eery m, n g , ( s ; 2 s Id.n m mqn 0

.

n

Therefore we will write

s

and

s ( (

n times '

;1 n n4 .i.e., the family defines a action on the space- Y, FF, . Thenggroup action generated by is the factor action on Y of the semigroupaction generated by T on X.


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. 4We analyze more closely the partition of X given by s S x : x gX .w xIn 10 it was shown tha t there is a one-to-one correspondence between

S-invariant sets and sets in the tail field F TynBB. From this we seenG 0that the measurable hull of is

Xs F Tyn, where is the pointnG 0

partition, and X generates the tail -algebra F TynBB. We denote byn G 0X X the factor space associated with the tail -algebra and by X the

associated factor measure. It is clear that every automorphic factor of . yn .X X XX, BB, ; T is contained in the tail X , F T BB, ; T , which is

nG 0

.isomorphic via the identity map to the space Y obta ined a bove; t hereforewe have just shown the following.

.P ROPOSITION 3.4. The factor Y, FF, ; is the maximal automorphic .factor of X, BB, ; T .

Ergodicity of the relation S is the same as exactness of the endomor-w xphism T 10 , so we have the following corollary. We remark that the

measures are not necessarily nonsingular for T.y

C OROLLARY 3.5. For a.e. y g Y, the measure is tail tri ial for T ony . yn .X, BB ; that is, for eery A g F T BB, A s 0 or 1.n G 0 y

. . Remark 3.6. 1 If Tsf=g and X, BB, s X =X , BB =BB , 1 2 1 2 1. .= with f an automorphism of X , BB , and g an exact endomor-2 1 1 1

. .phism of X , BB , , then Proposition 3.4 implies that Y, FF, ,2 2 2 .X , BB , . Furthermore, the fa ctor endomorphism induced by T on Y is1 1 1f.

. y1 n . n y1 .2 Since f or a ll n g , y s T y for a.e. y g Y, wen y1 w xcould define s (T ( instead of using the choice function of 8 .n

w xD EFINITION 3.7 8, 17 . For a nonsingular conservative countable-to-one

endomorphism T, we d efine the quotient relation R rS to be the group T T, .endowed with the action generated by on Y, FF .

The next proposition is a reflection of some duality properties ofrelations and subrelations discussed by Feldman, Sutherland, and Zimmerw x8, P roposition 1.5 . The only if direction of P roposition 3.8 is a lso sta ted

w x .in 2 . By , MM, we will denote the Borel space of the integers with thediscrete topology and counting measure. We give a short ergodic theoreti-

w xcal proof instead of using 8 .

P ROPOSITION 3.8. Suppose T is any countable-to-one ergodic nonsingular .endomorphism on X, BB, . Then T is exact if and only if the product map . .T: X= =X gi en by T x, m s Tx, m q 1 is ergodic with respectto s = .


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.Proof. Suppose that T is not exact. Then there exists a set A, . yn n .0 - A - 1, such that A s T T A for all n g. We construct a

j 4nontrivial invariant set for T of the form A s D T A = j . Clearlyj g . A ) 0 and the same is true for its complement, and it is easy to see

y1 .that T A sA. . We now suppose that T is exact, but that T is not ergodic. Thenyi F T BB is trivial, and there exists a nontrivial invariant set for T, sayi G 0

yi yi .B gBB =MM; in particular, B g F T BB =MM s F T BB =MM.i G 0 i G 0Then B is of the form X= C with C gMM, but invariance of C impliesC s .The contradiction implies the result .

Remark 3.9. T is orbit equivalent i.e., isomorphic as an amenablew x.equivalence relation 4 to a skew product over the relation R by aT

w x-valued index cocycle 8, D efinition 1.4 ; in this setting the cocycle

appears as the constant cocycle 1 for the endomorphism T. Clearly thetransformation T commutes with the action of the integers on X=

. .given by x, m ' x, m qk , k g , which is generated by the auto-kmorphism ' . Taking the ergodic decomposition of X= with re-1

.spect to T gives a Lebesgue space Y, FF, on which is well-defined and .generates an integer a ction i.e., is invertible . By the proof of Proposi-

w x.tion 3.8 and the discussion a bove or the duality results in 8 we see that .the automorphism on Y, FF, is isomorphic to the maximal automor-

.phic factor of T on X, BB, .

4. TYP E III MAXIMAL AU TOMOR PH ISMS

AND ENDOMORPHISMS

Throughout this section we assume that the endomorphism T on .X, BB, is n-to-one and is -finite. Using the notation above, we

.denote by on Y, FF, the maximal automorphic factor of T. Lettings y1 as before, the function d rd denotes the R adonNikodymderivative of the measure with respect to . The following identity

w x .follows from 3 and is an immediate consequence of Lemma 4.1 below :for a.e. y g Y,

d yn yny sE T BB y 'E T BB x . . .F F T T / /d nG0 nG0

.for any x gX such that x sy.The next three results are known identities about factors and condi-

w x.tional expectation operators cf. 3 .


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1 .LEMMA 4.1. For eery measurable function h with h(TgL X, , and a.e. y g Y, we hae

yn ynE h(T T BB y sE h T BB ( y d rd y . . . .F F T

/ /nG0 nG0yn

y1sE h T BB ( y . .FT /nG0

.We recall that the measure on X, BB disintegrates over the factor . . . . 1 .Y, FF, by B s H B d y , a n d f o r a n y h gL X, BB, , t h eY y

yn

. . . .y1function E h F T BB y s H h x d x . nG 0 y y

LEMMA 4.2. The disintegration of satisfies y1 Ty1 ; for a.e. y y

y g Y.

Proof. For any set B gBB, we apply Lemma 4.1 to the function h s Band evaluate the equality at y1y. This gives, for a.e. y,

d

y1

d T

y1

y1y1 T B y s x x d x . . . . . .H y B yy1d dXl y

y1 . . y1 . .Since both d rd y and d T rd x are assumed to bestrictly positive functions, a nd G 0 everywhere, the result follows imme-Bdiately.

We have the following chain rule for nonsingular endomorphisms de-

composed over a factor.

y 1 . .P R O P O S I TI O N 4.3. For a .e . x g X, d T rd x s y1 . . y1 . .y1d rd x d T rd x . In addition,

x . x .

d d x .

x s x (T x . . . .T y1d d T x .

U sing the terminology introduced by Krieger for integer actions and.originating with von Neumann factors , we define an ergodic automor-

.phism or endomorphism T on X, BB, t o be type III i f i t admits now x w x .y1invariant -finite measure ; 14 . In 10 , is defined to be a

T

.y1 . 1 .conditional coboundary for T if s h(T rE h for some mea-T

surable function h, and it is proved there that T admits an equivalent

.y1

-finite invariant measure if and only if is a conditional cobound-Tary. A conditional coboundary is a coboundary if and only if h is Ty1BBw xmeasurable 11 . It was shown earlier that an ergodic a utomorphism T is

w x.type III if and only if is not a coboundary cf. 14 .T


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. . .LEMMA 4.4. If x s d rd x for a.e. x gX, then isT T

. .a T-coboundary on X if and only if d rd x is a -coboundary on Yfor . If both are coboundaries, the transfer function is F TynBB-mea-n G 0surable.

.Proof. It suffices to show tha t if is measurable with respect toT

y1FF:BB, and is a coboundary, then the transfer function is y1FF . .measurable as well. We suppose that s h(T rh; i.e., h x s

T

. . . . .h(T r x . We show tha t for a ny x gX, and any y g S x , h x sT T

. . . .h y . We first suppose that Tx s Ty. Then h Tx sh Ty , and x sT

. . . y by a ssumption; therefore h x sh y . Suppose we have shown t hatT k k . .if T x s T y for any k Fn, then h x sh y . We now assume thatnq1 nq1 n . n . . .T x s T w. Then T Tx s T Tw , so h Tx sh Tw , and we have . . . . . . . .h x s h(T r x s h(T r w sh w . B y induct io n, we

T T

.have that h is constant on S x and by ergodicity of the relation S , itT Tfollows that h is constant -a .e. Therefore, the transfer function de-

x

pends only on x as claimed.

. . .

This direction is immediate since

x s Tx .

TH E O R E M 4.5. Suppose that T is a conser ati e ergodic n-to-one nonsin- .gular endomorphism of X, BB, with maximal automorphic factor , and

d rd Ty1 ' 1 for a.e. y. Then is of type III if and only if T is of y y

type III.

.Proof. Suppose T admits a -finite equivalent invariant measure

X ; . We can assume without loss of generality by Lemma 1.4 that is w x w x . .recurrent see 18 or 19 for definitions so that s h(T rh for h a

Ty1 .T BB-measurable and BB measurable function. B y P roposition 4.3,

d d x .

x s x (T x . . .T y1d d T x .

d h(T . .which implies that x s x for a.e. x.

d h

. .It follows from Lemma 4.4 that d rd x is a coboundary, which is

a contrad iction. . If admits a -finite equivalent invariant measure then d r. .d x is a coboundary. Then Proposition 4.3 and Lemma 4.4 imply that

T cannot be of type III.


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5. AN EXAMPLE OF AN ENDOMORPHISM WITH NO

EQUIVALENT INVARIANT MEASURE

We construct a two-to-one endomorphism T on the product space

q

4

4Xs Y= Y ' 0, 1 = 0, 1 and a B orel measure on Xnsy n ns0 n .with the following properties: 1 T is the product of the two-sided shift

.with the one-sided shift; 2 T is nonsingular, conservative, and ergodic .with respect to ; 3 T ad mits no -finite invariant measure equivalent to

.; 4 T is neither exact with respect to , nor is the product measure ofan exact with an automorphic measure.

We denote by BB=BBq the product Borel -algebra on Y= Yq. We

.define the measure on Y, BB to be the type III two-sided Bernoulliw xshift measure constructed by Ha machi in 9 . Then the measure on

q . y1 . .Y= Y will be of the form C s H C l y d y . We specifyY y q q.the measures on Y , BB according to the following algorithm. We fixy

. 0 1 4any g 0, 1 . We define two measures and on the space 0, 1 by0 . 0 . 1 . . 1 . . 0 s 1 s 1r2, and 0 s 1r 1 q , 1 s r 1 q . For each

y g Y, we define to be the infinite product measure given by sy y y

i . . That is, we consider y s . . . , y , y , y , . . . , y , . . . and the ithis0 y1 0 1 nfactor in the measure is j if and only if y sj. Each will be any i yinfinite product of factors of two different measures on Yq and, by results

w xof Ka kutani 13 , for a.e. y g Y, will be singular with respect to theyshift .

.We denote the invertible shift on Y by ; i.e., y sy . It followsi iq1that s y iq 1 s y i. We now restrict our attention to a

y is0 is1

4q y1

single fiber y = Y with a measure on DD, and we compute .y ySuppose C gDD is any cylinder set; then we can write

C s z g Yq: z s i , z s i , . . . , z s i , 40 0 1 1 n n

and

y1

4 C s z:

z s0,

z s i, . . . ,

z s i . 0 1 0 nq1 n 4j z : z s 1, z s i , . . . , z s i .0 1 0 nq1 n

y1 .Therefore Cy

y y y y0 1 2 nq 1s 0 i i i . . . .0 1 ny y y y0 1 2 nq1

q 1

i

i

i . . . .0 1 nn

yjq 1s i s C . . . j yjs0


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Since an infinite product measure is completely determined by its values

on cylinder sets, we see that the two measures are equal. Consequently,d y1rd s 1. To check that the endomorphism defined in this way isy y

.yn . .conservative, it is enough to see that d = rd y, z s

yn . . . .. nnd rd y for all n G 0, so y, z s n, y, z s d rT T. . n . .d y for all n G 0. Hamachi computes that d rd y s ns 0

w xa.e. 9, p. 279 , which gives the result. B y Theorem 4.5, T admits no -finite

equivalent invariant measure.

To prove that T is ergodic with respect to , we apply Proposition 3.4 .and use the fact that all T invariant sets must be in Y, BB since R rS isT T

just . The ergodicity of with respect to gives the result. The fact that

is not equivalent to any product measure also follows from Proposition3.4 and the uniqueness of the maximal automorphic factor; details of this

w xargument a re given in 5 .

R E F E R E N C E S

1. J. Aaronson, Ergodic theory for inner functions of the upper half plane, Ann. Inst. H. .Poincare Sect. B 14 1978 , 233253.2. J . Aaro nson, M. Lin, and B . Weiss, Mixing properties of ) Markov operators and ergodic

.transformations and ergodicity of Cartesian products, Israel J. Math. 33 1979 , 198224.

3. R . B utler and K. Schmidt, An information cocycle for groups of non-singular transforma- .tions, Z. Wahrsch. Verw. Gebiete 69 1985 347360.

4. A. Connes, J. Feldman, and B. Weiss, An amenable equivalence relation is generated by .a single transformation, Ergodic Theory Dynamical Systems 1 1981 , 431450.

5. K. Dajani and J. Hawkins, Rohlin factors, product factors, and joinings for n-to-one

.endomorphisms, Indiana Uni . Math. J. 42 1993 , 237258.6. K. Dajani and J. Hawkins, Examples of endomorphisms and their natural extensions,

.Proc. Amer. Math. Soc. 120 1994 , 12111217.

7. S. Eigen and C. Silva, n-Stack structures, in Measure and Measurable Dynamics, pp.

131140, Contemp. Math., Vol. 40, Amer. Math. Soc., Providence, 1989.

8. J . Feldman, C. Sutherland, and R . Z immer, Subrelat ions of ergodic equivalence relat ions, .Ergodic Theory Dynamical Systems 9 1989 , 239269.

9. T. H amachi, On a B ernoulli shift with non-identical factor measures, Ergodic Theory

.Dynamical Systems 1 1981 , 273

284. .10. J . H awkins, Amenable relations fo r endomorphisms, Trans. Amer. Math. Soc. 343 1994 ,

169191.

11. J. Hawkins and C. Silva, Remarks on recurrence and orbit equivalence of nonsingular

endomorphisms, in Dyn. Sys. Proc. Univ. Md., pp. 281290, Lecture Notes in Math.,

Vol. 1342, Springer-Verlag, New YorkrBerlin, 1988.

12. J. Hawkins and C. Silva, Noninvertible transformations admitting no absolutely continu- .ous -finite invariant measure, Proc. Amer. Math. Soc. 11 1991 , 455463.

.13. S. Kakutani, On equivalence of infinite product measures, Ann. of Math. 49 1948 ,

214224.14. W. Krieger, On t he Araki-Woods asymptotic rat io set a nd nonsingular t ransformations of

a measure space, in Lecture Notes in Math., Vol. 160, pp. 158177, Springer-Verlag,

New YorkrBerlin, 1970.


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15. W. P arry, E ntropy and G enerat ors in E rgodic Theory, Benjamin, E lmsford, New Y ork,

1969.

16. V. Rohlin, On the fundamental ideas of measure theory, in Amer. Math. Soc. Transl.,

Vol. 71, pp. 154, Amer. Math. Soc., Providence, 1952.

17. K. Schmidt, Strong ergodicity and quotients of equivalence relations, Proc. Centre Math.

.Anal. Austral. Nat. Uni . 16 1988 , 300311. .18. C. Silva, On -recurrent nonsingular endomorphisms, Israel J. Math. 61 1988 , 113.

19. C . Silva a nd P . Thieullen, The subadd itive ergodic theorem a nd recurrence properties of .Ma rkovian transformations, J. Math. Anal. Appl. 154 1991 , 8399.

20. P. Walters, Roots of n:1 measure-preserving transformations, J. London Math. Soc. 44 .1969 , 714.

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