Mh. Math. 103, 187-205 (1987) Mathemalik

~ d t e t i i t Mh. Math. 103, 187-205 (1987) Mathemalik

9 by Springer-Verlag 1987

Abelian Cocycles for Nonsingular Ergodic Transformations and the Genericity of Type III 1 Transformations*

By

J. R. Choksi, Montreal, J. M. Hawkins, Stony Brook, and V. S. Prasad, North York

(Received 17 September 1985; in revised form 27 January 1986)

Abstract. The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III 1 ergodic transformations form a dense G~ set with respect to the coarse topology. They also prove that for any locally compact second countable abelian group H, and any ergodic type III transformation T, it is generic in the space of H-valued cocycles for the integer action given by T that the skew product of Twith the cocycle is orbit equivalent to T. Similar results are given for ergodic measure-preserving transformations as well.

w 1. Introduction. This paper provides an answer to a question posed by D. Maharam in 1964 about nonsingular transformations, and it provides some results on the more recent subject of cocycles of amenable equivalence relations. In 1964, MAHARAM [M2] remarked that to each nonsingular invertible transformation T of a Lebesgue space (X, 9.I,/~) one can associate a skew product transformation 7-* on X* = X x [0, oo) defined by 7-* (x, t) = (Tx, t/co (T, x)) where co (T, x) =

_ dff T(x). Even though Tneed not preserve the measure/~, the skew + product T* always preserves the measure ff x dt on X* (where dt denotes Lebesgue measure). She remarked that if T admits an invariant measure equivalent to/~ (finite or infinite a-finite) then T* cannot be ergodic; she asked when and how often T* can be ergodic.

Various authors have since observed that T* is one example of a skew product of the integer action induced by T on X with a real- valued cocycle for that action, and that T* is ergodic if and only if Tis

* Research supported in part by: Nat. Sci. and Eng. Res. Council #A7163 and # U0080 F.C.A.C. Quebec, NSF Grants # MCS-8102399 and # DMS-8418431. 13"

188 J.R. CHOKSI, J. M. HAWKINS, V. S. PRASAD

ergodic type III~ (cf. w 2 for the definition of a type III 1 transformation) [K2], [$2]. We prove in w 3 of this paper that type III~ transformations form a residual set in the space of nonsingular transformations on X. Results suggesting that this might be the case appear in [K1] and [H3].

This result completes a study begun by A. Ionescu Tulcea when it was proved that the set of nonsingular transformations with no a- finite invariant measure form a residual set in 15 (J0, the group of nonsingular transformations of a Lebesgue space with the coarse topology; it was proved in 1979 by J. Choksi and S. Kakutani that the ergodic nonsingular transformations form a dense G~ in 15 (X) [I], [CK]. Our proof uses the methods ofcocycle theory developed in [M 1], [$2], and [PS].

In w 3 we prove a theorem which states that the operation of taking the skew product of a transformation in 15 (X) with a cocycle taking on values in a second countable locally compact abelian group H is jointly continuous. This leads to the result about the genericity of type III~ transformations; as a corollary we obtain a result about diffeomorphisms of manifolds which are of type III~.

In w 4 we consider ergodic elements of 15 (X) and cocycles taking on values in an arbitrary locally compact second countable abelian group H. K. PARTHASARATHY and K. SCHMIDT claimed in [PS] that for each fixed ergodic T~15 (X), with the topology of convergence in measure, a residual set of H-valued cocycles will give an ergodic skew product extension of T. Their proof contained an error; in the first part of 84 we give another proof of the result by combining our Theorem 1 ofw 3 with their methods. We next show that if Te 15 (X) is of type IIIz with ,~ e [0, 1], then it is generic that an/-/-valued cocycle gives a skew production extension which is orbit equivalent to the original transformation.

The type II case is also considered. Every ergodic extension will be type II, but orbit equivalence will follow in the type II1 case only if the group H is compact. The measure preserving (type II) case has been studied in this context before. Zimmer proved that if H is a compact separable group for which an ergodic transformation has an H-valued cocycle giving an ergodic skew product, then H is amenable [Z1]. HERMAN [H3] and V. GOLODETS and S. SINELSHCHIKOV [GS1], [GS2] show that any amenable locally compact separable group is the essential range of a cocycle for some ergodic type II transformation.

Abelian Cocycles for Nonsingular Ergodic Transformations 189

The authors would like to thank A. Ramsay for point ing out and helping fix an error in the literature upon which an earlier draft of this paper was based, and for bringing to the authors ' a t tent ion the work of Golodets and Sinelshchikov. We are also grateful to the referee for comments which enabled us to strengthen Theorem 6.

w 2. Definitions and Notation. Let (X, 9.I,/z) denote a Lebesgue space with /z a a-finite measure. All statements made regarding sets, functions, and t ransformations will hold except possibly on a set of measure zero. By (5 (X,/z) = (5 (X) we mean the group of invertible nonsingular t ransformations of (X, 9.1,/z) onto itself. On (5 (X) we put the coarse topology; that is, T 7

[[ Ur~f - Urflll ~ 0 for a l l f e L I

T in (5 (X) coarsely if and only if

(X,/z) where Ur f ( x ) = f ( T x ) (x) a/z

is the positive isometry induced on L 1 (X,/z) by T~ (5 (X). F r o m now + r

on we will write the Radon -Nikodym derivative of/ZT, -~-#, as mr (').

Equivalent measures on (X, 21) give the same coarse topology on (5 (X); with respect to this topology, (5 (X) is a complete metrisable group ([I], [HO]), and hence a Baire space.

We will also consider the closed subgroup of t5 (X) consisting of the /z-preserving transformations, which we will denote by 93l (X,/z) -- 93l (X). With the induced topology, ~ (X) is a complete metrisable group.

It will often be useful to consider the following equivalent charac ter i sa t ion of the coarse topology when the measure/z on X is finite ([HO]): the sequence 7', ~ T coarsely in (5 (X) if and only if: (i) mr. -~ COT in L 1 (X), and (ii)/Z(T2 1 E A T - 1 E ) --. 0 for all E e ~ i (and therefore COr~-i ~ COT-1 and /Z(T, E A T E ) ~ 0), where A denotes the symmetric difference.

Let H denote any locally compact second countable abelian group. Each element Te (5 (X) induces an integer action on (X, 21, #) by (n, x) ~ T n x for n ~ 77, x ~ X. A measurable map a: 7/x X--. H is a cocycle for the 7/act ion if (i) a (m, T n x) - a (m + n, x) + a (n, x) = 0 for/z-a.e, x e X, m, n e 7 / and (ii) for every n e ;7,

/z({x: T~x = x} n { x : a ( n , x ) ~ 0}) = 0 .


More generally, i f / " is any locally compact separable group acting nonsingularly on (X, 9/,/~) and K is any locally compact separable group, then a Borel map a : X x Y ~ K such that a(x, yl) ,2)= -- a (x, 71) a (x yl, ~'2) for all ~ , y2 e F,/~-a.e. x e X is a cocycle for the / ' -act ion on X. We will use the additive no ta t ion in this paper since the group H is always assumed to be abelian.

A cocycle a: 2 x X ~ Hi s a coboundary if there exists a measurable map b : X ~ H with a(n,x)= b ( T n x ) - b(x) for every neT/, #-a.e. x eX. The cocycles al and a2 are cohomologous if a 1 - a 2 is a coboundary (these definitions have obvious analogues in the non- abelian case).

By B(X, ~, H) we denote the group of equivalence classes of measurable maps f rom X to H with pointwise addition. A cocycle a: 7/x J ( ~ H for Teff i (X) is uniquely determined by the funct ion a (1, ") = f e B (X, ~, H), so we identify B (X,/~, H) with the group of cocycles for the Z action given by T: We give B (X, ~, H) the topology of convergence in measure, and with the metric d e ~ , f 2 ) = = S 0 (fl (x),f2 (x)) d/~ (x), where 0 is a bounded invariant metric on H,

x

B (X,/~, H) becomes a Baire space. I f H is not compact , we let / 1 = H u { ~ } , the one point

compact if icat ion of H. Suppose a e B (X, ~, H) is a cocycle for the 7/ action of an ergodic element Te (5 (X). We say r e H is an essential value of a eB(X,~ ,H) and we write t e E ( a ) if for every A e g/,

(A) > 0, and for every e > 0, there is an integer k such that

# ( A n T-kA n { x : a ( k , x ) eN~(0}) > 0

where N,(r) denotes the ball of radius e about ~. The set {r e / i t I z e E (a)} is called the essential range of a; E (a) n H is a closed subgroup of H, and cohomologous cocycles have the same essential range. For an ergodic integer action, ZIMMER showed that any locally compac t second countable group which is the essential range of a cocycle for the action is amenable [Z1]. For a discussion of which amenable groups can occur as the essential range of a finite measure- preserving ergodic integer (hyperfinite) action, the reader is referred to [Z1], [Z2], [GS1], and [H3].

I t is useful to consider the full group of the action generated by T:[T] = { V ~ A u t ( X , ~ ) : Vx = T"(~)x, n(x)~7/}. By an exhaust ion


argument given in [PS], it follows that if~ e E (a), then for any set B e 92, and any e > 0 we have:

# ( U ( B n T - k B n {x:a(k ,x )~N~(T) u N ~ ( - ~)}) = # ( B ) . k E Z

Equivalently, if for Ve [T], and a: g x X ~ H a cocycle, we define a (V, x) = a (n (x), x) if V(x) = T n(x) x, then T e E(a) implies that

/~ (sup ( B n V - 1 B n {x: a(V,x)eN,(~:) u N , ( - T)}) = # ( B ) . ve[z]

I f we consider a countable dense subset {Cn},~ of the measure algebra 92o with the metric ~ (A, B) = # (A A B), then we obtain the following sufficient condit ion for ~ ~ H to be in the essential range of a cocycle a: g x X ~ H.

Lemma 2.1. I f for every element A ~ { C,}, ~ and every e > 0 we have

# ( U ( A n T - k A n { x : a ( k , x ) 6 N ~ ( T ) u N ~ ( - T)} n k e Z

n {x : log~ , ( x )~N~(O)} ) > 0.9# (A),

or, equivalently i f

# ( s u p ( A n V - 1 A n {x:a(V,x)~N~(~) u N ~ ( - r)} n w[Tl

n {x: log O~v(X ) ~ N~ (0)}) > 0.9/~ (A),

then ~ ~ E (a) and (~, O) ~ E (a, log ~o~).

We note that the condit ion z ~ E (a) is not necessary, since it will not hold i fa = log o~T and ~ ~ E(a), ~ ~ O. The p roo f is a simplification of the p roo f of Lemma 2.2 given below. The authors use Lemma 2.1 and 2.2 to replace the statement of Lemma 4.4 of [PS] which appears wi thout proof, and which is not true as stated.

Lett ing ~+ denote the multiplicative group of positive real numbers , we define the ratio set of Te(fi (X), denoted r(T), by

r ( 7 ) = k d ~ / " We say that T i s of type: III~ if r ( 7 ) = [ 0 , oo),

III~ if r (7 ) = {2n}~z u {0} for 2~(0, 1), III 0 if r ( 7 ) = {0, 1} and II if r (7) = { 1 }. These are the only possibilities since r (7) c [R + u {0}, and r (7) n N § is a closed subgroup of N +. Tis of type II if and only if T admits a or-finite invariant measure ~ ~. We say T is type II~ if the


invariant measure is infinite and II1 if it is finite. The ratio set was defined by W. KRIEGER in [K2]. It is clear that 2 ~ E(log ~OT) n ~ if and

/ ~ r r . ' ,

only . s .n va,uod o ydo

under addition and - - as an ~ +-valued cocycle under multiplica-

tion. We now prove the following lemma.

Lemma 2.2. For any 2 ~ ~, e ~ ~ r (7) if and only if for every e > 0 and for every A ~ {Cn}n~n, a dense subset of gAo, we have

#( U (A n T - k A c~ {x: log ~ozk(x) ~N~(2) u N ~ ( - 2)}) > 0.9# (A) . k~Z

Proof (=~) This is true by Lemma 4.1 of [PS].

( ~ ) Since the set {2: e ~ r(T)} n ~ is a closed subgroup of the additive group ~, we can assume that 0 ~< 2 ~< 1. Given any measurable set B ~ 9.1, p (B) > 0 and any e ~ (0, 89 we first approximate B to within 0.001 # (B) by an element A ~910 such that A ~ {Cn},~; that is,

(BA A) < 0.001 ~ (B). By our hypothesis, we have

( U (A n T - h A n {x: [log o~, (x) l~ N~ (2)}) > 0.9 tz (A) , k~Z

which means we can choose an element V~ [T] such that

~(A n V-~A n {x: Ilog~ov(X)I~N~(2)} ) > 0.9/~ (A) .

We let A = A n V-1A n {x:llogcov(X)l ~N~(2)}. For x~_4 we have + v

x~A , V x ~ A , and e -~e -~ ~< ~ - - ( x ) ~< e;'e ~. We now consider

n B =/~, and we see that since ~ (A) > 0.9 # (A), we have

p(A) > 0.9~(B) (1 - 0.001) > 0.899/~ (B), so ~(/~) > 0 .898#(B) .

In order to show that .4 n B ~ V- ~ B has positive measure, we will first show that ~ ( V / ~ n B) > 0. Since

/~(VB) >~ e-ae-~O.898#(B) >~ (0.36)(0.60)(0.89) ~ (B) >

> 0.18~(B), and V ~ c A ,

it follows that # ( V / ~ n B ) = ,u ( V A n V B n B ) > 0. Since V is a

Abelian Cocycles for Nonsingular Ergodic Transformations t93

nonsingular automorphism, we have # (.4 c~ B ~ V- 1 B) > 0 as well. This means we have

F((A n B ) n V-1(A n B ) n {x" Ilog~ov(X)]~N~(,1)} ) > O,

so ,1~E(logo~) and e~r(T) . []

Using Haar measure which we will denote by m on the Borel ~-algebra ~3 of H, we form a measure space

(Xx H, 21x ~,#x m) = (X, 2[,#) .

For any T~(5(X) and a~B(X,#,H) we define the skew product extension denoted by T~, of T by a to be the map on (X, ~,/~) given by: (x, h) ~-. (Tx, h + a (1, x)) for every (x, h) ~)~. E (a) = / 1 if and only if this skew product is ergodic [$2, w 5].

For any T~ (5 (X), the skew product of T by log ~o r preserves the measure ~ x e-tdt on Xx ~ [M2]. Maharam asked as early as 1964 when the skew product is ergodic with respect to this measure. This skew product is ergodic precisely when T is of type III~. For arbitrary locally compact separable groups H, the skew product of T by a is ergodic when H is the essential range of the cocycle a of the ergodic Z action given by T. The groups H for which this can occur have been studied in [PS]I [GS1], [GS2], [Z1], and [Z2].

Let (X1,Nt,~l) and (X2,S~[2,#2) denote Lebesgue spaces. Two nonsingular hyperfinite countable group actions GI on X1 and G 2 o n X2 are orbit equivalent if there exists a measurable bijective map ~0:X1 ~ )(2 with ~1 ~0 - 1 ,,~ #2 and such that ~0 (G~ x) = G2(~0 x) for #l- a.e. x ~ X1. When G~ = G2 = Z, the actions are generated by single transformations T1 E (5 (X1) and T2 e (5 (X2); in this case we say Tt and T2 are orbit equivalent. The ratio set is invariant under orbit equivalence and provides a complete invariant for type III transformations when ,t e (0, 1) [K3].

There is a canonical way to associate an ergodic flow to any nonsingular ergodic transformation, and it has been proved by KRIEGER [K3] that in the type III case this flow (up to metric isomorphism) provides a complete invariant for orbit equivalence classes of ergodic transformations (in fact for countable hyperfinite ergodic nonsingular group actions). For any ergodic element of (5 (X), we first consider the 77 action on Xx ~ defined by the skew product of


T by log ~r. In general this action is not ergodic, so we consider a measurable partition of X x ~ which generates the ~-algebra ~[0 of invariant sets under Tlog~ up to sets of measure zero. The natural projection from X x ~ to (Xx ~)/~0 ~ Yis a factor map; the flow is obtained from the ~ action: (x, s) ~ (x, s + t) V t ~ [~, induced on the factor Lebesgue space Y. This factor action is called the ergodicflow associated to T, ([K3], cf. [HO] for a complete description).

3. Genericity of type III1 transformations. We assume that (X, 9.I,/~) is a Lebesgue probability space. We first prove

Theorem 1. The map q~: (5 (X) x B (X, #, H) ~ (5 (X x H), where q~ (T, a) = T~ is jointly continuous when (5 (X) and (5 (Xx H) have the coarse topology and B (X,/~, H), the space of measurable functions from X to H, has the topology of convergence in measure.

Proof. Choose A ~ ~[ where A = A x Kfo r some A ~ ~l and K c His a compact set, and so of finite Haar measure m. Since a (1,-) depends

d~r~ x d ~ r only on x and not on h, ~ - ( , h) = ~ - ( x ) . Let Sn, n e N, T be in

(5 (X), an, n e N, b be in B (X, #, H); a routine calculation then shows that

II g~so)oozm- ur~z~ll~ <-~(K)II US. Z A - UTZAI[1 +

where

and

+ ~ z s ;~ ~ ~- ,A (x) {~s. (x)" an (X) + ~ ( X ) " gn (x)} d~ (x) , x

an(x ) = r e ( K - an(1,x) \ K - b(1,x))

(3.1)

t3n(x) = m ( K - b(1,x) \ K - an(1,x)) 9 For each ~ > 0

~s~(x)an(x)d~(x) + ~o~(x)~n(x)dF,(x) ~ (3.:2) x x

m(K) ~ &({x: an(X) > ~ ) + ~ + m(K) ~ T(~x:~n(x) > ~}) + ~ 9

Now let Sn ~ T in (5 (X) and a, ~ b in B (X,/~, H); clearly the first term on the right-hand side of(3.1) tends to 0. We claim next that a n (x) and/3, (x) both tend in measure to 0. To prove this claim we show that an (x) + fin (x) = m (K - a n (1, x) A K - b (1, x)) tends to zero in meas-


ure, by showing that m (K - f , (x) A K) ~ 0 in measure iffn (x) ~ 0 in measure (heref~ (x) = an (1, x) - b (1, x)). Since a sequence converges in measure to zero if and only if every subsequence has a further subsequence tending to zero pointwise almost everywhere, and by the regularity of Haar measure, m (K - hn A K) ~ 0 whenever hn ~ 0 in H, the claim follows.

The finite measures # Sn, # T are absolutely cont inuous with respect to # uniformly in n, because COso ~ ~OT in L 1. Since an (x) and r (x) converge in measure to zero, the uni form absolute continuity implies that

~Sn{x:an(x) > d} + # T{x: f ln(x) > 6} ~ 0

as n ~ oo. It follows f rom this, (3.2) and (3.1) that as n ~ oe

II U<s.> oz - U z lll o . But since the linear span of {;~: for A = A x K, A ~ and K c H a compact set} is dense in L ~ (J~), it follows that

[[ U(s,),.f - ur~U[[1--* 0 for all f e L ' ( ~ .

This completes the p roo f of the joint continuity of qk []

I f we specialize to H = N, the real line, and take the skew product determined by log ~o r we obtain

Corollary 2. The map ~;: ~b (X, #) ~ 9J~ (X x ~, # x e -t dt) defined by ~(T) = T where T(x , t) = (Tx , t + logcov(x)) is continuous with respect to the coarse topologies on both (fi (X) and 93~(Xx [~).

Remark. Because for all S, T ~ | (X), log ~SoT-(x) = l o g ~ s ( T x ) + + log ~oT(x), it is easy to see that ~ above is a cont inuous monomor - phism. It is also easy to show, by using calculations similar to those found in Theorem 1 that ~ - ~ is cont inuous on its image ~; ((fi (X)) c c ffJ~ (Xx ~, # x e- 'd t ) , so ~ is a topological group embedding.

Corollary 2 is used to prove the main result of this section:

Theorem 3. The ergodic type III 1 transformations, ~iii, (X)form a dense G~ subset o f (fi (X) in the coarse topology.

Proof. Let ~ , ( X • ~) denote the set of ergodic ~,-preserving t ransformations in 9.R (X x ~), and ~IIT, (X) the set of ergodic type III~ t ransformat ions in (fi (X). ~ ( X x ~) is a coarse (dense) G~ set in


9J~ (X x N) ([S 1 ], [CK]). Thus it follows from the continuity of ~ proved in Corollary 2 that ~m, (X) = ~o -1 ( ~ (Xx ~)) is a G6 set in (5 (J0 with the coarse topology.

It is well-known (see e.g. [CK] Theorem 2) that the conjugates in (5 (X) of an antiperiodic transformation are coarse dense in (5 (J0. Since ~IIIl is non-empty [K2] and closed under conjugation, we get that it is a dense Ge in (5 (X) with the coarse topology. []

We now give a short alternative proof of Theorem 3 using the ratio set, motivated by [PS].

Let { Ck: k ~ N} be a sequence of sets of positive measure dense in 9.I0, the measure algebra of (X, 9.1,/~). For each/3 E N, k ~ N, p e N, let

0(3

Q(p,k, fl)= {Te(5(X): /~( U Ckc~ r - t c k c~ l= --oo

{x: Ilogcor~(x) - fll < i/p}) > (1/2) /z (Ck)} .

We show that for each p, k, fl, Q (p, k, fl) is open in the coarse topology. It is sufficient to show that the map 5o: (5 (X) ~ N defined by

oO

5O(T) =/z( U (CkC~ T-ICKC~ {x: [1ogcoT~(X) -- fll < i/p}) is lower l = - - o o

semi-continuous. It can be verified that for each l eE , the map T ~ Ck c~ T-t Ck c~ { x: [log cor~ (x) --/31 < l/p} is continuous as a map from (5 (JO to 9/0, and hence so is 50,: (5 ( X ) ~ JR, given by 5on (T) =

= #( 0 CkC~r-ICkC~{x:llogcor~(x)--fi[ < 1/p}).Since%(T),75o(T) l = - - n

as n--* oo, it follows that 5o (T) is lower semicontinuous. Let ff (X) denote the set of ergodic transformations in (5 (X); a

theorem of J. R. CHOKSI and S. KAKUTANI [CK] shows that r (X) is a (dense) G~ in (5 (J0. If {/3j:jeN} is dense in R, then it follows that

00

era, (X) = e (x) c~ ( (~ N ~ Q (p, k, flj)) is a G~ set in (5 (X). Denseness j = l k = l p = l

follows as in the first proof of Theorem 3. []

The proof of Theorem 1 yields the following variant of Theorem 3 for arbitrary cocycles. Let H be any locally coml~act abelian g roup .

Theorem 4. The following statements are equivalent." (i) There exists an element a e B (X, #, H) such that E (a) = 171 with

respect to some ergodic element T~ (5 (X).


(ii) There exists a dense Go set F c B (X, #, H) such that, for every a e F, E(a) = ffI with respect to a dense G o in (5 (X). Furthermore the set {(a, T) ~ B (X, #, H) x (5 (X): E (a) = / ) w.r.t. T} contains a dense G o in B(X, #, H) x (5 (X).

Proof. ( i i )~ (i) is trivial. Now suppose (i) holds. Then for all conjugates of T in (5 (X), E (a) = H, by invariance of the essential range under orbit equivalence ([$2, Thin. 3.16]). Therefore for a dense set in (5 (X), E (a) = / t . Since the coboundaries are dense in B (X,/~, H) [PS], then E(a) = / 2 for a dense set in B (X, #, H). Since the ergodics form a G0 in (5 (Xx H), the result follows from Theorem 1. []

Remark. One consequence of Theorem 6 (cf. w 4) is that condition (i) above is always satisfied, giving a stronger version of Theorem 4.

We now consider (M, ~,/~), a paracompact connected C r manifold with/~ a C r measure on M, and 1 ~< r ~< oe. The following result was proved in [H2] based on techniques from [H3]. Let Diff r (M) denote the set of C ~ diffeomorphisms of M.

Theorem 5. With the C ~ topology, the type III 1 diffeomorphisms form a Go set in Diff r (M).

Proof. As in Theorem 3 we need only show the continuity of the skew product map ~: Diffr (M) --, g)l (M x R) defined by T--, Twith T(x , t) = (Tx , t + log or(x)). This is obvious, since the C ~ topology is finer than the coarse topology. []

When M = T 1 = ~ / 2 with Lebesgue measure, Y. KATZNELSON has proved ([K1]) that in the C ~ closure of the ergodic diffeomorphisms o f t i, the type III1 diffeomorphisms are dense. This combined with Theorem 5 shows that in this set the type III~ diffeomorphisms form a dense Go set. A stronger result has already been obtained by HERMAN [H3].

w 4. Cocycle extensions of nonsingular ergodic transformations with values in abelian groups. In this section we consider the skew product of an ergodic transformation with an arbitrary cocycle taking Values in a locally compact second countable abelian group H, and we show that usually the ratio set is preserved under this operation. In particular, we prove that if T6(5 (X) is ergodic and of type III~, 0 ~< 2 ~< 1, then generically the skew product T~ will be orbit equivalent

198 J .R. CHOKSI, J. M. HAWKINS, V. S. PRASAD

to T. (That is, T~ will have the same associated ergodic flow as T, cf. w We obtain a similar result in the type II case, depending on whether the group H is compact or not.

Our me thod is to consider the essential range of H x N-valued cocycles on X o f t h e form (a, log OgT) with a e B (X,/z, H). We show that for each fixed element T~(5 (X) which is ergodic, a residual set in B (X,/z, H) satisfies (a, 0) e E (a, log O)T) for each a e H. We show this is a sufficient condi t ion for T~ to be ergodic and orbit equivalent to T. We state the main theorem of this section.

Theorem 6. Let Te (5 (X) be a type III ergodic transformation. Then it is generic in B (X, ~, H) that the transformation T~ on X x H is orbit equivalent to T.

Before proving the theorem, we establish some results which hold for all ergodic elements of (5 (X).

Lemma 7. ([K3], cf. [HO], Thm. 25 (1)). I f T is any ergodic transformation in (5 (X) and i f F denotes any type II1 ergodic countable hyperfinite group action on the Lebesgue space f2, then the F x 7/action on 0 x X is orbit equivalent to the 7/action generated by T on X.

The p roo f of this lemma is given in [HO]. In part icular we can oo

choose X2 = l-[ {0, 1}k and define the t ransformat ions k=l

J'o~ i if i r k We+ l ( m o d 2 ) i f i = k .

We let F be the group generated by the Ok's, k = 1, 2 , . . . . With respect to the Haar measure 9 on the compac t group O, F is an ergodic hyperfinite II 1 group action. Then by Lemma 7 the F x 7/act ion on s x X is orbit equivalent to the integer action given by T on X. We will write G -- F x 7/and g = f2 x Xwi th v = q x/z, and the G action we are considering is given by: for each g e G, g -- (~, n) and for each y ~ Y, y = x), g y = (y T" x).

By Z 1 (G, H) we will denote the group of cocycles (under pointwise addit ion) for the G-action taking on values in the group H. We say lim an = a whenever lim an (g, ") = a (g, -) in B (Y, v, H) (cf. w 2) for

every g e G. This gives Z ~ (G, H) the structure of a topological group. We remark that Z 1 (G, H) is topologically i somorphic to Z 1 (2, H), the


group of H-valued cocycles for the equivalent Z-action given by the transformation Te (5 (X), and the isomorphism leaves invariant the essential range of a cocycle [$2, Thm. 3.16].

Lemma 8. Suppose we have a cocycle a: G x Y ~ H such that (a, O) E E (a, log coa) for every a ~ H. Then the skew product extension G a is ergodic with respect to the measure ~ x m.

Proo f By [$2] it suffices to show that the H-valued G-cocycle a satisfies E(a) = H. It is clear from the hypotheses that this condition holds. []

Example 1. We outline the construction of a cocycle which satisfies the hypotheses of Lemma 8; this is based on an example in [PS]. Given any ergodic Te (5 (X), we can construct a cocycle for the associated G action satisfying E(a) = H.

By z~i: D --, 7/2 we denote the projection onto the i th coordinate, and the shift transformation S: D --, D will be defined by

= (o)2, co3 . . . . ) .

We now consider {ak}k~, a dense sequence in H and let {ak}k~ ~ be a sequence in H in which every ak occurs infinitely often. Define, for every j ~> 0, the H-valued map

~j(c~) = {~j ifzq (co) = 0 otherwise.

oo

Now we define the cocycle ~7 (~,, co) = y' (~j (S@ co) - ~j (S j co)). That j=0

this cocyc]e satisfies E(~) = H can easily be seen, since for each j, either/~j(S@ co) - ~j(SJco) = 0, or ~ flips t h e j + 1 St coordinate of co and the difference is @ Using the fact that the F action is finite measure preserving, the argument is now easily completed by applying Lemma 2.1 to finite unions of cyclinder sets. The desired cocycle is then defined by a ( ( y , n ) ( o , x ) ) = tT(y, co), and the skew product Ga : Y x / 4 0 is ergodic. We also note that if b is a G-cocycle which is cohomologous to a then Gb is isomorphic to Ga. []

We now give a short proof of the following result stated in [PS] before going on to prove the stronger main theorem of the paper.


Theorem 9. I f Tc (~ (X) & ergodic, then there is a dense G~ in Z 1 (Z, H) such that the skew product T a is ergodic.

Proof. By w 2 we have that Z 1 (Z, H) ~- B (X, ~, H); by Theorem 1 the map qST:B(X,~,H)-~ | (X• I4) is continuous and so the cocycles giving ergodic extensions are precisely the elements of the set r ({ergodics in (~ (X• H)}), a G~. The example above shows that there is a G-cocycle having the property that E(a) = H, and since Z 1 (G, I4) is isomorphic to Z 1 (Z, H), we get such a cocycle for the Y action of T. By [PS] the coboundaries form a dense set in Z 1 (Z, H); therefore adding coboundaries to this cocycle gives a dense family in Z1 (Z,H) such that E(a) = H. []

W e sharpen this result by proving a stronger version of Lemma 8. In fact the next lemma shows that Example 1 provides an example of a cocycle giving not only an ergodic but an orbit equivalent skew product for a type III transformation T.

Lemma 10. Let T~ (~ (X) be ergodic and type III. Suppose also that a e B (X, ~, H) satisfies (a, O) ~ E (a, log ~or)for every a ~ H (so that T~ is necessarily ergodic). Then T~ is orbit equivalent to T.

Proof. It is equivalent to show that under the above hypotheses, Ta has the same associated ergodic flow as T. To do this it suffices to show that the flow spaces for each Y action are isomorphic, or, that the invariant sets for the actions

Tlogo~T:XX ~ and (Ta)log~r:X•215 are in one-to-one correspondence.

We first define translation in the H direction on X • H • ~ by R~ (x, h, t) = (x, h + a, t) for each a c H, and in the H x ~ direction by S(~, ~) (x, h, t) = (x, h + a, t + fl) for all (a,/3) e H • ~. Now given any measurable set C c Xx H x ~ which is invariant under (T~)log%, we show that Ra C- - C (mod 0) for every a c H. We fix any a c H; by hypothesis, (a,O)cE(a, logo~v) so S<a,0)C= C(mod0) for any T~,log~)-invariant set C c X x H x ~ ([$2], Thin. 5.2). Since for each (x, h, t) e X • H • ~, we have

(Ta)log~T a (X, h, t) : (Tx , h @ a (1, x), t @ log ~or. (x, h)) =

= (Tx , h + a (1 ,x ) , t + log~r(x)) = T(a, logo~)(x,h,t) ,


these two t ransformat ions have the same invariant sets. Clearly R~ = S~,0), so R~ C = C (mod 0). This implies that C is of the form B• H with B c X x E a measurable invariant set for T~og~ .

Conversely, given any measurable set B c X • E invariant under Tlog~T, it is clear that the set B • H is invariant under (Ta)iog~T o. This one-to-one correspondence extends to an i somorphism between the flow spaces for T and T,, which are therefore orbit equivalent. []

We now prove Theorem 6.

Theorem 6. Let Te ff~ (X) be ergodic and of type IIIa, 0 ~< 2 <~ 1. Then it is generic in B (X, #, H) that the skew product extension ~ on X x H with a e B ( X , # , H ) is orbit equivalent to T.

Proof. By Lemma 10 it is enough to prove that there exists a dense Ga set of cocycles a e B ( X , # , H ) satisfying (a,O)eE(a, log~or) for every a e H. A dense set of these cocycles was given in the p roo f of Theorem 9.

Let { C k } ~ be a sequence of sets dense in the measure algebra ~I 0. Let {am}m~ ~ be a dense set of elements in H. We fix some positive integers (k, rn, n,p) and define the set

q~(k,m,n,p) = { a e B ( X , # , H ) : # ( u ( C k n T-t Ckn l~ 7/

~ { x : a ( l ~ x ) E N1/n(am) LJ N l / n ( - am) } n

{x: log o~z, (x) e N1/p (0)})) > 0.9/z (Ck)} 9 By applying L e m m a 2.1 we see that

(~ (~ (~qJ(k,m,n,p) = {a6B(X, tt, H): (am,O) eE(a, log~oT-)} , k n p

and by remarks in w 2,

(~ (~ (~ (~q~(k,m,n,p) = {a6B(X,# ,H): (a,O)eE(a, logo~.)Va~H} m k n p

Since for each fixed (k, m, n,p) the map

a~--~sup~( U ck n T-I Ck n {x: a(l,x) EN~/n(am)u N1/n(- am)} /0>~1 Iris<z0

c~ {x: log ~T' (X) E Nl/p (0))))

is lower semi-continuous, the set qJ (k, m, n,p) is open in B (X, #, H). This proves the existence of a dense G~ set of cocycles satisfying the hypotheses of L e m m a 10, and hence proves the theorem. [] 14 Monatshefte f/ir Mathematik, Bd. 103/3

202 J.R. CnOKSI, J. M. HAWKINS, V. S. PRASAD

We compare L e m m a 10 with the following 1emma.

Lemma 11. Let Te 15 (X) be ergodic, and suppose also that the skew product T~, a e B (X, #, H) is ergodic. Then 2 E r (T~) if and only if (0, log 2) e E(a, log cor).

Proof When 2 e N +, this is a direct consequence of the definitions involved. It is also true if 2 = 0 (the type III 0 case); the p roo f is more technical but is similar to the p roo f of Theorem 5.2 of [$2]. []

Therefore we have shown that (a, 0) e E(a, log or) for all a e H implies that (0, ,1) e E (a, log or) for all e a e r (7); this indicates that independent behavior of the cocycles a and log ~OT is generic.

It is well-known that a type II1 ergodic t ransformat ion is not equivalent to a type II~ one (cf. [HO]), even though they bo th have the ratio set { 1 } and the same associated flow (translation on R). It is clear that if a is a cocycle which makes Ta ergodic, then r (Ta) c r (7), so a type II t ransformat ion gives rise to a type II skew product T~ if and only if E (a) = H.

In the type II1 case, we might as well assume that T~15 (X) preserves the finite measure ~ on i". Then if a e B (X, #, H) satisfies E (a) = H it follows that T~ will preserve the ergodic measure ~ x m, where m denotes the Haar measure on H. This means that Ta will be of type II1 if and only if H is compact , and (under the hypothesis that E(a) = H), it will be an ergodic IIoo action otherwise.

On the other hand, if T e 15 (X) is I I~ , and T~ is ergodic, then To is also IIoo. The main theorems in this paper have obvious statements and proofs in the type II case according to the above discussion.

We conclude by construct ing an example illustrating that L e m m a 2.2 does not extend to the case when 2 = oo. This example was suggested by S. Hurder .

Example 2. Using the nota t ion given earlier, we let s = 00

= ~I {0, 1}k, and we give this compact space the a-algebra of Borel k=1

sets, denoted ~. As before we consider the group F which is generated by the t ransformat ions Ok, where each Ok changes the k th coordinate of each point co e s and leaves the other coordinates unchanged. Instead of put t ing Haar measure on s we define a different infinite p roduc t measure with respect to which the F-act ion is still nonsingular and


(3O

ergodic. We define the measure # = 1-I #k as follows" for each k = k = l

= 2q + 1, q = 0, 1, . . . , we define #k({0}) = &({1}) = 1/2. For each k = 2q, q = 1 ,2 , . . . , we define Ck({0}) = 1 - (l/k2), #k({1}) = 1/k 2. One can check that # is a non-atomic probability measure satisfying the above requirements. In fact, with respect to this measure, the F-action is ergodic type II.

We will show that r(F) = {1}; to do this, it is enough to give a measurable set C, # (63 > 0, satisfying the condition that whenever

# (C n 9, - 1 63 > 0 for some 7 e F, then (a)) = 1 for all points in the & intersection. This will imply that no 2 va 1 can be in the ratio set of the F-action. We define C = { f O G ~ : f . O Z q ---~ 0 ; q = 1,2, . . . ,} ; C is clearly measurable, since it is the countable intersection of the measurable cylinder sets B~q={~oeD:~O2q=O }. Also # ( C ) =

oo

= [ I (1 - (1/(2q)2)) > 0. Any y e / " having the property that q = l

- 1 co e C and a~ ~ C will have to be of the form 7 = ~< 6k2 Ok~... c~k,, . /

where each ki is odd; this means that ~ u y (o9) = 1. /

We claim that this action satisfies the hypotheses of Lemma 2.2; (we have not given an explicit t ransformation T, but it is well-known that there is a single transformation T whose full group is the same as the full group of / ' , cf. [HO]). If by i B k, we denote the set {co e X2: ok= i}, i = 0, 1", k . . . . 1, 2, , then we can use finite intersections of B~i's to form the elements of a countable dense subset of go, the measure algebra for g. Let A be an arbitrary element of that dense subset; then

A = {o)E~2: cok,-= il, O)k2 = i 2 , . . . COkp = ip} ,

i1 , i2 , . . .e{0,1}, and kl,k2...eN. Let M be some arbitrary large positive real number. We first find l e N even and large enough so that 1/(/2 " 1) < e -M and satisfying I > kp. We now define an element VeF as follows: for all coeA ~ = {meA: cot = 0}, Vco =Or. Then

d ~ V 1 -M V~oeA] = {~oeA:~ol= 1} _c A, and ~7- (co) - 12 ~ < e

Under the additional assumption that 1 - (1//2) > 0.9, we have that 14"


/~(A ~ = ( 1 - (1/12)),u(A)> 0 . 9 # ( A ) , so we have n o w fo u n d an e lement V s F satisfying

~(Ac~ V-1An{co~f2: l o g ~ V ( ~ o ) ] > M ) > 0.9/~(A) ;

this is because all po in ts in A 0 are in the above intersect ion. Since M and A were a rb i t ra ry , this p roves tha t the hypo theses o f L e m m a 2.2 are satisfied by the F act ion, even t h o u g h 0 6 r (P). [ ]

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[PSI PARTHASARATHY, K. R., SCHMIDT, K. : On the cohomology ofa hyperfinite action. Mh. Math. 84, 37~48 (1977).

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[Z1] ZIMMER, R. : Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Funct. Anal. 27, 350--372 (1978).

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J. R. CHOKSI Department of Mathematics McGill University Montreal, Quebec H3A 2K6 Canada

J. M. HAWKINS Department of Mathematics SUNY at Stony Brook Stony Brook, NY 11794 U.S.A.

V. S. PRASAD Department of Mathematics York University North York, Ontario M3J 1P3 Canada

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Mh. Math. 103, 187-205 (1987) Mathemalik