aa r X i v : . [ m a t h . D S ] D ec A PROPERTY OF ERGODIC FLOWS
MARIA JOIT¸ A AND RADU-B. MUNTEANU
Abstract.
In this paper we introduce a property of ergodic flows,called Property B. We prove that any ergodic hyperfinite equiva-lence relation of type III whose associated flow satisfies this prop-erty is not of product type. A consequence of this result is thatany properly ergodic flow with Property B is not approximatelytransitive. We use Property B to construct a non-AT flow which -up to conjugacy - is a flow built under a function with the dyadicodometer as base automorphism. Introduction
A remarkable result of Krieger [9] establishes a complete correspon-dence between orbit equivalence classes of ergodic hyperfinite equiv-alence relations of type III , conjugacy classes of properly ergodicflows and isomorphism classes of approximately finite dimensional fac-tors of type III . Product type equivalence relations are hyperfiniteequivalence relations, which, up to orbit equivalence, are generated byproduct type odometers. In order to show that there exist ergodicnon-singular automorphisms not orbit equivalent to any product typeodometer, Krieger [7] introduced a property of non-singular automor-phisms, called Property A. He proved that any product type odome-ter satisfies this property [8], and he also constructed an ergodic non-singular automorphism that does not have this property, and thereforeis not of product type. It was shown in [11] that there exist non sin-gular automorphisms which satisfy Property A but which are not ofproduct type.To characterize the ITPFI factors among all approximately finite di-mensional factors, Connes and Woods [1] introduced a property of er-godic actions, called approximate transitivity, shortly AT. They showedthat an approximately finite dimensional factor of type III is an ITPFIfactor if and only if its flow of weights is AT. Equivalently, their result This work was supported by a grant of the Romanian Ministry of Education,CNCS - UEFISCDI, project number PN-II-RU-PD-2012-3-0533.2010
Mathematics Subject Classification. says that an ergodic hyperfinite equivalence relation R of type III isof product type if and only if the associated flow of R is AT.In this paper we introduce a property of ergodic flows, called Prop-erty B. We show that any properly ergodic flow with this property isnot AT and we construct a flow which has this property. The non ATflow corresponding to the non ITPFI factor constructed in [4] does nothave Property B, and so the property of a flow to be not AT is notequivalent to Property B.The paper is organized as follows. In Section 2, we recall somenotations and definitions. In Section 3 we define Property B, we showthat this property is invariant for conjugacy of flows and we characterizethis property for a flow built under a function. In Section 4 we provethat a hyperfinite ergodic equivalence relation R of type III whoseassociated flow satisfies Property B is not of product type and we showthat a properly ergodic flow which has Property B is not AT. In Section5, we show that there exists a flow whith Property B. This flow is builtunder a function with the dyadic odometer as base automorphism.2. Preliminaries
Throughout this paper ( X, B , µ ) will be a standard σ -finite measurespace. A measurable flow on ( X, B , µ ) is a one parameter group of non-singular automorphisms { F t } t ∈ R of ( X, B , µ ) such that the mapping X × R ∋ ( x, t ) F t ( x ) ∈ X is measurable. Two flows { F t } t ∈ R and { F ′ t } t ∈ R on ( X, B , µ ) and ( X ′ , B ′ , µ ′ ) respectively, are conjugate if thereexists an isomorphism T : ( X, B , µ ) → ( X ′ , B ′ , µ ′ ) such that for all t ∈ R and for µ -almost all x ∈ X , F ′ t ( T ( x )) = T ( F t ( x )). We say that { F t } t ∈ R is ergodic if any F t -invariant measurable set is either null orconull.Let R be an equivalence relation on ( X, B , µ ). We say that R isa countable measured equivalence relation if the equivalence classes R ( x ), x ∈ X are countable, R is a measurable subset of X × X , andthe saturation of any set of measure zero has measure zero. R is calledergodic if any invariant set is either null or conull. Recall that if ν l and ν r are the left and the right counting measures on R we have that ν l ∼ ν r and δ ( x, y ) = dν l dν r ( x, y ) is the Radon-Nikodym cocycle of µ withrespect to R . We say that the measure µ is lacunary if there exist ε > δ ( x, y ) = 0 or | δ ( x, y ) | > ǫ , for ( x, y ) ∈ R . The full group[ R ] of R is the group of all nonsingular automorphisms V of ( X, B , µ )with ( x, V x ) ∈ R for µ -a.e. x ∈ X .A countable measured equivalence relation R is called finite if R ( x )are finite for almost all x ∈ X . We say that R is hyperfinite, if there PROPERTY OF ERGODIC FLOWS 3 are finite relations R n with R n ⊆ R n +1 and ∪R n = R , up to a setof measure zero. We recall that R is hyperfinite if and only if if thereexists a nonsingular automorphism T on ( X, B , µ ) such that, up toa set of measure zero, R is equal to the equivalence relation R T = { ( x, T n x ) , x ∈ X, n ∈ Z } generated by T , that is, R ( x ) = { T n x, n ∈ Z } , for µ -a.e. x ∈ X .Two countable measured equivalence relations R and R ′ on ( X, B , µ )and ( X ′ , B ′ , µ ′ ) respectively, are called orbit equivalent if there existsan isomorphism S : ( X, B , µ ) → ( X ′ , B ′ , µ ′ ), such that S ( R ( x )) = R ′ ( Sx ) for µ -a.e. x ∈ X .Let ( k n ) n ≥ be a sequence of positive integers, with k n ≥
2. Considerthe infinite product probability space (
X, µ ) = Q ∞ n =1 ( X n , µ n ), where X n = { , , . . . k n − } and µ n are probability measures on X n suchthat µ n ( x ) >
0, for all x ∈ X n . We recall that the tail equivalencerelation T on ( X, µ ) is defined for x = ( x n ) n ≥ and y = ( y n ) n ≥ by( x, y ) ∈ T iff there exists n ≥ x i = y i for all i > n. It easily can be observed that, up to a set of measure zero, T is gener-ated by the odometer defined on ( X, µ ). A countable measured equiv-alence relation is said of product type if it is orbit equivalent to the tailequivalence relation on an infinite product probability space as above,or equivalently, if it is orbit equivalent to the equivalence relation gen-erated by a product type odometer.An ergodic equivalence relation R is of type III if there is no σ -finite R -invariant measure ν equivalent to µ . The type III equivalencerelations are further classified in subtypes III λ , where 0 ≤ λ ≤
1. Up toorbit equivalence, for λ = 0, there is only one hyperfinite equivalenceof type III λ , and this is of product type.The orbit equivalence classes of ergodic hyperfinite equivalence re-lations of type III are completely classified by the conjugacy class oftheir associated flow. For more details we refer the reader to [3] and[12].In order to show that there exists ergodic non-singular automor-phisms not orbit equivalent to any product odometer, Krieger intro-duced a property of non-singular automorphisms, called Property A.This property can be defined for equivalence relations (see [10]), as fol-lows. Suppose that R is a hyperfinite equivalence relation on ( X, B , µ ).Let ν be a σ -finite measure on X , equivalent to µ , and δ ν the corre-sponding Radon-Nicodym cocycle. For x ∈ A , defineΛ ν,A, R ( x ) = { log δ ν ( y, x ) : ( x, y ) ∈ R and y ∈ A } MARIA JOIT¸ A AND RADU-B. MUNTEANU = { log dν ◦ φdν ( x ) : φ ∈ [ R ] , ( x, φ ( x )) ∈ R and φ ( x ) ∈ A } . For a σ -finite measure ν ∼ µ , A ∈ B of positive measure and s, ζ > K ν, R ( A, s, ζ ) = { x ∈ A : ( e s − ζ , e s + ζ ) ∩ Λ ν,A, R ( x ) = ∅} ∪{ x ∈ A : ( − e s + ζ , − e s − ζ ) ∩ Λ ν,A, R ( x ) = ∅} . Definition 1.
Let R be a hyperfinite equivalence relation on ( X, B , µ ) .Then R has Property A if there exists a σ -measure ν ∼ µ and η, ζ > such that: every set A ∈ B of positive measure contains a set B ∈ B of positive measure such that lim sup s →∞ K ν, R ( B, s, ζ ) > η · ν ( B ) . If R is a hyperfinite equivalence relation and T is a non-singularautomorphism such that R = R T , up to a null set, it easily can beobserved that R has Property A if and only if T has Property A (see[11]). We mention the following result (see [8] and [11]) that will beused in this paper. Proposition 2.1.
Assume that R has Property A. Then there exist η, δ > such that for all λ ∼ µ and all ǫ > , every measurable set A of positive measure contains a measurable set B of positive measurewith lim sup s →∞ K λ, R ( B, s, δ + ǫ ) > e − ǫ η · λ ( B ) . We recall that Krieger’s result from [8], can be reformulated in thefollowing way [11]:
Theorem 2.2.
Any ergodic equivalence relation of product type and oftype III has Property A. Property B
In this section we define a property of measurable flows that we callProperty B, we show that this is an invariant for conjugacy of flows,and we characterize this property for a flow built under a function.Let { F t } t ∈ R be a flow of automorphisms of ( X, B , µ ). For A ∈ B ofpositive measure and s, δ > F,δ,s ( A ) = { x ∈ A, ∃ t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , F t ( x ) ∈ A } . Definition 2.
We say that { F t } t ∈ R has Property B if there exists ameasurable set A ⊆ X of positive measure such that for all δ > s →∞ µ (Λ F,δ,s ( A )) = 0 . PROPERTY OF ERGODIC FLOWS 5
Proposition 3.1.
Let { F t } t ∈ R be a flow on ( X, B , µ ) satisfying Prop-erty B, and µ ′ a σ -finite measure equivalent to µ . Then, the flow { F t } t ∈ R on ( X, B , µ ′ ) has Property B.Proof. Let µ ′ ∼ µ be a σ -finite measure equivalent to µ and denoteby f the Radon-Nikodym derivative of µ ′ with respect to µ . Thus, µ ′ ( A ) = R A f dµ whenever A ∈ B .Let A a be measurable set satisfying (1). There exists a positiveinteger k such that µ ( A ∩ { x ∈ X, f ( x ) < k } ) >
0. Hence, B = A ∩ { x ∈ X, f ( x ) < k } is a subset of A of positive measure, and then,for every δ > s →∞ µ (Λ F,δ,s ( B )) = 0 . Since µ ( B ) > µ ∼ µ ′ it results that µ ′ ( B ) >
0. Notice that µ ′ (Λ F,δ,s ( B )) = Z Λ F,δ,s ( B ) f dµ < k Z Λ F,δ,s ( B ) dν = k · µ (Λ F,δ,s ( B )) . Consequently, lim sup s →∞ µ ′ (Λ F,δ,s ( B )) = 0 . and therefore, the flow { F t } t ∈ R on ( X, B , µ ′ ) has Property B. (cid:3) Proposition 3.2.
Let T : ( X ′ , B ′ , µ ′ ) → ( X, B , µ ) be an isomorphismand assume that µ ′ = µ ◦ T − . If { F t } t ∈ R is a flow on ( X, B , µ ) thatsatisfies Property B and { F ′ t } t ∈ R is a flow on ( X ′ , B ′ , µ ′ ) such that T ( F ′ t ( x )) = F t ( T x ) , for all t ∈ R and for µ ′ -almost all x ∈ X ′ , then F ′ t has Property B.Proof. Assume that there exists a measurable subset A of X of positivemeasure which satisfies (1). Let δ, s >
0. Up to sets of measure zero,the following equalities hold:Λ F ′ ,δ,s ( T − ( A ))= { x ∈ T − ( A ) , ∃ t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , F ′ t ( x ) ∈ T − ( A ) } = { x ∈ X ′ , T x ∈ A, ∃ t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , T ( F ′ t ( x )) ∈ A } = { x ∈ X ′ , T x ∈ A, ∃ t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , F t ( T x ) ∈ A } = T − (cid:0) { y ∈ A, ∃ t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , F t ( y ) ∈ A } = T − (Λ F,δ,s ( A )) . Hence, µ ′ (Λ F ′ ,δ,s ( T − ( A ))) = µ ′ ◦ T − (Λ F,s,δ ( A )) = µ (Λ F,δ,s ( A )) . MARIA JOIT¸ A AND RADU-B. MUNTEANU
It then follows that for every δ >
0, we havelim sup s →∞ µ ′ (Λ F ′ ,δ,s ( T − ( A ))) = lim sup s →∞ µ (Λ F,δ,s ( A )) = 0 , and therefore, the flow { F ′ t } t ∈ R on ( X ′ , B ′ , µ ′ ) has Property B. (cid:3) We can prove now the following result:
Proposition 3.3.
Property B is an invariant for conjugacy of flows.Proof.
Let ( X, B , µ ), ( X ′ , B ′ , µ ′ ) be two σ -finite measure spaces andassume that { F t } t ∈ R is a flow on ( X, B , µ ) which satisfies Property B.Let { F ′ t } t ∈ R be a flow on ( X ′ , B ′ , µ ′ ) which is conjugate to { F t } t ∈ R .Hence, there exists an isomorphism T : ( X ′ , B ′ , µ ′ ) → ( X, B , µ ) suchthat F t ( T x ) = T ( F ′ t ( x )) for µ ′ -almost all x ∈ X ′ and for all t ∈ R . As T is an isomorphism, µ ′ ∼ µ ◦ T − . Let µ ′′ be the measure on X ′ givenby µ ′′ = µ ◦ T − . Thus F t ( T x ) = T ( F ′ t ( x )) for µ ′′ almost all x ∈ X ′ andfor all t ∈ R . By Proposition 3.2, we have that { F ′ t } t ∈ R on ( X ′ , B ′ , µ ′′ )has Property B. As µ ′′ and µ ′ are equivalent measures, Proposition 3.1implies that { F ′ t } t ∈ R has Property B. (cid:3) Let T be an automorphism of ( X , B , µ ) and ξ : X → R be apositive measurable function. Consider Y = { ( x, t ) ∈ X × R , ≤ t < ξ ( x ) } and let ν be the measure on Y that is the restriction ofthe product measure µ × λ , where λ is the usual Lebesgue measureon R . Let { F t } t ∈ R be the flow built under the function ξ with baseautomorphism T ; it is defined on ( Y, ν ), and for t > F t ( x, s ) = ( x, t + s ) if 0 ≤ t + s < ξ ( x )( T ( x ) , t + s − ξ ( x )) if ξ ( x ) ≤ t + s < ξ ( T ( x )) + ξ ( x ) · · · . For a measurable set A ⊆ X we define∆ F,δ,s ( A ) = { x ∈ A, ∃ t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , F t ( x, ∈ A ×{ }} . With this notation we have the following result:
Proposition 3.4.
The flow { F t } t ∈ R has Property B, if and only if thereexists a measurable set A ⊆ X of positive measure such that, for all δ > , (2) lim sup s →∞ µ (∆ F,δ,s ( A )) = 0 . Proof.
Assume that { F t } t ∈ R has Property B. Then, there exists a mea-surable set A ⊆ Y such that, for every δ > s →∞ ν (Λ F, δ,s ( A )) = 0 . PROPERTY OF ERGODIC FLOWS 7
Since A has positive measure, there exists a measurable set A ⊆ X ofpositive measure, an integer m ≥
1, and a positive real α such that forall x ∈ A , λ ( A x ∩ [ m, m + 1]) > α , where A x = { y ∈ R : ( x, y ) ∈ A } .Let K = A × [ m, m + 1] ∩ A . Clearly, K ⊆ A , and then, for all δ > s →∞ ν (Λ F, δ,s ( K )) = 0 . (3)Let δ >
0. For any x ∈ ∆ F,δ,s ( A ), there exists y ∈ A and t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) such that F t ( x,
0) = ( y, x, a ) , ( y, b ) ∈ K , we have that F t − a + b ( x, a ) = F t + b ( x,
0) = F b ( F t ( x, F b ( y,
0) = ( y, b ) . It is straightforward to check that for s large enough, t − a + b ∈ ( e s − δ , e s +2 δ ) ∪ ( − e s +2 δ , − e s − δ ) whenever t ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ).Consequently, ∆ F,δ,s ( A ) × [ m, m + 1] ∩ K ⊆ Λ F, δ,s ( K ) , whence α · µ (∆ F,δ,s ( A )) ≤ ν (Λ F, δ,s ( K )) , and then (2) follows from (3).Conversely, consider A ⊆ X satisfying (2). Let A = A × [0 , ∩ Y .Proceeding in the same manner as above, for all δ >
0, and for s largeenough, we have Λ F,δ,s ( A ) ⊆ ∆ F, δ,s ( A ) × [0 , ∩ A. Then, by (2), we obtain that { F t } t ∈ R satisfies Property B. (cid:3) Property B implies not AT
In this section we show that if R is an ergodic hyperfinite equivalencerelation of type III whose associated flow has Property B, then R doesnot satisfy Krieger’s Property A and therefore is not of product type.A consequence of this result is that any properly ergodic flow withProperty B is not approximately transitive. Remark that if R is oftype III λ , λ = 0, then the associated flow of R does not have PropertyB.Consider an ergodic hyperfinite equivalence relation R of type III on( X, B , µ ) and let δ be the Radon-Nicodym cocycle of µ with respect to R . Replacing eventually µ with an equivalent measure we can assumethat µ is a lacunary measure (see for example [6], Proposition 2.3).Define ξ ( x ) = min { log δ ( x ′ , x ); ( x ′ , x ) ∈ R , log δ ( x ′ , x ) > } MARIA JOIT¸ A AND RADU-B. MUNTEANU and consider S the equivalence relation on X given by( x, y ) ∈ S if and only if ( x, y ) ∈ R and δ ( x, y ) = 1 . Let B ( S ) the σ -algebra of sets in B that are S -invariant. Let X be thequotient space X/ B ( S ), that is the space of ergodic components of S .We denote the quotient map from X onto X by π , where π ( x ) is theelement of X containing x . On X , consider the measure µ = µ ◦ π − .Note that ξ ( x ) is B ( S )-measurable and therefore, ξ can be regarded asa function on X . We have an ergodic automorphism T on X defined T ( π ( x )) = π ( x ′ ) where ( x, x ′ ) ∈ R and log δ ( x ′ , x ) = ξ ( π ( x )). Then,the associated flow { F t } t ∈ R of R can be realized as the flow built underthe ceiling function ξ with base automorphism T (see for example [5]or [6]). Lemma 4.1.
Let ( x, x ′ ) ∈ R , z = π ( x ) and z ′ = π ( x ′ ) . Then F log δ ( x ′ ,x ) ( z,
0) = ( z ′ , .Proof. Notice that it is enough to prove the lemma for log δ ( x ′ , x ) pos-itive. Since µ is a lacunary measure, there are only finitely many val-ues, say n , of log δ ( z, x ) between 0 and log δ ( x ′ , x ). Hence there exists x , x , . . . , x n in the orbit R ( x ) of x such that 0 < log δ ( x , x ) < · · · < log δ ( x n , x ) < log δ ( x ′ , x ). Thenlog δ ( x ′ , x ) = log δ ( x , x )+log δ ( x , x )+ · · · +log δ ( x n , x n − )+log δ ( x ′ , x n ) . If z i = π ( x i ), then z i = T i ( z ) and F ξ ( T i − ( z )) ( T i − ( z ) ,
0) = ( T i ( z ) , ≤ i ≤ n . Notice that log δ ( x ′ , x ) = ξ ( z ) + ξ ( T ( z )) + · · · + ξ ( T n ( z )).Therefore F log δ ( x ′ ,x ) ( z,
0) = F ξ ( z )+ ξ ( T ( z ))+ ··· + ξ ( T n ( z )) ( z, F ξ ( T ( z ))+ ··· + ξ ( T n ( z )) ( T ( z ) ,
0) = · · · = ( z ′ , . (cid:3) Theorem 4.2.
With the above notation, if the associated flow { F t } t ∈ R of R has Property B, then R does not have Property A.Proof. By Proposition 3.4, we can find a measurable set A ⊆ X ofpositive measure such that, for all δ > s →∞ µ (∆ F,δ,s ( A )) = 0 . Let C = π − ( A ) ⊆ X and δ >
0. Consider s > x ∈ K µ, R ( C, s, δ ).Thus, x ∈ C and there exists y ∈ C such that ( x, y ) ∈ R andlog δ ( y, x ) ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ). From Lemma 4.1, we havethat F log δ ( y,x ) ( π ( x ) ,
0) = ( π ( y ) , . PROPERTY OF ERGODIC FLOWS 9
Hence, π ( x ) ∈ ∆ F,δ,s ( A ) and then, x ∈ π − (∆ F,δ,s ( A )). Therefore, K µ, R ( C, s, δ ) ⊆ π − (∆ F,δ,s ( A )) , and consequently, µ ( K µ, R ( C, s, δ )) ≤ µ ◦ π − (∆ F,δ,s ( A )) = µ (∆ F,s,δ ( A )) . This clearly implies thatlim sup s →∞ µ ( K µ, R ( C, s, δ )) = 0and then, by Proposition 2.1, R does not have Property A. (cid:3) Remark 1.
Since any product type equivalent relation of type III sat-isfies Property A, it follows that an equivalent relation R whose asso-ciated flow has Property B is not of product type. Recall that any properly ergodic flow is the associated flow of certainergodic hyperfinite equivalence relation of type III and a hyperfiniteergodic equivalence relation is of product type, if and only if the as-sociated flow is approximately transitive. We have then the followingresult: Corollary 4.3.
Let { F t } t ∈ R be a properly ergodic flow on ( X, B , µ ) which satisfies Property B. Then { F t } t ∈ R is not approximately transi-tive. Remark 2.
There exists ergodic flows which are not AT and do notsatisfy Property B, as the following example shows.
Example 1. In [4] , Giordano and Handelman constructed a factor N whose flow of weights is not AT. We recall that the flow of weights of N can be realized as the flow built under a constant function and which hasa base automorphism that can be identified with the Poisson boundaryof the matrix valued random walk corresponding to the dimension spacegiven by the sequence of matrices (cid:20) x n x n (cid:21) , n ≥ . Since, up to isomorphism, N is the von Neumann algebra associatedto an ergodic hyperfinite equivalence relation R , the flow of weights of N is, up to conjugacy, the associated flow of R . According to [11] , theequivalence relation R has Property A, and then, from Theorem 4.2 weconclude that the associated flow of R does not satisfy Property B. The following result gives a sufficient condition for a nonsingular auto-morphism to be not AT.
Corollary 4.4.
Let T be a nonsingular automorphism of ( X, B , µ ) .Assume that there exists A ⊂ X of positive measure such that lim sup s →∞ { x ∈ A : ∃ n ∈ ( e s − δ , e s + δ ) ∪ ( − e s + δ , − e s − δ ) , T n x ∈ A } . Then T is not AT.Proof. Proposition 3.4 implies that { F t } t ∈ R , the flow built under theconstant function f = 1 with base automorphism T has Property B andthen by Corollary 4.3 it follows that { F t } t ∈ R is not AT. From Lemma2.5 of [1], we conclude that T is not AT. (cid:3) An ergodic flow which satisfies Property B
In this section we construct a properly ergodic flow which satisfiesProperty B and therefore is not AT. The flow that we construct is aflow built under a function with a product odometer (conjugate to thedyadic odometer) as base automorphism.Let ( z n ) n ≥ be the sequence of integers given by z n = 2 n −
1, for n ≥ X = Q n ≥ { , , . . . z n } endowed with theusual product σ − algebra and the product measure µ = ⊗ n ≥ µ n , where µ n are the probability measures on { , , . . . z n } given by µ n ( i ) = n ,for i = 0 , , . . . , z n and n ≥
1. Let T : X → X be the product odome-ter defined on X . We recall that T is the nonsingular automorphismdefined for almost every x ∈ X by(4) ( T x ) n = n < N ( x ) ,x n + 1 if n = N ( x ) ,x n if n > N ( x ) , where N ( x ) = min { n ≥ x n < z n } . Notice that T is measureconjugate to the dyadic odometer.Let ( K n ) n ≥ be the sequence given by K n = 1!2! · · · n ! , for n ≥ , and let f : X → R be the function defined for almost every x ∈ X bysetting(5) f ( x ) = K N +1 + x N +1 where x = ( x n ) n ≥ and N = N ( x ). Proposition 5.1.
Let n ≥ be a positive integer, m = [log n ] and l = n − m . For almost every x ∈ X , we have: PROPERTY OF ERGODIC FLOWS 11 (i)
If there exists an integer k ≥ such that (6) K n ≤ k − X i =0 f ( T i x ) < K n +1 , then x m +1 = l . (ii) If there exists an integer k ≥ such that (7) K n ≤ k X i =1 f ( T − i x ) < K n +1 , then x m +1 = l .Proof. (i) Let x ∈ X such that K n ≤ P k − i =0 f ( T i x ) < K n +1 , for someinteger k ≥
1. Let p = max { N ( T i x ); 0 ≤ i ≤ k − } . Hence, there exists j , 0 ≤ j < k such that N ( T j x ) = p . By (5) wehave that f ( T j x ) = K p +1 + x p +1 . From (4) we deduce that ( T i x ) n = x n for n > p and 1 < i ≤ k . Also, (4) implies that 2 · · · · p > k . Hence, K p +1 + x p +1 ≤ k − X i =0 f ( T i x ) < · · · · p · K p +1 + x p +1 < K p +1 + x p +1 +1 . We claim that n = 2 p +1 + x p +1 . Indeed, if n < p +1 + x n +1 we have K n +1 ≤ K p +1 + x n +1 ≤ P k − i =0 f ( T i x ), which contradicts (6). If n > p +1 + x p +1 , then K n ≥ K p +1 + x p +1 +1 > P k − i =0 f ( T i x ), which againcontradicts (6). Therefore n = 2 p +1 + x p +1 , and then m = p and x m +1 = l .(ii) Let x ∈ X such that K n ≤ P ki =1 f ( T − i x ) < K n +1 , for somepositive integer k . Let p = max { N ( T − i x ); 1 ≤ i ≤ k } and remark that ( T − i x ) n = x n for n > p and 1 ≤ i ≤ k . The prooffollows in the same way as in case (i) and we leave the details to thereader. (cid:3) Let { F t } t ∈ R be the flow built under the function f with base au-tomorphism T . Notice that { F t } t ∈ R is a properly ergodic flow. Thefollowing lemma follows directly from the definition of { F t } t ∈ R . Lemma 5.2. (i) If t > then F t ( x, ∈ X × { } if and only if thereexists an integer k ≥ such that t = P k − i =0 f ( T i x ) .(ii) If t < then F t ( x, ∈ X × { } if and only if there exists aninteger k ≥ such that t = − P ki =1 f ( T − i x ) . Proposition 5.3.
For any δ > , (8) lim s →∞ µ (∆ F,δ,s ( X )) = 0 . Proof.
By Lemma 5.2 we have(9) µ (∆ F,δ,s ) = µ ( x ∈ X ; ∃ k ∈ N , e s − δ < k − X i =0 f ( T i x ) < e s + δ ) [( x ∈ X ; ∃ k ∈ N , e s − δ < k X i =1 f ( T − i x ) < e s + δ )! . Proposition 5.1 implies that(10) µ ( x ∈ X ; ∃ k ∈ N , K n ≤ k − X i =0 f ( T i x ) < K n +1 )! ≤ [log n ]+1 , (11) µ ( x ∈ X ; ∃ k ∈ N , K n ≤ k X i =1 f ( T − i x ) < K n +1 )! ≤ [log n ]+1 . Notice that for s sufficiently large, ( e s − δ , e s + δ ) intersects at mosttwo consecutive intervals [ K n , K n +1 ). This, together (9), (10) and (11)implies (8). (cid:3) From Proposition 3.4 and Proposition 5.3 we can then conclude:
Corollary 5.4.
The flow { F t } t ∈ R constructed above satisfies PropertyB. References [1] A. Connes and E. J. Woods, ‘Approximately transitive flows and ITPFI factors’,
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Maria Joit¸a, Department of Mathematics,University of Bucharest,14Academiei St., 010014,Bucharest,Romania
E-mail address : [email protected] Radu-B. Munteanu, Department of Mathematics, Univ. of Bucharest,14 Academiei St., 010014,Bucharest,Romania
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