aa r X i v : . [ m a t h . D S ] N ov ENTROPY OF AT ( n ) SYSTEMS
RADU-B. MUNTEANU
Abstract.
In this paper we show that any ergodic measure pre-serving transformation of a standard probability space which isAT( n ) for some positive integer n has zero entropy. We show thatfor every positive integer n any Bernoulli shift is not AT( n ). Wealso give an example of a transformation which has zero entropybut does not have property AT( n ), for any integer n ≥ Introduction
In order to answer some questions from ergodic theory closely relatedto the theory of von Neumann algebras, T. Giordano and D. Handel-man [8] reformulated matrix valued random walks and their associatedgroup actions in terms of dimension spaces. Their approach leads toa notion of rank called AT( n ), for integers n ≥
1. This new conceptgeneralizes approximate transitivity (shortly AT), a property of ergodicactions introduced by A. Connes and E. J. Woods [1] in the theory ofvon Neumann algebras, which occurs for n = 1.Throughout this paper ( X, B , µ, T ) denotes a dynamical system,where T is a measure preserving automorphism of a standard prob-ability space ( X, B , µ ). For an integer n ≥
1, we say that the dynam-ical system ( X, B , µ, T ) (or simply T ) is AT( n ) if for any ε >
0, forany finite set of functions { f i } ki =1 from L ( X, µ ) there exist n func-tions { g m } m =1 ,...,n ∈ L ( X, µ ), a positive integer N , nonnegative reals { α ( m ) i,j } m =1 , ,...,ni =1 , ,...,k,j =1 , ,...,N and integers { t ( m ) j } m =1 ,...,nj =1 ,...,N , such that(1) k f i − n X m =1 N X j =1 α ( m ) i,j g m ◦ T t ( m ) j k < ε, for i = 1 , , . . . , k .Note it is sufficient to ask that equation (1) holds for k = n +1. Also,one can demand that k g m k = 1 for all m and that P nm =1 P Nj =1 α ( m ) i,j = k f i k , for all i .Remark that a rank n transformation (see [4]) is AT( n ) and everyAT( n ) system enjoys AT( n + 1) property. The techniques developed byT. Giordano and D. Handelman in [8] allowed the authors to construct examples of AT(2) transformation which are not AT. In [7], it wasproved that the measure preserving automorphism corresponding tothe Rudin-Shapiro substitution, which has rank 4 (and therefore isAT(4)), is not AT.Dynamical entropy is an invariant of measure theoretic dynamicalsystems introduced by A. N. Kolmogorov [9] and brought to its contem-porary form by Y. G. Sinai [11]. In 1970, D. Ornstein [10] showed thatKolmogorov-Sinai entropy completely classifies the Bernoulli shifts, abasic problem which couldn’t be solved for many decades. It was provedby A. Connes and E. J. Woods [1] that any dynamical system whichis AT has zero entropy. Different proofs of this result can be found in[2], [3] and [7]. It is natural to ask whether AT( n ) dynamical systemshave also zero entropy.In this paper we give a necessary condition for shift maps to be AT( n )(Theorem 2.1). For such transformations, this is a generalization of thenecessary condition for an action to be AT from [3].We use this condition to prove that, for any positive integer n ,Bernoulli shifts are not AT( n ), for n ≥
1. An important consequenceof this result is Corollary 3.3, which shows that any finite measure pre-serving transformation which is AT( n ) for some positive integer n , haszero entropy. In [3] A. Dooley and A. Quas proved that zero entropy isnot sufficient for AT; they gavr an example of a zero entropy transfor-mation which is not approximately transitive. In this paper we provethat the zero entropy and not AT transformtion from [3] is not AT( n ),for any n ≥ A necessary condition for shift maps to be AT ( n )Let Σ k = { , , . . . , k } Z be the shift space over the alphabet { , , . . . , k } .A cylinder set in Σ k is a set of the form [ y , y , . . . , y m ] n = { x ∈ Σ k ; x n = y , x n +1 = y , . . . , x m + n = y m } , where m, n ∈ Z , m ≥ y i ∈ { , , . . . k } . The cylinders of the form [ y , y , . . . , y n ] − n with n ≥ B k the σ -algebra generated by the cylinder sets of the shift space Σ k . The map S k : Σ k → Σ k defined by( S k x ) n = x n +1 for x = ( x n ) n ∈ Z is called the k -shift map.Let p = ( p (1) , . . . , p ( k )) be a probability vector with non-zero entries,i.e. p ( i ) > i = 1 , , . . . , k and P ki =1 p ( i ) = 1. Let µ p be the uniqueprobability measure on (Σ k , B k ), which on cylinder sets is given by µ p ([ y , y , . . . y m ] n ) = p ( y ) p ( y ) · · · p ( y m ) . NTROPY OF AT( n ) SYSTEMS 3 This measure is called the Bernoulli measure determined by p . The dy-namical system (Σ k , B k , µ p , S k ) is called the Bernoulli shift associatedto the probability vector p .Let Λ be a finite subset of the integers. A funny word on the alphabet { , , . . . , k } based on Λ is a finite sequence W = ( W n ) n ∈ Λ with W n ∈{ , , . . . , k } . For two funny words W, W ′ based on the same set Λ theirHamming distance is given by d Λ ( W, W ′ ) = 1 | Λ | card { n ∈ Λ : W n = W ′ n } . If x ∈ Σ k and Λ is a finite set in Z we denote by x | Λ the funny word( x n ) n ∈ Λ .The following theorem provides a necessary condition for shift mapsto be AT( n ). Theorem 2.1.
Let S k be the shift map on the space Σ k and ν be anon atomic shift invariant probability measure on Σ k . Assume that S k is AT ( n ) but not AT ( n − , for some n ≥ . Then for every ε > and every δ > there exist finite sets Λ , Λ , . . . , Λ n , with min {| Λ i | ; i =1 , , . . . , n } arbitrarily large, and funny words W i based on Λ i for i =1 , , . . . , n such that n X i =1 | Λ i | ν (cid:0) { x ∈ Σ k : d Λ i ( x | Λ , W i ) < ε } (cid:1) > − δ. Proof.
Let ( δ m ) m ≥ be a sequence decreasing to zero. The theoremwill result if we prove that for every ε > im ⊂ Z and funny words W im based on Λ im for i = 1 , , . . . n suchthat sup m ≥ min {| Λ im | , i = 1 , , . . . , n } = ∞ and n X i =1 | Λ im | ν (cid:0) { x ∈ Σ k : d Λ i ( x | Λ , W im ) < ε } (cid:1) > − δ m . Let ε >
0. For m ≥
0, denote by C m the set of all centered cylindersof the form [ y , y , . . . , y m ] − m which have positive measure. Since ν isnon atomic, for each C ∈ C m , one can find a measurable partition P C of C such that ν ( A ) < δ m , for every A ∈ P C . Let J m = { A ∈ P C : C ∈ C m } . Notice that { A : there exists m ≥ A ∈ J m } generates (upto null sets) the sigma algebra B k .For A ∈ J m let g A = ν ( A ) A the normalized indicator function cor-responding to A . Let m ≥ A ∈ J m . Since, by assumption, S k is AT( n ), there exists f ,m , f ,m , . . . , f n,m ∈ L (Σ k , ν ) of norm 1, RADU-B. MUNTEANU sequences of non-negative numbers { a A,j } j ∈ Z , { a A,j } j ∈ Z , . . . , { a nA,j } j ∈ Z with finitely many non-zero elements such that P j a A,j + P j a A,j + · · · + P j a nA,j = 1 and k g A − n X i =1 X j a iA,j f i,m ◦ S − jk k < εδ m . It then follows that for any A ∈ J m we have Z Σ k \ A n X i =1 X j a iA,j f i,m ◦ S − jk dν < εδ m . Let A ∈ J m . For i = 1 , , . . . , n define P iA = { j : Z Σ k \ A f i,m ◦ S − jk dν ≥ εδ m / } . It easily can be seen that P j ∈ P A a A,j + P j ∈ P A a A,j + · · · + P j ∈ P nA a nA,j <δ m / . By setting the a iA,j to be 0 for j ∈ P iA , and re-scaling the remain-ing a iA,j we obtain coefficients b A,j , b A,j , . . . , b nA,j with P j b A,j + P g b A,j + P j b nA,j = 1, satisfying k g A − n X i =1 X j b iA,j f i,m ◦ S − jk k < δ m , and such that Z Σ k \ A f i,m ◦ S − jk dν < εδ m b iA,j >
0. For i = 1 , , . . . , n , letΛ im = { j ∈ Z : there exists A ∈ J m , b iA,j > } . We claim that(2) sup m ≥ min {| Λ im | , i = 1 , , . . . , n } = ∞ . We will prove the claim by contradiction. Let us suppose thatmin {| Λ im | , ≤ i ≤ n } ≤ M < ∞ , for all m ≥ . Let i ( m ) be such that | Λ i ( m ) m | = min {| Λ im | , ≤ i ≤ n } , then | Λ i ( m ) m | ≤ M , for all m ≥
1. Let L m = { A ∈ J m : there exists j ∈ Z such that b i ( m ) A,j > } . Remark that |L m | ≤ | Λ i ( m ) m | . Since ν ( A ) < δ m for every A ∈ J m , itfollows that lim m →∞ ν ( ∪ A ∈L m A ) = 0. Hence, any f ∈ L (Σ k , ν ) can beapproximated arbitrarily close in L -norm by step functions of the form NTROPY OF AT( n ) SYSTEMS 5 P A ∈J m −L m α A g A with α A ≥ P A ∈J m \L m α A = k f k , by choosing m sufficiently large.Let f l ∈ L (Σ k , ν ), l = 1 , , . . . , n be functions of norm 1 and η > m ≥ δ m < η and such that existnon-negative numbers α iA with P A ∈J m \L m α lA = 1 satisfying k f l − X A ∈J m \L m α lA g A k < η , for l = 1 , , . . . , n . Note that if A ∈ J m \ L m then k g A − n X i =1 ,i = i ( m ) X j b iA,j f i,m ◦ S − jk k < δ m . We obtain then non-negative coefficients c i,lj , i, l ∈ { , , . . . n } , i = i ( m )with finitely many of them different from zero such that n X i =1 ,i = i ( m ) X j c i,lj = 1and k f l − n X i =1 ,i = i ( m ) X j c i,lj f i,m ◦ S − jk k < η. Since such an approximation can be done for any η >
0, it follows that S k is AT( n − i = 1 , , . . . , n we have X A ∈J m X j : b iA,j > Z Σ k \ A f i,m ◦ S − jk dν < εδ m | Λ im | εδ m | Λ im | > X A ∈J m X j : b iA,j > Z f i,m ◦ S − jk · Σ k \ A dν = Z f i,m X A ∈J m X g : b iA,j > Σ k \ A ◦ S jk dν. Let H im = 1 | Λ im | X A ∈J m X j : b iA,j > Σ k \ A ◦ S jk . For j ∈ Λ im , there exists a unique set A ∈ J m such that R A f i,m ◦ S − jk dν > − εδ m /
6. Let [ z ] = { x ∈ Σ k : x = z } be the unique cylin-der set from C containing A and define W im,j to be this z . Denoteby W im the funny word ( W im,j ) j ∈ Λ im based on Λ im . Since the above RADU-B. MUNTEANU inequality demonstrates that R H im f i,m dν < εδ m /
6, it follows that R { z : H im ( z ) >ε } f i,m ( x ) dν < δ m /
6. Let e f i,m be the function defined by e f i,m ( x ) = ( H im ( x ) > εf ( x ) / R { z : H im ( z ) <ε } f i,m dν otherwise . Clearly, H im ( x ) ≥ im card { j ∈ Λ im : W km,j = x j } . Therefore, the support of e f i,m is contained in { x ∈ Σ k : d ( x | Λ im , W im ) <ε } . Since k e f i,m − f i,m k < δ m for i = 1 , , . . . , n , for A ∈ J m we havethat k g A − n X i =1 X j b iA,j e f i,m ◦ S − jk k < δ m . Hence, summming over A ∈ J m we get k − n X i =1 X A ∈J m X j ν ( A ) b iA,j e f i,m ◦ S − jk k < δ m . Since the support of each e f i,m is contained in { x ∈ Σ k : d Λ im ( x | Λ im , W im ) <ε } , it follows that n X i =1 X A ∈J m X j ν ( A ) b iA,j e f i,m ◦ S − jk is supported on a set of measure at most n X i =1 | Λ im |{ x ∈ Σ k : d Λ im ( x | Λ im , W im ) < ε } . Therefore n X i =1 | Λ im | ν (cid:0) { x ∈ Σ k : d Λ im ( x | Λ , W im ) < ε } (cid:1) > − δ m and the proof of the theorem is complete. (cid:3) AT ( n ) systems have zero entropy In this section we show that for any positive integer n , Bernoullishifts are not AT( n ). We also show that AT( n ) systems have zeroentropy. Let us prove first the following lemma. Lemma 3.1.
A factor of an AT ( n ) system is AT ( n ) . NTROPY OF AT( n ) SYSTEMS 7 Proof.
Let π be a factor map from an AT( n ) system ( X, B , µ, T ) ontoanother dynamical system ( Y, F , ν, S k ). Let f , f , . . . , f n +1 ∈ L ( X, µ ).Since ( X, B , µ, T ) is AT( n ), there exists g , g , . . . , g n ∈ L ( X, µ ), apositive integer N , reals α ( m ) i,j ≥
0, for m = 1 , , . . . , n , i = 1 , , . . . , n +1, j = 1 , , ..., N and integers { t ( m ) j } m =1 ,...,nj =1 ,...,N , such that k f i ◦ π − n X m =1 N X j =1 α ( m ) i,j g m ◦ T t ( m ) j k < ε, for i = 1 , , . . . , n + 1. Taking expectation with respect to the T -invariant σ -algebra π − ( B ), we obtain k f i ◦ π − n X m =1 N X j =1 α ( m ) i,j E ( g m | π − ( B )) ◦ T t ( m ) j k < ε, for each i . Notice that for all i , we can write E ( g m | π − ( B ) = G m ◦ π for some measurable function G m on Y , and then, since S k ◦ π = π ◦ T ,we have k f i − n X m =1 N X j =1 α ( m ) i,j G m ◦ S t ( m ) j k k < ε. We can then conclude that the system ( Y, F , ν, S k ) is AT( n ). (cid:3) Proposition 3.2.
Let (Σ k , B k , µ p , S k ) be the Bernoulli shift associatedto the probability vector p = ( p (1) , . . . p ( k )) . Then, for any n ≥ , theshift map S k is not AT ( n ) .Proof. We prove the proposition by induction. It is well known that S k has positive entropy and therefore is not AT(1). Let us assume now thatfor some n ≥ S k is not AT( n − r = max { p ( i ); i = 1 , , . . . , k } .If W is a funny word based on Λ then µ p ( x ∈ Σ k : d Λ ( x | W , W )) ≤ (cid:18) m [ mε ] (cid:19) r m − [ mε ] , where | Λ | = m .Notice that if ε is sufficiently small then r (1 − ε ) ε (1 − ε ) ε ε r ε < . For m sufficiently large, we have (cid:18) m [ mε ] (cid:19) r m − [ mε ] < me ) m √ πm ( mεe ) mε √ πmε ( m (1 − ε ) e ) m (1 − ε ) p πm (1 − ε ) r m − [ mε ] RADU-B. MUNTEANU < p πmε (1 − ε ) (cid:18) r (1 − ε ) ε (1 − ε ) ε ε r ε (cid:19) m < − εn · m . It then follows that if | Λ | sufficiently large, µ p ( x ∈ Σ k : d Λ ( x | W , W ) <ε ) < − εn ·| Λ | . Then Theorem 2.1, implies that S k is not AT( n ). (cid:3) We can prove now the result announced in the beginning concerningthe entropy of AT( n ) systems. Corollary 3.3.
Let n be a positive integer and let T be an ergodic mea-sure preserving transformation of a standard probability space ( X, B , µ ) which is AT ( n ) . Then T has zero entropy.Proof. We prove this lemma by contradiction. Assume that the entropy h ( T ) of T is strictly positive. Consider a Bernoulli shift (Σ k , B k , µ p , S k )associated to a probability vector p = ( p (1) , p (2), . . . , p ( k )) such that h ( T ) ≥ h ( S k ) = P ki =1 p ( i ) log p ( i ). By Sinai’s theorem, (Σ k , B k , µ p , S k )is a factor of the system ( X, B , µ, T ). Then, Lemma 3.1, implies that(Σ k , B k , µ p , S k ) is AT( n ). This is a contradiction. (cid:3) In the last part of this setion we will show that there exists a zeroentropy dynamical system which is not AT( n ) for any ≥
1. Let α bean irrational number and let T be the transformation of the 2-torus T defined by T ( s, t ) = ( s + α, s + t + α ) (mod 1) . This transformation, studied by H. Furstenberg [5, 6], is measure pre-serving, uniquely ergodic and has zero entropy. It was proved by A.Dooley and A. Quas in [3] that T is not approximately transitive. Proposition 3.4.
The zero entropy transformation T defined abovedoes not have property AT( n ), for any positive integer n .Proof. For k ≥ P be the partition of T consisting of the sets A i = T × (cid:2) i − k , ik (cid:1) , for i = 1 , , . . . , k + 1. Let Σ k +1 = { , , . . . , k + 1 } Z and let π : T → Σ k +1 be the natutal map from T in Σ k +1 , defined by π ( z ) = ( x n ) n ∈ Z , where x n = i if T n ( z ) ∈ A i .If µ is the Haar measure µ of T , denote by ν the measure π − ◦ µ induced by µ on Σ k +1 .We show first that for all m = n (3) ν ( { y ∈ Σ k +1 : y m = i, y n = j } ) = 1( k + 1) Since ν is shift invariant it is enough to show that for every n ∈ Z ν ( { y ∈ Σ k +1 : y = i, y n = j } ) = k +1) . This is the Haar measureof the set of points ( s, t ) such that s ∈ A i and π ( T n ( s, t )) ∈ A j (here NTROPY OF AT( n ) SYSTEMS 9 π ( s, t ) = t for ( s, t ) ∈ T ). In other words, this is the set of all ( s, t )such that s ∈ A i and h t + n α + 2 ns i ∈ A j , where h z i denotes thefractional part of z ∈ R . It easily can be observed that the measure ofthis set is k +1) . Let Λ be an arbitrary subset of integers of cardinality n and fix x ∈ Σ k +1 . Define Λ j : Σ k +1 → C byΛ j ( y ) = y j = x j ,ε if y j = x j + 1 , · · · ε k if y j = x j + k, where ε = e πi/ ( k +1) . Let S = P j ∈ Λ Λ j . From (3) it results that E ( | S | ) = n and then, by Markov inequality we get ν (cid:18) | S | > k + 12 k + 2 · n (cid:19) = ν (cid:18) | S | > (2 k + 1) (2 k + 2) · n (cid:19) ≤ E ( | S | ) n · (2 k + 2) (2 k + 1) = (2 k + 2) (2 k + 1) · n . For y ∈ Σ k +1 define a i ( y ) = card { j ∈ Λ : y j = x j + i − } for i = 1 , , . . . k + 1 . Then S = a + εa + ε a + · · · + ε k a k +1 . Now, consider the sets B k +4 ,x, Λ = (cid:26) y ∈ Σ k +1 : d Λ ( y | Λ , x | Λ ) < k + 4 (cid:27) ,A = (cid:26) y ∈ Σ k +1 : | S | > k + 12 k + 2 · n (cid:27) and for i = 1 , , . . . , k + 1, A i = { y ∈ A : a i ( y ) > a j ( y ) for all j = i } . Notice that B k +4 ,x, Λ ⊂ A . Indeed if y ∈ B k +4 ,x, Λ , then a ( y ) ≥ k +34 k +4 · n , and consequently, | S ( y ) | > | a ( y ) | − | εa ( y ) + · · · ε k a k +1 ( y ) |≥ (cid:18) k + 34 k + 4 − k + 4 (cid:19) · n = 2 k + 12 k + 2 · n. Denote by R be the transformation of T defined by R ( s, t ) = ( s, t + k +1 ) (mod 1), which clearly commutes with T . It is easy to see that a i ( π ( s, t )) = a i +1 ( π ( R ( s, t ))) for all ( s, t ) ∈ T and i ∈ { , , . . . , k } and then R ( π − ( A i )) = π − ( A i +1 ) for every i ∈ { , , . . . , k } . Since R is ameasure preserving transformation it follows that ν ( A i ) = µ ( π − ( A i )) = µ ( R ( π − ( A i ))) = µ ( π − ( A i +1 )) = ν ( A i +1 ) for every i ∈ { , , . . . , k } and then, ν ( A ) ≤ k +1 · ν ( A ). Hence | Λ | · ν ( B k +4 ,x, Λ ) ≤ | Λ | · ν ( A ) ≤ k + 44 k + 4 k + 1and therefore(4) k · | Λ | · ν ( B k +4 ,x, Λ ) ≤ − k + 4 k + 1Theorem 2.1 from [3], implies that the system (Σ k +1 , B k +1 , ν, S k +1 )is not AT. Then (4) and Theorem 2.1 implies that the system is notAT(2). Applying successively the same Theorem 2.1, it results that thesystem (Σ k +1 , B k +1 , ν, S k +1 ) is not AT( k ). Using now Proposition 3.1,it follows that that T is not AT( k ). Since k ≥ (cid:3) Acknowledgement.
This work was supported by a grant of the Ro-manian Ministry of Education, CNCS - UEFISCDI, project numberPN-II-RU-PD-2012-3-0533.
References [1] A. Connes and E. J. Woods. Approximately transitive flows and ITPFI factors.
Ergod, Theory Dynam. Sys. (1985), 203–236.[2] M. C. David. Sur quelques problemes desc theorie ergodique non commutative .PhD thesis, 1979.[3] A. H. Dooley and A. Quas, Approximate transitivity for zero-entropy systems.
Ergod. Theory Dynam. Sys. (2005), 443–453.[4] S. Ferenczi. Systems of finite rank. Colloquium Mathematicae (1997), 35–65.[5] H. Furstenberg. Strict ergodicity and transformations of the torus. Amer. J.Math (1961), 573–601.[6] H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number The-ory , Princeton University Press, Princeton, NJ, 1981.[7] E. H. El Abdalaoui and M. Lemanczyk. Approximate transitivity property andLebesgue spectrum.
Monatshefte Math. (2010), 121–144.[8] T. Giordano and D. Handelman. Matrix-valued random walks and variationson property AT.
Munster J. of Math. (2008), 15–72.[9] A. N. Kolmogorov. A new invariant for transitive dynamical systems, D.A.N.SSSR (1958), 861–869.[10] D. S. Ornstein. Bernoulli shifts with the same entropy are isomorphic.
Advancesin Math. (1970), 337–352.[11] Y. G. Sinai. On the concept of entropy for a dynamic system. Dokl. Akad.Nauk SSSR (1959), 768–771.[12] C. Shannon. The mathematical theory of communication.
Bell Syst. Tech. J. (1948), 379–423. NTROPY OF AT( n ) SYSTEMS 11 Department of Mathematics, University of Bucharest, 14 AcademieiSt, 010014, Sector 1, Bucharest, Romania
E-mail address ::