Δ -Weakly Mixing Subset In Positive Entropy Actions Of A Nilpotent Group
aa r X i v : . [ m a t h . D S ] M a y ∆ -WEAKLY MIXING SUBSET IN POSITIVE ENTROPYACTIONS OF A NILPOTENT GROUP KAIRAN LIU
Abstract.
The notion of ∆-weakly mixing subsets is introduced for countabletorsion-free discrete group actions. It is shown that for a finitely generated torsion-free discrete nilpotent group action, positive topological entropy implies the ex-istence of ∆-weakly mixing subsets, and while there exists a finitely generatedtorsion-free discrete solvable group action which has positive topological entropybut without any ∆-weakly mixing subsets. Introduction
In this paper, let T be a countable discrete group with the unit θ T . By a T -system ( X, T ) we mean a compact metric space X endowed with a metric ρ , togetherwith T acting on X by homeomorphism, that is, there exists a continuous mapΨ : T × X → X , Ψ( T, x ) =
T x satisfying Ψ( θ T , x ) = x , Ψ( T, Ψ( S, x )) = Ψ(
T S, x )for each
T, S ∈ T and x ∈ X . When T is the group Z of integers, it is generatedby the element 1, in this case we let T : X → X , x Ψ(1 , x ) and denote thisdynamical system by (
X, T ). For a T -system ( X, T ) and m ∈ N , ( X m , T ) is alsoa T -system, where X m := X × X × . . . × X ( m -times), and T ( x , x , . . . , x m ) :=( T x , T x , . . . , T x m ) for any ( x , x , . . . , x m ) ∈ X m and T ∈ T .Recurrence is one of the central topics in the study of T -systems. In 1978, Fursten-berg and Weiss published a topological theorem generalizing Brikhoff’s recurrencetheorem and having interesting combinational corollaries, where T is an abeliangroup (see [11]). As a simple example due to Furstenberg shows that the statementis not valid when the assumption that T is commutative is omitted (see [10, P. 40]).In [26], Leibman proved the following conjecture, due to Yuzvinsky, formulated byHendrick: the multiple recurrence theorem holds true when T is nilpotent. In [17],Huang, Shao and Ye obtained a topological analogue of multiple ergodic averagesof weakly mixing minimal systems for nilpotent group actions.Recently, for Z -systems many researchers studied strong forms of multiple recur-rence. They introduced and investigated the ∆-transitivity and ∆-weakly mixing(see [12, 2, 4, 5, 24, 25, 31]). A Z -system ( X, T ) is ∆ -transitive if for every in-teger d ≥ X of X such that for each x ∈ X , { ( T n x, T n x, . . . , T dn x ) : n ∈ N } is dense in the d -th product metric space X d .Glasner showed that for a minimal system, weak mixing implies ∆-transitivity (see[12]). In [31], Moothathu proved ∆-transitivity implies weak mixing, but there existstrongly mixing systems which are not ∆-transitive. A Z -system is called ∆ -weakly Key words and phrases. ∆-transitivity, ∆-weakly mixing, topological entropy, nilpotent groups. mixing if ( X m , T ( m ) ) is ∆-transitive for any m ∈ N . In [16] Huang, Li, Ye and Zhoustudied ∆-transitivity and ∆-weak mixing and proved that for a Z -system ∆-weaklymixing is in fact equivalent to ∆-transitivity (see [16, Proposition 3.2]) but it is nolonger true for ∆-weakly mixing sets and ∆-transitive sets (see [16, Remark 3.5]).Inspired by the above ideas and results, we introduce ∆-transitivity and ∆-weakmixing for a countable torsion-free discrete group T -action. Recall that a groupis called torsion-free if any element has infinite order except the identity element.Let ( X, T ) be a T -system, where T is a countable torsion-free discrete group, and E be a closed subset of X with | E | ≥
2. We say that E is a ∆ -transitive subset of ( X, T ) provided that there is a residue subset A of E such that for any x ∈ A , d ≥ T , T , . . . , T d ∈ T \ { θ T } , the orbit closure of the d -tuple( x, x, . . . , x ) under the action T × T × · · · × T d contains E d , i.e. orb + (( x, x, . . . , x ) , T × T × · · · × T d ) ⊇ E d . where orb + (( x, x, . . . , x ) , T × T × · · · × T d ) := { ( T n x, T n x, . . . , T nd x ) : n ∈ N } , and E d := E × E × · · · × E ( d times), and a ∆ -weakly mixing subset of ( X, T ) if E m is a ∆-transitive subset of ( X m , T ) for any m ∈ N . If X is a ∆-transitive (reps.∆-weakly mixing) subset of ( X, T ) then we say that the T -system ( X, T ) is ∆-transitive (reps. ∆-weakly mixing). We will show that if ( X, T ) is ∆-weakly mixingthen X is perfect and ( X, T ) is weakly mixing (see Proposition 3.1).It is well known that for a Z -system ( X, T ), there always exists a T -invariantBorel probability measure on X , whereas for some groups T there do not exist anyinvariant Borel probability measures on a T -system, while the amenability of theacting group T ensures the existence of invariant Borel probability measures.For Z -systems, the variational principle of entropy and the Shannon-McMillan-Breiman (SMB) theorem are important in the study the entropy theory (see [34, 14,15, 8]). Comparing to Z -systems, the study of dynamical systems with amenablegroup actions lagged behind, while the situation is rapidly changed in recent years,many researchers studied the entropy theory of dynamics with amenable group ac-tions (see [9, 23, 6, 33, 37, 35, 13, 7, 29]). Lindenstrauss and Weiss obtained a gen-eralization of the SMB theorem (see [29, 38]). The variational principle of entropyalso holds for countable discrete amenable groups (see [32, 36]). In [19], Huang, Yeand Zhang established a local variational principle for the entropy of a given finiteopen cover of a countable discrete amenable group actions. Glasner, Thouvenot andWeiss proved an important disjointness theorem which asserts the relative disjoint-ness in the sense of Furstenberg, of zero-entropy extension from completely positiveentropy extensions. An application of this theorem is to obtain that the Pinskerfactor of a product system is equal to the product of the Pinsker factor of the com-ponent systems (see [13]). In [22] Kerr and Li showed that for a amenable groupaction positive entropy indeed implies Li-Yorke chaos.In [2], Blanchard and Huang introduced a local version of weakly mixing for Z -systems and showed that positive entropy implies the existence of weakly mixingsets. Recently, Huang, Li, Ye and Zhou proved that for a Z -system ( X, T ) positiveentropy indeed implies the existence of ∆-weakly mixing sets (see [16, Theorem C]). -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 3
In this paper, we generalize this result for dynamical systems of finitely generatedtorsion-free nilpotent group actions.
Theorem 1.1.
Let ( X, T ) be a T -system, where T is a finitely generated torsion-free discrete nilpotent group. If h top ( X, T ) > then there exist ∆ -weakly mixingsubsets of ( X, T ) . It should be noticed that after modifying the example introduced by Furstenbergin [10, P.40] we construct a dynamical system (
Z, F ) with positive topological en-tropy, where Z is a compact metric space and F is a finitely generated torsion-freediscrete solvable group, such that there is no ∆-transitive subsets for ( Z, F ) (seeProposition 4.5).Following the idea of Theorem A in [16], we also give an equivalent characteriza-tion for ∆-weakly mixing subsets as follows.
Theorem 1.2.
Let ( X, T ) be a T -system, where T is a countable torsion-free dis-crete group. If E is a closed subset of X with | E | > , then E is a ∆ - weaklymixing subset of ( X, T ) if and only if E is perfect and there exists an increasingsequence of Cantor sets C ⊂ C ⊂ . . . of E such that C = S ∞ i =1 C i is dense in E , and has the following property: for any subset A of C , d ∈ N , pairwise distinct T , T , .., T d ∈ T \ { θ T } and any continuous functions g j : A → E for j = 1 , , . . . , d there exists an increasing sequence { q k } ∞ k =1 of positive integers such that lim k →∞ T q k j x = g j ( x ) for every x ∈ A and j = 1 , , . . . , d . The term ”chaos” in connection with a map was introduced by Li and York andproved its value for interval maps (see [28]). In [3], the authors considered theLi-Yorke definition of chaos in the setting of general topological dynamics (
X, T )and proved that positive entropy implies Li-Yorke chaos. In sofic context, positivetopological entropy with respect to some sofic approximation sequence implies Li-Yorke chaos (see [21]). Comparing with the Li-York chaos we have the followingdefinition of chaos.We say that a T -dynamical system ( X, T ) is asynchronous chaotic if there existsan increasing sequence of Cantor sets C ⊂ C ⊂ . . . of X and δ > x, y ∈ C := S ∞ i =1 C i and T , T ∈ T \ { θ T } ,lim inf n →∞ ρ ( T n x, T n y ) = 0 , lim sup n →∞ ρ ( T n x, T n y ) ≥ δ, such C is called a asynchronous chaotic set .Following Theorem 1.1 and Theorem 1.2, we have the following corollary. Corollary 1.3.
Let ( X, T ) be a T -system, where T is a finitely generated torsion-free discrete nilpotent group. If h top ( X, T ) > , then ( X, T ) is asynchronous chaotic. For a countable torsion-free discrete amenable group actions, we do not knowwhether Corollary 1.3 still holds. More precisely, assume that T be a countabletorsion-free discrete amenable group and ( X, T ) is a T -system with h top ( X, T ) > X, T ) asynchronous chaotic? K. LIU
This paper is organized as follows. In Section 2, we recall some basic concepts anduseful properties. In Section 3, we will present some properties of ∆-weakly mixingsubsets of a T -system. In Section 4, we will present some concepts and properties ofnilpotent group, then give the proof of Theorem 1.1. Finally, we will prove Theorem1.2 and Corollary 1.3 in Section 5.2. Preliminaries
In this section, we will review the hyperspace 2 X of a compact metric space X with the Hausdorff metric, density of subsets of non-negative integers, extension,entropy of an amenable group action. We also present some basic results which willused later.2.1. Hyperspace of space.
For a compact metric space X with a metric ρ , theHausdorff metric of two non-empty compact subsets A, B of X is defined as: ρ H ( A, B ) = max { max x ∈ A min y ∈ B ρ ( x, y ) , max y ∈ B min x ∈ A ρ ( x, y ) } . The metric space (2 X , ρ H ) ( hyperspace of X ) is compact since ( X, ρ ) is compact,where 2 X is the collection of all non-empty compact subsets of X . For non-emptyopen subsets U , U , . . . , U n of X , let h U , U , . . . , U n i := (cid:26) A ∈ X : A ⊂ n [ i =1 U i and A ∩ U i = ∅ , i = 1 , , . . . , n (cid:27) . Collection of those h U , U , . . . , U n i form a basis for the Hausdorff topology of 2 X induced by ρ H , where U , U , . . . , U n are non-empty open subsets of X .A subset Q of 2 X is called hereditary if 2 A ⊂ Q for every set A ∈ Q . For ahereditary subset of 2 X , there is a consequence of the Kuratowski-Mycielski Theorem([1, Theorem 5.10]). Lemma 2.1.
Suppose that X is a perfect compact metric space. If a hereditarysubset Q of X is residual, then there exists an increasing sequence of Cantor subsets C ⊂ C ⊂ · · · of X such that C i ∈ Q for every i ≥ and C = S ∞ i =1 C i is dense in X . Density of subsets of non-negative integers.
Let Z , Z + and N denote thecollection of all integers, non-negative integers and positive integers respectively.The lower density and upper density of a subset F ⊆ Z + is defined respectively by D ( F ) = lim inf n →∞ | F ∩ { , , . . . , n − }| n and D ( F ) = lim sup n →∞ | F ∩ { , , .., n − }| n . We say that F has density D ( F ) if D ( F ) = D ( F ), where D ( F ) denote this commonvalue. There is a simple fact that we will use in the Section 4: for a real sequence -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 5 { a n } ∞ n =0 with 0 ≤ a n ≤ M for some positive real number M iflim inf N →∞ N N − X i =0 a n > D ( E ) > E := { n ∈ Z + : a n > } .2.3. Entropy of an amenable group action.
A countable discrete group T iscalled amenable if there exists a sequence of non-empty finite subsets { F n } + ∞ n =1 of T such that lim n → + ∞ | T F n ∆ F n || F n | = 0holds for every T ∈ T , and such { F n } + ∞ n =1 is called a Følner sequence of T . We knowthat finite groups, solvable groups and finitely generated groups of subexponentialgrowth are all amenable groups.Let ( X, T ) be a T -system, where T is a countable discrete amenable group withFølner sequence { F n } + ∞ n =1 . A finite cover of X is a finite family of Borel subsets of X , whose union is X . Denote by C oX the collection of all open finite covers of X . Let U ∈ C oX . We set N ( U ) to be the minimum among the cardinalities of all sub-familiesof U covering X and define H ( U ) = log N ( U ). For a finite non-empty subset S of T and U ∈ C oX , set U S = _ T ∈S T − U = (cid:26) \ T ∈S T − U T | U T ∈ U (cid:27) . The topological entropy of U , h top ( T , U ) = lim n →∞ | F n | H ( U F n )exists and is independent of the Følner sequence (see [30, Theorem 6.1]). The topological entropy of ( X, T ) is defined by h top ( X, T ) := sup U∈C oX h top ( T , U ) . Denote by M ( X ) the set of all Borel probability measures on X . µ ∈ M ( X ) iscalled T -invariant if T µ = µ for each T ∈ T . Denote by M ( X, T ) the set of all T -invariant elements in M ( X ). µ ∈ M ( X, T ) is called ergodic if µ ( S T ∈T T A ) = 0 or1 for any A ∈ B X . Denote by M e ( X, T ) the set of all ergodic elements in M ( X, T ).When the acting group T is amenable, M ( X, T ) = ∅ and M ( X ), M ( X, T ) areconvex compact metric spaces with weak ∗ -topology.A partition of X is a cover of X , whose elements are pairwise disjoint. Denoteby P X the set of all finite Borel partitions of X . Given α ∈ P X , a T -invariantsub- σ -algebra A ⊆ B X and µ ∈ M ( X ), define H µ ( α |A ) = X A ∈ α Z E (1 A |A ) log E (1 A |A ) dµ, K. LIU where E (1 A |A ) is the expectation of 1 A with respect to A . Define h µ ( T , α |A ) = lim n →∞ | F n | H µ ( α F n |A ) . Once again one can deduce the existence of this limit and its independence of thesequence { F n } + ∞ n =1 (see [20, 33]). When A = {∅ , X } , we write h µ ( T, α |{∅ , X } ) as h µ ( T, α ). The measure-theoretic entropy of ( X, T , µ ) is defined by h µ ( X, T ) = sup α ∈P X h µ ( T , α ) . The variational principle between topological entropy and measure-theoretic en-tropy also holds for countable infinite discrete amenable group actions (see [32, 36]): h top ( X, T ) = sup µ ∈M ( X, T ) h µ ( X, T ) = sup µ ∈M e ( X, T ) h µ ( X, T ) . For a T -system, one has the following property (see [19, Lemma 2.4] or [30,Theorem 6.1]). Proposition 2.2.
Let ( X, T ) be a T -system, where T is a countable infinite discreteamenable group. Then for any α ∈ P X and µ ∈ M ( X, T ) , we have h µ ( T , α ) = inf F⊂T , |F| < ∞ H µ ( α F ) | F | . Extensions between measure preserving systems.
We say that a prob-ability space ( X, B , µ ) is regular if there exists a metric on X suct that X is acompact metric space and B consists of all Borel subsets of X . In this paper, wealways assume that probability spaces to be regular. A measure preserving system ( X, B , µ, T ) consists of a probability space ( X, B , µ ) and a group T acting on X bytransformations preserving measure µ . A measure preserving system ( X, B , µ, T ) is regular if the underlying probability space ( X, B , µ ) is regular.A homomorphism of measure preserving systems π : ( X, B , µ, T ) → ( Y, D , ν, T )is given by a homomorphism π : ( X, B , µ ) → ( Y, D , ν ) satisfying(1) π − ( A ∪ A ) = π − ( A ) ∪ π − ( A ), A , A ∈ e D ,(2) π − ( e Y \ A ) = X \ π − ( A ), A ∈ e D ,(3) µ ( π − ( A )) = ν ( A ), A ∈ e D ,(4) π − ( T − A ) = T − ( π − ( A )), A ∈ e D , T ∈ T ,where e D is the abstract σ -algebra consisting of equivalence classes of sets in D (modnull sets). In this case we say that ( X, B , µ, T ) is an extension of ( Y, D , ν, T ) or that( Y, D , ν, T ) is a factor of ( X, B , µ, T ), and π is a factor map . The following resultis well known (see e.g. [10, Theorem 5.8]) Theorem 2.3.
Let ( X, B X , µ, T ) be a regular measure preserving system and π :( X, B X , µ, T ) → ( Y, B Y , ν, T ) be a factor map. Then there exists a measurable mapfrom Y to M ( X ) which we shall denote y → µ y which satisfies: -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 7 (1) for every f ∈ L ( X, B , µ ) , f ∈ L ( X, B , µ y ) for ν -a.e. y ∈ Y , and E ( f | Y )( y ) = Z f dµ y for ν -a.e. y ∈ Y ; (2) R { R f dµ y } dν ( y ) = R f dµ for every f ∈ L ( X, B , µ ) . We shall write ν = R µ y dν and refer to this as the disintegration of µ with respectto the factor ( Y, D , ν ) (or the disintegration of µ over ν ). For each S ∈ T and foralmost every y ∈ Y , µ Sy = Sµ y .Let ( X , B , µ ) and ( X , B , µ ) be two regular measure spaces and π : ( X , B , µ ) → ( Y, D , ν ) , π : ( X , B , µ ) → ( Y, D , ν )are two extensions of the same space ( Y, D , ν ). The measure space ( X × X , B ×B , µ × Y µ ) is called the product of ( X , B , µ ) and ( X , B , µ ) relative to ( Y, D , ν )and is denoted by X × Y X , where µ × Y µ is a measure on ( X × X , B × B )defined by: ( µ × Y µ )( A ) = Z µ ,y × µ ,y ( A ) dν ( y )for A ∈ B ×B , and µ i = R µ i,y dν ( y ) are the disintegrations of µ i over ν , for i = 1 , X, B , µ, T ) be a regular measure preserving system and π : ( X, B , µ, T ) → ( Y, D , ν, T ) be a factor map. We say that π is an ergodic extension of ( Y, D , ν, T ) relative to T ∈ T if the only T -invariant sets of B are images (modulo null sets) under π − of T -invariant sets of D , and a weakly mixing extension of ( Y, D , ν, T ) relative to T ∈ T if ( X × X, B×B , µ × Y µ, T ) is an ergodic extension of ( Y, D , ν, T ) relative to T .We say that π is an ergodic extension (or a weakly mixing extension ) of ( Y, D , ν, T )if π is an ergodic extension (or a weakly mixing extension) of ( Y, D , ν, T ) relative toevery T ∈ T \ { θ T } .Furstenberg proved the following proposition (see [10, Proposition 6.4]). Proposition 2.4.
Let ( X, B , µ, T ) be a regular measure preserving system, and let π : ( X, B , µ, T ) → ( Y, D , ν, T ) be a factor map. If π is a weakly mixing extensionof ( Y, D , ν, T ) , then e π = π ◦ π : ( X × X, B × B , µ × Y µ, T ) → ( Y, D , ν, T ) is also aweakly mixing extension of ( Y, D , ν, T ) , where π : X × X → X is the projection tothe first coordinate. Let ( X, T ) be a T -system, where T is a countable infinite discrete amenablegroup. For µ ∈ M ( X, T ) and T -invariant sub- σ -algebra A of B X , denote P µX ( T |A ) = { A ∈ B X : h µ ( T , { A, X \ A }|A ) = 0 } . It follows from [13, Lemma 1.1] or [18, Theorem 3.1] that P µX ( T |A ) must be a T -invariant sub- σ -algebra of B X containing A . We call this σ -algebra the Pinsker σ -algebra of ( X, B X , T , µ ) relative to A , and the corresponding factor the relativePinsker factor . When A is the trivial σ -algebra we get the Pinsker algebra and
Pinsker factor of X and we denote this σ -algebra by P µX ( T ). The following theoremis a classic result (see for example [13, Theorem 0.4] or [18, Lemma 4.2]). K. LIU
Theorem 2.5.
Let ( X, T ) be a T -system, where T is a countable infinite discreteamenable group, µ ∈ M e ( X, T ) , and π : ( X, B X , µ, T ) → ( Z, B Z , ν, T ) be the factormap to the Pinsker factor of ( X, B X , µ, T ) . Assume that π : X × X → X is theprojection to the first coordinate and e π = π ◦ π . Then P µ × Z µX × X ( T | e π − ( B Z )) = e π − ( B Z )(mod µ × Z µ ) . Proposition 2.6.
Let ( X, T ) be a T -system, where T is a countable infinite discreteamenable group, µ ∈ M e ( X, T ) , and π : ( X, B X , µ, T ) → ( Z, B Z , ν, T ) be the factormap to the Pinsker factor of ( X, B X , µ, T ) . Then π is a weakly mixing extension.Proof. Let π : X × X be the projection to the first coordinate and e π = π ◦ π :( X × X, B X × B X , µ × Z µ, T ) → ( Z, B Z , ν, T ).For any T ∈ T \ θ T , we shall show that e π is an ergodic extension of ( Z, B Z , ν, T )relative to T . Suppose E ∈ B X × B X such that T E = E . Let α = { E, X × X \ E } ,and F n = { T, T , . . . , T n } ⊂ T , n ∈ N . Then for any n ∈ N , α F n = α . ByProposition 2.2 we have h µ × Z µ ( T , α | e π − ( B Z )) ≤ h µ × Z µ ( T , α ) = inf F ⊂T , |F| < ∞ H µ × Z µ ( α F ) | F | ≤ H µ × Z µ ( α F n ) | F n | , for any n ∈ N . This implies h µ × Z µ ( T , α | e π − ( B Z )) = 0, thus E ∈ P µ × Z µX × X ( T | e π − ( B Z )) = e π − ( B Z )by Theorem 2.5. This finishes the proof. (cid:3)
3. ∆ -weakly mixing set
In this section, we assume that T is a countable torsion-free discrete group. Wewill present some properties of ∆-transitive subsets and ∆-weakly mixing subsets ofa T -system, by partially following the arguments in [16, section 3]. Proposition 3.1.
Let ( X, T ) be a ∆ -weakly mixing T -system, then X is perfectand ( X, T ) is weakly mixing.Proof. Recall that we require | E | ≥ E of ( X, T ). Thus | X | ≥ X, T ) is ∆-weakly mixing. Suppose X is not perfect,there exists an non-empty open set U of X such that U = { u } for some u ∈ X . Nowwe pick T ∈ T \ { θ T } , non-empty open subsets V , V of X , such that V ∩ V = ∅ .Since X is a ∆-transitive subset of ( X , T ) and { ( u, u ) } = U × U is an open subsetof X , one has { T n ( u, u ) : n ∈ N } is dense in X . Hence there exists n ∈ N , suchthat T n u ∈ V and T n u ∈ V , which contradicts with V ∩ V = ∅ . Thus X isperfect.For any non-empty open subsets U , U and V , V of X , we pick distinct T , T ∈T \ { θ T } . Since ( X, T ) is ∆-weakly mixing, there exists x = ( x , x ) ∈ X such that orb + (( x, x ) , T × T ) = X × X . So we can find n ∈ N such that T n x i ∈ U i and T n x i ∈ V i for i = 1 ,
2, this implies T n x ∈ T n T − n V ∩ U = ∅ , T n x ∈ T n T − n V ∩ U = ∅ . Thus ( X, T ) is weakly mixing. This finishes our proof. (cid:3) -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 9 Remark . Similarly, we can obtain that E is perfect if it is a ∆-weakly mixingsubset of a T -system ( X, T ).Let ( X, T ) be a T -system. For any d ∈ N , T , T , . . . , T d ∈ T , non-empty subsets V and U , U , . . . , U d of X , we define N ( V ; U , . . . , U d | T , T , . . . , T d ) := { n ∈ N : V ∩ T − n U ∩ . . . ∩ T − nd U d = ∅} . Lemma 3.3.
Let ( X, T ) be a T -system and E be a closed subset of X with | E | ≥ . Then E is ∆ -transitive if and only for any integer d ≥ , pairwise distinct T , T , . . . , T d ∈ T \ { θ T } , and non-empty open subsets V, U , U , . . . , U d of X inter-secting E , one has N ( V ∩ E ; U , . . . , U d | T , T , . . . , T d ) = ∅ . Proof.
Necessity. If E is a ∆-transitive subset of ( X, T ), then for any d ≥
1, pairwisedistinct T , T , . . . , T d ∈ T \{ θ T } , and non-empty open subsets V, U , U , . . . , U d of X intersecting E , there exists x ∈ E ∩ V such that for the diagonal d -tuple ( x, x, . . . , x )one has orb + (( x, x, . . . , x ) , T × T × · · · × T d ) ⊇ E d . This implies there exists n ∈ N such that T n i x ∈ U i , i = 1 , , . . . , d . Thus n ∈ N ( V ∩ E ; U , . . . , U d | T , T , . . . , T d ) = ∅ . Sufficiency. Let W be a countable topological base of X and U = { U ∈ W : U ∩ E = ∅} . Then U is also a countable set. Since T is a countable group, we can enumerate itas { θ T , T , T , . . . } . Let A := (cid:18) ∞ \ d =1 \ { U ,U ,...,U d }⊂U ∞ [ n =1 ( T − n U ∩ T − n U ∩ . . . ∩ T − nd U d ) (cid:19) ∩ E. Then A is a residue subset of E . For any d ≥
1, pairwise distinct elements T ′ , T ′ , . . . , T ′ d ∈ T \ { θ T } , and non-empty open subsets U ′ , U ′ , . . . , U ′ d of X intersect-ing E . Choose any v ∈ A , U h i ∈ U such that U h i ⊂ U ′ i for i = 1 , , . . . , d , and aninteger L large enough such that { T ′ , T ′ , . . . , T ′ d } ⊂ { T , T , . . . , T L } . Without lossof generality, we can assume T i = T ′ i for i = 1 , , . . . , d . Since v ∈ ∞ [ n =1 ( T − n U h ∩ T − n U h ∩ . . . ∩ T − nd U h d ∩ . . . ∩ T − nL U h L )where U h d +1 , U h d +2 , . . . , U h L are any L − d non-empty open subsets in U , there exists k ∈ N such that T ki v ∈ U h i ⊂ U ′ i for i = 1 , , . . . , d . This implies orb + { ( v, v, .., v ); T ′ × T ′ × . . . × T ′ d } ⊇ E d . Thus E is a ∆-transitive subset of ( X, T ). (cid:3) Using Lemma 3.3 we have the following result.
Proposition 3.4.
Let ( X, T ) be a T -system and E be a closed subset of X with | E | ≥ . Then E is a ∆ -weakly mixing subset of X if and only if for any d ≥ , pair-wise distinct T , T , . . . , T d ∈ T \ { θ T } , and non-empty open subsets U , U , . . . , U d and V , V , . . . , V d of X intersecting E , one has \ s ∈{ , ,...,d } d +1 N ( V s (1) ∩ E ; U s (2) , . . . , U s ( d +1) | T , T , . . . , T d ) = ∅ . Proof.
To prove the sufficiency, we shall to show E n is a ∆-transitive subset of( X n , T ) for any fixed n ∈ N . For any d ≥
1, distinct T , T , . . . , T d ∈ T \ { θ T } ,and non-empty open subsets V i , U ij of X intersecting E for i = 1 , , . . . , n and j =1 , , . . . , d . Let N = nd and choose pairwise distinct T ′ , T ′ , . . . , T ′ N ∈ T \ { θ T } suchthat T i = T ′ i for i = 1 , , . . . , n , since |T | = ∞ . We rewrite { U ij ; i = 1 , , . . . , n ; j =1 , , . . . , d } as { U , U , . . . , U N } , and let V ′ kd + j = V j for k = 0 , , . . . , n − j = 1 , , . . . , d . Then one has \ s ∈{ , ,...,N } N +1 N ( V ′ s (1) ∩ E ; U s (2) , . . . , U s ( N +1) | T ′ , T ′ , . . . , T ′ N ) = ∅ . In particular, there exists L ∈ N such that ( V i ∩ E ) ∩ ( T dj =1 T − Lj U ij ) = ∅ for any i = 1 , , . . . , n . Then E n is a ∆-transitive subset of ( X n , T ) by Lemma 3.3.Necessity. Suppose E is a ∆-weakly mixing subset of ( X, T ). Fix d ≥ L = |{ , , . . . d } d +1 | . Then E L is a ∆-transitive subset of ( X L , T ). We can rewrite { , , . . . , d } d +1 = { s , s , .., s L } . For pairwise distinct T , T , .., T d ∈ T \ { θ T } , andnon-empty open subsets U , U , . . . , U d and V , V , .., V d of X intersecting E , thereexists x k ∈ V s k (1) ∩ E for k = 1 , , . . . L such that L -tuple x = ( x , x , . . . , x L ) ∈ E L satisfies orb + (( x, x, . . . , x ) , T × T × . . . T d ) ⊇ E L × E L × . . . × E L ( d -times) . Thus there exists n ∈ N such that T n i x k ∈ U s k ( i +1) for k = 1 , , . . . , L and i =1 , , . . . , d , which implies that n ∈ \ s ∈{ , ,...,d } d +1 N ( V s (1) ∩ E ; U s (2) , . . . , U s ( d +1) | T , T , . . . , T d ) = ∅ . This ends our proof. (cid:3) Entropy and ∆ -weakly mixing subsets of nilpotent group actions In this section we will introduce the concept and some properties of nilpotentgroup. Finally, we will prove Theorem 1.1 by partially following from the argumentin the proof of Theorem C in [16] .4.1.
Nilpotent group-polynomial.
A group T with the unit θ T is called nilpotent if it has a finite sequence of normal subgroups ( a finite central series ): { θ T } = T ⊂T ⊂ . . . ⊂ T t = T , such that [ T i , T ] ⊂ T i − for i = 1 , , . . . t , where [ T i , T ] denotesthe subgroup generated by { [ T, S ] = T − S − T S : T ∈ T i , S ∈ T } . Any finitely gen-erated nilpotent group is a factor of finitely generated torsion-free nilpotent group, -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 11 thus every representation of a finitely generated nilpotent group can be lifted to arepresentation of a finitely generated torsion-free nilpotent group.If T is a finitely generated nilpotent torsion-free nilpotent group, then there existsa subset { S , S , . . . , S s } of T ( Maccev basis of T ) such that every element T ∈ T can be uniquely represented in the form T = S r ( T )1 S r ( T )2 . . . S r s ( T ) s , where the mapping r : T → Z s : r ( T ) = ( r ( T ) , r ( T ) , . . . , r s ( T )) , such that there exist polynomial mappings φ : Z s +1 → Z s , for any T ∈ T r ( T n ) = φ ( r ( T ) , n ) . The group of T -polynomials P T , is the minimal subgroup of the group T Z ofthe mappings Z → T which contains constant sequences and is closed with re-spect to raising to integral polynomial powers: if g, h ∈ P T , and p is an inte-gral polynomial (taking integer values at the integers), then gh, g p ∈ P T , where gh ( n ) = g ( n ) h ( n ) , g p ( n ) = g ( n ) p ( n ) , n ∈ Z . ThenΦ T : n → T n = S φ ( r ( T ) ,n ) S φ ( r ( T ) ,n ) . . . S φ ( r ( T ) ,n ) s s is T -polynomial, for any T ∈ T .4.2. PET induction.
In [27], the author introduced the weight ω ( g ) of a T -polynomial, then for any system (finite subset of P T ) A the weight vector ω ( A )is defined. The set of weight vectors is well ordered, we say the system A ′ percedes A if ω ( A ) grater than ω ( A ′ ). The PET-induction is an induction on the well or-dered set of systems, that is, if a statement is true for the system { θ T } and one canshow that it holds for a system A from the assumption that it is true for all systemspreceding A , then we can assert this statement holds for all systems.Using PET-induction, Leibman proved the following proposition (see [27, Corol-lary 11.7]). Proposition 4.1.
Let ( X, T ) be a T -system, where T is a finitely generated torsion-free discrete nilpotent group and µ ∈ M ( X, T ) . If π : ( X, B X , µ, T ) → ( Y, B Y , ν, T ) is a weakly mixing extension and µ = R µ y dν is the decomposition of the measure µ over ν , then D − lim n →∞ (cid:26)Z d Y i =1 f i ◦ T ni dµ y − d Y i =1 Z f i ◦ T ni dµ y (cid:27) = 0 in L ( Y ) , (4.1) for any d > , pairwise distinct T , T , . . . , T d ∈ T and f , f , . . . , f d ∈ L ∞ ( µ ) . Here, for a sequence of points { z n } ∞ n =1 in a topological space Z and z ∈ Z , D -lim n → + ∞ z n = z means that { z n } ∞ n =1 converges to z in density, that is, for every neighborhood V of z in Z , z n ∈ V for all n except a set of zero density. It is clearthat if (4.1) holds then for any ε > < δ <
1, the collection of n satisfying ν (cid:18)(cid:26) y ∈ Y : (cid:12)(cid:12)(cid:12)(cid:12)Z d Y i =1 f i ◦ T ni dµ y − d Y i =1 Z f i ◦ T ni dµ y (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27)(cid:19) > − δ. has density one.4.3. Proof of Theorem 1.1.
The following lemma (see [27, Theorem NM’]) andpropositions will be used in the proof of Theorem 1.1.
Lemma 4.2.
Let ( Y, D , ν, T ) be a measure preserving system, where T is a count-able discrete nilpotent group. Then for any d ∈ N , A ∈ D with ν ( A ) > and T , T , . . . , T d ∈ T , one has lim inf N →∞ N N − X n =0 ν (cid:18) d \ i =1 T − ni A (cid:19) > . Proposition 4.3.
Let ( X, T ) be a T -system, where T is a finitely generated torsion-free discrete nilpotent group and µ ∈ M ( X, T ) . Let π : ( X, B X , µ, T ) → ( Y, B Y , ν, T ) be a weakly mixing extension and µ = R µ y dν be the decomposition of the measure µ over ν . For any positive integers k and M , pairwise distinct T , T , . . . , T k ∈T \ { θ T } , if A , A , . . . , A M ∈ B X satisfies that Ω := { y ∈ Y : µ y ( A i ) > , f or all i = 1 , , . . . , M } has positive ν -measure, then we can find L ∈ N and c > such that Ω ′ := (cid:26) y ∈ Y : µ y (cid:18) A s (1) ∩ k \ i =1 T − Li A s ( i +1) (cid:19) > c, for all s ∈ { , , . . . , M } k +1 (cid:27) has positive ν -measure.Proof. For every p ∈ N , letΩ p = n y ∈ Y : µ y ( A i ) > p , for all i = 1 , , . . . , M o . It is clear that Ω = S ∞ p =1 Ω p . As ν (Ω) >
0, there exists some p ∈ N such that ν (Ω p ) >
0. By Lemma 4.2 there exists λ > N →∞ N N − X n =0 ν (cid:18) Ω p ∩ k \ i =1 T − ni Ω p (cid:19) > λ, then E := { n ∈ N : ν (Ω p ∩ T ki =1 T − ni Ω p ) > λ } has positive lower density.Fix 0 < ε < p k +1 and 0 < δ < λM k +1 . For any s ∈ { , , . . . , M } k +1 , let A ns = A s (1) ∩ k \ i =1 T − ni A s ( i +1) -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 13 and F s be the collection of n such that ν (cid:18)(cid:26) y ∈ Y : (cid:12)(cid:12)(cid:12)(cid:12) µ y ( A ns ) − µ y ( A s (1) ) k Y i =1 µ ( T Li y ) ( A s ( i +1) ) (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27)(cid:19) > − δ. (4.2)Then by Proposition 4.1 D ( F s ) = 1 for any s ∈ { , , . . . , M } k +1 . Thus F := T s ∈{ , ,...,M } k +1 F s has density 1 and E ∩ F = ∅ . We can pick L ∈ E ∩ F and let V = (cid:26) y ∈ Y : : (cid:12)(cid:12)(cid:12)(cid:12) µ y ( A Ls ) − µ y ( A s (1) ) k Y i =1 µ ( T Li y ) ( A s ( i +1) ) (cid:12)(cid:12)(cid:12)(cid:12) < ε, for all s ∈ { , , . . . , M } k +1 (cid:27) and H = V ∩ Ω p ∩ T − L Ω p ∩ · · · ∩ T − Lk Ω p . Then by (4.2) and L ∈ E one has µ ( V ) > − M k +1 δ and ν ( H ) > λ − M k +1 δ > . Now we pick 0 < c < p k +1 − ε such that for any y ∈ H , we have µ y ( A Ls ) > µ y ( A s (1) ) k Y i =1 µ ( T Li y ) ( A s ( i +1) ) − ε > p k +1 − ε > c, for any s ∈ { , , . . . , M } k +1 . This finishes our proof. (cid:3) Now let’s prove Theorem 1.1, by partially following from the arguments in theproof of Theorem C in [16]. Proof of the Theorem 1.1.
Since h top ( X, T ) >
0, there exists µ ∈ M e ( X, T ) suchthat h µ ( X, T ) >
0. Let π : ( X, B X , µ, T ) → ( Z, B Z , ν, T ) be the factor map to thePinsker factor of ( X, B X , µ, T ). By Proposition 2.6, we know π is a weakly mixingextension. Let µ = R µ z dν be the decomposition of the measure µ over ν . Since T is countable, we can enumerate T as { θ T , T , T , . . . } .Let λ = µ × Z µ . Then by Proposition 2.4, e π := π ◦ π : ( X × X, B X × B X , λ, T ) → ( Z, B Z , ν, T ) is also a weakly mixing extension, where π : X × X → X is theprojection to the first coordinate. Moreover, λ (∆ X ) = 0, where ∆ X = { ( x, x ) : x ∈ X } (see e.g. [18, Lemma 4.3]). Then we can pick ( x , x ) ∈ supp( λ ) \ △ X , andchoose disjoint closed neighborhood W i of x i such that diam( W i ) < for i = 1 , λ ( W × W ) = Z µ z ( W ) µ z ( W ) dν ( z ) > . Let Ω = { z ∈ Z : µ z ( W i ) > i = 1 , } . Then ν (Ω) >
0. We can find c > := { z ∈ Z : µ z ( W i ) > c for i = 1 , } has positive ν -measure. Nowwe denote E = { , } , E = E × E , . . . , E k = E k − × E k − × . . . × E k − ( k -times)for any k ≥
3. Let A i = W i for i ∈ E . Then by induction and Proposition 4.3 wecan construct a non-empty closed subset A σ of X for each σ ∈ E k , k ∈ N with thefollowing properties: (1) for any k >
1, there exists n k ∈ N , and a non-empty closed subset A σ of X for any σ = ( σ (1) , σ (2) , . . . , σ ( k )) ∈ E k , where σ ( i ) ∈ E k − , i = 1 , , . . . , k ,such that A σ ⊂ A σ (1) ∩ T − n k A σ (2) ∩ . . . ∩ T − n k k − A σ ( k ) ;(2) diam( A σ ) < − k , for all σ ∈ E k , k ∈ N ;(3) for any k ∈ N , there exists c k > { z ∈ Z : µ z ( A σ ) > c k , for all σ ∈ E k } has positive ν -measure.Let A = T ∞ k =1 S σ ∈ E k A σ . Now we shall show that A is a ∆-weakly mixing subsetof ( X, T ). Note that for any given k ∈ N , { A σ : σ ∈ E k } are pairwise disjointbecause of property (1). Thus A is a Cantor set. For any d ≥
1, pairwise dis-tinct T ′ , T ′ , . . . , T ′ d ∈ T \ { θ T } , and non-empty open subsets U , U , . . . , U d and V , V , . . . , V d of X intersecting A , by Proposition 3.4 it is suffice to show that d \ j =1 \ s ∈{ , ,...,d } d N ( V j ∩ A ; U s (1) , . . . , U s ( d ) | T , T , . . . , T d ) = ∅ . (4.3)We pick an integer L large enough such that { T ′ , T ′ , . . . , T ′ d } ⊂ { T , T , . . . , T L } and there exists pairwise distinct σ , σ , . . . , σ d , σ ′ , σ ′ , . . . , σ d ′ in E L , such that A σ i ⊆ U i and A σ i ′ ⊆ V i , for i = 1 , , . . . , d . Without loss of generality, we canassume T ′ i = T i for i = 1 , , . . . , d . Then there exists n L +1 ∈ N such that A σ ⊂ A σ (1) ∩ T − n L +1 A σ (2) ∩ . . . ∩ T − n L +1 L A σ ( L +1) = ∅ , for any σ = { σ (1) , σ (2) , . . . , σ ( L + 1) } ∈ E L +1 . In particularly, for any j = 1 , , . . . d and s ∈ { , , . . . , d } d , let σ js = { σ j ′ , σ s (1) , . . . , σ s ( d ) , η , . . . , η L − d } ∈ E L +1 , where η , η , . . . , η L − d is any L − d elements of E L . Then A σ js ⊂ A σ j ′ ∩ T − n L +1 A σ s (1) ∩ . . . ∩ T − n L +1 d A σ s ( d ) . Since A σ js ∩ A = ∅ , there exists v js ∈ A ∩ A σ j ′ ∩ T − n L +1 A σ s (1) ∩ . . . ∩ T − n L +1 d A σ s ( d ) . Thus v js ∈ A ∩ V j ∩ T − n L +1 U s (1) ∩ . . . ∩ T − n L +1 d U s ( d ) , for any j = 1 , , . . . , d and s ∈ { , , . . . , d } d . Thus (4.3) holds, which ends the proof. (cid:3) Remark . Furstenberg introduced the following example (see [10, P.40]). Let X = {− , } Z , T be the shift map: T ω ( n ) = ω ( n + 1) and R : X → X be definedby: Rω ( n ) = ( ω ( n ) , f or n = 0 − ω ( n ) , f or n = 0 . It is clear that R = id X . Let S = RT R . Then S n = RT n R for any n ∈ N .The group G generated by T and R is a solvable group. For any ω ∈ X , let U ω = { x ∈ X : x (0) = ω (0) } . Then T n ω ∈ U ω if and only if ω ( n ) = ω (0), and -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 15 S n ω ∈ U ω if and only if ω ( n ) = − ω (0). Hence ( ω, ω ) / ∈ orb + (( ω, ω ) , T × S ). Thusfor any closed subset E of X with | E | ≥
2, any ω ∈ E , we have orb + (( ω, ω ) , T × S ) + E . This implies that there is no ∆-transitive subsets in (
X, G ). For this group G ,there exists a finitely generated torsion-free discrete solvable group F with the unit e F such that there is a surjective homomorphism π : F → G . Let ( X, F ) be the F -system, where the group actions is defined as f ( ω ) = π ( f )( ω ) , for f ∈ F and ω ∈ X. Taking T F and S F ∈ F such that π ( T F ) = T and π ( S F ) = S , one has ( ω, ω ) / ∈ orb + (( ω, ω ) , T F × S F ) for any ω ∈ X . Thus there is no ∆-transitive subset in ( X, F ).From Remark 4.4, we can obtain the following proposition.
Proposition 4.5.
There exists a F -system ( Z, F ) , where F is a finitely generatedtorsion-free discrete solvable group such that h top ( Z, F ) > but there are no ∆ -transitive subsets in ( Z, F ) .Proof. Let (
X, F ) be the F -system as in the Remark 4.4, and Y = { , } F . F actson Y as ( gy )( h ) = y ( hg − ) for any g, h ∈ F and y ∈ Y . Now we consider theproduct F -system ( X × Y, F ). We will show that h top ( X × Y, F ) > X × Y, F ).Let U = { [0] , [1] } , where [ i ] = { y ∈ Y | y ( e F ) = i } for i = 0 ,
1. Then g − U = { [0] g , [1] g } , where [ i ] g = { y ∈ Y | y ( g ) = i } for i = 0 ,
1. Let { F n } ∞ n =1 be a Følnersequence of F . Then h top ( F, U ) = lim n →∞ H ( U F n ) | F n | = lim n →∞ log 2 | F n | | F n | = log 2 . Thus h top ( X × Y, F ) ≥ h top ( Y, F ) ≥ log 2 > E of ( X × Y, F ), then there exists ( ω, y ) ∈ E such that orb + ((( ω, y ) , ( ω, y )) , T F × S F ) ⊇ E . In particularly, ( ω, ω ) ∈ orb + (( ω, ω ) , T F × S F ), which contradicts the Remark 4.4.Thus F -system ( Z, F ) := ( X × Y, F ) has no ∆-transitive subsets. (cid:3) proof of theorem 1.2 In this section, firstly we present some properties of a ∆-weakly mixing subset ina T -system ( X, T ), and then we will prove Theorem 1.2 and Corollary 1.3. Thesearguments in this section partially follows from the proof of the Theorem A in [16].Let ( X, T ) be a T -system, where T is a countable torsion-free discrete group and E is a closed subset of X . For any ε > d ≥ T , T , . . . T d ∈T \ { θ T } , we say that a subset A of X is ( ε, T , T , . . . , T d ) -spread in E if there are0 < δ < ε , m ∈ N and pairwise distinct z , z , . . . z m ∈ X such that A ⊂ S mi =1 B ( z i , δ ) and for any maps g j : { z , z , . . . , z m } → E for j = 1 , , . . . d , there exists an integer L > ε such that T Lj B ( z i , δ ) ⊂ B ( g j ( z i ) , ε )for any i = 1 , , . . . , m and j = 1 , , . . . , d . Denote by H ( ε, T , T , . . . , T d ; E ) thecollection of all closed subsets of X that are ( ε, T , T , . . . , T d )-spread in E . Put H ( E ) = + ∞ \ d =1 \ { T ,T ,...,T d }⊂T \{ θ T } + ∞ \ k =1 H ( k , T , T , . . . , T d ; E ) . It is clear that H ( E ) is a hereditary subset of 2 X . Proposition 5.1.
Let ( X, T ) be a T -system, where T is a countable torsion-freediscrete group, and E be a perfect subset of X . If there exists an increasing sequenceof subsets C ⊂ C ⊂ . . . in H ( E ) ∩ E such that C = S ∞ i =1 C i is dense in E , then E is a ∆ -weakly mixing subset of X .Proof. For any d ≥
1, pairwise distinct T , T , . . . , T d ∈ T \ { θ T } , and non-emptyopen subsets U , U , . . . , U d , V , V , . . . , V d of X intersecting E , by Proposition 3.4,it is sufficient to show that d \ i =1 M \ k =1 N ( V i ∩ E ; U s k (1) , U s k (2) , . . . , U s k ( d ) ) = ∅ , (5.1)where M = |{ , , . . . , d } d | and enumerate { , , . . . , d } d as { s , s , . . . , s M } .Since U i ∩ E = ∅ , there exist u i ∈ U i ∩ E and ε > B ( u i , ε ) ⊂ U i for i = 1 , , . . . , d . Since V i ∩ E = ∅ for i = 1 , , . . . , d , E is perfect and C is densein E , we can pick v i , v i , . . . , v iM ∈ V i ∩ C for i = 1 , , . . . , d such that v ik = v i ′ k ′ whenever ( i, k ) = ( i ′ , k ′ ) ∈ { , , . . . , d } × { , , . . . , M } . Then we take an integer K large enough such that v ik ∈ C K for all i = 1 , , . . . , d and k = 1 , , . . . , M .Let a = min { ρ ( v ik , v i ′ k ′ ) : ( i, k ) = ( i ′ , k ′ ) , ≤ i, i ′ ≤ d, ≤ k, k ′ ≤ M } . Then a >
0. Take 0 < ε < min { a , ε } . Since C K ∈ H ( E ), there exist 0 < δ < ε , m ∈ N and pairwise distinct z , z , . . . , z m ∈ X such that C K ⊂ S mi =1 B ( z i , δ ) andfor any maps g j : { z , z , . . . , z m } → E , j = 1 , , . . . , d , there exists an integer L > ε such that T Lj B ( z i , δ ) ⊆ B ( g j ( z i ) , ε ) for any i = 1 , , . . . , m and j = 1 , , . . . , d .Now we can pick n ik ∈ { , , . . . , m } for i = 1 , , . . . , d and k = 1 , , . . . , M ,such that v ik ∈ B ( z n ik , δ ). Since δ < a , one has z n ik = z n i ′ k ′ whenever ( i, k ) =( i ′ , k ′ ) ∈ { , , . . . , d } × { , , . . . , M } . Thus M d ≤ m . For j = 1 , , . . . , d , we define h j : { z , z , . . . , z m } → E as h j ( z p ) = ( u s k ( j ) , if there exists z n ik such that z p = z n ik ,u d , others . Then there exists L ∗ ∈ N such that T L ∗ j B ( z p , δ ) ⊆ B ( h j ( z p ) , ε ) for any j =1 , , . . . , d and p = 1 , , . . . , m . In particularly, for any i = 1 , , . . . , d and k =1 , , . . . , M , T L ∗ j B ( z n ik , δ ) ⊂ B ( u s k ( j ) , ε ) ⊂ U s k ( j ) , -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 17 and so v ik ∈ T − L ∗ j U s k ( j ) . Thus for any 1 ≤ k ≤ M and 1 ≤ i ≤ dL ∗ ∈ N ( V i ∩ E ; U s k (1) , U s k (2) , . . . , U s k ( d ) | T , T , . . . , T d ) = ∅ , that is (5.1) holds. This ends the proof. (cid:3) Theorem 5.2.
Let ( X, T ) be a T -system, where T is a countable torsion-free dis-crete group. If E is a closed subset of X with | E | ≥ , then E is ∆ -weakly mixingif and only if E is perfect and E ∩ H ( E ) is a residue subset of E .Proof. Sufficiency. If E is perfect and 2 E T H ( E ) is a residue subset of 2 E , then wecan immediately obtain that E is a ∆-weakly mixing subset of ( X, T ) by Lemma2.1 and Proposition 5.1, since 2 E T H ( E ) is also a hereditary subset of 2 E .Necessity. Suppose E is a ∆-weakly mixing subset of ( X, T ). By Remark 3.2, E is perfect. To show that H ( E ) ∩ E is a residue subset of 2 E , it is suffice to showthat for any given ε > d ≥ T , T , . . . , T d ∈ T \ { θ T } , onehas H ( ε, T , T , . . . , T d ; E ) ∩ E is a dense open subset of 2 E .For any A ∈ H ( ε, T , T , . . . , T d ; E ), by the definition there exist δ ∈ (0 , ε ), m ∈ N ,and pairwise distinct points z , z , . . . , z m ∈ X such that A ∈ h B ( z , δ ) , B ( z , δ ) , . . . , B ( z m , δ ) i ⊆ H ( ε, T , T , . . . , T d ; E ) . Thus H ( ε, T , T , . . . , T d ; E ) ∩ E is an open subset of 2 E .Now we shall show that for any fixed n ∈ N and non-empty open subsets U ,U , . . . , U n of X intersecting E , h U , U , . . . , U n i ∩ H ( ε, T , T , . . . , T d ; E ) ∩ E = ∅ . (5.2)This implies H ( E ) ∩ E is dense in 2 E .Since E is a ∆-weakly mixing subset of ( X, T ) and U i ∩ E = ∅ for i = 1 , , . . . , n ,there exists u i ∈ U i ∩ E for i = 1 , , . . . , n such that the orbit closure of d -tuple( u, u, . . . , u ) under the action T × T × . . . × T d contains E n × E n × . . . × E n ( d -times), that is orb + (( u, u, . . . , u ) , T × T × . . . × T d ) ⊇ E n × E n × . . . × E n ( d -times) , where u = ( u , u , . . . , u n ). Since X is compact and E is a closed subset of X ,there exists m ∈ N and z , z , . . . , z m ∈ X such that E ⊂ S m i =1 B ( z i , ε ) and B ( z i , ε ) ∩ E = ∅ for any 1 ≤ i ≤ m . We can arrange the d -tuple on the set { , , . . . , m } n as the finite sequence { α , α , . . . , α L } , where α k = ( α k , α k , . . . , α kd )and α kj ∈ { , , . . . , m } n for k = 1 , , . . . , L and j = 1 , , . . . , d . For α , there exists n ∈ N such that T n j ( u i ) ∈ B ( z α j ( i ) , ε ) for i = 1 , , . . . , n and j = 1 , , . . . , d .Moreover, since T , T , . . . , T d are continuous, we can find a neighborhood W i of u i such that W i ⊂ U i and T n j ( W i ) ⊂ B ( z α j ( i ) , ε ) , for any i = 1 , , . . . , n and j = 1 , , . . . , d . Then replacing U i by W i for i =1 , , . . . , n , we can obtain n ∈ N and a neighborhood W i of u i such that W i ⊂ W i and T n j ( W i ) ⊂ B ( z α j ( i ) , ε ) for i = 1 , , . . . , n and j = 1 , , . . . , d . We continue inductively obtaining positive integers n , n . . . , n L and non-empty open subsets W ki of X intersecting E such that W Li ⊂ W L − i ⊂ . . . ⊂ W i ⊂ U i , T n k j ( W ki ) ⊂ B ( z α kj ( i ) , ε , (5.3)for i = 1 , , . . . , n , j = 1 , , . . . , d and k = 1 , , . . . , L .Now we take ω i ∈ W Li ∩ E , and 0 < δ < ε such that B ( ω i , δ ) ⊂ W Li for any i =1 , , . . . , n . Let W = S ni =1 B ( ω i , δ ) ⊂ S ni =1 B ( ω i , δ ). Then W ∈ h U , U , . . . , U n i .For any maps g j : { ω , ω , . . . , ω n } → E with j = 1 , , . . . , d , there exists 1 ≤ h ≤ L such that g j ( ω i ) ∈ B ( z α hj ( i ) , ε ) , (5.4)for any i = 1 , , . . . , n and j = 1 , , . . . , d . Combining (5.3) and (5.4), one has T n h j B ( ω i , δ ) ⊂ B ( z α hj ( i ) , ε ) ⊂ B ( g j ( ω i ) , ε ) , for i = 1 , , . . . , n and j = 1 , , . . . , d . Thus W ∩ E ∈ H ( ε, T , T , . . . , T d ; E ) ∩ E ,and (5.2) holds. This finishes the proof. (cid:3) Now let us prove Theorem 1.2.
Proof of Theorem 1.2.
Let E be a closed subset of X with | E | ≥
2. Suppose E is a∆-weakly mixing subset of ( X, T ). Then E is perfect and H ( E ) ∩ E is a residuesubset of 2 E by Theorem 5.2. Since H ( E ) ∩ E is also a hereditary subset of 2 E ,there exists an increasing sequence of Cantor sets C ⊂ C ⊂ . . . ⊂ E such that C i ∈ H ( E ) ∩ E and C = S ∞ i =1 C i is dense in E by Lemma 2.1.Let A be a subset of C , d ≥
1, pairwise distinct T , T , . . . T d ∈ T \ { θ T } , and g j : A → E be continuous maps for j = 1 , , . . . , d . For any k ∈ N , let A k = C k ∩ A .Then the closure A k of A k is also in H ( E ), since H ( E ) is hereditary.By the definition of H ( E ), there exists 0 < δ k < k , m k ∈ N , and z k , z k , . . . , z km k ∈ X , such that A k ⊂ m k [ i =1 B ( z ki , δ k )with A k ∩ B ( z ki , δ k ) = ∅ , i = 1 , , . . . , m k , and for any maps h j : { z k , z k , . . . , z km k } → E , j = 1 , , . . . , d , there exists L > k such that T Lj B ( z ki , δ k ) ⊆ B ( h j ( z ki ) , k ), for any i = 1 , , . . . , m k and j = 1 , , . . . , d .For any i = 1 , , . . . , m k , there exists u ki ∈ A k ∩ B ( z ki , δ k ). Now we define e g j : { z k , z k , . . . , z km k } → E as e g j ( z ki ) = g j ( u ki ), for any j = 1 , , . . . , d and i = 1 , , . . . , m k .Then there exists q k > k such that T q k j B ( z ki , δ k ) ⊂ B ( e g ( z ki ) , k ), for any j = 1 , , . . . , d and i = 1 , , . . . , m k . We show that the sequence { q k } is as required.For any x ∈ A , there exists K ∈ N such that for any k > K x ∈ A k . For any k > K , there exists z ki x ∈ { z k , z k , . . . , z km k } such that x ∈ B ( z ki x , δ k ). Then for any -WEAKLY MIXING SUBSET IN POSITIVE ENTROPY ACTIONS 19 j = 1 , , . . . , d we havelim k → + ∞ ρ ( T q k j x, g j ( x )) ≤ lim k → + ∞ (cid:0) ρ ( T q k j x, e g j ( z ki x )) + ρ ( e g j ( z ki x ) , g j ( x )) (cid:1) ≤ lim k → + ∞ k + lim k → + ∞ ρ ( g j ( u ki x ) , g j ( x ))= lim k → + ∞ ρ ( g j ( u ki x ) , g j ( x )) . Since ρ ( x, u ki x ) < k for any k > K , and g j is continuous for j = 1 , , . . . , d ,lim k → + ∞ ρ ( g j ( u ki x ) , g j ( x )) = 0 . Thus lim k → + ∞ T q k j x = g j ( x ) for any x ∈ A k , which ends the proof of necessity.Sufficiency. For any d ≥
1, pairwise distinct T , T , . . . , T d ∈ T \ { θ T } , non-emptyopen subsets U , U , . . . , U d and V , V , . . . , V d of X intersecting E , by Proposition3.4 we need to show that \ s ∈{ , ,...,d } d +1 N ( V s (1) ∩ E ; U s (2) , . . . , U s ( d +1) ) = ∅ . (5.5)Let M = |{ , , . . . , d } d | and enumerate { , , . . . , d } d as { s , s , . . . , s M } . Since U i ∩ E = ∅ , there exists u i ∈ U i ∩ E and ε > B ( u i , ε ) ⊂ U i for any i = 1 , , . . . , d . Since V i ∩ E = ∅ for i = 1 , , . . . , d , E is perfect, and C is dense in E , we can pick v i , v i , . . . , v iM ∈ ( V i ∩ E ) ∩ C for i = 1 , , . . . , d such that v il = v i ′ l ′ whenever ( i, l ) = ( i ′ , l ′ ) ∈ { , , . . . , d } × { , , . . . , M } .Let A := { v il : 1 ≤ i ≤ d, ≤ l ≤ M } . Then A is a subset of C . We define g j : A → E as g j ( v il ) = u s l ( j ) , for any i, j = 1 , , . . . , d and l = 1 , , . . . , M . Thenthere exists an increasing sequence { q k } + ∞ k =1 of positive integers such thatlim k →∞ T q k j v il = g j ( v il ) = u s l ( j ) ∈ U s l ( j ) , for any i, j = 1 , , . . . , d and l = 1 , , . . . , M . Thus we can pick k ∈ N large enoughsuch that v il ∈ T − q k j U s l ( j ) for any 1 ≤ i, j ≤ d and 1 ≤ l ≤ M that is (5.5) holds.This ends the proof. (cid:3) Now we will prove Corollary 1.3.
Proof of Corollary 1.3.
Since h top ( X, T ) >
0, there exists a ∆-weakly mixing subset E of ( X, T ) by Theorem 1.1. Then E is perfect by Remark 3.2, and by Theorem1.2 there exists increasing sequence of Cantor subsets C ⊂ C ⊂ . . . of E such that C = S ∞ i =1 C i is dense in E and satisfies the property in Theorem 1.2. Since | E | ≥ e , e ∈ E , and let δ = ρ ( e , e ) > x, y ∈ C and T , T ∈ T \ { θ T } , there are two cases.Case 1: T = T = T for some T ∈ T \ { θ T } . Let g : { x, y } → E with g ( x ) = g ( y ) = e , and g ′ : { x, y } → E with g ′ ( x ) = e , g ′ ( y ) = e . Then by the property inTheorem 1.2, there exist two increasing sequences { p k } + ∞ k =1 and { p ′ k } + ∞ k =1 of positiveintegers such that lim k →∞ ρ ( T p k x, T p k y ) = ρ ( g ( x ) , g ( y )) = 0 , lim k →∞ ρ ( T p ′ k x, T p ′ k y ) = ρ ( g ′ ( x ) , g ′ ( y )) = δ. Case 2: T = T . Let g : { x, y } → E with g ( x ) = g ( y ) = e , and g : { x, y } → E with g ( x ) = g ( y ) = e . Then by the property in Theorem 1.2, there exists anincreasing sequence { q k } + ∞ k =1 of positive integers such thatlim k →∞ ρ ( T q k x, T q k y ) = ρ ( g ( x ) , g ( y )) = 0 . Next let g ′ : { x, y } → E with g ′ ( x ) = g ′ ( y ) = e , and g ′ : { x, y } → E with g ′ ( x ) = g ′ ( y ) = e . Then by the property in Theorem 1.2, there exists an increasingsequence { q ′ k } + ∞ k =1 of positive integers such thatlim k →∞ ρ ( T q ′ k x, T q ′ k y ) = ρ ( g ′ ( x ) , g ′ ( y )) = δ. Summing up, one has lim inf n → + ∞ ρ ( T n x, T n y ) = 0 and lim sup n → + ∞ ρ ( T n x, T y ) ≥ δ for any x = y ∈ C and T , T ∈ T \ { θ T } . Thus ( X, T ) is asynchronouschaotic. (cid:3) References
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K. Liu: Department of Mathematics, University of Science and Technology ofChina, Hefei, Anhui, 230026, P.R. China
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