Weakly mixing, proximal topological models for ergodic systems and applications
aa r X i v : . [ m a t h . D S ] J u l WEAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS FORERGODIC SYSTEMS AND APPLICATIONS
ZHENGXING LIAN, SONG SHAO, AND XIANGDONG YE
Abstract.
In this paper it is shown that every non-periodic ergodic system hastwo topologically weakly mixing, fully supported models: one is non-minimal buthas a dense set of minimal points; and the other one is proximal. Also for indepen-dent interests, for a given Kakutani-Rokhlin tower with relatively prime columnheights, it is demonstrated how to get a new taller Kakutani-Rokhlin tower withsame property, which can be used in Weiss’s proof of the Jewett-Krieger’s theoremand the proofs of our theorems. Applications of the results are given. Introduction
A measurable system is a quadruple ( X, X , µ, T ), where ( X, X , µ ) is a Lebesgueprobability space and T : X → X is an invertible measure preserving transformation.A topological dynamical system is a pair ( X, T ), where X is a compact metric spaceand T : X → X is a homeomorphism.Let ( X, X , µ, T ) be an ergodic dynamical system. We say that ( ˆ X, ˆ X , ˆ µ, ˆ T ) is a topological model (or just a model ) for ( X, X , µ, T ) if ( ˆ X, ˆ T ) is a topological system,ˆ µ is an invariant Borel probability measure on ˆ X and the systems ( X, X , µ, T ) and( ˆ X, ˆ X , ˆ µ, ˆ T ) are measure theoretically isomorphic.The theory of topological models is an important part in dynamical systems andhas many applications. The well known Jewett-Krieger’s theorem asserts that everynon-periodic ergodic system has a topological model which is strictly ergodic. Lehrer[9] showed that we can further require the model to be topologically (strongly)mixing. We refer to [2, 3, 13, 15] for surveys and nice results on this topics. Wenote that topological models can also be used to obtain the pointwise convergenceof non-conventional ergodic averages, [6].We mention that the models obtained above are minimal. In this paper we studynon-minimal models for a given ergodic system, and obtain their applications. Hereare our main results of this paper. Note that an ergodic system is non-periodic if ithas no atom. Theorem 1.1. (1)
Every non-periodic ergodic system has a topological modelwhich is a non-minimal topologically weakly mixing system with a full supportand a dense set of minimal points.
Date : July 05, 2014.2000
Mathematics Subject Classification.
Primary: 37B05, 37A05.
Key words and phrases. topological model, weakly mixing, proximal, minimal point.Authors are supported by NNSF of China (11171320, 11371339). (2)
Every non-periodic ergodic system has a topological model which is a topo-logically weakly mixing system with a full support and a unique fixed point asits only minimal point.
Note that a topological system (
X, T ) with a unique fixed point as its only minimalpoint is proximal, i.e. for all x, y ∈ X , inf n d ( T n x, T n y ) = 0. Hence Theorem 1.1(2)means that every non-periodic ergodic system has a topological weakly mixing andproximal model with a full support.In Weiss’s new proof of the Jewett-Krieger’s theorem [12, 15] and Weiss’ theoremon the doubly minimal model [14], a technical complement should be discussed whenthe column heights of the Kakutani-Rokhlin tower are not relatively prime. In thispaper, we found that one can avoid this and thus simplify the proofs by using atechnical lemma, i.e. Lemma 3.2. This lemma will be used in the proofs of ourtheorems and we believe that it will be useful in other settings.We find two applications of our results. One gives an affirmative answer to aquestion in [10] by showing that if ( X, T ) is a topological system and ( M ( X ) , T M ) isthe induced system on the probability space, then the density of minimal points of( M ( X ) , T M ) does not implies ( X, T ) has the same property. The other one concernsthe existence of a proximal topological K-system which was constructed in [5]. Weobtain a lot of such examples simply using the proximal topological models of anymeasurable K-systems.
Acknowledgments:
We would like to thank Wen Huang for very useful sugges-tions. 2.
Preliminaries
In this section we recall some notions which we will use in the following sections.2.1.
A measurable system.
A measurable system is ergodic if all T -invariant setshave measures either 0 or 1. For an ergodic system, either the space X consists of afinite set of points on which µ is equidistributed, or the measure µ is atom-less. Inthe first case the system is called periodic , and it is called non-periodic in the latter.A homomorphism from ( X, X , µ, T ) to a system ( Y, Y , ν, S ) is a measurable map π : X → Y , where X is a T -invariant subset of X and Y is an S -invariant subsetof Y , both of full measure, such that π ∗ µ = µ ◦ π − = ν and S ◦ π ( x ) = π ◦ T ( x ) for x ∈ X . When we have such a homomorphism we say that the system ( Y, Y , ν, S ) isa factor of the system ( X, X , µ, T ). If the factor map π : X → Y can be chosen tobe bijective and π − is also measurable, then we say that the systems ( X, X , µ, T )and ( Y, Y , ν, S ) are (measure theoretically) isomorphic .2.2. A topological system.
A topological system (
X, T ) is transitive if for anynon-empty open sets
U, V there is some n ∈ Z + such that U ∩ T − n V = ∅ . When X has no isolated points, ( X, T ) is transitive if and only if there exists some point x ∈ X whose orbit O ( x, T ) = { T n x : n ∈ Z + } is dense in X and we call such apoint a transitive point . The system is minimal if the orbit of any point is dense in EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 3 X . A point x ∈ X is called a minimal point if ( O ( x, T ) , T ) is minimal. ( X, T ) is (topologically) weakly mixing if the product system ( X × X, T × T ) is transitive.A factor of a topological system ( X, T ) is another topological system (
Y, S ) suchthat there exists a continuous and onto map φ : X → Y satisfying S ◦ φ = φ ◦ T . Inthis case, ( X, T ) is called an extension of (
Y, S ). The map φ is called a factor map .2.3. Rokhlin tower.
We need some basic knowledge related to Kakutani-Rokhlintowers. We will use notations from [2, 3, 15].Let ( X, X , µ, T ) be a dynamical system. Let B ∈ X . An array c = { B, T B, . . . , T N − B } with { T j B } N − j =0 pairwise disjoint is called a Rokhlin tower or a column over B ofheight N . The set B is called the base of the tower, and T N − B is its roof . Let | c | = S N − j =0 T j B the carrier of c . A collection t of disjoint columns c k (with bases B k and heights N k ) is called a tower and let | t | = S k | c | . The union of the bases B = S k B k is the base of t , and the union of the roofs is the roof of t . The sets { T i x : 0 ≤ i < N k } for x ∈ B k are called the fibers of t .Here is the well known Rokhlin’s Lemma. Theorem 2.1 (Rokhlin’s Lemma) . Let ( X, X , µ, T ) be an ergodic system. Givenan ǫ > and a natural number N , there exists a Rokhlin tower c of height N withbase B ∈ X such that µ ( | c | ) > − ǫ . Refining a tower according to a partition.
Let t be a tower with columns { c k : k ∈ K } ( K is finite or countable) and base B = S k ∈ K B k . Given a partition(finite or countable) α , we define an equivalence relation on B as follows: x ∼ y iff x and y are in the same base B k and for every 0 ≤ j < N k , T j x and T j y are in the same elements of α , i.e. x and y have the same ( α, N k )-name. Now weconsider each equivalence class B k, a , with a an ( α, N k )-name, as a base of the column c k, a = { B k, a , T B k, a , . . . , T N k − B k, a } and say that the resulting tower t α = { c k, a : a ∈ α N k , k ∈ K } is the tower t refined according to α .2.5. Kakutani-Rokhlin tower.
For an ergodic system ( X, X , µ, T ), let B ∈ X with positive measure, then it is clear that S n ≥ T n B = X (mod µ ). Define the return time function r B : B → N ∪ {∞} by r B ( x ) = min { n ≥ T n x ∈ B } when this minimum is finite and r B ( x ) = ∞ otherwise. Let B k = { x ∈ B : r B ( x ) = k } and note that by Poincar´e’s recurrence theorem B ∞ is a null set. Let c k be thecolumn { B k , T B k ..., T k − B k } and we call the tower t = t ( B ) = { c k : k = 1 , ... } the Kakutani tower over B . If the Kakutani tower over B has finitely many columns(i.e. the function r B is bounded) we say that B has a finite height and we call theKakutani tower over B a Kakutani-Rokhlin tower or a K-R tower . The numbermax r B is called the height of B or the height of K-R tower. ZHENGXING LIAN, SONG SHAO, AND XIANGDONG YE
Symbolic dynamics.
Let S be a finite alphabet with m symbols, m ≥
2. Weusually suppose that S = { , , · · · , m − } . Let Ω = S Z be the set of all sequences x = . . . x − x x . . . = ( x i ) i ∈ Z , x i ∈ S , i ∈ Z , with the product topology. A metriccompatible is given by d ( x, y ) = k , where k = min {| n | : x n = y n } , x, y ∈ Ω. Theshift map σ : Ω −→ Ω is defined by ( σx ) n = x n +1 for all n ∈ Z . The pair (Ω , σ ) iscalled a shift dynamical system . Any subsystem of (Ω , σ ) is called a subshift system .Similarly we can replace Z by Z + = { , , , . . . } , and σ will be not a homeomorphismbut a surjective map.Each element of S ∗ = S k ≥ S k is called a word or a block (over S ). We use | A | = n to denote the length of A if A = a . . . a n . If ω = ( · · · ω − ω ω · · · ) ∈ Ω and a ≤ b ∈ Z , then ω [ a, b ] =: ω a ω a +1 · · · ω b is a ( b − a + 1)-word occurring in ω startingat place a and ending at place b . Similarly we define A [ a, b ] when A is a word. Aword A appears in the word B if there are some a ≤ b such that B [ a, b ] = A .For n ∈ N and words A , . . . , A n , we denote by A . . . A n the concatenation of A , . . . , A n . When A = . . . = A n = A denote A . . . A n by A n . If ( X, σ ) is asubshift system, let [ i ] = [ i ] X = { x ∈ X : x (0) = i } for i ∈ S , and [ A ] = [ A ] X = { x ∈ X : x x · · · x ( | A |− = A } for any word A .2.7. Partitions.
Let ( X, X , µ, T ) be a measurable system. A partition α of X is afamily of disjoint measurable subsets of X whose union is X . Let α and β be twopartitions of ( X, X , µ, T ). One says that α refines β , denoted by α ≻ β or β ≺ α , ifeach element of β is a union of elements of α . α ≻ β is equivalent to σ ( β ) ⊆ σ ( α ),where σ ( A ) is the σ algebra generated by the family A .Let α and β be two partitions. Their join is the partition α ∨ β = { A ∩ B : A ∈ α, B ∈ β } and extend this definition naturally to a finite number of partitions. For m ≤ n , define α nm = n _ i = m T − i α = T − m α ∨ T − m +1 α ∨ . . . ∨ T − n α, where T − i α = { T − i A : A ∈ α } .2.8. Symbolic representation.
Let ( X, X , µ, T ) be an ergodic system and α = { A j } ≤ j ≤ l a finite partition (we usually assume µ ( A j ) > j ). We sometimesthink of the partition α as a function ξ : X → Σ = { , , . . . , l } defined by ξ ( x ) = j for x ∈ A j . The pair ( X, α ) is traditionally called a process . Let Ω = Ω( l ) = { , , . . . , l } Z and let S be the shift. One can define a homomorphism φ α from X toΩ, given by φ α ( x ) = ω ∈ Ω, where ω n = ξ n ( x ) = ξ ( T n x ) . We denote the distribution of the stochastic process, ( φ α ) ∗ ( µ ), by ρ = ρ ( X, α ) andcall it the symbolic representation measure of (
X, α ). Let X α = supp( φ α ) ∗ µ = supp ρ. Then we get a homomorphism φ α : ( X, X , µ, T ) → ( X α , X α , ρ, S ). This homomor-phism is called the symbolic representation of the process ( X, α ). This will not be
EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 5 a model for ( X, X , µ, T ) unless W ∞ i = −∞ T − i α = X modulo null sets, but in any casethis does give a model for a non-trivial factor of X .2.9. Copying names.
An important way to produce partitions is by copying orpainting names on towers. If c = { T j B } N − j =0 is a column and a ∈ Σ N then copyingthe name a on the column c means that on | c | = S N − j =0 T j B we define a partition(may not be on the whole space) by letting A k = [ { T j B : a j = k } , k ∈ Σ = { , , . . . , l } . If there is a tower t with q columns c i = { T j B i } N i − j =0 , and q names a ( i ) ∈ Σ N i , i =1 , . . . , q , then copying these names on t means we copy each name a ( i ) on column c i , i.e. we define a partition on | t | by A k = [ { T j B i : a ( i ) j = k, i = 1 , . . . , q } , k ∈ Σ . These partitions can be extended to a partition α = { A , . . . , A l } of the whole spaceby assigning, for example, the value 1 to the rest of the space. Note that we will dothis in the sequel.2.10. A metric on partitions.
For the set of all finite partitions with the samecardinality, there is a complete metric.
Definition 2.2.
Let ( X, X , µ, T ) be a system. Let α = { A , . . . , A l } and β = { B , . . . , B l } be two l -set partitions ( l ≥ d ( α, β ) = µ ( α ∆ β ) = 12 l X j =1 µ ( A j ∆ B j ) . Note that d ( α, β ) will be different when the partitions are indexed in differentways.3. An improvement of a technical lemma in Weiss’s proof of theJewett-Krieger’ Theorem
In this section we will prove a lemma which is an improvement of a technicallemma in Weiss’s proof of the Jewett-Krieger’s Theorem. Using this lemma onemay simplify Weiss’s arguments in some sense.In Weiss’s new proof of the Jewett-Krieger’ theorem [12, 15, 2] and in the proof ofWeiss’ theorem on the doubly minimal model [14], one needs the following technicallemma to get a new K-R tower from a given one:
Lemma 3.1. [2, 3, 15]
Let ( X, B , µ, T ) be a non-periodic ergodic system, and let t ( C ) be a K-R tower (i.e. max r C < ∞ ). Then for all N sufficiently large, thereexists a set C ⊂ C such that N ≤ r C ( y ) ≤ N + 4 max r C , ∀ y ∈ C . That is, the corresponding K-R tower t ( C ) satisfies range r C ⊂ [ N, N + 4 max r C ] . ZHENGXING LIAN, SONG SHAO, AND XIANGDONG YE
When one uses this lemma, one hopes that the column heights of the K-R towerare relatively prime, which is not guaranteed in Lemma 3.1. Hence in Weiss’s newproof of the Jewett-Krieger’s theorem, one first assumes that the system has norational spectrum, in which case automatically the column heights of every K-Rtower are relatively prime. Then one deals with other cases. The following lemmawill avoid this kind of discussion.
Lemma 3.2.
Let ( X, B , µ, T ) be a non-periodic ergodic system. Let t ′ be a K-R towerwith bases C i and heights h i , ≤ i ≤ k , and let N = max i { h i } and C = S ki =1 C i .Assume that h , h , . . . , h k are relatively prime. Then for any n large enough, thereis a K-R tower t with base D such that: (1) D ⊂ C ; (2) r D ( y ) ∈ [ n, n + 6 N ] , ∀ y ∈ D ; (3) the column heights of t ( D ) are relatively prime.Proof. First we will find a set ˆ D ⊂ C with the following two properties:(i) n + N ≤ r ˆ D ( y ) ≤ n + 5 N, ∀ y ∈ ˆ D and(ii) µ ( ˆ D ∩ C i ) > ≤ i ≤ k .Then according to the second property of ˆ D , we adjust some part of ˆ D to get D such that the column heights of t ( D ) are relatively prime. Step 1: The construction of ˆ D . Now we describe how to get ˆ D . To that aim,we first construct a Kakutani tower t ( ˆ B ) with height larger than 10( n + 3 N ) and µ ( ˆ B ∩ C i ) > i . But at this point we may have max r ˆ B = ∞ (i.e. t ( ˆ B )may not be a K-R tower). So we need to modify it such that the resulting tower isa K-R tower t ( ˆ D ).By Rokhlin Lemma, there is a B ⊂ C such that the Rokhlin tower c = { B, T B, . . . , T M − B } satisfies that M > n + 3 N ) and µ ( B ) < min i { µ ( C i ) } k ( n +3 N ) + k .Let n = 0. Now find the smallest n ∈ N with ( a ): n − n ≥ n + 3 N ) ; ( b ): µ ( T n B ∩ ( S kj =1 C j \ ( S n +3 N ) j =0 T j B ))) > d ∈ { , , . . . , k } such that µ ( T n B ∩ ( C d \ ( ∪ n +3 N ) j =0 T j B ))) > . Let B = T n B ∩ ( C d \ ( ∪ n +3 N ) j =0 T j B )) . Inductively, assume that for 1 ≤ i ≤ k − n , . . . n i , distinctnumbers d , . . . , d i ∈ { , . . . , k } and measurable sets B , . . . , B i .Let n i +1 be the smallest natural number satisfying: ( a i +1 ): n i +1 − n i ≥ n + 3 N ) ; ( b i +1 ): µ ( T n i +1 B ∩ ( S kj =1 C j \ ( S ij =1 C d i ∪ S is =0 S n s +10( n +3 N ) j = n s T j B ))) > EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 7
Hence there is some d i +1 ∈ { , , . . . , k } \ { d , . . . , d i } such that µ ( T n i +1 B ∩ ( C d i +1 \ ( i [ j =1 C d i ∪ i [ s =0 n s +10( n +3 N ) [ j = n s T j B ))) > . Let B i +1 = T n i +1 B ∩ ( C d i +1 \ ( i [ j =1 C d i ∪ i [ s =0 n s +10( n +3 N ) [ j = n s T j B )) . Note that B i +1 = T n i +1 B ∩ ( C d i +1 \ ( S is =0 S n s +10( n +3 N ) j = n s T j B )). This inductiveprocess can be done for i = 2 , , . . . , k since µ ( B ) < min i { µ ( C i ) } k ( n +3 N ) + k , which means µ ( S k − i =0 S n +3 N ) j =0 T n i + j B ) < µ ( C s ), 1 ≤ s ≤ k .Now by induction we obtain subsets B , . . . , B k . Let ˆ B = S ki =1 B i . We claim that: the height of each column in the Kakutani tower t ( ˆ B ) is larger than n +3 N ) , i.e. r ˆ B ( y ) ≥ n + 3 N ) , ∀ y ∈ ˆ B . To prove the claim, we need to prove that for any l > ≤ u, v ≤ k , µ ( T l B u ∩ B v ) > l ≥ n + 3 N ) . Since µ ( T l B u ∩ B v ) >
0, there is asubset P ⊂ T l B u ∩ B v with positive measure. If u = v , then l ≥ n + 3 N ) since B u ⊂ B . If u < v , then l ≥ n + 3 N ) since T − l P ⊂ B u ⊂ B, P ⊂ B v and µ (( S n u +10( n +3 N ) j = n u T j B ) ∩ B v ) = 0. Finally assume u > v . Since n u is the first numbersatisfing the inductive condition ( a u ), we have µ ( S n v − j = n v − +10( n +3 N ) +1 T j B ∩ B u ) = 0.We also have µ ( S n v − +10( n +3 N ) n v − B ∩ B u ) = 0, so µ (( S n v − j = n v − T j B ) ∩ B u ) = 0. Since n u − n u − ≥ n +3 N ) and T − l P ⊂ B u ∩ T − l B v , we conclude that l ≥ n +3 N ) .By the construction we also see that µ ( ˆ B ∩ C i ) > i ∈ { , . . . , k } . Since n + 3 N and n + 3 N + 1 are relatively prime, we may partition each column of t ( ˆ B )into blocks of sizes n + 3 N and n + 3 N + 1. And then we move the base level of eachblock to the nearest level that belongs to C . Collect the union of the base level andˆ B , and we get a set ˆ D ⊂ C satisfying( I ) : The height of t ( ˆ D ) ranges in [ n + N, n + 5 N ];( II ) : µ ( ˆ D ∩ C i ) > ≤ i ≤ k .The set of heights of t ( ˆ D ) may not be relatively prime, and we need modify it towhat we need. Step 2: The construction of D . For each i ∈ { , . . . , k } , let E i ⊂ ˆ D ∩ C i bea measurable subset with positive measure.Then we get k sets E , E , . . . , E k withthe corresponding heights ˆ h , ˆ h , . . . , ˆ h k respectively. Since T is non-periodic andergodic, µ ( E i \ T ˆ h i E i ) > i . Let ǫ = min i { µ ( E i \ T ˆ h i E i ) } .Let F ⊂ E \ T ˆ h E be a subset satisfying 0 < µ ( F ) < ǫ k +1 . Inductivelyassume for 1 ≤ i ≤ k − F , . . . , F i satisfying 2 µ ( F j ) ≤ ZHENGXING LIAN, SONG SHAO, AND XIANGDONG YE µ ( F j +1 ) < ǫ k − j +1 for each 1 ≤ j ≤ i −
1. Note that(3.1) i X j =1 µ ( F j ) ≤ i X j =1 ǫ k − j +2 = ǫ k − i +2 − / i − / < ǫ k − i +1 . Thus µ (( E i +1 \ T ˆ h i +1 E i +1 ) \ ( S ij =1 T ˆ h j F j )) > ǫ (1 − k − i +1 ). Hence one can find F i +1 ⊂ ( E i +1 \ T ˆ h i +1 E i +1 ) \ ( i [ j =1 T ˆ h j F j )satisfying 2 µ ( F i ) ≤ µ ( F i +1 ) < ǫ k − i +1 .In such a way by induction we get k sets F , F , . . . , F k . For each i , we have thefollowing properties: ( i ): F i ⊂ C i , which implies T h i F i ⊂ C ; ( ii ): F i ⊂ E i \ T ˆ h i E i , which implies T ˆ h i F i ⊂ ˆ D \ F i ; ( iii ): For j ≥ i , T ˆ h i F i ∩ F j = ∅ and µ ( T ˆ h i F i ) = µ ( F i ) > Σ i − s =1 µ ( F s ). ( iv ): µ ( T ˆ h i F i ∩ ( ˆ D \ ( S kj =1 F i ))) > F i . The secondpart of (iii) follows from the inequality µ ( F j +1 ) ≥ µ ( F j ), i.e. µ ( T ˆ h i F i ) = µ ( F i ) ≥ µ ( F i − ) ≥ µ ( F i − ) + 2 µ ( F i − ) ≥ . . . > Σ i − s =1 µ ( F s ) . And (iv) is deduced from (iii) readily.Finally we put D = ( ˆ D \ ( S ki =1 F i )) ∪ ( S ki =1 T h i F i ). By the properties of { F i } , weconclude:(1) D ⊂ C .(2) t ( D ) is a K-R tower, and the height of t ( D ) ranges in [ n, n + 6 N ].(3) The collection of the column heights of the K-R tower t ( D ) contains { ˆ h i , ˆ h i − h i } ki =1 , which are relatively prime since { h i } ki =1 are relatively prime.(1) is followed by the definition of D , and (2) is from ( i ) above. By ( iv ) and thedefinition of D for each i ∈ { , . . . , k } there is some column of t ( D ) with heightˆ h i − h i . By (3.1), we have that µ ( E i )2 ≥ ǫ > Σ ki =1 µ ( F i ), which implies that for each i ∈ { , . . . , k } there is some column of t ( D ) with height ˆ h i . Hence we have (3).The tower t ( D ) is as required. The proof is completed. (cid:3) Proof of Theorem 1.1-(1)
A subset S of Z + is syndetic if it has a bounded gaps, i.e. there is N ∈ N suchthat { i, i + 1 , · · · , i + N } ∩ S = ∅ for every i ∈ Z + . S is thick if it contains arbitrarilylong runs of positive integers, i.e. there is a strictly increasing subsequence { n i } of Z + such that S ⊃ S ∞ i =1 { n i , n i + 1 , . . . , n i + i } . Some dynamical properties can beinterrupted by using the notions of syndetic or thick subsets. For example, a classicresult of Gottschalk and Hedlund [4] stated that x is a minimal point if and only if N ( x, U ) = { n ∈ Z + : T n x ∈ U } EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 9 is syndetic for any neighborhood U of x , and by Furstenberg [1] a topological system( X, T ) is weakly mixing if and only if N ( U, V ) = { n ∈ Z + : U ∩ T − n V = ∅} is thick for any non-empty open subsets U, V of X .A set S is called thickly syndetic if for every N the positions where length N runs begin form a syndetic set. A subset S of Z + is piecewise syndetic if it is anintersection of a syndetic set with a thick set. It is known that a topological system( X, T ) is an M - system (i.e. the set of minimal point of ( X, T ) is dense) if andonly if there is a transitive point x such that N ( x, U ) is piecewise syndetic for anyneighborhood U of x (see for example [7, Lemma 2.1]). We will use this fact in thesequel.To prove Theorem 1.1-(1), we begin with the following observation. Lemma 4.1.
Let ( X, X , µ, T ) be a non-periodic ergodic system. Then there is atower whose set of the column heights is infinite.Proof. Given a tower with base C , if the set of column heights is infinite then weare done; or we put it to be { h , . . . , h n } . Let C i be the corresponding column-base with the height h i , and we may assume that h < . . . < h n (by putting thecolumn-bases with the same height together to form a new column-base). Choose ameasurable set E ⊂ T h n C n such that 0 < µ ( E ) < min ≤ i ≤ n { µ ( C i ) } .Let C = C \ E and we have a tower with base C . If the set of the columnheights is infinite then we are done. Or we have a bigger height set than the towerwith base C , and let it be { h , . . . , h n , h n +1 . . . , h n } . Let C i be the correspondingcolumn-base with the height h i , and we assume that h < . . . < h n < . . . < h n .Choose a measurable set E ⊂ T h n C n such that 0 < µ ( E ) < min ≤ i ≤ n { µ ( C i ) } .Let C = C \ E and continue the process above. If after finite steps we get atower with infinitely many heights, then we are done. Or we will have a sequenceof towers with deceasing bases { C k } , n < . . . < n k and measurable sets E j with0 < µ ( E j ) < j min ≤ i ≤ n j { µ ( C ij ) } for 1 ≤ j ≤ k such that µ ( C k +1 ) ≥ µ ( C k ) − µ ( E k ) > (1 − k ) µ ( C k )for all k ∈ N . Let C = ∞ \ k =1 C k Then µ ( C ) > C has infinitely many heights. The proofis completed. (cid:3) We follow the standard procedure to prove Theorem 1.1-(1). Namely, first for agiven partition ˆ α we construct a partition α close to ˆ α such that the correspondingsymbolic representation ( X α , X α , ρ, S ) is a non-minimal topologically weakly mixingsystem with a dense set of minimal points. Then we use the inverse limit by a moredelicate argument. Finally we show the resulting system is the one which we need. Proposition 4.2.
Let ( X, X , µ, T ) be a non-periodic ergodic system and let ˆ α be a fi-nite partition of X . For each ǫ > , there is a partition α such that the correspondingsymbolic representation ( X α , X α , ρ, S ) is a non-minimal topologically weakly mixingsystem with a dense set of minimal points, and d ( α, ˆ α ) < ǫ. Proof.
By Lemma 4.1, there is a tower consisting of infinitely many columns withdifferent heights. Precisely, let t ( C ) be a tower as in Lemma 4.1 with columns { c k : k ∈ N } and base C = S k ∈ N C k . Let h k be the height of column c k , and assumethat h < h < . . . . Note that for k large enough | c k | will be very small. We willadjust some | c k | to get what we need.Let α = ˆ α = { ˆ A , . . . , ˆ A k } . For each m ∈ N , let ω m = v v . . . v k m , where v a km − a km − ... + am = ( a , a , . . . , a m ) , for each ( a , . . . , a m ) ∈ { , , . . . , k } m . That is, each v i is a word of length m and ω m is a word which contains all the m -name in { , , . . . , k } m . Note that | ω m | = mk m .Before going on, let us recall the notion of copying a name on the column. Let c = { T j B } h − j =0 be a column and a ∈ { , . . . , k } N with N ≤ h . Then copying thename a on the column c means that we copy the name a on the first N levels of c .That means, for the new partition { A , . . . , A k } one has that T i − B ⊂ A a i , ≤ i ≤ N, where a = ( a , . . . , a N ) ∈ { , . . . , k } N . Step 1:
Since P k | c k | < ∞ , there are columns c n , c n such that µ ( | c n ∪ c n | ) < ǫ and h n > h n > k . Let ξ = ω h n − k ∈ { , . . . , k } h n , where 1 j = (1 , , . . . ,
1) withthe length j . And let ξ = ω h n − k ∈ { , . . . , k } h n . For i = 1 ,
2, copy the name ξ i to the column c n i , and we get a partition α . Note that d ( α , α ) = d ( ˆ α, α ) < ǫ .The first step of adjustment is finished. Step m : In general, for each m ∈ N , choose columns c n m , . . . , c n m +1 m such that µ ( m +1 [ i =1 | c n im | ) < ǫ m , and assume that h n m +1 m > . . . > h n m > m k m . For 1 ≤ i ≤ m , let ξ n im = ω i − ω i − . . . ω i − | {z } m times h nim − m (2 i − k i − = ( ω i − ) i h nim − m (2 i − k i − ∈ { , . . . , k } h nim . And let ξ n m +1 m = ω m h nm +1 m − mk m ∈ { , . . . , k } h nm +1 m . Now for 1 ≤ i ≤ m + 1, copy the name ξ n im to the column c n im , and we get a newpartition α m . Note that d ( α m − , α m ) < ǫ m .Moreover, note that we do copying on the m columns c n m , . . . , c n mm to make surethat the symbolic representation of the resulting partition is not minima but has adense set of minimal points, and we do copying on c n m +1 m to make sure the symbolic EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 11 representation of the resulting partition is weakly mixing. Of course we can do it ona single column, but this will cause complication when dealing with the situation inProposition 4.4.For all m ∈ N , we make the above adjustment, and we obtain a new partition α = { A , . . . , A k } . It is clear that d ( ˆ α, α ) ≤ ∞ X i =1 d ( α i − , α i ) < ∞ X i =1 ǫ i = ǫ. Properties of α : Now we prove that ( X α , X α , ρ, S ) is non-minimal, weakly mixing,and the set of minimal points is dense.To show ( X α , S ) is weakly mixing, it suffices to show that for each m ∈ N , E , F , E , F ∈ W m − i =0 T − i α with positive measures, the following holds µ × µ ( E × F ∩ ( T × T ) − m E × F ) > . This depends on the adjustment on c n m +1 m .Denote ω m by ω m = u u . . . u k m , where { u j } k m j =1 = { , , . . . , k } m . Let thenames of E , E , F and F be e , e , f , f ∈ { , . . . , k } m respectively. Then e e = u s and f f = u t for some 1 ≤ t, s ≤ k m . By the construction of α , it follows that T m ( s − C n m +1 m ⊂ E ∩ T − m E , and T m ( t − C n m +1 m ⊂ F ∩ T − m F . Thus, ( E × F ) ∩ ( T × T ) − m ( E × F ) ⊃ T m ( s − C n m +1 m × T m ( t − C n m +1 m . In particular, we conclude that µ × µ (( E × F ) ∩ ( T × T ) − m ( E × F )) ≥ µ × µ ( T m ( s − C n m +1 m × T m ( t − C n m +1 m ) > . To see ( X α , S ) is a non-minimal M -system, we show that each transitive point w ∈ X α is piecewise syndetically but not syndetically recurrent. Let x ∈ X suchthat φ α ( x ) = w . It is easy to see that w = 1 ∞ .Let w = ( . . . , a − , a − , a , a , a , . . . ). Then for each m ∈ N ,[ w ] m − − m +1 = { p ∈ X α : p [ − m + 1 , m −
1] = ( a − m +1 , . . . , a , a , . . . , a m − ) } is a neighborhood of w . Let A ∈ W m − i = − m +1 T − i α with the name ( a − m +1 , . . . , a m − ).Since w = 1 ∞ , it is clear that when m large enough we have ( a − m +1 , . . . , a m − ) =1 m − .As defined before, ω m − = v v . . . v k m − , where { v i } k m − i =1 = { , . . . , k } m − .Then ( a − m +1 , . . . , a m − ) = v r for some r . For each j ≥ m , by the definition of x ,one can find l j such that T l j x ∈ C n mj . By the construction of α , for 1 ≤ i ≤ j , T i (2 m − k m − +(2 m − r − m C n mj ⊂ A . That means, for each j ≥ m , { l j + i (2 m − k m − + (2 m − r −
1) + m } ≤ i ≤ j ⊂ N ( w, [ w ] m − − m +1 ) , which implies N ( w, [ w ] m − − m +1 ) is piecewise syndetic.On the other hand, for each j > m and m − < i < h n mj − j (2 m − k m − , T l j + j (2 m − k m − + i x ∈ T m − d = − m +1 T − d A . As ( a − m +1 , . . . , a m − ) = 1 m , we have that T m − d =0 T − d A ∩ A = ∅ and hence T l j + j (2 m − k m − + i x A , which implies for each j > m , { l j + j (2 m − k m − + i } m −
Let α = { A , . . . , A a } , α ′ = { A ′ , . . . , A ′ a } , and β = { B , . . . , B b } bepartitions with α ≻ β . Then there is a natural way to get a partition β ′ such that α ′ ≻ β ′ . Moreover, if d ( α, α ′ ) < ǫ , then we also have d ( β, β ′ ) < ǫ . To see it we note that α ≻ β defines a function φ : { , . . . , b } → { ,...,a } \ ∅ such that A x ⊂ B y iff x ∈ φ ( y ). Let β ′ = { B ′ , . . . , B ′ b } , B ′ s = [ t ∈ φ ( s ) A ′ t . Notice that if d ( α, α ′ ) < ǫ , then we also have d ( β, β ′ ) < ǫ , since it is easy to checkthat ( A ∪ B )∆( A ∪ B ) ⊂ A ∆ A ∪ B ∆ B . Proposition 4.4.
Let ( X, X , µ, T ) be a non-periodic ergodic system. Then thereexists an increasing sequence of finite partitions { γ n } such that σ ( γ n ) ր X and foreach n ∈ N the corresponding symbolic representation ( X γ n , X γ n , ρ n , S ) is a non-minimal topologically weakly mixing system with a dense set of minimal points.Proof. The basic idea of the proof is the same as in the proof of of Proposition 4.2.Since we have to deal with countably many partitions, we need to do some smallmodifications with the proof.Let { β n } be an increasing sequence of finite partitions such that σ ( β n ) ր X .First we fix the same tower t ( C ) as in the proof of Proposition 4.2 and let { ǫ n } bea sequence with P ∞ n =1 ǫ n < ∞ .For β , we adjust α as in Step 1 of the proof of Proposition 4.2 to get a newpartition γ . We replace β by β W γ , and thus we have γ ≺ β . Then continueour induction. To be precise, we rewrite the Step m . Step m ′ : We replace β m by β m W γ m − m − (still denote it by β m ), and thus we have γ m − m − ≺ β m . Let β j = { B j , B j , . . . , B jk j } for 1 ≤ j ≤ m . Since β ≺ . . . ≺ β m , wemay assume that B ji ⊂ B j − i , for each 2 ≤ j ≤ m , 1 ≤ i ≤ k j − . EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 13
Recall that the word ω m depends on the cardinality of the partition in the proofof Proposition 4.2. Unlike the situation there, now the cardinalities of partitionsare increasing. Let ω m = ω m ( k ) as in the proof of Proposition 4.2, and denote ω m,j = ω m ( k j ) for 1 ≤ j ≤ m . That is, ω m,j is a word which contains all the m -name in { , , . . . , k j } m .For each m ∈ N , choose columns c n m , . . . , c n m ( m +1) m such that µ ( m ( m +1) [ i =1 | c n im | ) < ǫ m . The columns should be disjoint from the columns in Step k , k < m . We assume thatfor each 1 ≤ j ≤ m , h n j ( m +1) m > . . . > h n ( j − m +1)+1 m > m k mj . For each 1 ≤ j ≤ m ,1 ≤ i ≤ m , s = ( m + 1)( j −
1) + i , let ξ n sm = ω i − ,j ω i − ,j . . . ω i − ,j | {z } m − j times h nsm − ( m − j )(2 i − k i − j = ( ω i − ,j ) m − j h nsm − ( m − j )(2 i − k i − j ∈ { , . . . , k j } h nsm . And let ξ n j ( m +1) m = ω m,j h nj ( m +1) m − mk mj ∈ { , . . . , k j } h nj ( m +1) m . Now for 1 ≤ i ≤ m ( m + 1), copy the name ξ n im to the column c n im and we obtain anew partition γ mm with d ( β m , γ mm ) < ǫ m .Inductively, we could construct a sequence of partitions { γ nn } n with the propertythat d ( β m , γ mm ) < ǫ m for each m ∈ N . Now we need to build the required partition { γ n } from { γ nn } n . First we construct partitions { γ nk } n ∈ N , ≤ k ≤ n via { γ nn } n . Then γ k = lim n γ nk is what we are looking for. γ γ γ γ γ γ . . . . . . . . . . . . ↓ ↓ ↓ ↓ γ γ γ . . . Applying Lemma 4.3 to β , γ , and γ we obtain γ . Similarly, applying Lemma4.3 to β , γ , and γ we obtain γ , and we get γ by applying Lemma 4.3 to β , γ and γ . Inductively, we construct γ nk for k < n by applying Lemma 4.3 and β n ≺ γ n − k ,for k < n . Since β ≺ β ≺ . . . ≺ β n , we have γ n ≺ γ n ≺ . . . ≺ γ nn accordingly.Since d ( γ nn , β n ) < ǫ n and β n ≻ γ n − n − , we know that for each 1 ≤ k ≤ n −
1, we have d ( γ nk , γ n − k ) < ǫ n . That means for each k , { γ nk } n ≥ k is a Cauchy sequence. So thereis a partition γ k such that γ nk → γ k , as n → ∞ . Let X nk denote the correspondingsymbolic system of γ nk . The array shows the induction. X X X X X X ... ... ... ... By the construction, for each k , the sequence { γ nk } n ≥ k has the same property as { α n } in Proposition 4.2. According to the proof of Proposition 4.2, the correspondingsymbolic system X k = X γ k of γ k is non-minimal topologically weakly mixing with adense set of minimal points.Since for each n ∈ N , γ n ≺ γ n ≺ . . . ≺ γ nn , we conclude that { γ k } k is increasing.As σ ( β k ) ր X , and d ( γ k , β k ) < P ∞ s = k ǫ s , we deduce σ ( γ k ) ր X too. (cid:3) Now Theorem 1.1(1) follows from Proposition 4.4 and the following lemma.
Lemma 4.5.
Let ( X, T ) be the inverse limit of { ( X n , T n ) } n , where each ( X n , T n ) is a non-minimal topologically weakly mixing system with a dense set of minimalpoints. Then ( X, T ) is also a non-minimal topologically weakly mixing system witha dense set of minimal points.Proof. By the definition of the inverse limit, it is easy to see that (
X, T ) is notminimal as the factor of a minimal system is minimal.To show the density of minimal points in X assume U is a nonempty open set. Let π n : X −→ X n be the projection. Then by the topology of X , there are n ∈ N andan open non-empty set U n ⊂ X n such that π − n U n ⊂ U . Let x n ∈ U n be a minimalpoint and A be its orbit closure. Then there is a minimal set B of X such that π n ( B ) = A . This implies that there is a minimal point x of X such that π n ( x ) = x n which implies that x ∈ π − n ( x n ) ⊂ π − n ( U n ) ⊂ U , and hence the set of minimal pointof X is dense. The similar argument can be applied to show that ( X, T ) is weaklymixing. The proof of is completed. (cid:3) Proof of Theorem 1.1-(2)
In this section, we will prove Theorem 1.1-(2). First we will construct a modelwhich is weakly mixing with a full support but its set of minimal points is notdense. Since in this case the closure of the set of minimal points has measure zero,we collapse it to a point and get the system required.First we need the following lemma (see [2, 3, 15] for a proof).
Lemma 5.1.
Let X be a non-periodic ergodic system. For any positive integers N , N with ( N , N ) = 1 , there exists a set C of finite height such that the K-Rtower t ( C ) satisfies range r C ⊂ { N , N } . To show Theorem 1.1-(2) we start with the following proposition and then followthe standard procedure to finish the proof. To control the thickly syndetic sets, theconstruction here is more involved than that in Proposition 4.2.
EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 15
Proposition 5.2.
Let ( X, X , µ, T ) be a non-periodic ergodic system and ˆ α a finitepartition of X . Then for each ǫ > , there is a partition α such that the correspond-ing symbolic representation ( X α , X α , ρ, S ) is a weakly mixing system whose set ofminimal points is not dense, and d ( α, ˆ α ) < ǫ. Proof.
The proof will be conducted by an inductive procedure. We first choose asequence of positive real numbers { ǫ n } ∞ n =0 with P ∞ n =0 ǫ n < ǫ . Then we start from α − = ˆ α and construct { α n } so that d ( α n , α n +1 ) < ǫ n +1 for n ≥ −
1. It is easy to seethat the limiting partition α satisfies d ( ˆ α, α ) < ǫ . To do so let α = { A , A , . . . , A k } .On one hand, α is constructed so that ( X α , T ) is topologically weakly mixing. Onthe other hand, almost every point will enter A thickly syndetically so that the setof minimal points is not dense. Now we begin our construction. Step : Let ˆ α = { ˆ A , . . . , ˆ A k } . Let ω be the name containing all pairs of names ofnon-trivial elements in W i =0 T − i ˆ α , where “non-trivial elements” in this proof meansthe elements with positive measures.Let M = min { µ ( B ) : B ∈ W i =0 T − i ˆ α } and 0 < ǫ < min { ǫ , M } . Choose l ∈ N such that l > max { ǫ , k } . Now for a fixed N > max { l ǫ , M } , by Lemma 5.1there is a tower t ( C ) = { c , c } such that heights of columns c , c are N , N + 1respectively and the corresponding bases are C , C . It is clear C = C ∪ C . Put e = µ ( C ). Copy the name ω on the partial column { T i C } ≤ i ≤| ω |− . Then incolumn c , copy 2 to the position il for all 0 ≤ i ≤ N − l , and in column c , copy 2to the position il for all 0 ≤ i ≤ N l .In such a way we have constructed a new partition α = { A , A , . . . , A k } . Notethat d ( α , ˆ α ) < ǫ , since the measure changed is less than(2 k + N l ) µ ( C ) < (2 k + N l ) 1 N < ǫ ǫ ǫ . Let A i , A i , A i , A i ∈ α . Assume that positions of 2-name ( i , i ) , ( i , i ) ap-pearing in ω are s and t . Then T s C ⊆ A i , T s +1 C ⊆ A i , T r C ⊆ A i and T r +1 C ⊆ A i . Hence T s C × T r C ⊂ ( A i ∩ T − A i ) × ( A i ∩ T − A i ) = ( A i × A i ) ∩ ( T × T ) − ( A i × A i ) . In particular, µ × µ (cid:0) ( A i × A i ) ∩ ( T × T ) − ( A i × A i ) (cid:1) ≥ µ ( C ) = e > . Now assume that inductively we have constructed partitions { α i } ni =0 , two se-quences of positive integers { l i } ≤ i ≤ n , { s i } ≤ i ≤ n , two sequences of positive numbers { ǫ i } ≤ i ≤ n , { e i } ≤ i ≤ n , with ǫ i +1 < min { ǫ i , e i } . Also assume that we have obtaineda sequence of K-R towers with relatively prime heights { t ( C ( j ) i ) } ≤ i ≤ j ≤ n such that C ⊃ C (1)1 ⊃ . . . ⊃ C ( n ) n , and the height of t ( C ( n ) i ) ranges in [ N i , N i + 6 N i − ] withsome positive integers { N j } ≤ j ≤ n .Let α i = { A i , A i , . . . , A ik } for 1 ≤ α i ≤ n . The sequence { α i } ni =1 satisfies thefollowing properties: for each i ≤ n (1) i : We have d ( α i − , α i ) < ǫ i . Let W ij =0 T − j α i − = { U , . . . , U η } with U j beingnontrivial. Then there is a subset { B , . . . , B η } ⊂ W ij =0 T − j α i such that the α i − -name of U h and the α i -name of B h are the same for each 1 ≤ h ≤ η .Moreover, for all E , F , E , F ∈ { U , . . . , U η } , one has that µ × µ (( T × T ) s i ( E × F ) ∩ ( E × F )) > e i > . In particular, for all E , F , E , F ∈ W i − j =0 T − j α i − , one has that µ × µ (( T × T ) s i ( E × F ) ∩ ( E × F )) > e i > . (2) i : C ( i ) i ⊂ C ( i ) i − ⊂ . . . ⊂ C ( i )1 and for j ≤ i − µ ( | t ( C ( i − j ) | ∆ | t ( C ( i ) j ) | ) < ǫ .Refine the towers t ( C ( i ) j ) according to α i for each 1 ≤ j ≤ i . For each1 ≤ j ≤ i , if a column c with base C in the resulting tower t ( C ( i ) j ) has the α i -name ( a , a , . . . , a h ) ∈ Σ h , then the name satisfies(5.1) a sl j + t = 2 for each 0 ≤ t ≤ j, ≤ s ≤ h − j − l j , i.e. T sl j + t C ⊂ A i . t ( C (1)1 ) t ( C (2)1 ) t ( C (3)1 ) · · · → t ( C ∗ ) t ( C (2)2 ) t ( C (3)2 ) · · · → t ( C ∗ ) t ( C (3)3 ) · · · → t ( C ∗ ) · · · · · · · · · Note that (1) i will be used to show that X α is weakly mixing, and (2) i will beused to show that the minimal points are not dense in X α . Step n + 1 : Now we make the induction for the n + 1 case. First we need to define aword ω n +1 which contains all pairs of names of non-trivial elements in W n +1 i =0 T − i α n .We do it as follows.Refine the tower t ( C ( n ) n ) according to α n , and let the resulting tower be t ( C ( n ) n ) = { c jn } j . Note that the height of each column is in [ N n , N n + 6 N n − ]. Let W n +2 = { B , B , . . . , B t } ⊂ { , . . . , k } n +2 be the set of all names of nontrivial elements of W n +1 i =0 T − i α n . Each ( n + 2)-word B j ( j ∈ { , . . . , t } ) in W n +2 either appears in somecolumn c i j n of t ( C ( n ) n ), or appears in the concatenation of two columns of t ( C ( n ) n ) (i.e.there are c a , c b in t ( C ( n ) n ) such that the name appears in c a c b ). In the second case wealso use c i j n to denote the concatenation of two columns. Let ˜ B j be the name of c i j n .Now fix a large number s n +1 > N n , and construct the word ω n +1 as follows: Foreach pair ( j , j ) ∈ { , . . . , t } , make sure that words c i j n and c i j n appear in ω n +1 , andthe distance from the word B j to the word B j is s n +1 . Since the column heightsof t ( C ( n ) n ) are relatively prime and s n +1 is large enough, one can use α n -names ofcolumns { c in } i to fill gaps between each pair ˜ B i , ˜ B j . EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 17
Let M n +1 < min B ∈ W n +2 i =0 T − i α n { µ ( B ) } , ǫ n +1 < min { ǫ n , e n } . Then let l n +1 > max {| ω n +1 | + 10 N n + 3 N n , nǫ n +1 } and N n +1 > max { n +1) l n +1 ǫ n +1 , n +3 M n +1 } . By Proposi-tion 3.2, we have a new K-R tower t ( C ( n +1) n +1 ) with relatively prime column heightsand C ( n +1) n +1 ⊂ C ( n ) n , and its height ranges in [ N n +1 , N n +1 + 6 N n ]. Refine t ( C ( n +1) n +1 )according to α n , and let the resulting tower be { c jn +1 } j . Let the base of c jn +1 be C jn +1 , and let its height be H j . Let e n +1 = min i { µ ( C in +1 ) } . Now we do the followingadjustment for each column c jn +1 .Denote the name c jn +1 by ( c , c , . . . , c H j ) ∈ Σ H j . First, copy the name ω n +1 to( c h , . . . , c | ω n +1 | + h − ), where h > n + 3 is the first number such that T h − C jn +1 ⊂ C ( n ) n .Secondly, we choose a R ∈ N such that l n +1 − N n ≤ R < l n +1 , R − ( | ω n +1 | + h − > N n , and T R − C jn +1 ⊂ C ( n ) n . Since the column heights of the tower t ( C ( n ) n ) are relatively prime, we can replace( c | ω n +1 | +1 , . . . , c R − ) by the names encountered in the tower t ( C ( n ) n ).Finally, copy 2 to c sl n +1 + r for each 0 ≤ r ≤ n + 1 , ≤ s ≤ H − n − l n +1 . Then accordingto the new name we have a new partition α n +1 . Properties of α n +1 : Note that by the construction of α n +1 if we refine the tower t ( C ( n +1) n +1 ) according to α n +1 , then the resulting tower will still be { c jn +1 } j . Sincecolumn heights of the tower t ( C ( n ) n ) are relatively prime, we have made sure that thefirst l n +1 length part of the name along the column in { c jn +1 } j consists only of thename encountered in the tower t ( C ( n ) n ). These change the levels where the bases ofthe t ( C ( n ) n ) name blocks occur. Thus it defines a new base which we called C ( n +1) n ,and therefore a new K-R tower t ( C ( n +1) n ). Since d ( α n , α n +1 ) < ǫ n +1 , changes fromthe tower t ( C ( n ) n ) to t ( C ( n +1) n ) are very small (less than ǫ n +1 ). Since we copy 2 to c sl n +1 + r for each 0 ≤ r ≤ n + 1 , ≤ s ≤ H − n − l n +1 , each α n +1 -name of t ( C ( n +1) n ) eitherhas the same name with some column in t ( C ( n ) n ), or has more 2 appeared than somecolumn name in t ( C ( n ) n ). Anyway, for each name with the length h in t ( C ( n +1) n ), inthe positions sl n + t, ∀ ≤ t ≤ j, ≤ s ≤ h − n − l n the names are 2.By (2) n , C ( n ) n ⊂ C ( n ) n − ⊂ . . . ⊂ C ( n )1 , above changes from the tower t ( C ( n ) n ) to thetower t ( C ( n +1) n ) will induce corresponding changes such that the tower t ( C ( n ) j ) willbecome some new tower t ( C ( n +1) j ) for each 1 ≤ j ≤ n −
1, where C ( n +1) n +1 ⊂ C ( n +1) n ⊂ . . . ⊂ C ( n +1)1 . By the same reason as showed for t ( C ( n +1) n ), equality (5.1) holds foreach j ≤ n + 1. Thus we have (2) n +1 .Now we verify that α n +1 satisfies (1) n +1 . By the construction, the measure changed from α n to α n +1 is less than µ ( C n +1 )( | ω n +1 | + ( n + 2) N n +1 +6 N n l n +1 ) < N n +1 ( l n +1 + ( n + 2) 2 N n +1 l n +1 ) < n + 1 ǫ n +1 + 2( n + 1) l n +1 < ǫ n +1 ǫ n +1 ǫ n +1 . Thus we conclude that d ( α n , α n +1 ) < ǫ n +1 . And the second part of (1) n +1 is guar-anteed by the construction of ω n +1 .Let D i , D i , D j , D j ∈ W n +1 i =0 T − i α n , and let their names be B i , B i , B j , B j ∈ W n +2 respectively, where 1 ≤ i , i , j , j ≤ t . Then by the definition of ω n +1 , pairs( B i , B j ) and ( B i , B j ) appear in the word ω n +1 . Given arbitrary column c in +1 withthe base C in +1 , let p be the position of B i in this column and let r be the distancefrom the position of B i to the position of B i . Then we have: T p − C in +1 ⊂ D i , T p − s n +1 C in +1 ⊂ D j , T p − r C in +1 ⊂ D i , T p − r + s n +1 C in +1 ⊂ D j . It follows that T p − C in +1 × T p − r C in +1 ⊂ ( D i ∩ T − s n +1 D j ) × ( D i ∩ T − s n +1 D j )= ( D i × D i ) ∩ ( T × T ) − s n +1 ( D j × D j )Hence µ × µ (( D i × D i ) ∩ ( T × T ) − s n +1 ( D j × D j )) ≥ µ × µ ( T p − C in +1 × T p − r C in +1 ) ≥ e n +1 > . Thus (1) n +1 holds. Properties of α : So by the induction we have a sequence of partitions { α n } andassume that limit partition is α = { A , A , . . . , A k } . It is clear d ( ˆ α, α ) < ∞ X i =0 ǫ i < ǫ. Also by the condition (2) n , for n ≥ { t ( C ( j ) n ) } j ≥ n has a limit tower t ( C ∗ n ) with base C ∗ n . And by (2) n , C ∗ ⊃ C ∗ ⊃ . . . .Now we show α is the partition required. First we claim that α satisfies thefollowing properties:(1) For each m ≥ E , F , E , F ∈ W m − j =0 T − j α , we have that µ × µ (( T × T ) s m ( E × F ) ∩ ( E × F ) > . (2) Refine the towers t ( C ∗ j ) according to α for each j ≥
1. If column c in theresulting tower t ( C ∗ j ) has the α -name ( a , a , . . . , a h ) ∈ Σ h and let its basebe C , then the name satisfies(5.2) a sl j + t = 2 for each 0 ≤ t ≤ j, ≤ s ≤ h − j − l j , i.e. T sl j + t C ⊂ A . EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 19
Condition (2) is guaranteed by (2) n . It is left to verify the condition (1). Bycondition (1) m there are E ′ , F ′ , E ′ , F ′ ∈ W m − j =0 T − j α m − such that they have thesame names with E , F , E , F respectively. By (1) m µ × µ (( T × T ) s m ( E ′ × F ′ ) ∩ ( E ′ × F ′ )) > e m . Then by d ( α m , α ) < P ∞ j = m +1 ǫ j , one has that µ × µ (( T × T ) s m ( E × F ) ∩ ( E × F ) > e m − ∞ X j = m +1 ǫ j > . Now using conditions (1) and (2) we will show α is what we need. Let X α be thecorresponding symbolic representation of α , and φ : X → X α be the factor map. Let[ i ] = { w ∈ X α : w = i } for i ∈ { , , . . . , k } . Let w = φ ( x ) ∈ [1] be a transitivepoint of ( X α , T ).By property (1), ( X α , T ) is weakly mixing. By property (2), N ( w, [2] ) is thicklysyndetic, which implies that N ( w, [1] ) is not piecewise syndetic. Hence the set ofminimal points of ( X α , T ) is not dense. (cid:3) Similar to Lemma 4.5 we have the following easy observation.
Lemma 5.3.
Let ( X, T ) be the inverse limit of { ( X n , T n ) } n , where each ( X n , T n ) is a non-minimal topologically weakly mixing system whose set of minimal points isnot dense. Then ( X, T ) is also a non-minimal topologically weakly mixing systemwhose set of minimal points is not dense. Using the similar argument that we obtain Proposition 4.4 from Proposition 4.2,and adjusting the proof of Proposition 5.2, we deduce the following result.
Proposition 5.4.
Every non-periodic ergodic system has a topological model whichis a weakly mixing system with a full support and the set of minimal points is notdense.Proof.
The idea of the proof is similar to the one used in the proof of Proposition 4.4.We will show that there exists an increasing sequence of finite partitions { γ n } suchthat σ ( γ n ) ր X and for each n ∈ N the corresponding symbolic representation( X γ n , X γ n , ρ n , S ) is a weakly mixing system with a full support and the set of minimalpoints is not dense. Then by Lemma 5.3, we finish the proof.Let ( X, X , µ, T ) be the ergodic system. Let { β n } n ≥ be an increasing sequenceof finite partitions such that σ ( β n ) ր X . And let { ǫ n } be a sequence of positivenumbers with P ∞ n =0 ǫ n < ∞ . We will modify the proof of Proposition 5.2 carefullyto get what we need.As in the proof of Proposition 5.2 we choose a tower t ( C ), and adjust β byStep 0 to get a new partition γ . We replace β by β W γ (still denote it by β ),and it is clear γ ≺ β . We assume that the first element (resp. second element) of β is a subset of the first element (resp. the second element) of γ . As in Step 1 of the proof of Proposition 5.2, we modify β to deduce a newpartition γ . We then construct a tower t ( C ) using Lemma 3.2, and form a newtower t ( C ). By Fact in the proof of Proposition 4.4, we construct γ ≺ γ . Refining γ to t ( C ), we know that γ satisfying (1) , (2) in Step 1 of the proof of Proposition 5.2since β ≻ γ .Inductively, we replace β n by β n W γ n − n − . And we assume that the first element(resp. second element) of β n is a subset of the first element (resp. second element)of γ n − n − .We modify β n by Step n to get a new partition γ nn such that d ( β n , γ nn ) < ǫ n , andby the same argument we know that γ nn satisfies the same properties listed in (1) n and (2) n for the tower t ( C ( n ) n ). Now construct γ nk ≺ γ nn by Lemma 4.3. Since thefirst and second elements of β n are subsets of the first and second elements of γ n − n − respectively, and β n ≻ γ n − n − , we conclude that γ nk satisfies the same properties listedin (1) n and (2) n for the tower t ( C ( n ) j ), k ≤ j ≤ n . By the proof of Proposition 5.2,the partition γ k = lim n γ nk satisfies properties as (1) , (2) there. Hence according tothe proof of Proposition 5.2, X γ k is a weakly mixing system with a full support andthe set of minimal points is not dense.Following the same discussion as in the proof of Proposition 4.4, we know that { γ k } is increasing and σ ( γ k ) ր X . The proof is completed. (cid:3) Now using Proposition 5.4, we are able to finish the proof of Theorem 1.1-(2).
Proof of Theorem 1.1-(2).
For a given ergodic system ( X, X , µ, T ), by Proposition 5.4,there is a topological model ( Y, S ) of X with an ergodic measure ρ , which isweakly mixing, non-minimal and the set of minimal point Min( Y ) is not densein supp ( ρ ) = Y . Note that ρ (Min( Y )) = 0, since Min( Y ) is an S -invariant set.Define an equivalence relation ′ ∼ ′ in Y as follows: x ∼ y if x, y ∈ Min( Y ).Then the quotient system ( ˆ X = Y / ∼ , ˆ T ) is a system that is measure theoreticallyisomorphism to ( Y, S ) since ρ (Min( Y )) = 0. Hence ( ˆ X, ˆ T ) is also a topologicalmodel of ( X, X , µ, T ). Note that ( ˆ X, ˆ T ) is a topologically weakly mixing systemwith a full support and a unique fixed point as its only minimal point. Thus theproof is completed. (cid:3) Applications
In this section we give two applications of the results we obtained. Let (
X, T ) bea topological dynamics and M ( X ) is the collection of all Borel probability measureson X with the weak ∗ topology. Then T induces a map T M on M ( X ) naturally bysending µ ∈ M ( X ) to T µ . An unsolved question in [10] is that if there is a weaklymixing proximal system (
X, T ) such that ( M ( X ) , T M ) has dense minimal points.We give an affirmative answer to this question. That is, Theorem 6.1.
There is a weakly mixing proximal system ( X, T ) such that ( M ( X ) , T M ) has dense minimal points. To show this result we need a lemma from [10].
Lemma 6.2.
Let
X, Y be two compact metric spaces, µ ∈ M ( X ) and ν ∈ M ( Y ) . EAKLY MIXING, PROXIMAL TOPOLOGICAL MODELS 21 (1) If A = S ni =1 A i , where A , . . . , A n are Borel subsets of X with µ ( A i ) > and µ ( A i ∩ A j ) = 0 for all ≤ i < j ≤ n , then µ A = P ni =1 µ ( A i ) µ ( A ) µ A i . (2) Let ǫ > and A be a Borel subset of X with µ ( A ) > . If B is a Borel subsetof X such that µ ( B ) > and µ ( A ∆ B ) < µ ( A ) · ǫ , then d ( µ A , µ B ) ≤ ǫ . (3) If π : ( X, µ ) → ( Y, ν ) is measurable and πµ = ν , then πµ π − A = ν A for eachBorel subset A of Y .Proof of Theorem 6.1: Let (Σ , T ) be the dyadic adding machine with a uniqueergodic measure µ . By Theorem 1.1 (Σ , T, µ ) is isomorphic to ( Y, S, ν ), where(
Y, S ) is a weakly mixing proximal topological system and ν has full support. Wenow show that the set of periodic points of ( M ( Y ) , S M ) is dense.Let π : (Σ , T, µ ) → ( Y, S, ν ) be an isomorphism, that is, there are invariantBorel subsets X ⊂ X and X ⊂ Y with µ ( X ) = ν ( X ) = 1 and an invertiblemeasure-preserving transformation π : X → X such that π ( T x ) = Sπ ( x ) for all x ∈ X .Let ǫ > U be a non-empty open subset of Y . Since ν has full support, wehave ν ( U ) >
0. Thus, there are finitely many pairwise disjoint cylinders A , . . . , A k of X such that µ ( π − U ∆ A ) < ν ( U ) · ǫ with A = S ki =1 A i , which implies ν ( U ∆ π ( A ∩ X )) < ν ( U ) · ǫ . Using Lemma 6.2 (2), d ( ν U , ν π ( A ∩ X ) ) ≤ ǫ . Since T | C | C = C foreach cylinder C of X , where | C | stand for the length of C , we conclude that µ C isperiodic. In particular, each µ A i is periodic. By Lemma 6.2 (3), each ν π ( A i ∩ X ) is alsoperiodic. By Lemma 6.2 (1), ν π ( A ∩ X ) = P ki =1 p i ν π ( A i ∩ X ) , where p i = µ ( A i ) /µ ( A ).Thus, ν π ( A ∩ X ) is periodic. It follows that ν U is approached by periodic points of( M ( Y ) , S M ).Now take y ∈ Y and let { U n } ∞ n =1 be a sequence of open neighborhoods of y suchthat diam( U n ) →
0. For any f ∈ C ( Y, R ), we have (cid:12)(cid:12)(cid:12)(cid:12)Z Y f ( z ) d ν U n − f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z U n | f ( z ) − f ( y ) | d ν U n → . A simple calculation shows ν Un → δ y , and hence δ y is a limit point of P ( S M ).This implies that each element of M n ( Y ) = { n P ni =1 δ x i : x i ∈ X } is approached byelements of P ( S M ). Since S ∞ n =1 M n ( Y ) is dense in M ( Y ), it follows that ( M ( Y ) , S M )is a P -system. This ends the proof.Another application of our result is the following. A topological analogy of K-systems, called topological K-system was studied in [8]. In [5] the authors constructeda proximal topological K-system which is weakly mixing. Using Theorem 1.1, wecan get a lot of such examples which are strongly mixing. Theorem 6.3.
There exist strongly mixing proximal topological K-systems.Proof.
Let (
X, T, µ ) be a measurable K-system. By Theorem 1.1 (
X, T, µ ) is iso-morphic to a proximal system (
Y, S ) with a measure ν of full support. Thus ( Y, S )is strongly mixing, since a K-system is strongly mixing in the measurable sense. Atthe same time we know that (
Y, S ) is topological K by [8, Theorem 3.4]. (cid:3)
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Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sci-ences and Department of Mathematics, University of Science and Technology ofChina, Hefei, Anhui, 230026, P.R. China.
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