Family-independence for topological and measurable dynamics

For a family F (a collection of subsets of Z_+), the notion of F-independence is defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.). It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent if and only if it has completely positive entropy; and a m.d.s. is weakly mixing if and only if it is {IP}-independent. For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing if and only if it is {IP}-independent. Moreover, a non-trivial proximal topological K system is constructed, and a topological proof of the fact that minimal topological K implies strong mixing is presented.


Introduction
By a topological dynamical system (t.d.s.) (X, T ) we mean a compact metrizable space X together with a surjective continuous map T from X to itself.For a t.d.s.(X, T ) and nonempty open subsets U and V of X let N(U, V ) = {n ∈ Z + : U ∩T −n V = ∅}, where Z + denotes the set of non-negative integers.It turns out that many recurrence properties of t.d.s.can be described using the return times sets N(U, V ), see [1,11,14,28,29].For example, for a t.d.s.(X, T ) it is known that T is (topologically) strongly mixing iff N(U, V ) is cofinite, T is (topologically) weakly mixing iff N(U, V ) is thick [11] and T is (topologically) mildly mixing iff N(U, V ) is an (IP − IP) * set [29,20], for each pair of nonempty open subsets U and V .Huang and Ye [29] showed that a minimal system (X, T ) is weakly mixing iff the lower Banach density of N(U, V ) is 1, and (X, T ) is mildly mixing iff N(U, V ) is an IP * set, for each pair of nonempty open sets U and V .
By a measurable dynamical system (m.d.s.) we mean a quadruple (X, B, µ, T ), where (X, B, µ) is a Lebesgue space (i.e., X is a set, B is the σ-algebra of Borel subsets on X for some Polish topology on X, and µ is a probability measure on B) and T : X → X is measurable and measure-preserving, that is: µ(B) = µ(T −1 B) for each B ∈ B. For a t.d.s (X, T ), there are always invariant Borel probability measures on X and thus for each such measure µ, (X, B X , µ, T ), with B X the Borel σ-algebra on X, is a m.d.s..For a m.d.s.(X, B, µ, T ), let B + = {B ∈ B : µ(B) > 0} and N(A, B) = {n ∈ Z + : µ(A ∩ T −n B) > 0} for A, B ∈ B + .It is known that T is ergodic iff N(A, B) = ∅ iff N(A, B) is syndetic; T is weakly mixing iff the lower Banach density of N(A, B) is 1 iff N(A, B) is thick; and T is mildly mixing iff N(A, B) is an IP * set iff N(A, B) is an (IP − IP) * set for all A, B ∈ B + iff for each IP set F and A ∈ B + , µ( n∈F T −n A) = 1.Finally, it is known that T is intermixing iff N(A, B) is cofinite for all A, B ∈ B + , see [36,37] and references therein.
In ergodic theory there exists a rich and powerful entropy theory.The analogous notion of topological entropy was introduced soon after the measure theoretical one, and was widely studied and applied.Notwithstanding, the level of development of topological entropy theory lagged behind.In recent years however this situation is rapidly changing.A turning point occurred with F. Blanchard's pioneering papers [4,5] in the 1990's.
In recent years a local entropy theory has been developed, see [21] for a survey.More precisely, in [4] Blanchard introduced the notions of completely positive entropy (c.p.e.) and uniformly positive entropy (u.p.e.) as topological analogues of the K-property in ergodic theory.In [5] he defined the notion of entropy pairs and used it to show that a u.p.e.system is disjoint from all minimal zero entropy systems.The notion of entropy pairs can also be used to show the existence of the maximal zero entropy factor for any t.d.s., namely the topological Pinsker factor [8]. Blanchard et al. [7] also introduced the notion of entropy pairs for an invariant Borel probability measure.Glasner and Weiss [18] introduced the notion of entropy tuples.In order to gain a better understanding of the topological version of a K-system, Huang and Ye [31] introduced the notion of entropy tuples for an invariant Borel probability measure.They showed that if (X, T ) is a t.d.s. and k ≥ 2, then a non-diagonal tuple (x 1 , . . ., x k ) in X k is an entropy tuple iff for every choice of neighborhoods U i of x i there is a subset F of Z + with positive density such that i∈F T −i U s(i) = ∅ for each s ∈ {1, . . ., k} F .We mention that at the same time a theory on sequence entropy tuples and tame systems were developed [15,25,16].It is Kerr and Li who captured the idea behind the results on entropy tuples, sequence entropy tuples and tame systems and treated them systematically using a notion called independence in [34,35], which first appeared in Rosenthal's proof of his groundbreaking ℓ 1 theorem [44,45].
Let (X, T ) be a t.d.s..For a tuple A = (A 1 , . . ., A k ) of subsets of X, we say a subset F ⊆ Z + is an independence set for A if for any nonempty finite subset J ⊆ F , we have j∈J T −j A s(j) = ∅ for any s ∈ {1, . . ., k} J .We call a tuple x = (x 1 , . . ., x k ) ∈ X k (1) an IE-tuple if for every product neighborhood U 1 × • • • × U k of x the tuple (U 1 , . . ., U k ) has an independence set of positive density; (2) an IT-tuple if for every product neighborhood U 1 × • • • × U k of x the tuple (U 1 , . . ., U k ) has an infinite independence set; (3) an INtuple if for every product neighborhood U 1 × • • • × U k of x the tuple (U 1 , . . ., U k ) has arbitrarily long finite independence sets.Kerr and Li [34] showed that (1) entropy tuples are exactly non-diagonal IE-tuples; (2) sequence entropy tuples are exactly non-diagonal IN-tuples, and in particular a t.d.s.(X, T ) is null iff it has no nondiagonal IN-pairs; (3) a t.d.s.(X, T ) is tame iff it has no non-diagonal IT-pairs.For similar results concerning m.d.s.see [35].
Thus the notion of independence is very useful to describe dynamical properties.For a family F , the notion of F -independence can be defined both for topological dynamics (t.d.s.) and measurable dynamics (m.d.s.).So a natural question is: for a given family F which dynamical property is equivalent to F -independence?In this paper we try to answer this question.
It is shown that there is no non-trivial {syndetic}-independent m.d.s.; a m.d.s. is {positive-density}-independent iff it has completely positive entropy; and a m.d.s. is weakly mixing iff it is {infinite}-independent iff it is {IP}-independent.For a t.d.s. it is proved that there is no non-trivial minimal {syndetic}-independent system; a t.d.s. is weakly mixing iff it is {infinite}-independent iff it is {IP}-independent.
Moreover, a non-trivial proximal topological K system is constructed, and a topological proof (using independence) of the fact that minimal topological K implies strong mixing is presented.In a forthcoming paper [26] we will deal with the problem of how to localize the notion of F -independence.
The paper is organized as follows.In Section 2 we investigate the relationship between a given family F and the associated block family bF .In Section 3, the basic properties of F -independence for a t.d.s. are discussed.Particularly we show that F and bF define the same notion of independence.In Section 4, the basic properties of F -independence for a m.d.s. are discussed.In Section 5, we investigate classes of F -independent systems for t.d.s. and show that there is no non-trivial minimal {syndetic}-independent t.d.s..Moreover, a non-trivial proximal topological K system is constructed.In Section 6, we investigate classes of F -independent systems for m.d.s. and show that a m.d.s. is {positive-density}-independent iff it has completely positive entropy.We also show that there is no non-trivial {syndetic}independent m.d.s..In Section 7, we give a topological proof of the fact that minimal topological K implies strong mixing.An interesting combinatorial result, which is needed for the proof of non-existence of no-trivial minimal {syndetic}-independent t.d.s., is established in the Appendix.

Preliminary
The idea of using families to describe dynamical properties goes back at least to Gottschalk and Hedlund [22].It was developed further by Furstenberg [11,12].For a systematic study and recent results, see [1,14,28,29].
Let us recall some notations related to a family (for details see [1]).Let P = P(Z + ) be the collection of all subsets of Z + .A subset a proper subset of P, i.e. neither empty nor all of P. It is easy to see that F is proper if and only if Z + ∈ F and ∅ / ∈ F .Any subset A of P generates a family [A] = {F ∈ P : F ⊇ A for some A ∈ A}.If a proper family F is closed under taking finite intersection, then F is called a filter.For a family F , the dual family is There is an important property being well studied: the Ramsey property.We say that a family F has the Ramsey property if whenever F 1 ∪ F 2 ∈ F , one has either F 1 ∈ F or F 2 ∈ F .One can show that a proper family F has the Ramsey property if and only if F * is a filter [1, page 26].
Denote by F inf the family of all infinite subsets of Z + and by F c the dual family F * inf .Note that F c is the collection of all cofinite subsets of Z + .All the families considered in this paper are assumed to be proper and contained in F inf .
Let F be a subset of Z + .The lower density and upper density of F are defined by , we then say that the density of F is d(F ).The upper Banach density of F is defined by where I is taken over all nonempty finite intervals of Z + .We denote by F pd the family generated by sets with positive density, by F pud the family of sets with positive upper density, and by F pubd the family of sets with positive upper Banach density.
Note that a subset F of Z + is said to be thick if for any n ∈ N there exists some is the intersection of a thick set and a syndetic set.We denote by F t , F s and F ps the families of thick sets, syndetic sets and piecewise syndetic sets respectively.
A subset F of Z + is called a central set if there exists a t.d.s.(X, T ), a point x ∈ X, a minimal point y ∈ X which is proximal to x and a neighborhood U y of y such that F ⊇ N(x, U y ) [12,Section 8.3].Here y is proximal to x means that for a compatible metric d of X, one has inf n∈Z + d(T n x, T n y) = 0. We denote by F cen the family of all central sets.
A subset F of Z + is called an IP-set if there exists a sequence We denote by F ip the family generated by all IP-sets.Definition 2.1.Let F be a family.The block family of F , denoted by bF , is the family consisting of sets S ⊆ Z + for which there exists some F ∈ F such that for every finite subset W of F one has m + W ⊆ S for some m ∈ Z.
Clearly F ⊆ bF and b(bF ) = bF .It is also clear that bF inf = F inf and bF c = F t .
Example 2.2.It is clear that bF pd ⊆ bF pud ⊆ F pubd .It is a result of Ellis that F pubd ⊆ bF pd [12,Theorem 3.20] (one can also give a topological proof for it, using an argument similar to that in the proof of Lemma 4.5).Thus one has bF pd = bF pud = F pubd .
It is easy to see that F is syndetic and that for every finite subset W of F one has m + W ⊆ S 1 ∩ S 2 for some m ∈ Z + .Therefore bF s ⊇ F ps , and hence bF s = F ps .
Example 2.4.It is clear that F cen ⊆ F ps and hence bF cen ⊆ bF ps = F ps .Let S ∈ F ps .Denote by X the smallest closed shift-invariant subset of {0, is easy to see that for every finite subset W of F one has m + W ⊆ S for some m ∈ Z.This means that S ∈ bF cen .Therefore bF cen ⊇ F ps , and hence bF cen = F ps .
The following result shows the relation between the block family and the broken family introduced in [9, Defintion 2].Proposition 2.5.Let F be a family.Let S ⊆ Z + .Then S ∈ bF if and only if there exist an F = {p 1 < p 2 < . . .} ∈ F and a (not necessarily strictly) increasing sequence {b j } ∞ j=1 of integers such that S ⊇ ∞ j=1 {b j + {p 1 , p 2 , . . ., p j }}.Proof.The "if" part is trivial.
Suppose that S ∈ bF .Let F = {p 1 < p 2 < . . .} ∈ F witnessing this.Then for each j ∈ N we find some b j ∈ Z with b j + {p 1 , . . ., p j } ⊆ S. Note that b j + p 1 ≥ 0 for every j ∈ N. Thus we can find an increasing subsequence This proves the "only if" part.The next result follows from Proposition 3.7 and Lemma 3.9, which we shall prove in the next section.Proposition 2.6.If F has the Ramsey property, then so does bF .
We remark that if bF has the Ramsey property, it is not necessarily true that F has the Ramsey property.For example, F pud and F pubd have the Ramsey property, while F pd does not.

Independence: topological case
In this section, for a given family F , we define F -independence for t.d.s., and discuss 1-independence for various families.Recall first the notion of independence set introduced in [34, Definition 2.1].Definition 3.1.Lt (X, T ) be a t.d.s..For a tuple A = (A 1 , . . ., A k ) of subsets of X, we say that a subset F ⊆ Z + is an independence set for A if for any nonempty finite subset J ⊆ F , we have j∈J T −j A s(j) = ∅ for any s ∈ {1, . . ., k} J .
We shall denote the collection of all independence sets for A by Ind(A 1 , . . ., A k ) or IndA.The basic properties of independence sets are listed below.
Lemma 3.2.The following hold: (1) If F ∈ Ind(A 1 , . . ., A k ) and Definition 3.3.Let F be a family.We say that F has the dynamical Ramsey property, if for any t.d.s.(X, T ), any k ∈ N and closed subsets It was shown in [34,Lemmas 3.8 and 6.3] that the families F pd and F inf have the dynamical Ramsey property.
Similar to the definition of u.p.e. of order n (see [31]), we have Standard arguments as in [5] show the following: Proposition 3.5.Let F be a family with the dynamical Ramsey property, and let (X, T ) be a t.d.s..The following are true: (1) If A = (A 1 , . . ., A k ) is a tuple of closed subsets of X with IndA ∩ F = ∅, then there exists ) Let (Y, S) be a t.d.s. and π : X → Y be a factor map, i.e., π is continuous surjective and equivariant.Let k ∈ N. Then π × • • • × π maps the set of F -independent k-tuples of X onto the set of F -independent k-tuples of Y .
Recall that two t.d.s.(X, T ) and (Y, S) are said to be disjoint [11] if X × Y is the only nonempty closed subset Z of X × Y satisfying (T × S)(Z) = Z and projecting surjectively to X and Y under the natural projections X × Y → X and X × Y → Y respectively.Following the arguments in the proofs of [5,Proposition 6] and [8,Theorem 2.1] we have Theorem 3.6.Let F be a family with the dynamical Ramsey property.The following are true: (1) Each t.d.s.being F -independent of order 2 is disjoint from any minimal system without non-diagonal F -independent pair.(2) Each t.d.s.admits a maximal factor with no non-diagonal F -independent pairs.Different families might lead to the same notion of independence.In fact, it follows from Lemma 3.2(2)(3) that Ind(A 1 , . . ., A k ) ∩ F = ∅ if and only if Ind(A 1 , . . ., A k ) ∩ bF = ∅.Thus we have: Proposition 3.7.Let F be a family.Then: (1) The families F and bF define the same notion of independence.
(2) F has the dynamical Ramsey property if and only if bF does.
Theorem 3.8.Let F 1 , F 2 be two families having the dynamical Ramsey property.
Then each F 1 -independent pair is an F 2 -independent pair and viceversa if and only if bF 1 = bF 2 .
Proof.The "if" part follows from Proposition 3.7.Now assume that each F 1 -independent pair is an F 2 -independent pair.We are going to show that bF 1 ⊆ bF 2 .
Let F ∈ F 1 .Denote by X the smallest closed shift-invariant subset of {0, where T denotes the shift.Since F 1 has the dynamical Ramsey property, there exists (x, y) This proves the "only if" part.From Theorem 3.8 one sees that if a family bF has the dynamical Ramsey property, then among the families which has the dynamical Ramsey property and defines the same independence as F does, bF is the largest one.
Now suppose that F = bF , and for any t.d.s.(X, T ) and closed subsets it follows that for any finite subset W of F ′ there exists some m ∈ Z with m + W ⊆ F 1 .Thus F 1 ∈ bF = F .Therefore F has the Ramsey property.
From Proposition 3.7 and Lemma 3.9 we get: Proposition 3.10.Let F be a family.If F has the dynamical Ramsey property, then bF has the Ramsey property.
We remark that if F has the dynamical Ramsey property, it is not necessarily true that F has the Ramsey property.For example, F pd has the dynamical Ramsey property, but not the Ramsey property.
It is easy to see that F ps has the Ramsey property.It is also known that F cen has the Ramsey property [3,Corollary 2.16].The celebrated Hindman theorem [24] says that F ip has the Ramsey property.This leads to the following questions: Question 3.11.Is there any family which has the Ramsey property but not the dynamical Ramsey property?Question 3.12.Do the families F ps and F ip have the dynamical Ramsey property?
To end the section we shall discuss 1-independence for various families.Denote by F rs the family generated by {nZ + : n ∈ N}.The following notion was introduced in [30].Let (X, T ) be a t.d.s..We say that (X, T ) has dense small periodic sets, if for any nonempty open subset U of X there exist a nonempty closed A ⊆ U and k ∈ N such that T k A ⊆ A. To state our result we need a local version of this notion.That is, for a point x in a t.d.s.(X, T ), x is called quasi regular if for each neighborhood U of x, there exist a nonempty closed A ⊆ U and k ∈ N such that T k A ⊆ A. The closed set of quasi regular points of T is denoted by QR(T ).Theorem 3.13.Let (X, T ) be a t.d.s.. Then (1) x ∈ X is F ip -independent iff x ∈ Rec(T ), where Rec(T ) denotes the set of recurrent points of T .Thus, (X, T ) is and M(X, T ) denotes the set of all invariant Borel probability measures on X.Thus, (X, T ) is F pubd -independent of order 1 iff M(T ) = X, iff there exists a µ ∈ M(X, T ) with full support.
(4) x ∈ X is F ps -independent if x ∈ AP(T ), where AP(T ) denotes the set of minimal points of T .Thus, (X, T ) is F ps -independent of order 1 iff AP(T ) = X.(5) x ∈ X is F rs -independent iff x ∈ QR(T ).Thus, (X, T ) is F rs -independent of order 1 iff QR(T ) = X.
Proof.(1).Assume that x ∈ X is F ip -independent and U is a closed neighborhood of x.Then Ind(U) ∩ F ip = ∅, and hence there are an IP-set F and y ∈ X such that Conversely, assume that x ∈ Rec(T ) and U is an open neighborhood of x.Then there exists a y ∈ Rec(T ) ∩ U.By [12, Theorem 2.17], the set N(y, U) contains an IP-set.Thus Ind(U) ∩ F ip = ∅.
(2).The first statement follows easily from the definition.The statement that the F inf -independence of order 1 for (X, T ) implies Rec(T ) = X follows from the fact that if (X, T ) is non-wandering in the sense that (3).This was proved in [34, Proposition 3.12].(4).Assume that x ∈ X is F ps -independent and U is a closed neighborhood of x.Then Ind(U) ∩ F ps = ∅, and hence there are a piecewise syndetic set F and y ∈ X such that Conversely, assume that x ∈ AP(T ) and U is an open neighborhood of x.Then there is y ∈ AP(T ) ∩ U.By a well-known result of Gottschalk, N(y, U) contains a syndetic set.Thus Ind(U) ∩ F ps = ∅.
(5).It is clear that if x ∈ QR(T ) then x is an F rs -independent point.Assume now that x is an F rs -independent point.Let U be a closed neighborhood of x.Then there is exists a k ∈ N such that kZ + is in Ind(U).Take Remark 3.14.The family bF rs does not have the Ramsey property.
Proof.Let (X, T ) be a non-trivial totally minimal t.d.s., i.e., X is minimal under T k for every k ∈ N.For example, any minimal (X, T ) with X being a connected topological space is totally minimal [49,II(9.6) We claim that N(y, X i ) ∈ bF rs for each i = 1, 2. Assume the contrary that N(y, X 1 ) ∈ bF rs .This means that there are d ∈ N and a sequence {n i } i∈N in Z + such that for each i, T n i +dj (y) ∈ X 1 for each 0 ≤ j ≤ i. Replacing {n i } i∈N by a subsequence if necessary, we may assume that T n i (y) converges to some z ∈ X.Then z ∈ X 1 and T dj (z) ∈ X 1 for each j ∈ N, contradicting the assumption that (X, T ) is totally minimal.The same argument shows that N(y, X 2 ) ∈ bF rs .Since Z + = N(y, X) = N(y, X 1 ) ∪ N(y, X 2 ), we conclude that bF rs does not have the Ramsey property.

Independence: measurable case
In this section, for a given family F , we define F -independence for m.d.s., and discuss 1-independence for various families.First we define independence sets for m.d.s., similar to that for t.d.s. in Definition 3.1.Definition 4.1.Lt (X, B, µ, T ) be a m.d.s..For a tuple A = (A 1 , . . ., A k ) of sets in B, we say that a subset F ⊆ Z + is an independence set for A if for any nonempty finite subset J ⊆ F , we have We shall still denote the collection of all independence sets for A by Ind(A 1 , . . ., A k ) or IndA.Note that Lemma 3.2.(1)-( 3) holds also for m.d.s.. Proposition 4.2.Let F be a family with the dynamical Ramsey property.For any m.d.s.(X, B, µ, T ), any k ∈ N and Then Y is a closed subset of Σ k+2 , and contains the constant function k + 1.It is also easily checked that σ(Y ) = Y , where σ denotes the shift map.Thus (Y, σ) is a t.d.s..Note that Ind(A Next we define F -independence for m.d.s., similar to that for t.d.s. in Definition 3.4.Definition 4.3.Let F be a family and k ∈ N. We say that a m.d Note that Proposition 3.7.(1)holds also for m.d.s..

(X, T ). Let
Then the function k → a k on N is subadditive in the sense that a k+j ≤ a k + a j for all k, j ∈ N. Thus the limit lim k→+∞ a k k exists and is equal to inf k∈N a k k (see for example [50, Theorem 4.9]).We call this limit the independence density of A and denote it by I(A) (see the discussion before Proposition 3.23 in [34] for the case of actions of discrete amenable groups).The following lemma was proved by Glasner and Weiss in the second paragraph of the proof of Theorem 3.2 in [19], using Birkhoff's ergodic theorem.We give a topological proof here.Proof.For each k ∈ N we claim that there exists Suppose that this is not true.Then I(A) − 1 k > 0. Furthermore, for any F ∈ IndA we can find a strictly increasing sequence which contradicts m = k 2 + 1.This proves our claim.Now some subsequence of . Therefore F has density I(A).We now discuss 1-independence for various families.Using Birkhoff's ergodic theorem, Bergelsen proved part (1) of the following theorem [2, Theorem 1.2].Here we give a different proof.
Theorem 4.6.Let (X, B, µ, T ) be a m.d.s..The following hold: (1) For any A ∈ B with µ(A) > 0, there exists F ∈ F pd ∩ Ind(A) with density at least µ(A).In particular, (X, B, µ, T ) is F pd -independent of order 1. (2) (X, B, µ, T ) is F s -independent of order 1 iff T is a.e.periodic, iff (X, B, µ, T ) is F rs -independent of order 1, iff for each A ∈ B, a.e.every point of A returns to A syndetically, iff for each A ∈ B, a.e.every point of A returns to A along nZ + for some n ∈ N.
Proof.(1).For each k ∈ N let a k be defined as before Lemma 4.5 for A = (A).Then (2).By Theorem 6.8 the first condition implies the second one.Clearly the second condition implies the third one and the fifth one, the third one implies the first one, and the fifth one implies the fourth one.Thus it suffices to show that the fourth condition implies the first one.
Let A ∈ B with µ(A) > 0 and assume that a.e.every point of X returns to A syndetically.For each n ∈ N set 0 and thus there exists n ∈ N with µ(A n ) > 0. Denote by N the union of the measure zero ones among j∈J T −j A for J running over nonempty finite subsets of Z + .Then µ(N) = 0, and hence µ( (3).The condition implies that µ(A n ) = 1.Thus the conclusion follows from the last paragraph.

5.
Classes of topological F -independence 5.1.General discussion.In this subsection we characterize F inf (resp.F ip ) independent t.d.s. in Theorem 5.1, construct a nontrivial topological K system with a unique minimal point in Example 5.7, and discuss F rs -independence at the end.
A t.d.s.(X, T ) is said to be (topologically) transitive if for any nonempty open subsets U and V of X, N(U, V ) is nonempty; it is called weakly mixing if (X × X, T × T ) is transitive.The equivalence of the conditions (1), ( 2) and (3) in the following theorem was proved in [34,Theorem 8.6].Here we strengthen it by adding the conditions (4) and (5).
Proof.It is clear that (5)⇒(4)⇒( 2) and ( 5)⇒(3)⇒(2).The implication (2)⇒(1) follows from the fact that if for any nonempty open subsets U and V of X one has N(U, U) ∩ N(U, V ) = ∅ , then (X, T ) is weakly mixing [39,Lemma].( [39,Lemma] was proved only for invertible t.d.s., but it is easy to modify the proof to make it work for any t.d.s..) Thus it suffices to show that (1)⇒ (5).Now assume that (X, T ) is weakly mixing.Then each For any given nonempty open subsets U 1 , . . ., U n of X, we are going to find an IP-set F in Ind(U 1 , . . ., U n ).
Petersen [40] showed there exists a t.d.s. which is strictly ergodic, strongly mixing, and has zero topological entropy.Thus in such a system every tuple is F ipindependent, while no non-diagonal tuple is F pd -independent.
A t.d.s. is called an E-system if it is transitive and has an invariant Borel probability measure with full support; it is called an M-system if it is transitive and the set of minimal points is dense; it is called totally transitive if (X, T k ) is transitive for every k ∈ N. By Theorems 3.13 and 5.1 we have Corollary 5.2.Let (X, T ) be a t.d.s..The following hold: (1) If (X, T ) is F pd -independent of order 2, then it is an E-system.
(2) If (X, T ) is F ps -independent of order 2, then it is an M-system.
( Next we show that there is an invertible topological K system with only one minimal point.Recall that a t.d.s.(X, T ) is said to be proximal if the orbit closure of every point in (X × X, T × T ) has nonempty intersection with the diagonal.Following [34] we shall call F pubd -independent tuples of a t.d.s. as IE-tuples.To construct the example we need Lemma 5.4.Let (X, T ) be a t.d.s..We have: (1) Suppose that (X, T ) has a transitive point x.Then T is topologically K if and only if for each j ∈ N, (x, T x, . . ., T j−1 x) is an IE-tuple.(2) (X, T ) has only one minimal point if and only if (X, T ) is proximal.
Proof.(1).This follows from the fact that the set of IE j-tuples is closed in X j for each j ∈ N.
(2).The "only if" part is trivial.Assume that (X, T ) is proximal.Take x ∈ X. Say, (y, y) is in the intersection of the diagonal and the orbit closure of (x, T x).Then T y = y.Let z ∈ X.Then the orbit closures of y and z have nonempty intersection, which of course has to be {y}.It follows that if z is minimal, then z = y.For a t.d.s.(X, T ), recall its natural extension ( X, T ) defined as follows.X is the closed subspace of n∈N X consisting of (x 1 , x 2 , . . . ) with T (x n+1 ) = x n for all n ∈ N, and T is defined as T (x 1 , x 2 , . . . ) = (T (x 1 ), x 1 , x 2 , . . .).Note that T is a homeomorphism and the projection π : X → X sending (x 1 , x 2 , . . . ) to x 1 is a factor map.It is well known that (X, T ) and ( X, T ) share many dynamical properties.
Here we need a special case.
Lemma 5.5.Let (X, T ) be a t.d.s..The following are true: (1) Let F be a family and k ∈ N. Then (X, T ) is F -independent of order k if and only if ( X, T ) is so.(2) (X, T ) is proximal if and only if ( X, T ) is so.
Proof.(1).The "if" part follows from the fact that if a t.d.s. is F -independent of order k, then so is every factor.Suppose that (X, T ) is F -independent of order k.Let U 1 , . . ., U k be nonempty open subsets of X.Then there exist nonempty open subsets , J be a nonempty finite subset of F , and s ∈ {1, . . ., k} J .Then j∈J T −j V s(j) = ∅.Take y ∈ j∈J T −j V s(j) .We can find x = (x 1 , x 2 , . . . ) ∈ X such that x m = y.Then x ∈ j∈J T −j U s(j) .Thus F ∈ Ind(U 1 , . . ., U k ).This proves out claim. Since Therefore ( X, T ) is also F -independent of order k.This proves the "only if" part.
Proof.By Lemma 5.5 it suffices to show that there exists a non-trivial t.d.s. which is topological K and proximal.We use the idea in the proof of Theorem 4.2 in [27].The main idea is to construct a recurrent point x ∈ Σ 2 with the following two properties: (I) for any j ∈ N, (x, σ(x), . . ., σ j−1 (x)) is an IE-tuple of (X, σ), where X is the orbit closure of x, and (II) for each n ∈ N, 0 n appears in x syndetically.By Lemma 5.4 it is clear that (X, σ) is topological K with a unique minimal point 0. First we give the detailed construction of the recurrent point x.
Say, m = φ(k).Since (Y, σ) has a unique minimal point 0, there exists ℓ k ∈ N with ℓ k ≥ t m such that 0 n k appears in y with gaps bounded above by ℓ k .Set b k = 2ℓ k n k , and set It is clear that x := lim k→+∞ A k is a recurrent point of σ in Σ 2 .Denote by X the orbit closure of x in Σ 2 .We claim that x satisfies (I) and (II).(I).Given j ∈ N, we show that (x, σ(x), . . ., σ j−1 (x)) is an IE-tuple of (X, σ).Suppose that V ′ 0 , V ′ 1 , . . ., V ′ j−1 are neighborhoods of x, σ(x), . . ., σ j−1 (x) respectively.Then there is some m ∈ N with m > j such that V i ⊆ V ′ i for all 0 ≤ i ≤ j − 1, where Since (z m,1 , . . ., z m,m ) is an IE-tuple of Y , there exists some d > 0 such that for any n ∈ N we can find a finite subset J ⊆ Z + with |J| ≥ n contained in an interval with length at most d|J| such that for any s ∈ {1, 2, . . ., m} J one has i∈J σ −i U s(i) = ∅, where U j = (z m,j [0, t m ]) Y for 1 ≤ j ≤ m.Since y is a transitive point of Y , we have σ N (y) ∈ i∈J σ −i U s(i) for some N ∈ Z + .Then y [N +i,N +i+tm] = z m,s(i) [0, t m ] for all i ∈ J. Take k ≥ N + max J + t m with φ(k) = m.Then b k ≥ k ≥ N + i + t m for all i ∈ J, and for the map ψ ∈ {0, 1, . . ., m − 1} 2nmJ defined by ψ(2n m i) = s(i) − 1 for all i ∈ J. Therefore 2n m J is an independence set for (V 0 , . . ., V m−1 ).Clearly 2n m J is contained in an interval with length at most 2n m d|J| = 2n m d|2n m J|.Thus by Lemma 4.5 (x, σ(x), . . ., σ j−1 (x)) is an IE-tuple of (X, σ).(II).We now show that for each n ∈ N, 0 n appears in x syndetically.It suffices to prove that for each k ∈ N, 0 n k appears in x syndetically with gaps bounded above by 2b k .
Fix k ∈ N. Say, φ(k) = m.By the construction Note that f m (a) = 0 for every a ∈ Λ tm+1 Assume that 0 n k appears in A ℓ with gaps bounded above by 2b k , where ℓ ≥ k + 1.Now we are going to prove that this is also true for ℓ + 1. Set m ′ = φ(ℓ).First note that If m ′ ≥ k + 1, then by the induction assumption and the construction of C m ′ ,i we know that 0 n k appears in A ℓ+1 with gaps bounded above by 2b k .If m ′ ≤ k, then by the induction assumption and the discussion similar to the case of A k+1 , we know that 0 n k appears in A ℓ+1 with gaps bounded above by 2b k .Hence 0 n k appears in x syndetically with gaps bounded above by 2b k , as x = lim ℓ→+∞ A ℓ .Definition 5.8.We say that a t.d.s (X, T ) is Bernoulli if it is conjugate to (A Z + , σ), where A is a compact metrizable space with |A| ≥ 2 and σ is the shift.Theorem 5.9.A Bernoulli system is F rs -independent.
Proof.Let (X, T ) be a Bernoulli system.Without loss of generality we may assume that (X, T ) = (A Z + , σ) as above.Let U 1 , . . ., U n be nonempty open subsets of X for some n ∈ N. Then there exist some k ∈ N and nonempty subsets Recall that a t.d.s.(X, T ) is called strongly mixing if for any nonempty open subsets U and V of X, N(U, V ) is a cofinite subset of Z + .In [4, Example 5] Blanchard constructed examples of invertible t.d.s. which are F rs -independent of order 2 and are not strongly mixing.In fact, the Property P defined in [4] is exactly the same as F rs -independence of order 2. It is easily checked that the condition in [4, Proposition 4] actually implies F rs -independence.Thus Blanchard's examples are actually F rs -independent.Thus F rs -independence does not imply strong mixing and hence does not imply Bernoulli.
A factor map π : (X, T ) → (Y, S) between t.d.s. is said to be an almost one-to-one extension if the set {x ∈ X : π −1 (π(x)) = {x}} is dense in X.
For a sequence K = {k n } n∈N in N with k n+1 being divisible by k n for each n ∈ N, the adding machine (X K , T K ) associated to K is defined as follows.X K is the projective limit of lim ← − n→+∞ Z/k n Z, as a metrizable compact abelian group, and T K is the addition by 1.For a t.d.s.(X, T ), recall that x ∈ X is called a regular minimal point [22, Definition 3.38] if for each neighborhood U of x, there exists k ∈ N such that N(x, U) ⊇ kZ + .It is known that if x is a regular minimal point, then its orbit closure is an almost one-to-one extension of some adding machine, see for instance [30,Proposition 3.5].Now we show Proposition 5.10.Let (X, T ) be a minimal t.d.s..The following are equivalent: (1) (X, T ) has dense small periodic sets.
(2) (X, T ) is an almost one-to-one extension of some adding machine.
(3) X has a regular minimal point.
(1)⇒(2).For any nonempty open subset U of X, let B be a nonempty closed subset of U with T k B ⊆ B for some k ∈ N. Take x ∈ B. Then the argument in the proof of [30,Proposition 3.5] shows that the orbit closure A of x under T k is a nonempty clopen subset of U and there exists some ℓ ∈ N such that {A, T A, . . ., T ℓ−1 A} is a clopen partition of X and T ℓ A = A.
Fix a compatible metric on X. Starting with some nonempty open subset U of X with diam(U) < 1, we obtain A and ℓ as above, and set A 1 = A and ℓ 1 = ℓ.Inductively, assuming that we have found subsets A 1 ⊇ A 2 ⊇ • • • ⊇ A k and positive integers ℓ 1 , ℓ 2 , . . ., ℓ k such that diam(A j ) < 1/j, {A j , T A j , . . ., T ℓ j −1 A j } is a clopen partition of X, and T ℓ j A j = A j for all 1 ≤ j ≤ k.We shall find A k+1 and ℓ k+1 with the same property.Let U be a nonempty open subset of A k with diam(U) < 1/(k + 1).We obtain A and ℓ as above, and set A k+1 = A and ℓ k+1 = ℓ.Now the argument in the proof of [30,Proposition 3.5] shows that (X, T ) is an almost one-to-one extension of some adding machine.5.2.Non-existence of non-trivial minimal F s -independent t.d.s.It was shown in [31,Theorem 3.4] that there exist non-trivial minimal topological K systems (the existence of nontrivial minimal u.p.e.systems was proved earlier by Glasner and Weiss [17]).We have Theorem 5.11.There is no non-trivial minimal t.d.s. which is F s -independent of order 2.
To prove Theorem 5.11, we need some preparation.Crucial to the proof of Theorem 5.11 is the following combinatorial result, which is also of independent interest.We postpone its proof to the Appendix.Recall the notion introduced before Lemma 5.6.Theorem 5.12.Let p, ℓ ∈ N with p ≥ 2. For any integer m ≥ 4ℓ + 2, given any sequence {A n } n∈Z + of subsets of Λ m p with |A n | ≤ ℓ for each n ∈ Z + , there exists x ∈ Σ p such that x[n, n + m − 1] ∈ A n for every n ∈ Z + .
We remark that under the conditions of Theorem 5.12, the set {x ∈ Σ p : x[n, n + m − 1] ∈ A n for all n ∈ Z + } is small in both topological and measure-theoretical sense: it is a closed subset of Σ p with empty interior and has measure 0 for the product measure on Σ p associated to any probability vector (t 0 , . . ., t p−1 ) with p−1 j=0 t j = 1 and t j > 0 for all 0 ≤ j ≤ k − 1. Lemma 5.13.For every minimal subshift X ⊆ Σ 2 , Ind([0] X , [1] X ) does not contain any syndetic set.
Proof.We argue by contradiction.Assume that X ⊆ Σ 2 is a minimal subshift and Ind([0] X , [1] X ) contains a syndetic set F .Say, F = {n 0 < n 1 < . . .} with ℓ = max j∈Z + (n j+1 − n j ).Let m be as in Theorem 5.12 for p = 2 and ℓ.Take a ∈ Λ mℓ 2 such that a appears in some element of X.For each j ∈ Z + , set A j to be the subset of Λ m 2 consisting of elements of the form (a(k), a(k + n j+1 − n j ), a(k + n j+2 − n j ), . . ., a(k + n j+m−1 − n j )) for 1 ≤ k ≤ ℓ.Then |A j | ≤ ℓ for all j ∈ Z + .By Theorem 5.12 we can find x ∈ Σ 2 such that x[j, j + m − 1] ∈ A j for every j ∈ Z + .Since F ∈ Ind([0] X , [1] X ), we can find y ∈ X with y(n j ) = x(j) for all j ∈ Z + .As X is minimal, there exists some i ≥ n 1 such that y[i, i We are ready to prove Theorem 5.11.
Proof of Theorem 5.11.We shall show that if (Y, S) is a minimal t.d.s., and V 0 , V 1 are disjoint closed subsets of X with nonempty interior, then Ind(V 0 , V 1 ) ∩ F s does not contain any syndetic set.
It is well known that we can find a minimal t.d.s.(X 1 , T 1 ) and a factor map π : (X 1 , T 1 ) → (Y, S) such that X 1 is a closed subset of a Cantor set (see for eample [6, page 34]).It is easy to see that Ind(V 0 , Define a coding φ : X 1 → Σ 2 such that for each x ∈ X 1 , φ(x) = (x 0 , x 1 , . ..),where By Lemma 5.13 we know that Ind([0] X , [1] X ) does not contain any syndetic set.Then Ind(V 0 , V 1 ) does not contain any syndetic set either.

Finite product.
In this subsection we investigate the question for which family F , the product of finitely many F -independent t.d.s.remains F -independent.
It is known that if F = F pd the question has a positive answer [31, Theorem 8.1] [34, Theorem 3.15].We now show that the question has a positive answer for F = F rs , F ps .It is clear that We need the following lemma.It is also needed for the proof of Theorem 7.1 later.Proof.Take N ∈ N such that We need the following simple lemma.For a subset K of Z + , denote by X K the set of limit points of the sequence {σ n 1 K } n∈Z + in {0, 1} Z + , where σ denotes the shift map on {0, 1} Z + .Note that (X K , σ) is a t.d.s.. Lemma 5.15.The following statements hold: (1) Let S 1 , S 2 ∈ F pubd .Then there are two subsets (3) Let S 1 , S 2 ∈ F rs .Then there are two subsets Proof.(1).Set X i = X S i .Recall the independence density defined before Lemma 4.5.
We have ).Note that we can find arbitrarily long finite interval J 2 in Z + and a set . By Lemma 5.14, when ) is syndetic and central.
Theorem 5.16.The product of finitely many F s -(resp.F rs , F pd ) independent t.d.s. is F s -(resp.F rs , F pd ) independent.
Proof.We shall prove the case F = F s , and the proof for the other cases is similar.Let (X i , T i ) be an F s -independent t.d.s. for i = 1, 2. Let U 1 , . . ., U n and V 1 , . . ., V n be nonempty open subsets of X 1 and X 2 respectively.Then there are syndetic sets S 1 ∈ Ind(U 1 , . . ., U n ) and S 2 ∈ Ind(V 1 , . . ., V n ).By Lemma 5.15 there are two subsets The theorem follows by induction.
Since a family F has the Ramsey property if and only if its dual family F * has the finite intersection property, we have Theorem 5.17.Let F be a family with the Ramsey property.Then the product of finitely many F * -independent t.d.s.remains F * -independent.
In [51, page 278] Weiss constructed two weakly mixing t.d.s.whose product is not transitive.(Weiss's example was only stated to be Z-weakly mixing, but is easily checked to be Z + -weakly mixing.)In view of Theorem 5.1, this implies that the product of F inf -independent (F ip -independent resp.)t.d.s. may fail to be F infindependent (F ip -independent resp.).
6. Classes of measurable F -independence 6.1.General discussion.In this subsection we characterize F inf -(resp.F ip , F pubd ) independent m.d.s. in Theorems 6.1 and 6.2.
Recall that a m.d.s (X, T ) is said to be ergodic if for any A, B ∈ B with positive measures, N(A, B) is nonempty; it is called weakly mixing if T ×T is ergodic.Similar to the topological case (Theorem 5.1) we have Theorem 6.1.For a m.d.s.(X, B, µ, T ) the following are equivalent: (1) (X, B, µ, T ) is weakly mixing. ( Proof.It is clear that (5)⇒( 4)⇒( 2) and ( 5)⇒(3)⇒(2).The implication (2)⇒(1) follows from the fact that if for any A, B ∈ B with positive measures one has . Thus the proof of (1)⇒(5) in Theorem 5.1 also applies here.
It was proved in [31,Theorem 8.3] and [34,Theorem 3.16] that a t.d.s. is topological K if and only if its every finite cover by non-dense open subsets has positive entropy.Moreover, it is shown in [31,Theorem 9.4] that there exists t.d.s. which is F pubd -independent of order 2 but is not F pubd -independent of order 3. Now we show that in the measurable setup the situation is different.
We refer the reader to [38,Chapter 4] for the basics of the entropy theory.A m.d.s.(X, B, µ, T ) is said to have completely positive entropy if for every non-trivial countable measurable partition α of X with 0 < H(α) < ∞ one has h µ (T, α) > 0. The Rohlin-Sinai theorem says that an invertible m.d.s. has completely positive entropy if and only if it is an K-automorphism [42] [38,Theorem 4.12].
For µ, T ) is an invertible m.d.s. and that P = {P 0 , P 1 , . . ., P a−1 } is a finite measurable partition of X. Construct a set Y P ⊆ Ω a as follows: Glasner and Weiss showed that an invertible m.d.s.(X, B, µ, T ) has completely positive entropy if and only if for every finite measurable partition P = {P 0 , P 1 , . . ., P a−1 } of X with min 0≤j≤a−1 µ(P j ) > 0 the set Y P has interpolating sets of positive density.In our terminology, clearly interpolating sets of P are exactly the independence sets of the tuple (P 0 , P 1 , . . ., P a−1 ).Now we extend the result of Glasner and Weiss to general m.d.s.. Theorem 6.2.Let (X, B, µ, T ) be a m.d.s.. Then the following are equivalent: (1) (X, B, µ, T ) is F pubd -independent.
To prove Theorem 6.2, we need some preparation.For a Lebesgue space (X, B, µ) and a measurable partition α of X, we denote by α the σ-algebra generated by the items of α; for a family {B j } j∈J of sub-σ-algebras of B, we denote by j∈J B j the sub-σ-algebra of B generated by j∈J B j .For a m.d.s.(X, B, µ, T ), a measurable partition α of X, and 0 ≤ n ≤ m ≤ ∞, we denote m j=n T −j α and m j=n T −j α by α m n and αm n respectively.The following lemma is [38,Lemma 4.6] for non-invertible m.d.s.. Lemma 6.3.Let (X, B, µ, T ) be a m.d.s., and let α and β be countable measurable partitions of X with Since we conclude that lim n→+∞ where the second equality comes from the above paragraph.One also has Taking lim sup on both sides, by the above paragraph we get That is, lim inf n→+∞ For a m.d.s.(X, B, µ, T ), denote by P(T ) the Pinsker σ-algebra of T [50, page 113], consisting of A ∈ B such that h µ (T, {A, X \ A}) = 0.For a Lebesgue space (X, B, µ) and sub-σ-algebras B 1 and B 2 of B, we write The next theorem appeared in [41, 12.3].For the convenience of the reader, we give a proof here.Theorem 6.4.Let (X, B, µ, T ) be a m.d.s.. Then P(T ) = µ α n∈Z + α∞ n for α running over countable measurable partitions of X with H(α) < ∞.The next result appeared implicitly in [41, 13.2].For completeness, we give a proof here.Theorem 6.5.Let (X, B, µ, T ) be a m.d.s.. Then the following are equivalent:
Proof.Assume that (X, B, µ, T ) is a non-trivial m.p.s.being F pubd -independent of order 2 and has entropy 0. Clearly (X, B, µ, T ) is ergodic.By Rosenthal's extension of the Jewett-Krieger theorem to non-invertible m.d.s.[43], there exists a t.d.s. ( X, T ) with a unique invariant Borel probability measure µ such that µ has full support and the m.denotes the Borel σ-algebra of X, in the sense that there are X 0 ∈ B, X 0 ∈ B b X and a measure-preserving bijection φ : X 0 → X 0 with µ(X 0 ) = µ( X 0 ) = 1, T X 0 ⊆ X 0 , T X 0 ⊆ X 0 , and φ • T = T • φ.By the variational principle [50,Theorem 8.6], one has h top ( T ) = h b µ ( T ) = h µ (T ) = 0. Since (X, B, µ, T ) is non-trivial, ( X, T ) is a non-trivial t.d.s.. Thus we can find two disjoint closed subsets A, B of X with µ(A) > 0, µ(B) > 0. Set U = { X \ A, X \ B}.Then U is an open cover of X, and for any F ∈ Ind(A, B) we have Hence d(F ) = 0. Thus Ind(A, B)∩F pud = ∅.It follows from Example 2.2 and Proposition 3.7 that Ind(A, B) ∩ F pubd = ∅.Thus ( X, B b X , µ, T ) is not F pubd -independent of order 2. Then by Remark 4.4 (X, B, µ, T ) is not F pubd -independent of order 2 either.
(3)⇒(1): We claim first that (X, B, µ) is non-atomic in the sense that µ({x}) = 0 for every x ∈ X.In fact, since T has completely positive entropy, it is ergodic.If µ({x}) > 0 for some x ∈ X, then we can find some n ∈ N such that x, T x, . . ., T n−1 x are pairwise distinct, T n x = x, and µ(x n .If n > 1, denoting by β the partition of X into {x} and its complement, we have h µ (T, β) = 0. Thus n = 1, which means that (X, B, µ, T ) is trivial.Therefore (X, B, µ) is nonatomic.
Given a tuple (A 1 , . . ., A k ) of sets in B with positive measures, we are going to show that Ind(A 1 , . . ., A k ) ∩ F pubd = ∅.Without loss of generality, we may assume that A 1 , . . ., A k are pairwise disjoint.
(2)⇒(3): Assume that (X, B, µ, T ) is F pubd -independent of order 2. Note that the definitions of independence sets and entropy apply to more general measuretheoretical dynamical systems in which the probability space does not have to be a Lebesgue space.In this sense (X, P(T ), µ, T ) is also F pubd -independent of order 2 and has entropy 0. Since (X, B, µ) is a Lebesgue space, it is easy to see that B is separable under the semi-metric d(A, B) = µ(A∆B).Then P(T ) is also separable under this semi-metric.It follows that there is a m.d.s.(Y, J , ν, S) (i.e., (Y, J , ν) is a Lebesgue space) such that the measure algebra triples associated to (X, P(T ), µ, T ) and (Y, J , ν, S) in Remark 4.4 are isomorphic [12,Proposition 5.3].Then (Y, J , ν, S) is also F pubd -independent of order 2 and has entropy 0. By Lemma 6.6 (Y, J , ν, S) is trivial.Thus P(T ) consists of measurable subsets of X with measure 0 or 1.That is, (X, B, µ, T ) has completely positive entropy.6.2.Non-existence of F s -independent m.d.s.It is somewhat surprising that there is no non-trivial m.d.s. which is F s -independent.In the following, we aim to show that for any non-periodic m.d.s.(X, B, µ, T ), there exists A ∈ B with µ(A) > 0 such that Ind(A) does not contain a syndetic set.
A m.d.s.(X, B, µ, T ) is called non-periodic or free if µ({x ∈ X : T n x = x}) = 0 for every n ∈ N. It is easy to see that an ergodic m.d.s.(X, B, µ, T ) is non-periodic if and only if (X, B, µ) is non-atomic in the sense that µ(x) = 0 for every x ∈ X. Theorem 6.8.Let (X, B, µ, T ) be a non-periodic m.d.s.. Then for any ε > 0 there exists A ∈ B with µ(A) > 1 − ε such that Ind(A) does not contain any syndetic set.
Proof.Endow X with a Polish topology such that B is the corresponding Borel σ-algebra.Replacing the Polish topology on X by a finer one if necessary [33, Theorem 13.11 and Lemma 13.3], we may assume that T is continuous.Let ε > 0. We claim that there is a compact subset K of X such that µ(K) > 1 − ε and A n := j∈Z + n−1 i=0 T −j−i K has measure 0 for every n ∈ N. Assuming this claim let us show how it implies the theorem. Since By the regularity of µ [33,Theorem 17.11], we can find a compact set A contained in K \ ( n∈N A n ) such that µ(A) > 1 − ε.We shall show that Ind(A) does not contain any syndetic set.
Let F ∈ Ind(A) be nonempty.Replacing F by F − min F if necessary, we may assume that 0 ∈ F .One has µ( j∈J T −j A) > 0 and hence j∈J T −j A = ∅ for every nonempty finite subset J of F .Since A is compact, we conclude that j∈F T −j A is nonempty.Take x ∈ j∈F T −j A. Then x ∈ A and T j x ∈ A ⊆ K for every j ∈ F .For each n ∈ N one has x ∈ A n , and hence for some j n ∈ Z + none of T jn x, T jn+1 x, . . ., T jn+n−1 x is in K. Then [j n , j n + n − 1] ∩ F = ∅.Therefore F is not syndetic.
We are left to prove the above claim.Since the main idea of the proof is well illustrated in the case µ is ergodic, we consider this case first.
So assume that µ is ergodic.Since (X, B, µ, T ) is non-periodic, by the comment before Theorem 6.8, (X, B, µ) is non-atomic.Replacing X by supp(µ) if necessary, we may assume that µ has full support.Take x ∈ X and set W = {T n x : n ∈ Z + }.Then T W ⊆ W , and W is nonempty and countable.Since µ is non-atomic, one has µ(W ) = 0, and hence µ(X \ W ) = 1.By the regularity of µ, we can find a compact This finishes the proof in the case µ is ergodic.Now we consider the general case, using the ergodic decomposition of (X, B, µ, T ).Denote by P (X) the set of all probability Borel measures on X, and endow it with the σ-algebra generated by the functions µ ′ → µ ′ (A) on P (X) for all A ∈ B [33, Section 17.E].
Endow Y with a Polish topology such that J is the corresponding Borel σ-algebra.Replacing the Polish topology on X by a finer one if necessary, we may assume that π is continuous.
Denote by F (X) the set of all closed subsets of X, and endow it with the Effros Borel structure, i.e., the σ-algebra generated by the sets {Z ∈ F (X) : Z ∩U = ∅} for all open subsets U of X.The map φ : P (X) → F (X) sending each µ ′ to supp(µ ′ ) is measurable [33,Exercise 17.38].By the Kuratowski-Ryll-Nardzewski selection theorem [33,Theorem 12.13] we can find a measurable map ψ : F (X) → X such that ψ(Z) ∈ Z for each nonempty Z ∈ F (X).
Note that supp(µ y ) ⊆ π −1 (y) for every y ∈ Y ′ .Thus the map ϕ n : Y ′ → X sending y to T n (ψ(φ(µ y ))) is measurable and injective for each n ∈ Z + .Recall that a measurable space is a standard Borel space if the σ-algebra is the Borel σalgebra for some Polish topology on the set.A measurable subset of a standard Borel space together with the restriction of the σ-algebra to the subset is also a 6.3.Finite product.By contrast to the topological case, it is well known that the product of two weakly mixing m.d.s. is still weakly mixing [12,Proposition 4.6].In view of Theorem 6.1, this means that the products of finitely many F inf -independent (F ip -independent resp.)m.d.s. are F inf -independent (F ip -independent resp.).
Meanwhile, it is known that the product of finitely many invertible completely positive entropy m.d.s. has completely positive entropy [38,Theorem 4.14].As the topological case, every m.d.s. has a natural extension [10, Page 240], which is always invertible.The natural extension of a completely positive entropy m.d.s. has completely positive entropy [41, 13.8] (one can also deduce this from Theorem 6.5 and the fact that the natural extension of a m.d.s. is the inverse limit of a sequence of m.d.s.being identical to the original one).It follows that the product of finitely many completely positive entropy m.d.s. has completely positive entropy.In view of Theorem 6.2, this means that the product of finitely many F pd -independent m.d.s.remains F pd -independent.Thus we make the following conjecture.Conjecture 6.11.For any family F , the product of finitely many F -independent m.d.s.remains F -independent.

Topological proof of minimal topological K systems are strongly mixing
In this section we prove Theorem 7.1 and Corollary 7.3.For a cover V of a compact space X by open subsets, we denote by N(V) the minimal cardinality of subcovers of V. Let F be a family.A t.d.s.(X, T ) is called F -scattering if for each F = {a 1 < a 2 < . ..} ∈ F and each finite cover U of X by non-dense open subsets, one has lim n→+∞ N( n i=1 T −a i U) = ∞.It was shown in [28, Theorem 5.5] using ergodic theory that topological K systems are F inf -scattering.Combining this with the fact that a minimal F inf -scattering t.d.s. is strongly mixing [29,Theorem 5.6], one knows that a minimal topological K system is strongly mixing [28,Theorem 5.10].Now we give a topological proof of the fact that a topological K system is F inf -scattering.
Recall that for any F = {a 1 < a 2 < . ..} ∈ F inf and any open cover U of X, the topological sequence entropy of T with respect to F is defined as Theorem 7.1.Let (X, T ) be a t.d.s., n ≥ 2, (x 1 , . . ., x n ) be an F pubd -independent tuple of X with points pairwise distinct, and U 1 , . . ., U n be pairwise disjoint closed neighborhoods of x 1 , . . ., x n respectively.Set U = {U c 1 , . . ., U c n }.Then for any F ∈ F inf , one has h F top (T, U) > 0. Consequently, a topological K system is F infscattering.
Proof.Since (x 1 , . . ., x n ) is an F pubd -independent tuple, there exists an S ∈ Ind(U 1 , . . ., U n ) with positive upper Banach density d.Let F = {a 1 < a 2 < . . .} in F inf .Then by Lemma 5.14 for any k ∈ N, setting q k to be the smallest integer no less than 2k/d, we can find p k ∈ Z and W k ⊆ {a 1 , a 2 , . . ., a q k } with |W k | = k and p k + W k ⊆ S.
Thus, W k ∈ Ind(U 1 , . . ., U n ).This implies that h F top (T, U) ≥ lim sup k→+∞ 1 q k log N( Now for any finite open cover V of X by non-dense open subsets, we may find some n ≥ 2, pairwise distinct x 1 , . . ., x n in X and pairwise disjoint closed neighborhoods U 1 , . . ., U n of x 1 , . . ., x n respectively such that V refines U = {U c 1 , . . ., U c n }.If (X, T ) is topological K, then each tuple in X is F pubd -independent.Thus for any F = {a 1 < a 2 < . . .} in F inf , by the above paragraph we have h F top (T, V) ≥ h F top (T, U) > 0. This implies that N( m i=1 T −a i V) → ∞ as m → +∞, i.e., (X, T ) is F inf -scattering.Let F be a family.A t.d.s.(X, T ) is called F -transitive if for any nonempty open subsets U and V of X, one has N(U, V ) ∈ F ; it is called mildly mixing if its product with any transitive t.d.s. is transitive.It was shown in [31,Theorem 7.5] that a u.p.e.system is mildly mixing.From [31,Theorem 7.3] or [34,Theorem 3.16] one knows that a t.d.s. is u.p.e. if and only if it is F pubd -independent of order 2. Denote by ∆ the family in Z + generated by the sets F − F := {a − b : a, b ∈ F, a − b > 0} for all F ∈ F inf .By [29,Theorem 6.6] every ∆ * -transitive system is mildly mixing.Now we strengthen the above result to show that every t.d.s.being F pubd -independent of order 2 is ∆ * -transitive.For this we need the following proposition, which appeared in [12, page 84] (see also [52,Proposition 2.3]) and also follows directly from Lemma 5.14.Proof.Let (X, T ) be F pubd -independent of order 2. Then for any nonempty open subsets U and V of X, there exists an F ∈ Ind(U, V ) ∩ F pubd .Clearly F − F ⊆ N(U, V ).By Proposition 7.2 one has N(U, V ) ∈ ∆ * .
To end the appendix we make the following remark.Denote by F ss the family consisting of S ⊆ Z + satisfying that for each F ∈ F inf and each k ∈ N there exists p k ∈ Z with |F ∩ (S + p k )| ≥ k.It is clear that for any S ∈ F ss one has S − S ∈ ∆ * .Remark 7.4.One obvious corollary of Lemma 5.14 is that F pubd ⊆ F ss .We remark that there exists an S ∈ F ss containing no arithmetic progression of length 3 (and thus having zero upper Banach density by Roth's theorem [46]).
Each S k is countable.Enumerate k≥3 S k as {A 1 , A 2 , . ..}.Now let {t i } i∈N be a sequence in N and set S = i∈N (A i + t i ).Now assume that F is an infinite subset of Z.For each k ≥ 3, inductively we can find b 1 , b 2 , . . ., b k ∈ F such that b j − b i > b i − b s > 0 for all 1 ≤ s < i < j ≤ k.This implies that there exists p k ∈ Z with |F ∩ (S + p k )| ≥ k.Thus S is in F ss .
If we choose t i to grow rapidly enough, it is easy to check that S does not contain any arithmetic progression of length 3.
For a m.d.s.(X, B, µ, T ) and A, B ∈ B, we denote by N(A, B) the set {n ∈ Z + : µ(A T −n B) > 0}.Acknowledgements.W.H. and X.Y. are partially supported by a grant from NSFC and 973 Project (2006CB805903).W.H. is partially supported by FANEDD Grant 200520.H.L. is partially supported by NSF grant DMS-0701414.Part of this work was carried out during visits of H.L. to W.H. and X.Y. in the summers of 2008 and 2009.H.L. is grateful to them for their warm hospitality.

Lemma 3 . 9 .
Let F be a family.If F has the Ramsey property, then for any t.d.s.(X, T ) and closed subsets Y, Y 1 , Y 2 of X with Y = Y 1 ∪ Y 2 and Ind(Y ) ∩ F = ∅, one has either Ind(Y 1 ) ∩ F = ∅ or Ind(Y 2 ) ∩ F = ∅.The converse holds if furthermore F = bF .Proof.Suppose that F has the Ramsey property.Consider a t.d.s.(X, T ) and closed subsets

Remark 4 . 4 .
Given a probability space (X, B, µ), one may consider the equivalence relation defined on B by A ∼ B exactly when µ(A∆B) = 0, where A∆B = (A \ B) ∪ (B \ A) is the symmetric difference of A and B. The set of equivalence classes in B, denoted by B, has the induced operation of taking complement and countable union.Furthermore, µ descends to a function μ on B. The pair ( B, μ) is called a measure algebra [12, Section 5.1] [13, Section 2.1].Given a measurable and measurepreserving map T : X → X, one also gets an induced map T −1 : B → B preserving μ, complement and countable union.For any family F and k ∈ N, it is clear that whether a m.d.s.(X, B, µ, T ) is F -independent of order k or not depends only on the triple ( B, μ, T −1 ).Consider a m.d.s.(X, B, µ, T ) or a t.d.s.

3
with a(0) = 0.As 0 n k appears in y with gaps bounded above by ℓ k , 0 n k appears in C m,fm(y[0,tm]) C m,fm(y[1,tm+1]) . . .C m,fm(y[b k −tm,b k ]) with gaps bounded above by 2n m ℓ k ≤ 2n k ℓ k = b k .Thus 0 n k appears in A k+1 with gaps bounded above by b k + n k ≤ 2b k .

Lemma 5 . 14 .
For any d > 0, k ∈ N, and finite subset F ⊆ Z + with d|F | > k, there exists N = N(d, k, F ) ∈ N such that for any nonempty finite interval I ⊆ Z + and S ⊆ I with |S| |I| ≥ d and |I| ≥ N one has |S ∩ (F + p)| ≥ k for some p ∈ Z.
d.s.(X, B, µ, T ) and ( X, B b X , µ, T ) are isomorphic, where B b X
, . . ., x k ) ∈ X k is called an F -independent tuple if for any neighborhoods U 1 , . . ., U k of x 1 , . . ., x k respectively, one has Ind(U 1 , . . ., U k ) ∩ F = ∅.A t.d.s. is said to be F -independent of order k, if for each tuple of nonempty open subsets U 1 , . . ., U k , Ind(U 1 , . . ., U k ) ∩ F = ∅, and a t.d.s. is said to be [30,rder 1, then it has dense small periodic sets.If it is F rs -independent of order 2, then it is totally transitive, and hence is disjoint from all minimal systems by[30, Theorem 3.4].
Definition 5.3.We say that a t.d.s. is topological K if it is F pubd -independent.By [31, Theorem 8.3] and [34, Theorem 3.16] a t.d.s. is topological K if and only if its every finite cover by non-dense open subsets has positive topological entropy.