Localization of mixing property via Furstenberg families

This paper is devoted to studying the localization of mixing property via Furstenberg families. It is shown that there exists some $\mathscr{F}_{pubd}$-mixing set in every dynamical system with positive entropy, and some $\mathscr{F}_{ps}$-mixing set in every non-PI minimal system.

1. Introduction. Throughout this paper a topological dynamical system (or dynamical system, system for short) is a pair (X, T ), where X is a non-empty compact metric space with a metric d and T is a homeomorphism from X to itself.
Let (X, T ) be a dynamical system. For two subsets U and V of X, we define the hitting time set of U and V by We say that (X, T ) is topologically transitive (or just transitive) if for every two nonempty open subsets U and V of X, the set N (U, V ) is not empty; weakly mixing if the product system (X × X, T × T ) is transitive; strongly mixing if for every two non-empty open subsets U and V of X, the set N (U, V ) is cofinite, that is there exists N ∈ N such that {N, N + 1, N + 2, . . . } ⊂ N (U, V ).
It was shown by Furstenberg that if (X, T ) is weakly mixing, then it is weakly mixing of all finite orders, that is for any n ≥ 2 and any non-empty open subsets U 1 , U 2 , . . . , U n and V 1 , V 2 , . . . , V n of X, In [30], Xiong and Yang characterized mixing properties by Xiong chaotic sets. More specifically, a subset K of X is called a Xiong chaotic set with respect to an increasing sequence {p i } of positive integers if for any subset E of K and for any continuous map g : E → X there is a subsequence {q i } of {p i } such that lim i→∞ T qi (x) = g(x) for every x ∈ E. They showed that a non-trivial dynamical system (X, T ) is weakly mixing if and only if there exists some c-dense F σ -subset K of X that is Xiong chaotic with respect to the sequence 1, 2, 3, . . . ; strongly mixing if and only if for any increasing sequence of positive integers there is a c-dense F σ -subset K of X that is Xiong chaotic with respect to this sequence, where by 2 JIAN LI c-dense we mean K ∩ U is an uncountable set for any non-empty open subset U of X.
Inspired by Xiong chaotic sets, Blanchard and Huang introduced the concept of weakly mixing sets, which can be regarded as the localization of weak mixing [7]. A closed subset A ⊂ X is called a weakly mixing set if there exists a dense Mycielski subset (union of countable many Cantor sets) B of A such that for any subset E of B and any continuous map g : E → A there exists a sequence {q i } of positive integers such that lim i→∞ T qi (x) = g(x) for every x ∈ E. An alternative definition of weakly mixing sets is using hitting time sets. Let A be a closed subset of X with at least two point. Then A is a weakly mixing subset of X if and only if for any n ≥ 1 and any open subsets U 1 , U 2 , . . . , U n and V 1 , V 2 , . . . , V n of X intersecting A, It is shown in [7] that a topological dynamical system with positive entropy contains many weakly mixing sets. Recently, there is a series of work on the study of weak mixing sets of finite order and relative dynamical properties, see [23,26,27,28].
As the hitting time set N (U, V ) is a subset of positive integers, we can use some class of sets of positive integers to classify transitive systems. A collection F of subsets of positive integers is called a Furstenberg family (or just family), if it is hereditary upward, i.e., F 1 ⊂ F 2 and F 1 ∈ F imply F 2 ∈ F. A dynamical system (X, T ) is called F-transitive if for any two non-empty open subsets U and V of X, N (U, V ) ∈ F. We say that (X, T ) is F-mixing if (X × X, T × T ) is F-transitive.
In [17], Huang, Shao and Ye extended Xiong-Yang's result to F-mixing systems, that is a non-trivial dynamical system (X, T ) is F-mixing if and only if for every S ∈ κF (the dual family of F) there exists a dense Mycielski subset K of X which is Xiong chaotic with respect to S.
In this paper, we are devoted to studying the localization of mixing property via Furstenberg families. The paper is organized as follows. In Section 2, we recall some notions and aspects of topological dynamics. In Section 3, we introduce the concept of F-mixing sets and characterize them by Xiong chaotic sets. In Section 4, we study dynamical systems with positive entropy. We show that if an ergodic invariant measure µ has positive entropy then the collection of F pubd -mixing sets is residual in the collection of µ-entropy sets, where F pubd is the Furstenberg family of sets with positive upper Banach density. As a consequence, if a dynamical system has positive entropy then the collection of F pubd -mixing sets is dense in the collection of entropy sets. In Section 5, we study factor maps between dynamical systems. We show that if an intrinsic factor map is F-point mixing, then the collection of F-mixing sets is residual in the fiber space. We also apply this result to show that a minimal non-PI system contains some F ps -mixing set, where F ps is the Furstenberg family of piecewise syndetic sets.
2. Preliminary. In this section, we provide some basic notations, definitions and results which will be used later.
2.1. Furstenberg family. For the set of positive integers N, denote by P = P(N) the collection of all subsets of N. A subset F of P is called a Furstenberg family (or just family), if it is hereditary upward, i.e., F 1 ⊂ F 2 and F 1 ∈ F imply F 2 ∈ F. A family F is proper if it is a non-empty proper subset of P, i.e., neither empty nor all of P. It is easy to see that a family F is proper if and only if N ∈ F and ∅ ∈ F. Any non-empty collection A of subsets of F generates a family F(A) = {F ⊂ N : For a family F, the dual family is κF = {F ⊂ N : Clearly, κF is a family or a proper family if F is. It is also not hard to see that κκF = F. Let F inf be the family of all infinite subsets of N. It is easy to see that its dual family κF inf is the family of all cofinite subsets, denoted by F cf .
Convention: All the families considered in this paper are assumed to be proper and contained in F inf .
We say that a subset F of N is an IP-set if there is a subsequence {p i } of N such that {p i1 + · · · + p i k : i 1 < · · · < i k , n ∈ N} ⊂ F . The family of IP-sets is denoted by F ip .
Let F be a subset of N. The upper Banach density of F is defined by where I is taken over all non-empty finite intervals of N. The family of sets with positive upper Banach density is denoted by A subset of F of N is thick if it contains arbitrarily long runs of positive integers, i.e., for every n ∈ N there exists some k n ∈ N such that {k n , k n +1, . . . , k n +n} ⊂ F ; syndetic if there is N ∈ N such that {i, i + 1, . . . , i + N } ∩ F = ∅ for any i ∈ N; piecewise syndetic if it is the intersection of a thick set and a syndetic set. The family of syndetic sets, thick sets and piecewise syndetic sets are denoted by F s , F t and F ps , respectively.
2.2. Topological dynamics. Let (X, d) be a compact metric space. For x ∈ X and ε > 0, denote B(x, ε) = {y ∈ X : d(x, y) < ε}. For n ≥ 2, denote the nth product space X n = X × X × · · · × X (n-times) and the diagonal ∆ n (X) = {(x, x, . . . , x) ∈ X n : x ∈ X}. Let M (X) be the set of regular Borel probability measures on X. The support of a measure µ ∈ M (X), denoted by supp(µ), is the smallest closed subset C of X such that µ(C) = 1.
Let (X, T ) be a topological dynamical system. If Y is a non-empty closed subset of X and T Y ⊂ Y , then (Y, T ) forms a subsystem of (X, T ). For n ≥ 2, denote the n-th product system by (X n , T (n) ) = (X × X × · · · × X, T × T × · · · × T ) (n-times).
The orbit of a point x ∈ X, {x, T (x), T 2 (x), . . . }, is denoted by Orb(x, T ). We say that a point x ∈ X is a periodic point of (X, T ) if T n (x) = x for some n ∈ N; a recurrent point of (X, T ) if there exists an increasing sequence {k n } of positive integers such that T kn (x) → x as n → ∞; a transitive point of (X, T ) if Orb(x, T ) is dense in X. Denote by Trans(X, T ) the set of all transitive points of (X, T ). It is well known that if (X, T ) is transitive, then the Trans(X, T ) is a dense G δ subset of X.
Denote by M (X, T ) and M e (X, T ) respectively the set of all invariant probability measures and all ergodic invariant probability measures on (X, T ). A dynamical system (X, T ) is called an E-system if it is transitive and there is an invariant probability measure µ with full support, i.e., supp(µ) = X; an M-system if it is transitive and the set of minimal points is dense in X.
Let (X, T ) and (Y, S) be two dynamical systems. If there is a continuous surjection π : X → Y which intertwines the actions (i.e., π • T = S • π), then we say that π is a factor map, (Y, S) is a factor of (X, T ) or (X, T ) is an extension of (Y, S).

2.3.
Dynamical properties via families. The idea of using families to describe dynamical properties goes back at least to Gottschalk and Hedlund [12]. It was developed further by Furstenberg [11]. For a systematic study and recent results, see [1], [16], [18] and [19].
Let (X, T ) be a dynamical system. For a point x ∈ X and a subset U of X, we define the entering time set of x into U by It is well known that the following lemmas hold (see, e.g., [1,11]). Lemma 2.1. Let (X, T ) be a dynamical system and x ∈ X. Then 1.
x ∈ X is a minimal point if and only if it is an F s -recurrent point. 2.
x ∈ X is a recurrent point if and only if it is an F ip -recurrent point.
Recall that a dynamical system (X, T ) is F-transitive if for two non-empty open subsets U and V of X,

2.4.
Hyperspace. Let (X, d) be a compact metric space. Let (2 X , d H ) be the hyperspace of (X, d), i.e., the collection of all non-empty closed subsets of X endowed with the Hausdorff metric d H . We refer the reader to [2] or [21] more details on the hyperspace. Let (X, T ) be a dynamical system. It induces naturally a dynamical system (2 X ,T ) on the hyperspace, whereT (A) = T (A) for A ∈ 2 X . Let Perf(X) denote the collection of all non-empty perfect subsets of X. For every n ∈ N we define a subset L n (X) of 2 X as follows: E ∈ L n (X) if and only if there exist some k ∈ N and non-empty open subsets is an open subset of 2 X and Perf(X) = ∞ n=1 L n (X). Then Perf(X) is a G δ subset of 2 X . In addition, if X is perfect, then Perf(X) is a dense G δ subset of 2 X (see [2] or [7]).
Let R be a relation of n-tuples on X, i.e., R ⊂ X n . A subset K of X with at least n points is said to be R-dependent, or a dependent set of R if for any pairwise distinct n elements x 1 , . . . , x n of K, we have ( 3. F-mixing sets. Xiong and Yang [30] characterized weak mixing by Xiong chaotic sets. Huang, Shao and Ye [17] extended to Xiong-Yangs result to F-mixing. Inspired by Xiong chaotic sets, Blanchard and Huang [7] introduced the localizaiton of weak mixing, weakly mixing sets, and characterized them by Xiong chaotic sets. In this section, we introduce the concept of F-mixing sets and also characterize them by Xiong chaotic sets. Definition 3.1. Let (X, T ) be a dynamical system and F be a Furstenberg family. Suppose that A is a closed subset of X with at least two points. The set A is said to be F-mixing if for any k ∈ N, any open subsets Remark 3.2. By the definition, it is not hard to see that an F-mixing set must be perfect.
We have the following characterization of F-mixing sets. The proof is in main part the same as the proof of Theorem 5.2 and the Appendix of [17] in the case of F-mixing systems and the proof of Proposition 4.2 of [7] in the case of weak mixing sets. For the sake of completeness, we provide a proof here. Theorem 3.3. Let (X, T ) be a dynamical system and F be a Furstenberg family. Suppose that A is a closed subset of X with at least two points. Then A is an Fmixing set if and only if for every S ∈ κF (the dual family of F) there are Cantor (ii) for any n ∈ N and any continuous function g : C n → A there exists a subse- uniformly on x ∈ C n ; (iii) for any subset E of K and any continuous map g : for every x ∈ E.

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Proof. We first prove the sufficiency. Fix S ∈ κF. Choose K = ∞ n=1 C n satisfying requirements with respect to x and S. Fix k ∈ N and non-empty open subsets . .} be a dense countable subset of A and Y n = {y 1 , y 2 , . . . , y n }. Set a 0 = 0 and U 0 = X.
for any β ∈ {1, 2, . . . , a n } an there exists m(β) ∈ S such that Proof of Claim 1. Let a 1 = 1 and U Assume that for 1 ≤ j ≤ n − 1, we have {a j } n−1 j=1 and {U j,1 , U j,2 , . . . , U j,aj } n−1 j=1 satisfying conditions (1)- (6). Since A has no isolated points and the number of elements of Y n is n, we can take 2a n−1 ≤ a n ≤ 2a n−1 + n and non-empty open subsets U , where t n = (a n ) an . Since A is F-mixing, Choose m(β 1 ) ∈ S ∩ F 1 and then there exist non-empty open subsets U Assume that for 1 ≤ j ≤ t n −1 one has m(β 1 ), m(β 2 ), . . . , m(β j ) ∈ S and non-empty open subsets U Then there exist non-empty open subsets U Claim 2. For every n ∈ N, C n satisfies the requirement of (ii) with respect to S.
Proof of Claim 2. Fix a continuous map g : C n → A and ε > 0. Since C n is a compact set, there exists m ≥ n such that if x, y ∈ C n and d(x, y) ≤ 1 m then d(g(x), g(y)) < ε/2. For every i ∈ {1, 2, . . . , 2 m−n a n }, we choose x i ∈ U m,i ∩ C n . Note that when z i ∈ U m,i , one has d(z i , x i ) ≤ 1/m. By the choice of m, one has d(g(z i ), g(x i )) < ε/2 when z i ∈ U m,i ∩ C n .
For each x ∈ C n , there exists some j ∈ {1, 2, . . . , 2 N −n a n } such that x ∈ U N,j . By the definition of i(j), one has x ∈ U m,i(j) , then d(g(x), g(x i(j) )) < ε/2. Since This ends the proof of Claim 2.
Claim 3. C satisfies the requirement of (iii) with respect to S.
Proof of Claim 3. Fix a subset E of K and a continuous map g : E → A. For any n ≥ 1, let It should be noticed that E n may be empty when n is small. Obviously E 1 ⊂ E 2 ⊂ · · · ⊂ E and where 1 ≤ i n,1 < i n,2 < · · · < i n,bn ≤ a n . For each n ∈ N , let β n ∈ {1, 2, . . . , a n } an be any sequence with β n (i n,j ) = max k ∈ {1, 2, . . . , a n } : Let q n = m(β n ) ∈ S be as in the Claim 1.
For every ε > 0, there exists some N ∈ N such that diam(O n ) < ε for every n ≥ N . Fix x ∈ C and choose t > N such that O t is a neighborhood of g(x). As g is continuous, (2). By (4) and the definition of K, for each j ∈ N there is some i x,j ∈ {1, 2, . . . , a nt+j } such that Thus, β nt+j (i x,j ) ≥ t for any j ∈ N. Moreover, by (6) and the definition of {q n } one has for j ∈ N, Hence the whole proof is finished. 4. Dynamical systems with positive topological entropy. In this section, we study dynamical systems with positive topological entropy. We refer the reader to the textbook [29] for the theory of entropy. Let (X, T ) be a dynamical system. For an open cover U of X, denote the topological entropy of U by h top (T, U). The topological entropy of (X, T ) is where the supremum is taken over all open covers U.
Let µ be an invariant measure on (X, T ). For a finite measurable partition α of X, denote the µ-entropy of α by h µ (T, α). The entropy of µ is where α ranges over all finite measurable partitions of (X, B X ).
The variational principle tells us that h top (T ) = sup µ∈M e (X,T ) h µ (T ).
A measure-theoretical dynamical system is a Kolmogorov system if and only if it has uniformly positive entropy (i.e., each finite non-trivial partition has positive entropy) if and only if it has completely positive entropy (i.e., each of its nontrivial factors has positive entropy) if and only if it is disjoint from every zeroentropy system. The notion of Kolmogorov system plays an important role in ergodic theory. To get a topological analogy, Blanchard [4] introduced the notions of complete positive entropy and uniformly positive entropy in topological dynamics. He then naturally defined the notion of entropy pairs and used it to show that a uniformly positive entropy system is disjoint from all minimal zero entropy systems [5]. Later on, in [6] the authors were able to define entropy pairs for an invariant measure and showed that for each invariant measure the set of entropy pairs for this measure is contained in the set of entropy pairs. To obtain a better understanding of the topological version of a Kolmogorov system, Huang and Ye [20] introduced the notions of entropy n-tuples (n ≥ 2) both in topological and measure-theoretical settings. See [15] for a survey on the local entropy theory.
Let K ⊂ X be a non-empty set and U be an open cover or a partition of X. We say that U is admissible with respect to K if U ⊃ K for any U ∈ U. Denote by E n (X, T ) the set of all topological entropy n-tuples and by E µ n (X, T ) the set of all entropy n-tuples for µ.
In order to find where the entropy is concentrated, the notion of entropy sets was introduced by Dou-Ye-Zhang [8] and Blanchard-Huang [7], independently. In this paper, we follow the definition in [7] which requires entropy sets to be closed. Denote by E s (X, T ) the collection of all entropy sets and by E µ s (X, T ) the collection of all µ-entropy sets. It follows immediately from definitions that a closed subset K of X with at least two points is an entropy set (resp. µ-entropy set) if and only if for every n ≥ 2, every n-distinct n points k 1 , . . . , k n ∈ K one has (k 1 , . . . , k n ) ∈ E n (X, T ) (resp. (k 1 , . . . , k n ) ∈ E µ n (X, T ). Define H(X, T ) = E s (X, T ) and H µ (X, T ) = E µ s (X, T ). A point x ∈ X is called an entropy point if {x} ∈ H(X, T ); a µ-entropy point if {x} ∈ H µ (X, T ). The set of all entropy points and µ-entropy points of (X, T ) is denoted by E 1 (X, T ) and E µ 1 (X, T ), respectively. Theorem 4.4 ([7]). Let (X, T ) be a dynamical system. If µ ∈ M e (X, T ) with h µ (X, T ) > 0, then E µ 1 (X, T ) = supp(µ). It is not hard to see that It is shown in [7] that if an ergodic invariant measure µ ∈ M e (X, T ) has positive entropy, then (H µ (X, T ),T ) is an E-system. We could extend the result as follows. ). For every n ≥ 1, let R n = Trans(supp(λ µ n ), T (n) ). Then R n is a dense G δ subset of R (n) π , and it is a G δ subset of X n since R (n) π is closed in X n . By Proposition 2.4, D(R n ) is a G δ subset of 2 X . For every n ≥ 1, R n is dense in supp(λ µ n ), then D(R n ) is dense in H µ (X, T ). So D(R n ) is a dense G δ subset of H µ (X, T ). By [7,Theorem 4.5], Perf(X) ∩ H µ (X, T ) is also a dense G δ subset of H µ (X, T ).
Theorem 4.6. Let (X, T ) be a dynamical system and µ ∈ M e (X, T ) with h µ (T ) > 0. Suppose that µ = µ y dν is the disintegration of µ over the Pinsker factor (Y, ν, S). Then there is a Borel subset Y 0 of Y with ν(Y 0 ) = 1 such that for every y ∈ Y 0 , supp(µ y ) is an F pubd -mixing set. Moreover, the collection of F pubd -mixing sets is residual in H µ (X, T ).
Proof. For every n ≥ 1, let R n = Trans(supp(λ µ n ), T (n) ). We do not know the structure of the collection of F pubd -mixing sets, but we can handle a subclass Q µ of µ-entropy sets. A perfect µ-entropy set A is in Q µ if and only if for every ε > 0 there exists a subset E of A such that E is {R n }-dependent and d H (E, A) < ε. By the proof of Theorem 4.5, we know that D(R n ) ∩ Perf(X) is a dense G δ subset of H µ (X, T ). Then Q µ is residual in H µ (X, T ). We are going to show that every element in Q µ is an F pubd -mixing set.

JIAN LI
Proof. By Theorems 2.9 and 2.10 of [7], µ∈M e (X,T ) H µ (X, T ) is dense in H(X, T ). Then the result follows from Theorem 4.6.
Remark 4.8. It is interesting to know that whether the collection of F pubd -mixing sets is residual in H(X, T ). It is shown in [7,Theorem 4.4] that the collection of weakly mixing sets is a G δ subset of 2 X . But we do not know whether the collection of F pubd -mixing sets is a G δ subset of 2 X . 5. F-(point) mixing extensions. In this section, we study factor maps between dynamical systems. We show that if an intrinsic factor map is F-point mixing, then the collection of F-mixing sets is residual in the fiber space. We also apply this result to non-PI factors maps between minimal systems.
Let π : (X, T ) → (Y, S) be a factor map between two dynamical systems. For every n ≥ 2, denote . , x n ) ∈ X n : π(x 1 ) = · · · = π(x n )}. Inspiring by the idea of entropy sets, we define the fiber space of π as It is clear that H π isT -invariant and it is closed by Lemma 5.1. Then (H π ,T ) is a subsystem of (2 X ,T ).
Proof. Let {A n } be a sequence in H π with A n → A as n → ∞. For every two points a, b ∈ A, there are a n , b n ∈ A n such that a n → a, b n → b as n → ∞. Since A n is in H π , we have π(a n ) = π(b n ). By continuity of π, π(a) = π(b). Then there exists y ∈ Y such that A ⊂ π −1 (y), which implies that A ∈ H π .
Remark 5.2. It is clear that for every x ∈ X, {x} ∈ H π . Then (X, T ) can be regarded as a system of (H π ,T ). If π : X → Y is a homeomorphism, then H π is homeomorphic to X. This case is not interesting in our consideration. We say that a factor map π : (X, T ) → (Y, S) is intrinsic if π is not a homeomorphism.
We say that the factor map π : (X, T ) → (Y, S) is weak mixing of order n ∈ N\{1} if (R (n) π , T (n) ) is transitive; weak mixing of all finite orders if it is weakly mixing of order m for every m ≥ 2. Note that there exists some factor maps which is weak mixing of order 2 but not order 3 (see [14] and [26]). Lemma 5.3. Let π : (X, T ) → (Y, S) be an intrinsic factor map between two dynamical systems. If π is weakly mixing of all finite orders, then H π ∩ Perf(X) is a dense G δ subset of H π .
Let F be a Furstenberg family. We say that the factor map π : (X, T ) → (Y, S) is F-mixing of all finite orders, if (R (n) π , T (n) ) is F-transitive for every n ≥ 2.
Theorem 5.4. Let π : (X, T ) → (Y, S) be an intrinsic factor map and F be a Furstenberg family. If π is F-mixing of all finite orders, then (H π ,T ) is F-mixing.
Theorem 5.5. Let π : (X, T ) → (Y, S) be an intrinsic factor map and F be a Furstenberg family. If π is F-point mixing of all finite orders, then the collection of F-mixing sets is residual in H π . Furthermore, if π is open, then there is a dense G δ subset Y 0 of Y , such that π −1 (y) is an F-mixing set for every y ∈ Y 0 .
Corollary 5.6. Let (X, T ) be a dynamical system.
1. If (X, T ) is a weakly mixing E-system, then the collection of F pubd -mixing sets is residual in 2 X . 2. If (X, T ) is a weakly mixing M-system, then the collection of F ps -mixing sets is residual in 2 X .
Proof. We only prove (1), since the proof of (2) is similar. Assume that (X, T ) is a weakly mixing E-system. It is clear that for every n ≥ 2, (X n , T (n) ) is also an E-system. By Theorem 2.3, Trans F pubd (X n , T (n) ) is a dense G δ subset of X n . Then the result follows from applying Theorem 5.5 to the factor map from (X, T ) to the trivial system.
Theorem 5.7. Let π : (X, T ) → (Y, S) be a factor map between minimal systems. If π is a non-PI extension, then there is a dense subset Y 0 of Y such that for every y ∈ Y 0 there exists some F ps -mixing subset of π −1 (y).