This paper is devoted to studying the localization of Δ-mixing property via Furstenberg families. The notion of Δ-weakly mixing sets is extended to F-Δ-weakly mixing sets with respect to a sequence, and the characterization of Δ-weakly mixing sets is also generalized.

NSF of Anhui Province1608085QA12NSF of Education Committee of Anhui ProvinceKJ2016A506KJ2017A454Excellent Young Talents Foundation of Anhui ProvinceGXYQ2017070Doctoral Scientific Research Foundation of Chaohu UniversityKYQD-201605Scientific Research Project of Chaohu UniversityXLY-2015011. Introduction

Throughout this paper, a topological dynamical system (or dynamical system, system for short) is a pair (X,f), where X is a compact metric space with metric ρ and f is a continuous map of X to itself.

In some literature on nonautonomous discrete systems, the topological dynamical system is usually called the autonomous discrete system; for example, see [1].

Let (X,f) be a dynamical system. For two subsets U, V of X, we define the hitting time set of U and V by (1)NU,V≔n∈N:U∩f-nV≠∅.We say that (X,f) is transitive if, for every two nonempty open subsets U and V of X, the hitting time set N(U,V) is nonempty, weakly mixing if the product system (X×X,f×f) is transitive, and strongly mixing if, for every two nonempty open subsets U and V of X, there exists N∈N such that {N,N+1,…}⊂N(U,V). In his seminal paper [2], Furstenberg showed that weak mixing implies n-fold transitivity for every positive integer n as follows.

Let (X,f) be a topological dynamical system. Then (X×X,f×f) is transitive if and only if, for every n∈N, the product system (X×X×⋯×X,f×f×⋯×f) (n-times) is also transitive.

For convenience, we denote (X×X×⋯×X,f×f×⋯×f) (n-times) by (Xn,f(n)).

A closed subset C of X is called Δ-transitive if, for every d≥2, there exists a residual subset C′ of C such that, for every x∈C′, the orbit closure of the diagonal d-tuple x(d), that is, x(d)=(x,x,…,x︸d-times), under the action f×f2×⋯×fd contains Cd. A closed subset C of X with at least two points is Δ-weakly mixing if, for every n≥1, Cn is Δ-transitive in the n-th product system (Xn,f(n)). In [3], Huang et al. show that Δ-weakly mixing sets exhibit a nice characterization as follows.

Theorem 1 (see [<xref ref-type="bibr" rid="B9">3</xref>]).

Let (X,f) be a dynamical system and E a closed subset of X but not a singleton. Then E is Δ-weakly mixing if and only if there exists a strictly increasing sequence of Cantor sets C1⊂C2⊂⋯ of E such that C≔∪k=1∞Ck is dense in E and

for any d∈N, any subset A of C, and any continuous functions gj:A→E for j=1,2,…,d, there exists a strictly increasing sequence {qk}k=1∞ of positive integers such that (2)limk→∞fj·qkx=gjx

for every x∈A and j=1,2,…,d;

for any d∈N, k∈N, any closed subset B of Ck, and continuous functions hj:B→E for j=1,2,…,d, there exists a strictly increasing sequence {qk}k=1∞ of positive integers such that (3)limk→∞fj·qkx=hjx

uniformly on x∈B and j=1,2,…,d.

In this paper, we will extend the notion of Δ-weakly mixing sets via Furstenberg family and show that F-Δ-weakly mixing sets with respect to a sequence also share the same characterization under some considerable conditions.

2. Preliminary

In this section, we provide some definitions which will be used later.

Let P denote the collection of all subsets of N. A subset F of P is called a Furstenberg family (or family for short ), if it admits the property of hereditary upward; that is to say, (4)F1⊂F2 andF1∈FimplyF2∈F.A family F is called proper if it is neither empty nor all of P.

For a family F, the dual family of F is defined by (5)κF≔F∈P:F∩F′≠∅, for every F′∈F.Let Finf be the family of all infinite subsets of N. It is easy to see that its dual family κFinf, denoted by Fcf, is the family of all cofinite subsets.

Any nonempty collection A of subsets of N naturally generates a family (6)A=F⊂N:A⊂F for someA∈A.We say that F is countable generated if F=[A] for a collection A consisting of countable subsets of N.

The idea of using families to describe dynamical properties goes back at least to Gottschalk and Hedlund [4] and was developed further by Furstenberg [5]. For recent results, see [6–9].

Let (X,f) be a dynamical system, F a family, and a≔{ai}i=0∞ a strictly increasing sequence of nonnegative integers with a0=0.

For every d≥2 and subsets U1,U2,…,Ud of X, we define the hitting time set of U1,U2,…,Ud by (7)NaU1,U2,…,Ud≔n∈N:⋂i=1df-ai-1nUi≠∅

We say that A is Δ-F-weakly mixing with respect to a if, for every n,d≥2, and nonempty open subsets Ui,j of X intersecting A for i=1,2,…,n and j=1,2,…,d, we have (8)⋂i=1nNaUi,1∩A,Ui,2,…,Ui,d∈F.

Let (X,ρ) be a compact metric space. Denote 2X by the collection of all nonempty closed subsets of X and endow 2X with the Hausdorff metric (9)ρHA,B=maxmaxx∈Aminy∈Bρx,y,maxy∈Bminx∈Aρx,yfor any A,B∈2X. Then the metric space (2X,ρH) is compact whenever X is compact. For any nonempty subsets S1,…,Sn⊂X, denote (10)S1,…,Sn≔A∈2X:A⊂⋃i=1nSi andA∩Si≠∅ for each i=1,…,n,then the following family (11)U1,…,Un:U1,…,Unarenon-emptyopensubsets ofX,n∈Nforms a basis for a topology of 2X, which is called the Vietoris topology. It is well known that the Hausdorff topology induced by the Hausdorff metric ρH coincides with the Vietoris topology for 2X.

3. Key Lemmas

In this section, we provide some lemmas which will be used later.

Let (X,f) be a dynamical system, F a family, a≔{ai}i=0∞ a strictly increasing sequence of nonnegative integers with a0=0, and E a closed subset of X. For ε>0, d≥1, and any S∈kF, we say that a subset A of X is (S,ε,ρ)-spread with respect to a, if there exist δ∈(0,ε), n∈N, and distinct points x1,x2,…,xn∈X such that A⊂∪i=1nB(xi,δ) and for any maps (12)gj:x1,x2,…,xn→Ewhere j=1,2,…,d, there exists k∈S such that 1/k<ε and (13)faj·knBx1,δ×⋯×Bxn,δ⊂Bgjx1,ε×⋯×Bgjxn,εfor j=1,2,…,d.

For any S∈kF, denote by SS,a(ε,ρ,E) the collection of all closed sets that are (S,ε,ρ)-spread in E with respect to a.

Remark 2.

It is not hard to check that the SS,a(ε,ρ,E) is hereditary; that is, if A is (S,ε,ρ)-spread in E with respect to a and B is a nonempty closed subset of A, then B is also (S,ε,ρ)-spread in E with respect to a.

Let (14)SS,aE=⋂d=1∞⋂k=1∞SS,a1k,ρ,E,SF,aE=⋂S∈kFSS,aE.

Lemma 3.

If E is a Δ-F-weakly mixing subset of X but not a singleton, then SS,a(E)∩2E is a dense open subset of 2E for every S∈kF.

Proof.

Fix ε>0, d∈N, and S∈kF. We will divide our discussion into three claims to show that SS,a(E)∩2E is a dense open subset of 2E.

Claim 4. SS,a(ε,ρ,E) is open in 2X.

Proof of Claim 4. Let A∈SS,a(ε,ρ,E). For any S∈kF, let δ>0 and x1,x2,…,xn∈X as in the definition of set (S,ε,ρ)-spread in E with respect to a. For i=1,2,…,n, put Ui=B(xi,δ). Then, by Remark 2, every closed subset B⊂∪i=1nUi is (S,ε,ρ)-spread in E with respect to a. It follows that (15)A∈U1,…,Un⊂SS,aε,ρ,E.Hence SS,a(ε,ρ,E) is open in 2X.□

Claim 5. E is a perfect set.

Proof of Claim 5. If E is not perfect, then there exists an isolated point x of E. Note that the set {x} is open in E. It follows that there are two nonempty open subsets U1,U2⊂X such that (16)U1∩E=x,U2∩E≠∅and U1∩U2=∅ since the set E is not a singleton. So, for any m∈S, such that fa1m{x}∩U1≠∅, one has fa1m{x}∩U2=∅. This is a contradiction, and thus E is perfect.□

Claim 6. SS,a(ε,ρ,E)∩2E is dense in 2E.

Proof of Claim 6. Fix n nonempty open subsets U1,…,Un of X intersecting E. We want to show that (17)U1,…,Un∩SS,aε,ρ,E∩2E≠∅.Since E is compact, there exists a finite subset {x1,x2,…,xm} of E such that ∪i=1mB(xi,ε/2)⊃E. For convenience, denote Bi=B(xi,ε/2) for i=1,2,…,m. Since E is Δ-F-weakly mixing with respect to a, the set E is perfect, so we may assume that m≥1/ε. The collection of n-tuples on the set {1,2,…,m}n can be arranged as the following finite sequence: (18)γ1,1,…,γ1,n,γ2,1,…,γ2,n,…,γl,1,…,γl,n.For (γ1,1,…,γ1,n), as E is Δ-F-weakly mixing with respect to a, there exists k1∈S with k1>m, such that (19)Ui∩E∩f-a1k1Bγ1,1∩f-a2k1Bγ1,2∩⋯∩f-adk1Bγ1,d≠∅for i=1,2,…,n. By continuity of f, we can choose a nonempty subset Wi1 of Ui intersecting E such that faj·k1Wi1⊂Bγ1,j for j=1,2,…,d. Similarly, for (γ2,1,…,γ2,n), since S is a strictly increasing sequence of positive integers, there exist k2∈S with k2>k1 and nonempty subsets Wi2 of Wi1 intersecting E such that faj·k2Wi2⊂Bγ2,j for i=1,2,…,n and j=1,2,…,d. After repeating this process l times, we obtain l distinct positive integers k1,k2,…,kl∈S with k1>k2>⋯>kl, and nonempty open subsets Wi1,Wi2,…,Wil intersecting E such that (20)Wil⊂Wil-1⊂⋯⊂Wi1⊂Ui,i=1,2,…,n, and(21)faj·krnW1r×W2r×⋯×Wnr⊂Bγr,j×Bγr,j×⋯×Bγr,jfor r=1,2,…,l, and j=1,2,…,d.

For each i=1,2,…,n, pick xi∈Wil∩E. Since E is perfect, it is reasonable to assume that those xi’s are distinct. It is clear that (22)x1,…,xn∈U1,…,Un∩2E.Choose 0<δ<ε such that B(xi,δ)⊂Wil for i=1,2,…,n. For any map (23)gj:x1,x2,…,xn→Efor j=1,2,…,d, there exists an n-tuple (γ0,1,…,γ0,n) such that Vγ0,j⊂B(gj(xi),ε) for i=1,2,…,n, and j=1,2,…,d. Thus there is k∈{k1,…,kr}(⊂S) such that (24)faj·kBxi,δ⊂faj·kWil⊂Vγ0,j⊂Bgjxi,εfor i=1,2,…,n, and j=1,2,…,d. This implies {x1,…,xn} is (S,ε,ρ)-spread in E with respect to a and hence SS,a(ε,ρ,E)∩2E is dense in 2E.□

Lemma 7.

Let (X,f) be a dynamical system and F a Furstenberg family with kF being countable generated. If E is a Δ-F-weakly mixing subset with respect to a but not a singleton, then SF(E)∩2E is a residual subset of 2E.

Proof.

Since kF is countable generated, there exists a sequence {Si}i=1∞ of P such that (25)kF=S⊂N:Si⊂S for some i∈N.By Lemma 3, it follows that (26)SF,aE∩2E=⋂S∈kFSS,aE∩2E=⋂i∈NSSi,aE∩2E.And this completes the proof of Lemma 7.

Lemma 8.

If C∈SS,a(E), then, for any closed subset B of C, d∈N, and continuous functions hj:B→E for j=1,2,…,d, there exists a strictly increasing sequence {qk}k=1∞⊂S such that (27)limk→∞faj·qkx=hjxuniformly on x∈B and j=1,2,…,d.

Proof.

Let B be a closed subset of C, d∈N. Let hj:B→E, j=1,2,…,d, be continuous functions. Then hj is uniform continuous and this follows that, for any k∈N, there exists η>0 such that (28)ρx,y<ηimpliesρhjx,hjy<1k,j=1,2,…,d.Choose l>k with 1/l<η. Let {xk,1,…,xk,nk} and 0<δk<1/l be as in the definition of B which is (S,1/l,ρ)-spread in E. By the definition of (S,1/l,ρ)-spread subset, there exists qk∈S with qk>k such that faj·qk(B(xk,i,δk))⊂B(hj(xk,i),1/l) for i=1,2,…,nk and j=1,2,…,d. Without lose of generality, it can be assumed that {qk}k=1∞ is increasing.

We are going to show that the sequence {qk}k=1∞ is as required.

For any x∈B, there exists yk,nkx such that ρ(x,yk,nkx)<δk<1/k<η. Then (29)ρfaj·qkx,hjx≤ρfaj·qkx,hjyk,nkx+ρhjyk,nkx,hjx<1l+1k<2kfor j=1,2,…,d. Thus (30)limk→∞faj·qkx=hjxfor j=1,2,…,d. This completes the proof of Lemma 8.

Lemma 9.

If C1⊂C2⊂⋯ is a strictly increasing sequence of sets in SS,a(E), then, for any subset A of C≔∪i=1∞Ci, d∈N and continuous functions gj:A→E for j=1,2,…,d, there exists a strictly increasing sequence {qk}k=1∞⊂S such that (31)limk→∞faj·qkx=gjxfor every x∈A and j=1,2,…,d.

Proof.

Let A⊂C, d∈N, and gj:A→E, j=1,2,…,d, be continuous functions. For k≥1, take Ak=A∩Ck. Since SS,a(E) is hereditary, the closure Ak¯ of Ak is also in SS,a(E) for all k≥1.

For any k∈N, the set Ak is (1/k,ρ)-spread in E. Let yk,1,…,yk,nk and 0<δk<1/k be as in the definition of Ak which is (1/k,ρ)-spread in E. Then there exists qk∈S with qk>k such that (32)faj·qknByk,1,δk×⋯×Byk,n,δk⊂Bgjyk,1,1k×⋯×Bgjyk,n,1kfor j=1,2,…,d. Without lose of generality, it can be assumed that {qk}k=1∞ is increasing.

We are going to show that the sequence {qk} is as required.

Fix any x∈A. There exists M1∈N such that x∈Ak for all k≥M1. By the continuity of gj, for any ε>0, there exists M2∈N with M2>ε such that (33)ρx,y<1M2impliesρgjx,gjy<εforj=1,2,…,d.For every k>max{M1,M2}, there exists yk,nkx such that ρ(x,yk,nkx)<δk<1/k. Then (34)ρfaj·qkx,gjx≤ρfaj·qkx,gjyk,nkx+ρgjyk,nkx,gjx<1k+ε<2εfor j=1,2,…,d. Thus (35)limk→∞faj·qkx=gjxfor j=1,2,…,d. This completes the proof of Lemma 9.

4. The Characterization of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M418"><mml:mrow><mml:mi mathvariant="script">F</mml:mi></mml:mrow></mml:math></inline-formula>-<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M419"><mml:mrow><mml:mi>Δ</mml:mi></mml:mrow></mml:math></inline-formula>-Mixing Sets

In this section, we will show the following theorem which generalizes the result in [3].

Theorem 10.

Let (X,f) be a dynamical system, F a Furstenberg family with kF being countable generated, E a closed subset of X but not a singleton, and a≔{ai}i=0∞ a strictly increasing sequence of nonnegative integers with a0=0. Then E is Δ-F-weakly mixing with respect to a if and only if, for any S∈kF, there exists an increasing sequence of Cantor sets C1⊂C2⊂⋯ of E such that C≔∪k=1∞Ck is dense in E and

for any d∈N, any subset A of C, and any continuous functions gj:A→E for j=1,2,…,d, there exists a strictly increasing sequence {qk}k=1∞ of S such that (36)limk→∞faj·qkx=gjx

for every x∈A and j=1,2,…,d;

for any d∈N, k∈N, any closed subset B of Ck, and continuous functions hj:B→E for j=1,2,…,d, there exists a strictly increasing sequence {qk}k=1∞ of S such that (37)limk→∞faj·qkx=hjx

uniformly on x∈B and j=1,2,…,d.

A subset Q of 2X is called hereditary if 2A⊂Q for every set A∈Q. The following lemma, which is a consequence of the Kuratowski-Mycielski Theorem, is cited from [3].

Lemma 11.

Let X be a perfect compact space. If a hereditary subset Q of 2X is residual then there exists an increasing sequence of Cantor sets C1⊂C2⊂⋯ of X such that

Ci∈Q for every i≥1;

C=∪i=1∞Ci is dense in X.

Proof of Theorem <xref ref-type="statement" rid="thm4.1">10</xref>.

Necessity. Since E is F-Δ-weakly mixing with respect to a, by Lemma 7, SF,a(E)∩2E is a residual subset of 2E. Now, by Lemma 11, there exists a strictly increasing sequence of Cantor sets C1⊂C2⊂⋯ of E such that Ci∈SF,a(E) for every i≥1 and C=∪i=1∞Ci is dense in E. The conclusion follows from Lemmas 8 and 9.

Sufficiency. Fix any S∈kF. Let C be the set satisfying the requirement. Let n,d≥2 and Ui,j be nonempty open subsets of X intersecting E for i=1,2,…,n, and j=1,2,…,d. It is not hard to see that E is perfect. It follows that there exist pairwise distinct points xi,j∈Ui,j∩E for i=1,2,…,n, and j=1,2,…,d.

For j=1,2,…,d-1, define gj:A={xi,1:i=1,2,…,n}→E by gj(xi,1)=xi,j+1 for i=1,2,…,n. Choose ε>0 such that B(xi,j,ε)⊂Ui,j for all i,j. It is clear that gj are continuous; thus we can find k∈S such that ρ(faj·kxi,1,gj(xi,1))<ε for all i,j. Then faj·k(xi,1)∈Ui,j+1 for all i,j, which implies (38)S∩NaU1,1∩A×⋯×Un,1∩A︸n-times,U1,2×⋯×Un,2︸n-times,…,U1,d×⋯×Un,d︸n-times≠∅.Therefore, the set E is F-Δ-weakly mixing with respect to a.

It is not hard to see that a subset A of X is Δ-weakly mixing if and only if Δ-Finf-weakly mixing with repect to {n}, and since Fcf={S⊂N:Si⊂S for some i∈N}, where Si={i,i+1,i+2,…} for each i∈N, it follows that Fcf is coutable generated. Thus our result extended [3, Theorem A].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank the referee and the editor for their valuable suggestions, and the authors sincerely thank Doctor Zhi-Jing Chen for his kind help and concern all the time. The second and third authors are supported by NSF of Anhui Province (no. 1608085QA12), NSF of Education Committee of Anhui Province (nos. KJ2016A506 and KJ2017A454), Excellent Young Talents Foundation of Anhui Province (no. GXYQ2017070), Doctoral Scientific Research Foundation of Chaohu University (no. KYQD-201605), and Scientific Research Project of Chaohu University (no. XLY-201501).

BalibreaF.On problems of topological dynamics in non-autonomous discrete systemsFurstenbergH.Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximationHuangW.LiJ.YeX.ZhouX.Positive topological entropy and Δ-weakly mixing setsGottschalkW. H.HedlundG. A.FurstenbergH.AkinE.GlasnerE.Classifying dynamical systems by their recurrence propertiesHuangW.YeX.Topological complexity, return times and weak disjointnessHuangW.YeX.Dynamical systems disjoint from any minimal system