Blanchard and Huang introduced the notion of weakly mixing subset, and Oprocha and Zhang gave the concept of transitive subset and studied its basic properties. In this paper our main goal is to discuss the weakly mixing subsets and transitive subsets in set-valued discrete systems. We prove that a set-valued discrete system has a transitive subset if and only if original system has a weakly mixing subset. Moreover, we give an example showing that original system has a transitive subset, which does not imply set-valued discrete system has a transitive subset.

1. Introduction

Throughout this paper a topological dynamical system (abbreviated to TDS) is a pair (X,f), where X is a compact metric space with metric d and f:X→X is a continuous map. When X is finite, it is a discrete space and there is no any nontrivial convergence. Hence, we assume that X contains infinitely many points. Let ℕ denote the set of all positive integers and let ℤ+=ℕ∪{0}.

Topological transitivity, weak mixing, and sensitive dependence on initial conditions (see [1–4]) are global characteristics of topological dynamical systems. Let (X,f) be a TDS. (X,f) is (topologically) transitive if for any nonempty open subsets U and V of X there exists an n∈ℕ such that fn(U)∩V≠∅. (X,f) is (topologically) weakly mixing if for any nonempty open subsets U1,U2,V1, and V2 of X, there exists an n∈ℕ such that fn(U1)∩V1≠∅ and fn(U2)∩V2≠∅. It follows from these definitions that weak mixing implies transitivity.

In [5], Blanchard introduced overall properties and partial properties. For example, sensitive dependence on initial conditions, Devaney chaos (see [6]), weak mixing, mixing, and more belong to overall properties; Li-Yorke chaos (see [7]) and positive entropy (see [1, 8]) belong to partial properties. Weak mixing is an overall property; it is stable under semiconjugate maps and implies Li-Yorke chaos. We have a weakly mixing system that always contains a dense uncountable scrambled set (see [9]). In [10], Blanchard and Huang introduced the concepts of weakly mixing subset, derived from a result given by Xiong and Yang [11] and showed “partial weak mixing implies Li-Yorke chaos” and “Li-Yorke chaos can not imply partial weak mixing.”

Motivated by the idea of Blanchard and Huang's notion of “weakly mixing subset,” Oprocha and Zhang [12] extended the notion of weakly mixing subset, gave the concept of “transitive subset,” and discussed its basic properties. In recent years, many authors studied the dynamical properties for set-valued discrete systems. Román-Flores [13], Banks [14], Peris [15], Wang and Wei [16], and Acosta et al. [17] investigated the properties of topological transitivity and weak mixing for set-valued discrete systems. Fedeli [18], Guirao et al. [19] and Hou et al. [20] studied Devaney chaos for set-valued discrete systems. Lampart and Raith [21] discussed topological entropy for set-valued maps. Liu et al. [22] and Wang et al. [23] studied sensitivity of set-valued discrete systems. Wu and Xue [24] discussed shadowing property for induced set-valued dynamical systems. Also, we continue to discuss transitive subsets, weakly mixing subsets for set-valued discrete systems, and investigate the relationship between set-valued discrete system and original system on transitive subset, weakly mixing subset. More precisely, a set-valued discrete system has a transitive subset if and only if original system has a weakly mixing subset and we give an example showing that original system has a transitive subset which does not imply set-valued discrete system has a transitive subset. Moreover, we prove that a transitive point of set-valued discrete system is a transitive subset of original system.

2. Preliminaries

A TDS (X,f) is point transitive if there exists a point x0∈X with dense orbit, that is, orb(x0)¯=X, where orb(x0)¯ denotes the closure of orb(x0). Such a point x0 is called transitive point of (X,f). If X is a compact metric space without isolated points, then topologically transitive and point transitive are equivalent (see [2]). A TDS (X,f) is minimal if orb(x,f)¯=X for every x∈X; that is, every point is transitive point. A point x is called minimal if the subsystem (orb(x,f)¯,f) is minimal.

The distance from a point x to a nonempty set A in X is defined by
(1)d(x,A)=infa∈Ad(x,a).

Let 2X be the family of all nonempty compact subsets of X. The Hausdorff metric on 2X is defined by
(2)dH(A,B)=max{supa∈Ad(a,B),supb∈Bd(b,A)}foreveryA,B∈2X.

It follows from Michael [25] and Engelking [26] that 2X is a compact metric space. The Vietoris topology τυ on 2X is generated by the base
(3)υ(U1,U2,…,Un)={F∈2X:F⊆⋃i=1nUi,F∩Ui≠∅∀i≤n⋃i=1nUi},
where U1,U2,…,Un are open subsets of X.

Let 2f be the induced set-valued map defined by
(4)2f:2X⟶2X,2f(F)=f(F)foreveryF∈2X.
Then 2f is well defined. (2X,2f) is called a set-valued discrete system.

Let X be T1 space; that is, single point set is closed. Then 2A={F∈2X:F⊆A} is a closed subset of 2X for any nonempty closed subset A of X (see [25]).

Definition 1 (see [<xref ref-type="bibr" rid="B4">10</xref>]).

Let (X,f) be a TDS and let A be a closed subset of X with at least two elements. A is said to be weakly mixing if for any k∈ℕ, any choice of nonempty open subsets V1,V2,…,Vk of A and nonempty open subsets U1,U2,…,Uk of X with A∩Ui≠∅, i=1,2,…,k, there exists an m∈ℕ such that fm(Vi)∩Ui≠∅ for 1≤i≤k. (X,f) is called partial weak mixing if X contains a weakly mixing subset.

Definition 2 (see [<xref ref-type="bibr" rid="B19">12</xref>]).

Let (X,f) be a TDS and A be a nonempty subset of X. A is called a transitive subset of (X,f) if for any choice of nonempty open subset VA of A and nonempty open subset U of X with A∩U≠∅, there exists an n∈ℕ such that fn(VA)∩U≠∅.

Remark 3.

(1) (X,f) is topologically transitive if and only if X is a transitive subset of (X,f).

(2) By [12], A is a transitive subset if and only if A¯ is a transitive subset, where A¯ denotes the closure of A.

According to the definitions of transitive subset and weakly mixing subset, we have the following.

Result 1.

If A is a weakly mixing subset of (X,f), then A is a transitive subset of (X,f).

Result 2.

If a∈X is a transitive point of (X,f), then {a} is a transitive subset of (X,f).

Result 3.

If A=orb(x,f) is a periodic orbit of (X,f) for some x∈X, then A is a transitive subset of (X,f).

Example 4.

Tent map
(5)f(x)={2x,if0≤x≤12,2(1-x),if12≤x≤1,
is shown in Figures 1 and 2, which is known to be transitive on I=[0,1] (see [6]). We prove that [1/4,3/4] is a transitive subset of (X,f).

Let S(fk) denote the set of extreme value points of fk for every k∈ℕ. Then S(fk)={1/2k,2/2k,…,(2k-1)/2k}. Since S(f)={1/2}, f(1/2)=1, f(0)=0, and f(1)=0, we have
(6)fk(x)={1,ifx=12k,32k,…,2k-12k,0,ifx=0,22k,42k,…,2k-22k,1.
Let Ikj=[j/2k,(j+1)/2k] for 0≤j≤2k-1. Then fk(Ikj)=[0,1]. For any nonempty open set U of [1/4,3/4], without loss of generality, we take U=(x0-ε,x0+ε) for a given ε>0 and x0∈int[1/4,3/4], where int[1/4,3/4] denotes the interior of [1/4,3/4]. When l∈ℕ and l>log2(1/ε), then there exist j∈ℤ+ and 0≤j≤2l-1 such that Ilj⊆U. Furthermore, we have fl(U)=[0,1]. Thus, for any nonempty open set U of [1/4,3/4] and nonempty open set V of [0,1] with V∩[1/4,3/4]≠∅, there exists a k∈ℕ such that fk(U)∩V≠∅. This shows that [1/4,3/4] is a transitive subset of (I,f).

Definition 5 (see [<xref ref-type="bibr" rid="B7">27</xref>]).

Let (X,τ) be a topological space and A be a nonempty set of X. A is a regular closed set of X if A=int(A)¯, where int(A) denotes the interior of A.

We easily prove that A is a regular closed set if and only if int(VA)≠∅ for any nonempty set VA of A.

Theorem 6 (see [<xref ref-type="bibr" rid="B2">14</xref>, <xref ref-type="bibr" rid="B20">15</xref>]).

Let X be a compact space, and let 2X be equipped with the Vietoris topology. If f:X→X is a continuous map, then 2f:2X→2X is continuous and (X,f) is weakly mixing ⇔(2X,2f) is weakly mixing ⇔(2X,2f) is topologically transitive.

3. Transitive Subsets and Weakly Mixing Subsets of Set-Valued Discrete Systems

For a TDS (X,f) and two nonempty subsets U,V⊆X, we use the following notation:
(7)N(U,V)={n∈ℕ:fn(U)∩V≠∅}.

Theorem 7.

A is a weakly mixing subset of (X,f) if and only if 2A is a weakly mixing subset of (2X,2f).

Proof

Necessity. We prove for any k∈ℕ, any choice of nonempty open subsets 𝒱12A,𝒱22A,…,𝒱k2A of 2A and nonempty open subsets 𝒰1,𝒰2,…,𝒰k of 2X with 2A∩𝒰i≠∅ for i=1,2,…,k, that there exists an m∈ℕ such that
(8)(2f)m(𝒱i2A)∩𝒰i≠∅fori=1,2,…,k.
For nonempty open subset 𝒱i2A of 2A, there exist open subsets 𝒱i of 2X such that 𝒱i2A=𝒱i∩2A for i=1,2,…,k. Without loss of generality, let
(9)𝒱i=ν(V1i,V2i,…,Vni),𝒰i=ν(U1i,U2i,…,Uni)𝒱i=ν(V1i,V2i,…,Vni)fori=1,2,…,k.Vji and Uji are nonempty open subsets of X for i=1,2,…,k,j=1,2,…,n. Furthermore,
(10)𝒱i2A={F∈2A:F⊆⋃j=1n(Vji∩A),F∩(Vji∩A)≠∅∀1≤j≤n⋃j=1n(Vji∩A)}.
Let (Vji)A=Vji∩A. Then we have (Vji)A≠∅ for j=1,2,…,n. Moreover, 𝒰i∩2A≠∅, then Uji∩A≠∅ for j=1,2,…,n.

We consider any nonempty open subsets (V11)A,…,(Vn1)A,…,(V1k)A,…,(Vnk)A of A and any nonempty open subsets U11,…,Un1,…,U1k,…,Unk of X with A∩Uji≠∅ for i=1,2,…,k, j=1,2,…,n. Since A is a weakly mixing subset of (X,f), then there exists an m∈ℕ such that
(11)(Vji)A∩f-m(Uji)≠∅fori=1,2,…,k,j=1,2,…,n.
Take xji∈(Vji)A∩f-m(Uji) for i=1,2,…,k, j=1,2,…,n. We have xji∈Vji∩A and fm(xji)∈Uji for i=1,2,…,k, j=1,2,…,n. Let Bi=⋃j=1n{xji}. Then Bi∈𝒱i2A and fm(Bi)∈𝒰i. Furthermore, we have fm(Bi)∈fm(𝒱i2A). Therefore, (2f)m(𝒱i2A)∩𝒰i≠∅ for i=1,2,…,k.

Sufficiency. We show that for any k∈ℕ, any choice of nonempty open subsets V1A,V2A,…,VkA of A and nonempty open subsets U1,U2,…,Uk of X with A∩Ui≠∅ for each i=1,2,…,k, there exists an m∈ℕ such that fm(ViA)∩Ui≠∅ for i=1,2,…,k.

For nonempty open subset ViA for i=1,2,…,k, there exists an open subset Vi of X such that ViA=Vi∩A for i=1,2,…,k. Let
(12)𝒱i2A=ν(Vi)∩2A,𝒰i=ν(Ui)fori=1,2,…,k.
Then 𝒱i2A is a nonempty open subset of 2A and 𝒰i is a nonempty open set of 2X with 2A∩𝒰i≠∅ for i=1,2,…,k. Since 2A is a weakly mixing subset of (2X,2f), there exists an m∈ℕ such that
(13)𝒱i2A∩(2f)-m(𝒰i)≠∅fori=1,2,…,k.
Take Fi∈𝒱i2A∩(2f)-m(𝒰i). We have Fi∈ν(Vi)∩2A and Fi∈(2f)-m(𝒰i) for i=1,2,…,k. Therefore, Fi⊆(Vi∩A)∩f-m(Ui). Furthermore, we have ViA∩f-m(Ui)≠∅ for i=1,2,…,k. This shows A is a weakly mixing subset of (X,f).

Theorem 8.

Let A be a nonempty closed set of X. If 2A is a transitive subset of (2X,2f), then A is a transitive subset of (X,f).

Proof.

We show that for any choice of nonempty open subset VA of A and nonempty open subset U of X with A∩U≠∅, there exists an n∈ℕ such that fn(VA)∩U≠∅.

For nonempty open subset VA of A, there exists a nonempty open subset V of X such that VA=V∩A. Let 𝒰=ν(U), 𝒱=ν(V), and 𝒱2A=ν(V)∩2A; then 𝒱2A is a nonempty open subset of 2A. Moreover, U∩A≠∅ implies that ν(U)∩2A≠∅. Since 2A is a topologically transitive subset of (2X,2f), there exists an n∈ℕ such that 𝒱2A∩(2f)-n(𝒰)≠∅. Furthermore, there exists F∈𝒱2A∩(2f)-n(𝒰) such that F∈𝒱2A and fn(F)∈𝒰, which implies F⊆VA and fn(F)⊆U. Therefore, we have fn(VA)∩U≠∅.

Lemma 9.

Let A be a regular closed set of X but not a singleton. A is a weakly mixing subset of (X,f) if and only if for any choice of nonempty open subsets V1A,V2A of A and nonempty open subsets U1,U2 of X with A∩Ui≠∅, i=1,2, there exists an n∈ℕ such that fn(ViA)∩Ui≠∅ for i=1,2.

Proof.

Necessity is obvious by the definition of weakly mixing subset. We need only to prove sufficiency.

Let V1A,V2A be two nonempty open subsets of A and let U1,U2 be two nonempty open subsets of X with A∩Ui≠∅, i=1,2. Since A is a regular closed set of X, then int(V2A)≠∅. We consider two nonempty open subsets V1A, U1∩A of A and two nonempty open subsets int(V2A), U2 of X; there exists an n∈ℕ such that fn(V1A)∩int(V2A)≠∅ and fn(U1∩A)∩U2≠∅. Furthermore, we have V1A∩f-n(int(V2A))≠∅ and U1∩A∩f-n(U2)≠∅.

Let V=V1A∩f-n(int(V2A)) and U=U1∩f-n(U2)≠∅. Then V is a nonempty open subset of A and U is a nonempty open subset of X with A∩U≠∅. By assumption, N(U,V)≠∅. For any m∈N(U,V), we have fm(V)∩U⊆fm(V1A)∩U1, which implies fm(V1A)∩U1≠∅. Since fn(V)⊆int(V2A), fn(U)⊆U2, it follows that
(14)fn(fm(V)∩U)⊆fm+n(V)∩fn(U)⊆fm(int(V2A))∩fn(U)⊆fm(V2A)∩U2.
Hence, fm(V2A)∩U2≠∅. Furthermore, we have m∈N(V1A,U1)∩N(V2A,U2) and N(U,V)⊆N(V1A,U1)∩N(V2A,U2). This shows that for any k∈ℕ, any choice of nonempty open subsets V1A,V2A,…,VkA of A and nonempty open subsets U1,U2,…,Uk of X with A∩Ui≠∅ for i=1,2,…,k, we have ⋂i=1kN(ViA,Ui)≠∅. This means that there exists an n∈ℕ such that fn(ViA)∩Ui≠∅ for i=1,2,…,k. Therefore, A is a weakly mixing subset of (X,f).

Lemma 10.

Let A be a regular closed set of X but not a singleton. A is a weakly mixing subset of (X,f) if and only if for any choice of nonempty open subset VA of A and nonempty open subsets U,W of X with A∩U≠∅ and A∩W≠∅, there exists an n∈ℕ such that fn(VA)∩U≠∅ and fn(VA)∩W≠∅.

Proof.

Necessity is obviously by the definition of weakly mixing subset. We need only prove sufficiency.

By Lemma 9, we only prove that for any choice of nonempty open subsets V1A,V2A of A and nonempty open subsets U1,U2 of X with A∩Ui≠∅, i=1,2, there exists an m∈ℕ such that fm(ViA)∩Ui≠∅ for i=1,2.

Let V1 and V2 be two open sets of X satisfying V1A=V1∩A and V2A=V2∩A. Since A is a regular closed set, then int(V1A)≠∅ and int(V2A)≠∅. We consider nonempty open subset V1A of A and nonempty open subsets int(V2A), U2 of X, according to the assumption that there exists an n∈ℕ such that
(15)PA=V1A∩f-n(int(V2A))≠∅,QA=V1A∩f-n(U2)≠∅.
Moreover, P=V1∩f-n(int(V2A)) and Q=V1∩f-n(U2) are nonempty open sets of X with P∩A≠∅ and Q∩A≠∅. We consider nonempty open subset PA of A and nonempty open subsets Q, U1 of X; there exists an m∈ℕ such that fm(PA)∩Q≠∅ and fm(PA)∩U1≠∅. As fm(PA)∩U1⊆fm(V1A)∩U1, we have fm(V1A)∩U1≠∅. Since fm(PA)∩Q≠∅, then fm(f-n(V2A))∩f-n(U2)≠∅, which implies fm(V2A)∩U2≠∅. Therefore, by Lemma 9, A is a weakly mixing subset of (X,f).

Theorem 11.

Let A be a regular closed subset of X but not a singleton. If 2A is a transitive subset of (2X,2f), then A is a weakly mixing subset of (X,f).

Proof.

Suppose A is a regular closed subset of X but not a singleton. Then 2A is a closed subset of 2X but not a singleton. Let VA is a nonempty open subset of A, and let U and W be two nonempty open subsets of X with A∩U≠∅ and A∩W≠∅. According to Lemma 10, we only prove there exists an n∈ℕ such that fn(VA)∩U≠∅ and fn(VA)∩W≠∅.

For nonempty open subset VA of A, there exists an open subset V of X such that VA=V∩A. Let 𝒰=ν(V) and 𝒱=ν(U,W), then 𝒰 and 𝒱 are open subsets of 2X with 2A∩𝒰≠∅ and 2A∩𝒱≠∅. We consider nonempty open subset 𝒰∩2A of 2A and nonempty open subset 𝒱 of 2X. Since 2A is a transitive subset of (2X,2f), there exists an n∈ℕ such that
(16)(𝒰∩2A)∩(2f)-n(𝒱)≠∅.
Take B∈(𝒰∩2A)∩(2f)-n(𝒱). We have B⊆VA, (2f)n(B)∩U≠∅, and (2f)n(B)∩W≠∅, that is, B⊆VA, fn(B)∩U≠∅, and fn(B)∩W≠∅, which implies fn(VA)∩U≠∅ and fn(VA)∩W≠∅. This shows A is a weakly mixing subset of (X,f).

By Theorems 7 and 11, we have the following corollary.

Corollary 12.

Let A be a regular closed subset of X but not a singleton. Then the following properties are equivalent:

Ais a weakly mixing subset of (X,f);

2A is a weakly mixing subset of (2X,2f);

2A is a transitive subset of (2X,2f).

Lemma 13.

Let A be a transitive point of (2X,2f). Then x is a transitive point of (X,f) for every x∈A.

Proof.

Suppose that A is a transitive point of (2X,2f). Then for any open set υ(U1,U2,…,Um) of (2X,2f), there exists k∈ℤ+ such that
(17)(2f)k(A)∈υ(U1,U2,…,Um).
In particular, take U1=U2=⋯=Um=U; there exists l∈ℤ+ such that (2f)l(A)∈υ(U). Furthermore, for any x∈A, we have fl(x)∈U. Since U is any nonempty open set of X, it follows that x is a transitive point of (X,f).

Theorem 14.

Let A be a transitive point of (2X,2f). Then A is a transitive subset of (X,f).

Proof.

Suppose that A is a transitive point of (2X,2f). Then A is a nonempty closed set of X. Let VA be a nonempty open set of A and let U be a nonempty open set of X with A∩U≠∅. We prove that there exists an n∈ℕ such that fn(VA)∩U≠∅.

Since A is a transitive point of (2X,2f), by Lemma 13, x is a transitive point of (X,f) for every x∈A. Let VA=V∩A, where V is an open set of X. Let x∈VA. Then orb(x,f)¯=X. It means that there exists an n∈ℕ such that fn(x)∈U. Furthermore, we have fn(VA)∩U≠∅. Therefore, A is a transitive subset of (X,f).

Example 15.

Let S1 be the unit circle and let Tλ:S1→S1 be a translation map such that
(18)Tλ(θ)=θ+2λπ,λ∈ℝ.
If λ is an irrational number, then A=[0,π] is a transitive subset of (S1,Tλ), but 2A is not a transitive subset of (2S1,2T).

It is well known that if λ=q/p is a rational number, then all points are periodic of period q, and so the set of periodic points is, obviously, dense in S1. Moreover, by Jacobi’s Theorem [6], if λ is an irrational number, then each orbit {Tλn(θ):n∈ℕ} is dense in S1. Since S1 is a compact metric space. Hence, (S1,Tλ) is topologically transitive.

Let A=[0,π]. Then A is a nonempty closed subset of S1. Let VA be a nonempty subset of A and U is an open subset of S1 with A∩U≠∅. Take x∈VA. Since {Tλn(x):n∈ℕ} is dense in S1, there exists an m∈ℕ such that Tλm(x)∈U. Furthermore, we have Tλm(VA)∩U≠∅. Therefore, A is a transitive subset of (S1,Tλ); that is, [0,π] is a transitive subset of (S1,Tλ).

Let K=[0,1]∈2A. Then diam(K)=diam(2Tλ(K))=1, where diam(K) denotes the diameter of K. Put ε>0 such that 1-ε>ε. Let 𝒱=B(K,ε/2) and 𝒰=B({1},ε/2). Then 𝒱2A=𝒱∩2A is a nonempty open subset of 2A and 𝒰 is an open subset of S1 with 2A∩𝒰≠∅. Moreover, for any F∈𝒱2A and any G∈𝒰, we have diam(F)≥1-ε and diam(G)≤ε. Furthermore, diam((2Tλ)n(F))≥1-ε>ε for all n∈ℕ. Therefore, (2Tλ)n(𝒱A)∩𝒰=∅ for all n∈ℕ. It means that 2A is not a transitive subset of (2S1,2Tλ).

Example 16.

Let I=[0,1]. Define g:I→I by
(19)g(x)={12+2x,if0≤x≤14,32-2x,if14≤x≤12,1-x,if12≤x≤1.
Then (2I,2g) has a weakly mixing subset (Figures 3 and 4).

Let J=[0,1/2] and K=[1/2,1]. Then g(J)=K and g(K)=J. Hence, g2|K is equal to the tent map f of Example 4. Furthermore, by [8], (K,g2) is mixing. Hence, K is a weakly mixing subset of (K,g2). We prove that K is a weakly mixing subset of (I,g).

For any m∈ℕ, any choice of nonempty open subsets V1K,…,VmK of K and nonempty open subsets U1,…,Um of I with K∩Ui≠∅, i=1,2,…,m, we have K∩Ui are nonempty open subsets of K for all i=1,2,…,m. Since (K,g2) is weak mixing, by [28], there exists an n∈ℕ such that (g2)n(ViK)∩(K∩Ui)=g2n(ViK)∩(K∩Ui)≠∅ for i=1,2,…,m. Furthermore, we have g2n(ViK)∩Ui≠∅ for i=1,2,…,m. Hence, K is a weak mixing subset of (I,g). By Theorem 7, 2K is a weakly mixing subset of (2I,2g).

Acknowledgments

The author would like to thank the referees for many valuable and constructive comments and suggestions for improving this paper. This work is supported by the Natural Science Foundation of Henan Province (122300410427), China.

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