# The Topological Sensitivity with respect to Furstenberg Families

Authors: Tengfei Wang; Kai Jing; Jiandong Yin
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this work, a dynamical systemX,f means that X is a topological space and f:X⟶X is a continuous map. The aim of the article is to introduce the conceptions of topological sensitivity with respect to Furstenberg families, n-topological sensitivity, and multisensitivity and present some of their basic features and sufficient conditions for a dynamical system to possess some sensitivities. Actually, it is proved that every topologically ergodic but nonminimal system is syndetically sensitive and a weakly mixing system is n-thickly topologically sensitive and multisensitive under the assumption that X admits some separability.


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## Body

## 1. Introduction

For a compact systemX,f which means that f is a continuous self-map on a compact metric space X,d, sensitive dependence on initial condition (sensitive for simplicity) for X,fwas firstly introduced by Ruelle [1] as if there exists δ>0 such that for each x∈X and every open neighborhood Vx of x, there is a nonnegative integer n such that supdfnx,fny:y∈Vx>δ. One can write this in a slightly different way (see [2]) as follows. For a nonempty V⊂X and δ>0, let(1)SfV,δ=n∈ℕ0:∃x,y∈V,s.t.dfnx,fny>δ,where ℕ0 denotes the set of nonnegative integers. Then, a compact system X,f is sensitive if and only if there exists δ>0 such that SfV,δ≠∅ for each nonempty open (opene for simplicity) subset V of X.Sensitivity is a key conception used to characterize the unpredictability of a compact system and a chief component of some chaotic properties such as the chaos in the sense of Devaney [3] and Banks et al. [4]. In [5, 6], the authors introduced the linear chaos and linear topological dynamics, and one can in [7] for the concept of multivalued linear and nonlinear topological dynamics. More detailed information of the related studies of sensitivity are introduced in [2, 8–13] and [14]. For the sake of distinguishing the following topological version of sensitivity, we use the classical sensitivity to stand for the sensitivity of compact systems in this paper.From now on, we call the pairX,f a dynamical system if f is a continuous self-map on a topological space X. In [15], the author introduced the topological version of sensitivity (topological sensitivity for short) for dynamical systems as follows.Definition 1.
(see [15]). Let X,f be a dynamical system. An open cover U of X is called a sensitivity cover (s-cover for short) for X,f if for every opene subset G of X there exist x,y∈G and n∈ℕ0 such that x,y∉∪f−nU×f−nU:U∈U.Definition 2.
(see [15]). A dynamical system X,f (or simply the map f) is called topologically sensitive if X,f has an s-cover.In other words, a dynamical system is topologically sensitive if it admits an open coverU satisfying that, for every opene set G in X, there are x,y∈G and n∈ℕ0 such that x,y∉f−nU×f−nU for each U∈U.Topological sensitivity of dynamical systems generalizes the classical sensitivity of compact systems since there exist in [5] dynamical systems on metric noncompact spaces which are topologically sensitive but not classically sensitive. The author of [15] presented some sufficient conditions for a dynamical system to be topologically sensitive. For example, it was proven in [15] that each transitive map possessing a dense set of almost periodic points and admitting an eventually periodic point on an infinite Urysohn space is topologically sensitive; every weakly mixing map is topologically sensitive if there are two opene subsets in the phase space such that the intersection of their closures is empty. Moreover, the weakly positively expansive maps were also considered in [15].In this paper, for the sake of dealing with the conception of topological sensitivity of dynamical systems in a unified way, we introduce the conceptions of topological sensitivity with respect to families ofℕ0, n-topological sensitivity, and multisensitivity for dynamical systems and give some sufficient conditions for a dynamical system to possess distinct sensitivities. Some of the presented results improve or generalize the main results of [15] to a great extent.


## 2. Preliminaries

In this section, we recall some notations, notions and basic theories of nonnegative integers and dynamical systems.


### 2.1. Subsets of Nonnegative Integers

Throughout this paper, denote byℕ0 the set of nonnegative integers, ℕ the set of positive integers, ℤ the set of integers, and ℝ the set of real numbers, respectively.LetP be the collection of all subsets of ℕ0. A subset ℱ of P is called a Furstenberg family (family for short) of ℕ0 provided it is hereditary upward, that is, F1⊂F2 and F1∈ℱ imply F2∈ℱ. A family ℱ of ℕ0 is proper if it is a proper subset of P, namely, ℱ≠∅ and ℱ≠P; translation invariant if for each n∈ℕ0, F+n=i+n:i∈F∈ℱ for each F∈ℱ; and a filter if it is proper and closed under intersection, i.e., F0,F1∈ℱ implies F0∩F1∈ℱ.Letℱ be a family of ℕ0, write ℱ=A∈P:there existsF∈ℱsuch thatF⊂A, namely, ℱ is the smallest family generated by the family ℱ.A setF⊂ℕ0 is called thick if for each m∈ℕ there exists tm∈ℕ0 such that tm,tm+1,…,tm+m⊂F and syndetic if there exists m∈ℕ such that, for every t∈ℕ0, t,t+1,…,t+m∩F≠∅. By their definitions, it is obvious that every thick subset of ℕ0 intersects each syndetic subset of ℕ0.A setF⊂ℕ0 is called piecewise syndetic if it can be written as the intersection of a thick set and a syndetic set and thickly syndetic if for any n∈ℕ, there exists a syndetic set s1n<s2n<⋯ such that(2)∪j=1∞sjn,sjn+1,…,sjn+n⊂F.The upper density of a subsetS of ℕ0 is defined as(3)limsupn⟶∞1nS∩0,1,2,…,n−1,where A denotes the cardinality of the set A. Similarly one can define the lower density of S⊂ℕ0.The upper Banach density ofS⊂ℕ0 is defined as(4)limsupI⟶∞S∩II.The supremun is taken over all segments of ℕ0. One can see [16] for more details of families.In general, we useℱinf to denote the family consisting of all infinite subsets of ℕ0 and use ℱs, ℱt, ℱts, ℱps, ℱpud, ℱpld, and ℱpubd to denote, respectively, the families of syndetic subsets, thick subsets, thickly syndetic subsets, piecewise syndetic subsets, the subsets with positive upper density, the subsets with positive lower density, and the subsets with positive upper Banach density of ℕ0. Then, ℱs, ℱt, ℱts, ℱps, ℱpud, ℱpld, and ℱpubd are proper and translation invariant. About the sets with positive upper Banach density, there is a result in [16] stated as follows which will be used in the proof of Lemma 2 in the paper.Proposition 1.
(see [16]). For each S∈ℱpubd, S−S is syndetic. Here, S−S=i−j≥0:i,j∈S.


### 2.2. Basic Notions of Dynamical Systems

LetX,f be a dynamical system. Let A¯ denote the closure of A⊂X in X and A−B denotes the difference set of A,B, i.e., A−B=a:a∈A,a∉B.For nonempty open (opene in brief and hereinafter) subsetsU,V of X and x∈X, let Ux denote the collection of all open neighborhoods of x and write NU,V=n∈ℕ0:U∩f−nV≠∅ and Nx,U=n∈ℕ0:fnx∈U.A pointx∈X is a transitive point of f if orbx, the orbit of x under f, is dense in X. A dynamical system X,f is minimal if each point in X is transitive. In a dynamical system X,f, x∈X is called a minimal point if the dynamical system orbx¯,forbx¯ is minimal. A subset M of X is called minimal if every point of M is minimal.A pointx∈X is called an almost periodic point of f if Nx,U∈ℱs for each U∈Ux. Denote by Af the set of all almost periodic points of f. In [17], the authors introduced the concept of positive upper Banach density recurrent points for compact systems. Now, we introduce this notion for dynamical systems.Definition 3.
LetX,f be a dynamical system. A point x∈X is called a positive upper Banach density recurrent point of f if Nx,U∈ℱpubd for each U∈Ux.Denote byBD∗f the set of all positive upper Banach density recurrent points of f. Obviously, Af⊂BD∗f.Letℱ be a family of ℕ0. A dynamical system X,f is ℱ-transitive if for each pair of opene subsets U,V of X, NU,V∈ℱ; ℱ-central if for each opene subset V of X, NV,V∈ℱ. Especially, a dynamical system X,f is transitive if for each pair of opene subsets U,V of X, NU,V≠∅; topological ergodic if NU,V∈ℱs for each pair of opene subsets U,V of X; weakly mixing if the product system X×X,f×f is transitive. In Section 3, we will prove that a weakly mixing system possesses some analogous properties for the hitting time set of any two opene subsets to compact systems.A dynamical systemX,f is called an M-system if it is transitive and the set of its minimal points is dense in X.A dynamical systemX,f is semiconjugate to a dynamical system Y,g if there exists a continuous surjection π:X⟶Y such that π°f=g°π. Meanwhile, π is called a semiconjugation from f to g. Moreover, a semiconjugation π from f to g is semiopen if πU has a nonempty interior for each opene U⊂X. Especially, if π is a homeomorphism from X to Y, then X,f is said to be conjugate to Y,g.Now, based on Definitions1 and 2, we introduce a more general version of topological sensitivity for a dynamical system. Actually, we introduce the notion of topological sensitivity with respect to a family of ℕ0 stated as below.Definition 4.
LetX,f be a dynamical system, ℱ be a family of ℕ0, and U be an open cover of X. U is called an ℱ-sensitivity cover for X,f if for each opene subset G of X,(5)NfG,U≔n∈ℕ0:∃x,y∈Gs.t.x,y∉∪f−nU×f−nU:U∈U∈ℱ.Now, by using the notion ofℱ-sensitivity covers, we introduce the topological version of sensitivity with respect to a Furstenberg family for dynamical systems.Definition 5.
Letℱ be a family of ℕ0. A dynamical system X,f is called ℱ-topologically sensitive if X,f admits an ℱ-sensitivity cover.Obviously, a dynamical systemX,f is topologically sensitive if and only if it is ℱ-topologically sensitive with respect to the family ℱ=P−∅ and the notion of {family}-topological sensitivity of dynamical systems generalizes that of {family}-sensitivity of compact systems.Example 1.
Letf be the self-map on ℝ defined by fx=x+1, and let us consider the following metric on ℝ: ρx,y=ex−ey. Then, this metric is equivalent to the usual metric d on ℝ, namely, they generate the same topology of ℝ. From [15], it is known that ℝ,f is topologically sensitive. In fact, it is not hard to check that ℝ,f is ℱ-topologically sensitive for the families ℱ≔ℱs,ℱt,ℱts,ℱps,ℱpld,ℱpubd.
Take an open coverU=Bp,e/2:p∈ℝ of ℝ. For any opene subset G of ℝ and x,y∈G with x≠y, let Fx,y=ln1/ex−ey+2+i:i∈ℕ0. Obviously, each Fx,y is cofinite, i.e., ℝ−Fx,y is finite. Put ℱ=Fx,y:x,y∈G,x≠y,G⊂ℝis nonempty and open. For any opene subset G of ℝ and x,y∈G with x≠y and ni≔ln1/ex−ey+2+i∈Fx,y, we have(6)x,y∉∪f−niU×f−niU:U∈U.
That is,fnix,fniy⊈Bp,e/2 for each p∈ℝ and i∈ℕ0. If not, there exist some opene subset G of ℝ and x,y∈G with x≠y and q∈ℝ and j∈ℕ0 such that ρfnjx,q<e/2 and ρfnjy,q<e/2; then,(7)ρfnjx,fnjy≤ρfnjx,q+ρfnjy,q<e2+e2=e.
On the contrary,(8)ρfnjx,fnjy=ex+nj−ey+nj=eln1/ex−ey+2+j⋅ex−ey>e1+j≥e,which is a contradiction. Therefore, for every opene set G⊂ℝ and every pair x,y∈G×G with x≠y, Fx,y⊂NfG,U and NfG,U∈ℱ. It implies that U=Bp,e/2:p∈ℝ is an ℱ-sensitivity cover of f.


## 2.1. Subsets of Nonnegative Integers

Throughout this paper, denote byℕ0 the set of nonnegative integers, ℕ the set of positive integers, ℤ the set of integers, and ℝ the set of real numbers, respectively.LetP be the collection of all subsets of ℕ0. A subset ℱ of P is called a Furstenberg family (family for short) of ℕ0 provided it is hereditary upward, that is, F1⊂F2 and F1∈ℱ imply F2∈ℱ. A family ℱ of ℕ0 is proper if it is a proper subset of P, namely, ℱ≠∅ and ℱ≠P; translation invariant if for each n∈ℕ0, F+n=i+n:i∈F∈ℱ for each F∈ℱ; and a filter if it is proper and closed under intersection, i.e., F0,F1∈ℱ implies F0∩F1∈ℱ.Letℱ be a family of ℕ0, write ℱ=A∈P:there existsF∈ℱsuch thatF⊂A, namely, ℱ is the smallest family generated by the family ℱ.A setF⊂ℕ0 is called thick if for each m∈ℕ there exists tm∈ℕ0 such that tm,tm+1,…,tm+m⊂F and syndetic if there exists m∈ℕ such that, for every t∈ℕ0, t,t+1,…,t+m∩F≠∅. By their definitions, it is obvious that every thick subset of ℕ0 intersects each syndetic subset of ℕ0.A setF⊂ℕ0 is called piecewise syndetic if it can be written as the intersection of a thick set and a syndetic set and thickly syndetic if for any n∈ℕ, there exists a syndetic set s1n<s2n<⋯ such that(2)∪j=1∞sjn,sjn+1,…,sjn+n⊂F.The upper density of a subsetS of ℕ0 is defined as(3)limsupn⟶∞1nS∩0,1,2,…,n−1,where A denotes the cardinality of the set A. Similarly one can define the lower density of S⊂ℕ0.The upper Banach density ofS⊂ℕ0 is defined as(4)limsupI⟶∞S∩II.The supremun is taken over all segments of ℕ0. One can see [16] for more details of families.In general, we useℱinf to denote the family consisting of all infinite subsets of ℕ0 and use ℱs, ℱt, ℱts, ℱps, ℱpud, ℱpld, and ℱpubd to denote, respectively, the families of syndetic subsets, thick subsets, thickly syndetic subsets, piecewise syndetic subsets, the subsets with positive upper density, the subsets with positive lower density, and the subsets with positive upper Banach density of ℕ0. Then, ℱs, ℱt, ℱts, ℱps, ℱpud, ℱpld, and ℱpubd are proper and translation invariant. About the sets with positive upper Banach density, there is a result in [16] stated as follows which will be used in the proof of Lemma 2 in the paper.Proposition 1.
(see [16]). For each S∈ℱpubd, S−S is syndetic. Here, S−S=i−j≥0:i,j∈S.


## 2.2. Basic Notions of Dynamical Systems

LetX,f be a dynamical system. Let A¯ denote the closure of A⊂X in X and A−B denotes the difference set of A,B, i.e., A−B=a:a∈A,a∉B.For nonempty open (opene in brief and hereinafter) subsetsU,V of X and x∈X, let Ux denote the collection of all open neighborhoods of x and write NU,V=n∈ℕ0:U∩f−nV≠∅ and Nx,U=n∈ℕ0:fnx∈U.A pointx∈X is a transitive point of f if orbx, the orbit of x under f, is dense in X. A dynamical system X,f is minimal if each point in X is transitive. In a dynamical system X,f, x∈X is called a minimal point if the dynamical system orbx¯,forbx¯ is minimal. A subset M of X is called minimal if every point of M is minimal.A pointx∈X is called an almost periodic point of f if Nx,U∈ℱs for each U∈Ux. Denote by Af the set of all almost periodic points of f. In [17], the authors introduced the concept of positive upper Banach density recurrent points for compact systems. Now, we introduce this notion for dynamical systems.Definition 3.
LetX,f be a dynamical system. A point x∈X is called a positive upper Banach density recurrent point of f if Nx,U∈ℱpubd for each U∈Ux.Denote byBD∗f the set of all positive upper Banach density recurrent points of f. Obviously, Af⊂BD∗f.Letℱ be a family of ℕ0. A dynamical system X,f is ℱ-transitive if for each pair of opene subsets U,V of X, NU,V∈ℱ; ℱ-central if for each opene subset V of X, NV,V∈ℱ. Especially, a dynamical system X,f is transitive if for each pair of opene subsets U,V of X, NU,V≠∅; topological ergodic if NU,V∈ℱs for each pair of opene subsets U,V of X; weakly mixing if the product system X×X,f×f is transitive. In Section 3, we will prove that a weakly mixing system possesses some analogous properties for the hitting time set of any two opene subsets to compact systems.A dynamical systemX,f is called an M-system if it is transitive and the set of its minimal points is dense in X.A dynamical systemX,f is semiconjugate to a dynamical system Y,g if there exists a continuous surjection π:X⟶Y such that π°f=g°π. Meanwhile, π is called a semiconjugation from f to g. Moreover, a semiconjugation π from f to g is semiopen if πU has a nonempty interior for each opene U⊂X. Especially, if π is a homeomorphism from X to Y, then X,f is said to be conjugate to Y,g.Now, based on Definitions1 and 2, we introduce a more general version of topological sensitivity for a dynamical system. Actually, we introduce the notion of topological sensitivity with respect to a family of ℕ0 stated as below.Definition 4.
LetX,f be a dynamical system, ℱ be a family of ℕ0, and U be an open cover of X. U is called an ℱ-sensitivity cover for X,f if for each opene subset G of X,(5)NfG,U≔n∈ℕ0:∃x,y∈Gs.t.x,y∉∪f−nU×f−nU:U∈U∈ℱ.Now, by using the notion ofℱ-sensitivity covers, we introduce the topological version of sensitivity with respect to a Furstenberg family for dynamical systems.Definition 5.
Letℱ be a family of ℕ0. A dynamical system X,f is called ℱ-topologically sensitive if X,f admits an ℱ-sensitivity cover.Obviously, a dynamical systemX,f is topologically sensitive if and only if it is ℱ-topologically sensitive with respect to the family ℱ=P−∅ and the notion of {family}-topological sensitivity of dynamical systems generalizes that of {family}-sensitivity of compact systems.Example 1.
Letf be the self-map on ℝ defined by fx=x+1, and let us consider the following metric on ℝ: ρx,y=ex−ey. Then, this metric is equivalent to the usual metric d on ℝ, namely, they generate the same topology of ℝ. From [15], it is known that ℝ,f is topologically sensitive. In fact, it is not hard to check that ℝ,f is ℱ-topologically sensitive for the families ℱ≔ℱs,ℱt,ℱts,ℱps,ℱpld,ℱpubd.
Take an open coverU=Bp,e/2:p∈ℝ of ℝ. For any opene subset G of ℝ and x,y∈G with x≠y, let Fx,y=ln1/ex−ey+2+i:i∈ℕ0. Obviously, each Fx,y is cofinite, i.e., ℝ−Fx,y is finite. Put ℱ=Fx,y:x,y∈G,x≠y,G⊂ℝis nonempty and open. For any opene subset G of ℝ and x,y∈G with x≠y and ni≔ln1/ex−ey+2+i∈Fx,y, we have(6)x,y∉∪f−niU×f−niU:U∈U.
That is,fnix,fniy⊈Bp,e/2 for each p∈ℝ and i∈ℕ0. If not, there exist some opene subset G of ℝ and x,y∈G with x≠y and q∈ℝ and j∈ℕ0 such that ρfnjx,q<e/2 and ρfnjy,q<e/2; then,(7)ρfnjx,fnjy≤ρfnjx,q+ρfnjy,q<e2+e2=e.
On the contrary,(8)ρfnjx,fnjy=ex+nj−ey+nj=eln1/ex−ey+2+j⋅ex−ey>e1+j≥e,which is a contradiction. Therefore, for every opene set G⊂ℝ and every pair x,y∈G×G with x≠y, Fx,y⊂NfG,U and NfG,U∈ℱ. It implies that U=Bp,e/2:p∈ℝ is an ℱ-sensitivity cover of f.


## 3. Topological Sensitivity with respect to Families of Dynamical Systems

In this section, we give some basic properties ofℱ-topological sensitivity and prove that the ℱ-topological sensitivity is invariant under semiconjugations and that a dynamical system X,f is ℱs-topologically sensitive (resp. ℱt, ℱps, ℱts, ℱpud, ℱpld, ℱpubd-topologically sensitive) if and only if so is X,fn for each n∈ℕ if and only if so is X,fn for some n∈ℕ.Theorem 1.
Suppose thatX,f and Y,g are two dynamical systems and ℱ is a family of ℕ0. If X,f and Y,g are semiconjugate and the semiconjugation φ from f to g is semiopen and Y,g is ℱ-topologically sensitive, then so is X,f.Proof.
LetV be an ℱ-sensitivity cover for Y,g. Set U=φ−1V:V∈V, then U is an open cover of X. For each opene subset G of X, φG has a nonempty interior since φ is semiopen. Take an opene set W⊂φG. By the ℱ-topological sensitivity of Y,g,(9)NgW,V=n∈ℕ0:∃x,y∈Ws.t.x,y∉∪g−nV×g−nV:V∈V∈ℱ.
Thus, for eachn∈NgW,V, there exist yn,y˜n∈W such that, for every V∈V, yn∉g−nV or y˜n∉g−nV. Take xn,x˜n∈G such that yn=φxn and y˜n=φx˜n. Then, for every U∈U, xn∉f−nU or x˜n∉f−nU. So, NgW,V⊂NfG,U. The definition of families implies NfG,U∈ℱ for each opene subset G of X. Therefore, X,f is ℱ-topologically sensitive.In the following, we will show that theℱs-topological sensitivity as well as ℱt, ℱps, ℱts, ℱpud, ℱpld, and ℱpubd-topological sensitivity of a dynamical system is invariant under iterations.Theorem 2.
LetX,f be a dynamical system. Then, the following statements are equivalent:(1)
X,f is ℱs-topologically sensitive (resp. ℱt, ℱps, ℱts-topologically sensitive)(2)
X,fn is ℱs-topologically sensitive (resp. ℱt, ℱps, ℱts-topologically sensitive) for each n∈ℕ(3)
X,fn is ℱs-topologically sensitive (resp. ℱt, ℱps, ℱts-topologically sensitive) for some n∈ℕProof.
We only prove the case ofℱs-topological sensitivity since the proofs of other cases are similar:1⇒2: assume that U is an ℱs-sensitivity cover for X,f and n∈ℕ. For every t∈0,…,n−1, let Ut=f−tU:U∈U. Set V as the intersection of all the covers U0,…,Un−1, i.e., V is the open cover of X consisting of all (nonempty) sets with the form of U0∩U1∩⋯∩Un−1 with Ut∈Ut for t=0,1,…,n−1. Now, take arbitrarily an opene subset G of X, then NfG,U∈ℱs. For each m∈NfG,U, there exist xm,ym∈G such that xm∉f−mU or ym∉f−mU for every U∈U. Clearly, m=tmn+sm for some tm∈ℕ0 and 0≤sm≤n−1. Hence, xm∉f−tmn−smU or ym∉f−tmn−smU for every U∈U which implies that xm∉fn−tmW or ym∉fn−tmW for every W∈W. Obviously, tm:m∈NfG,U∈ℱs. This turns out that X,fn is ℱs-topologically sensitive.2⇒3 is obvious.3⇒1 is easy, so we leave it to the reader.Theorem 3.
LetX,f be a dynamical system. Then, the following statements are equivalent:(1)
X,f is ℱpud-topologically sensitive (resp. ℱpld, ℱpubd-topologically sensitive)(2)
X,fn is ℱpud-topologically sensitive (resp. ℱpld, ℱpubd-topologically sensitive) for each n∈ℕ(3)
X,fn is ℱpud-topologically sensitive (resp. ℱpld, ℱpubd-topologically sensitive) for some n∈ℕProof.
We only prove the case ofℱpud-topological sensitivity since the proofs of other cases are similar:1⇒2: assume that U is an ℱpud-sensitivity cover for X,f and n∈ℕ. For every t∈0,…,n−1, let Ut=f−tU:U∈U. Set V as the intersection of all the covers U0,…,Un−1, i.e., V is the open cover of X consisting of all (nonempty) sets with the form of U0∩U1∩⋯∩Un−1 with Ut∈Ut for t=0,1,…,n−1. Now, for any opene subset G of X, NfG,U∈ℱpud. For each m∈NfG,U, there exist xm,ym∈G such that xm∉f−mU or ym∉f−mU for every U∈U. Clearly, m=tmn+sm for some tm∈ℕ0 and 0≤sm≤n−1. Hence, xm∉f−tmn−smU or ym∉f−tmn−smU for every U∈U which implies that xm∉fn−tmW or ym∉fn−tmW for every W∈W. Then,(10)nlimsupm⟶∞NfnG,W∩0,1,…,tm−1tm≥limsupm⟶∞NfG,U∩0,1,…,ntm+smtmn+sm+1>0.2⇒3 is obvious.3⇒1 is not difficult, so we leave it to the reader.Theorem 4.
If a dynamical systemX,f is topologically sensitive and there is no isolated points in X and f is semiopen, then NfG,U is infinite for each opene subset G of X, where U is a sensitivity cover for X,f.Proof.
In fact, if there exists some opene setG⊂X such that NfG,U is finite, put n=maxk:k∈NfG,U, then n∈NfG,U. So, there exist x,y∈G such that(11)x,y∉∪f−nU×f−nU:U∈U.
Notefn+1x∈X, and there exists some V∈U such that fn+1x∈V. Since fn+1 is continuous, there exists Ux∈Ux such that fn+1Ux⊂V. Without loss of generality, assume Ux⊂G. It is clear that fn+1Ux has a nonempty interior as f which is semiopen. So, there exists an opene set U˜⊂fn+1Ux and NfU˜,U≠∅ which gives that there are y1,y2∈U˜ and m∈NfU˜,U such that(12)y1,y2∉∪f−mU×f−mU:U∈U.
Letx1,x2∈Ux be such that y1=fn+1x1 and y2=fn+1x2; then,(13)fn+1x1,fn+1x2∉∪f−mU×f−mU:U∈U.
So,(14)x1,x2∉∪f−m−n−1U×f−m−n−1U:U∈U,which implies that m+n+1∈NfG,U. That is a contradiction.The following theorem reveals the relations between the topological sensitivity and the classical sensitivity and generalizes the result of Theorem 2.3 of [15]. At first, we recall that τd denotes the topology of X generated by the metric d of X if X,d is a metric space. Let ℱ be a family of ℕ0. Review that a continuous self-map f on a metric space X,d is called ℱ-sensitivity if there exists δ>0 such that, for each opene set U⊂X,(15)SfU,δ=n∈ℕ0:∃x,y∈Usuch thatdfnx,fny>δ∈ℱ.Theorem 5.
Letf be a continuous self-map on a metric space X,d and let us consider the following conditions:(1)
f:X,d⟶X,d is ℱ-sensitive(2)
f:X,τd⟶X,τd is ℱ-topologically sensitive
Then, (1) implies (2). In addition, ifX is compact, then (2) is equivalent to (1).Proof.
The proof of Theorem5 is similar to that of Theorem 2.3 in [15], so we omit it.For proving the next theorem, we need firstly the following lemmas whose proofs are similar to those of the corresponding results of compact systems, but for the completeness, we include them in the paper.Lemma 1.
Assume thatℱ is a proper and translation invariant family of ℕ0. Then, a dynamical system X,f is ℱ-transitive if and only if it is transitive and ℱ-central.Proof.
Sinceℱ is translation invariant, ℱ⊂ℱinf. If X,f is ℱ-transitive, then it is transitive. So, X,f is ℱ-central.
Conversely, sinceX,f is transitive, for any opene subsets U,V of X, there exists i∈ℕ0 such that W=U∩f−iV≠∅. It is easy to prove that NU,V⊇NW,W+i. Noting that X,f is ℱ-central and ℱ is translation invariant, we have NU,V∈ℱ, so X,f is ℱ-transitive.Lemma 2.
A dynamical systemX,f is topologically ergodic if it is topologically transitive and the set of positive upper Banach density recurrent points of f is dense in X.Proof.
Assume thatV is an opene subset of X, by the given hypothesis, V∩BD∗f≠∅. Choose x∈V∩BD∗f; then, Nx,V has positive upper Banach density. Take n1,n2∈Nx,V with n1<n2; then, fn1x∈V, fn2x∈V. Set y=fn1x∈V; then, x∈f−n1y. Thus,(16)fn2−n1V∩V⊃fn2−n1y∩V⊃fn2x∩V≠∅.
Namely,n2−n1∈NV,V, note n2−n1∈Nx,V−Nx,V which implies that Nx,V−Nx,V⊂NV,V, by Proposition 1, and NV,V is syndetic. Therefore, X,f is ℱs-central. Note that ℱs is proper and translation invariant, by Lemma 1, X,f is topologically ergodic.Next, we give one of the main results of the paper as follows. Firstly, we review that a topological spaceX is called a Uryshon space if for every pair of distinct points x,y∈X, and there are two opene sets U and V in X such that x∈U and y∈V and U¯∩V¯=∅. For more information about Uryshon spaces, one can refer to [18].Theorem 6.
LetX,f be a dynamical system, where X is a T3 space. If X,f is topologically transitive but nonminimal and the set of positive upper Banach density recurrent points of f is dense in X, then X,f has an ℱs-sensitivity cover with two elements. Therefore, X,f is ℱs-topologically sensitive.Proof.
SinceX,f is not minimal, take a∈X such that X−orba¯≠∅, pick b∈X−orba¯, then there exist Ub∈Ub and a neighborhood U of orba¯ such that Ub¯∩U¯=∅. Let us consider the open cover W=X−U¯b,X−U¯ of X. For any opene subset G of X, by Lemma 2, NG,Ub is syndetic. Suppose M1∈ℕ is a gap of j0∈NG,Ub. By the continuities of fi, i=1,…,M1, there exists Ua∈Ua such that fiUa⊂U for all i=1,…,M1. By Lemma 2 again, NG,Ua is syndetic. Suppose M2∈ℕ is a gap of NG,Ua. Take NG,Ua there exists x∈G such that fj0x∈Ua, so fj0+ix∈U for all i=1,…,M1. Clearly, there exists 0≤i0≤M1 such that j0+i0∈NG,Ub, so there is y∈G such that fj0+i0y∈Ub. Note that fj0+i0y∉X−U¯b, fj0+i0x∉X−U¯, since j0∈NG,Ua, NG,Ua+i0⊂NfG,W. Since NG,Ua is syndetic and ℱs is a translation invariant Furstenberg family, NfG,W∈ℱs. It follows that W is an ℱs-sensitivity cover for X,f. Therefore, X,f is ℱs-topologically sensitive.Remark 1.
Theorem6 generalizes Theorem 2.5 of [15] to a great extent since in [17] and there exists an example showing that the set Af of almost periodic points of f may properly be contained in BD∗f.In order to prove the next result, we firstly prove two lemmas whose proofs are same as those of the corresponding results of compact systems, but for the completeness of the paper, we present their complete proofs here.Lemma 3.
LetX,f be a dynamical system. Then, X,f is weakly mixing if and only if the family ℱ is a filter of ℕ0, where ℱ=NU,V:U,Vare opene subsets ofX.Proof.
If the familyℱ is a filter, then for any opene subsets U1,U2,V1,V2 of X,(17)NU1×U2,V1×V2=NU1,V1∩NU2,V2∈ℱ.
Especially,NU1×U2,V1×V2≠∅. So, X,f is weakly mixing.
Conversely, supposeX,f is weakly mixing and NU1,V1,NU2,V2∈ℱ. By the definition of weak mixing, there exists some m∈NU1,U2∩NV1,V2. Set A=U1∩f−mU2, B=V1∩f−mV2. For any k∈NA,B, we have(18)A∩f−kB=U1∩f−mU2∩f−kV1∩f−mV2=U1∩f−kV1∩f−mU2∩f−kV2.
This meansU1∩f−kV1≠∅ and U2∩f−kV2≠∅. Then, NA,B⊂NU1,V1∩NU2,V2. So, ℱ is a filter of ℕ0.Lemma 4.
LetX,f be a dynamical system. If X,f is weakly mixing, then for each pair of opene subsets U,V of X, NU,V∈ℱt.Proof.
SupposeU,V are opene subsets of X. By Lemma 3, for each n∈ℕ,(19)NU,V∩NU,f−1V∩⋯∩NU,f−nV≠∅.
So,NU,V is thick.Theorem 7.
Suppose thatX,f is a weakly mixing and topologically ergodic dynamical system, where X admits two opene subsets U and V such that U¯∩V¯=∅. Then, X,f has an ℱps-sensitivity cover with two elements. Therefore, X,f is ℱps-topologically sensitive.Proof.
ObviouslyW=X−U¯,X−V¯ is an open cover of X. Since X,f is weakly mixing, by Lemma 3, ℱ is a filter of ℕ0, where ℱ=NP,Q:P,Qare opene subsets ofX and each element of ℱ is thick. Suppose that G is an opene subset of X, then there exist a pair of opene subsets A,B of X such that NA,B⊂NG,U∩NG,V is thick. Since X,f is topologically ergodic, NA,B is syndetic which implies that NA,B is piecewise syndetic. Take arbitrarily n∈NA,B and note that NA,B⊂NG,U∩NG,V, then G∩f−nU≠∅ and G∩f−nV≠∅. So, there exists x,y∈G such that fnx∈U and fny∈V which yields that fnx∉X−U¯ and fny∉X−V¯. Therefore, W is an ℱps-sensitivity cover for X,f which drives that X,f is ℱps-topologically sensitive.


## 4.N-Topological Sensitivity of Dynamical Systems

In this section, we introduce the notion ofn-sensitivity for a dynamical system. One can see [19] for the same conception for compact systems and see [20] for the difference between n-sensitivity and n+1-sensitivity of compact systems.Definition 6.
LetX,f be a dynamical system and U be an open cover of X and n∈ℕ. U is called an n-topological sensitivity cover for X,f if for every opene subset G of X, there exist m∈ℕ0 and n different points x1,x2,…,xn∈G such that, for all x,y∈x1,x2,…,xn with x≠y,(20)x,y∉∪f−mU×f−mU:U∈U.Definition 7.
A dynamical systemX,f (or simply the map f) is called n-topologically sensitive if X,f has an n-topological sensitivity cover.Definition 8.
LetX,f be a dynamical system, ℱ be a family of ℕ0, and U be an open cover of X. U is called an n-ℱ-sensitivity cover for X,f if for each opene subset G of X,(21)NfnG,U≔m∈ℕ0:∃x1,x2,…,xn∈Gs.t.xi,xj∉∪f−mU×f−mU:U∈U,i≠j∈ℱ.Now, by using the notion ofn-ℱ-sensitivity cover, we introduce the topological version of n-sensitivity with respect to a Furstenberg family.Definition 9.
Letℱ be a family of ℕ0. A dynamical system X,f is called n-ℱ-topologically sensitive if X,f admits an n-ℱ-sensitivity cover.In the following, we present some basic properties ofn-topological sensitivity of dynamical systems.Theorem 8.
LetX,f and Y,g be two dynamical systems and n∈ℕ:(1)
IfX,f and Y,g are semiconjugate and the semiconjugation φ from f to g is semi-open and Y,g is n-topologically sensitive, then so is X,f(2)
X,f is n-topologically sensitive if and only if X,fm is n-topologically sensitive for each m∈ℕ if and only if X,fm is n-topologically sensitive for some m∈ℕProof.
(1)
The proof is similar to that of (1) of Theorem1, so we omit it(2)
IfU is an n-topological sensitivity cover for X,fm for some m∈ℕ, then U is also an n-topological sensitivity cover for X,f
Now, assume thatf is n-topologically sensitive and take an n-topological sensitivity cover U for X,f. Give arbitrarily m∈ℕ. For every k∈0,…,m−1, let Uk=f−kU:U∈U. Set V as the intersection of all the covers U0,…,Um−1, i.e., V is the open cover of X consisting of all (nonempty) sets with the form U0∩U1∩⋯∩Um−1 with Uk∈Uk for k=0,1,…,m−1. We claim that V is an n-topological sensitivity cover for X,fm. In fact, let G be an opene subset of X, then there exist s∈ℕ0 and n different points x1,x2,…,xn in G such that, for any xi,xj∈x1,x2,…,xn with xi≠xj,(22)xi,xj∉∪f−sU×f−sU:U∈U.
Letq∈ℕ0 and 0≤p<m such that s=qm+p. Now, for any xi,xj∈x1,x2,…,xn with xi≠xj,(23)xi,xj∉∪f−qm−pU×f−qm−pU:U∈U,which implies that(24)xi,xj∉∪fm−qV×fm−qV:V∈V.So, V is an n-topological sensitivity cover for X,fm and X,fm is n-topologically sensitive.
IfX,fm is n-topologically sensitive for each m∈ℕ, then it is obvious that X,fm is n-topologically sensitive for some m∈ℕ.Aboutn-topological sensitivity, we also have the following result that is similar to Theorem 5.Theorem 9.
Letf be a continuous self-map on a metric space X,d, and let us consider the following conditions:(1)
f:X,d⟶X,d is n-sensitive(2)
f:X,τd⟶X,τd is n-topologically sensitive
Then, (1) implies (2). In addition, ifX is compact, then (2) is equivalent to (1).Proof.
The proof is similar to that of Theorem5, so we omit it.Theorem 10.
Assume thatX,f is a weakly mixing dynamical system, where X is an infinite Uryshon space and there is no isolated points in X. Then, X,f is n-thickly topologically sensitive for each n∈ℕ.Proof.
Letn∈ℕ. By the given conditions, there exist n different points x1,x2,…,xn∈X such that, for each i=1,2,…,n, xi admits a neighborhood Uxi satisfying U¯xi∩U¯xj=∅ for all i,j∈1,2,…,n with i≠j. Let A=1,…,n and(25)W=X−∪i∈A−1U¯xi,X−∪i∈A−2U¯xi,…,X−∪i∈A−nU¯xi.
We claim thatW is an n-thickly topological sensitivity cover for X,f. In fact, for any opene subset G of X, since X,f is weakly mixing, from Lemmas 3 and 4, it follows that(26)H=NG,Ux1∩NG,Ux2∩⋯∩NG,Uxn∈ℱt.
For eachm∈H, there exist n distinct points y1,y2,…,yn∈G such that fmyi∈Uxi for each i∈A; then, for any x,y∈y1,y2,…,yn with x≠y,(27)x,y∉∪f−mW×f−mW:W∈W.
It follows thatX,f is n-thickly topologically sensitive.Remark 2.
(1)
Theorem10 generalizes Proposition 2.7 of [15] to a great extent.(2)
Fix a familyℱ of ℕ0, one can introduce the notion of n-ℱ-topological sensitivity for dynamical systems by imitating the definition of ℱ-topological sensitivity. For instance, fix a family ℱ of ℕ0 and n∈ℕ; a dynamical system X,f is called n-ℱ-topologically sensitive if there exists an open cover U of X such that, for each opene G⊂X,(28)NfnG,U≔m∈ℕ0:∃x1,x2,…,xn∈Gs.t.xi,xj∉∪f−mU×f−mU:U∈U,i≠j∈ℱ.
Then, one can obtain the similar results forn-ℱ-topological sensitivity as Theorems 8–10.


## 5. Multitopological Sensitivity of Dynamical Systems

In this section, we introduce the notion of multisensitivity for dynamical systems. Review that a compact systemX,f is multisensitive if there exists δ>0 such that, for any finitely many opene subsets G1,…,Gl of X, ∩i=1lSfGi,δ≠∅ (see [2] for more details of multisensitivity of compact systems). One can refer to [10, 12, 21] for the recent developments of multisensitivity for compact systems and see [22, 23] for the new results of multisensitivity for nonautonomous systems. Now, motivated by the notion of multisensitivity of compact systems, we introduce the conception of multisensitivity for dynamical systems as follows.Definition 10.
LetX,f be a dynamical system. An open cover U of X is called a multisensitivity cover for X,f if for any finitely many opene subsets G1,…,Gn of X, ∩i=1nNfGi,U≠∅.Definition 11.
A dynamical systemX,f (or simply the map f) is called multisensitive if X,f has a multisensitivity cover.In the following, we present some basic features of multisensitivity of dynamical systems.Theorem 11.
LetX,f and Y,g be two dynamical systems:(1)
IfX,f and Y,g are semiconjugate and the semiconjugation φ from f to g is semiopen and Y,g is multisensitive, then so is X,f(2)
X,f is multisensitive if and only if X,fn is multisensitive for each n∈ℕ if and only if X,fn is multisensitive for some n∈ℕProof.
(1)
The proof is similar to that of (1) of Theorem1, so we omit it(2)
IfU is a multisensitivity cover for X,fn for some n∈ℕ, then it is clear that U is also a multisensitivity cover for X,f
Now, assume thatf is multisensitive and take a multisensitivity cover U for X,f. Give arbitrarily n∈ℕ. For every k∈0,…,n−1, let Uk=f−kU:U∈U. Set V as the intersection of all the covers U0,⋯Un−1, i.e., V is the open cover of X consisting of all (nonempty) sets with the form of U0∩U1∩⋯∩Un−1 with Uk∈Uk for k=0,1,…,n−1. We claim that V is a multisensitivity cover for X,fn. In fact let G1,…,Gs be any finitely many opene subsets of X, by the multisensitivity of X,f, there exist xi,x˜i∈Gi and m∈ℕ0 such that(29)xi,x˜i∉∪f−mU×f−mU:U∈U,for each 1≤i≤s. Let q∈ℕ0 and 0≤p<n such that m=qn+p. Now, we have for all 1≤i≤s,(30)xi,x˜i∉∪f−qn−pU×f−qn−pU:U∈U=∪fn−qf−pU×fn−qf−pU:U∈U,which implies(31)xi,x˜i∉∪fn−qV×fn−qV:V∈V,for each 1≤i≤s. So, q∈NfnGi,V for every 1≤i≤s, namely, ∩i=1sNfnGi,V≠∅. Therefore, V is a multisensitivity cover for X,fn and X,fn is multisensitive.
IfX,fn is multisensitive for each n∈ℕ, clearly X,fn is multisensitive for some n∈ℕ.
Summarize the above proof process, and we prove result (2).About multisensitivity, we have the following Theorem12 that is similar to Theorems 5 and 9.Theorem 12.
Letf be a continuous self-map on a metric space X,d, and let us consider the following conditions:(1)
f:X,d⟶X,d is multisensitive(2)
f:X,τd⟶X,τd is multisensitive
Then, (1) implies (2). In addition, ifX is compact, then (2) is equivalent to (1).Proof.
The proof is similar to that of Theorem9, so we omit it.Theorem 13.
Suppose thatX,f is a weakly mixing dynamical system and X admits two opene subsets U and V of X such that U¯∩V¯=∅. Then, X,f is multisensitive.Proof.
By the conditions there exist two opene setsU,V⊂X such that U¯∩V¯=∅. Set W=X−U¯,X−V¯. Then, W is a multisensitivity cover of X,f. In fact, for any finitely many opene sets U1,…,Un of X, since X,f is weakly mixing, by Lemma 3, NUi,U and NUi,V are thick for each 1≤i≤n and A≔∩i=1nNUi,U∩NUi,V≠∅. Then, for each k∈A and each 1≤i≤n, there exist xi,yi∈Ui such that fkxi∈U,fkyi∈V which yields that fkxi∉X−U¯ and fkyi∉X−V¯. So, A⊂NfUi,W≠∅ for all 1≤i≤n. It follows that ∩i=1nNfUi,W≠∅. Therefore, W is a multisensitivity cover for X,f and X,f is multisensitive.Theorem 14.
LetX,f be an M-system satisfying; there exist two minimal sets A,B⊂X and two open sets U,V⊂X such that A⊂U, B⊂V, and U¯∩V¯=∅, then X,f is ℱts-topologically sensitive.Proof.
We directly prove thatW=X−U¯,X−V¯ is an ℱts-sensitivity cover for X,f. Let W be any opene subset of X. For each k∈ℕ, since f,f2,…,fk are continuous and A is f-invariant, f−iU is an open set containing A for all 1≤i≤k. By the assumption of X,f being an M-system, there is a minimal point zi∈f−iU−A such that orbzi¯∩A=∅ for each 1≤i≤k. Let UAi=f−iU−orbzi¯, 1≤i≤k, then UA=∩i=1kUAi is an open set containing A. So, A⊂UA⊂f−iU for each 1≤i≤k. Since X,f is transitive, NW,UA≠∅. Hence, there are x∈W and m∈ℕ such that fmx∈fmW∩UA. By the continuity of fm, there is an open set Wx such that x∈Wx⊂W and fmWx⊂UA. So, there exists a minimal point x˜∈Wx of f such that fmx˜∈UA. Note that fmx˜ is also a minimal point of f, then there exists a syndetic set nj of ℕ0 such that fnj+mx˜∈UAi for all j≥1 and each 1≤i≤k, so fnj+m+ix˜∈fiUAi⊂U for each 1≤i≤k. This shows that NW,U is a thickly syndetic set. By the same argument, NW,V is also thickly syndetic. Thus, NW,U∩NW,V is thickly syndetic, which implies that W is an ℱts-sensitivity cover for X,f and X,f is ℱts-topologically sensitive.Lemma 5.
IfX,f is multisensitive and there is no isolated points in X, then ∩i=1nNfUi,U is infinite for any finitely many opene sets U1,…,Un of X, where U is a multisensitivity cover for X,f.Proof.
In fact, if there exist finitely many opene setsG1,…,Gk of X such that ∩i=1kNfGi,U is finite, put(32)m=maxn:n∈∩i=1kNfGi,U,then m∈NfGi,U for all i=1,…,k. Thus, there exist xi,yi∈Gi such that(33)xi,yi∉∪f−mU×f−mU:U∈U,for every 1≤i≤k. Note that, for every 1≤j≤m, there exists Wi,j∈U such that fjxi∈Wi,j. By the continuities of fj, 1≤j≤m, there is an open neighbourhood Ui,j⊂Gi of xi such that fjUi,j⊂Wi,j for each 1≤j≤m and 1≤i≤k. Now, for 1≤i≤k, let Gi˜=∩j=1mUi,j⊂Gi, and it is easy to check that fjGi˜⊂Wi,j∈U for every 1≤j≤m and 1≤i≤k. Thus, ∩i=1kNfGi˜,U>m and ∩i=1kNfGi,U>m. That is a contradiction.Theorem 15.
LetX,f be a dynamical system, where f is a surjection. If X,f is multisensitive, then X,f is thickly topologically sensitive. Moreover, if X,f is point-transitive, then the converse holds.Proof.
Firstly assume thatX,f is multisensitive with a multisensitivity cover U of X and k∈ℕ. For any opene set G⊂X, we choose opene sets Gi⊂f−iG for each i=1,…,k. By Lemma 5, ∩i=1kNfGi,U is infinite, so we can choose some nk∈∩i=1kNfGi,U with nk>k. Note that nk∈NfGi,U for each i=1,…,k, and there exist xi,yi∈Gi which gives that fnkxi,fnkyi∈fnk−iG such that(34)xi,yi∉∪f−nkU×f−nkU:U∈U.
Then, for eachi=1,…,k, there exist x˜i,y˜i∈G such that fnk−ix˜i=fnkxi,fnk−iy˜i=fnkyi, and(35)x˜i,y˜i∉∪f−nk−iU×f−nk−iU:U∈U,which implies nk−i∈NfG,U for all i=1,…,k. Therefore, X,f is thickly topologically sensitive.
Now, we assumeX,f is thickly topologically sensitive with a thickly topological sensitivity cover U of X and x is a transitive point of X,f. Let k∈ℕ and U1,…,Uk be opene subsets of X. Then, for each i=1,…,k, there exists ni∈ℕ such that fnix∈Ui. Hence, there exists V∈Ux such that fniV∈Ui for each i=1,…,k. By the given assumptions, there is s∈ℕ0 such that(36)s,s+1,…,s+n1+n2+⋯+nk⊂NfV,U.
So,s+ni∈NfV,U for all i=1,…,k which yields that there exist xi,yi∈V such that(37)xi,yi∉∪f−s+niU×f−s+niU:U∈U,for all i=1,…,k. Therefore, fnixi,fniyi∈fniV⊂Ui for each i=1,…,k and(38)fnixi,fniyi∉∪f−sU×f−sU:U∈U,which implies that s∈NfUi,U for every i=1,…,k, i.e., s∈∩i=1kNfUi,U. Therefore, X,f is multisensitive.Remark 3.
For the related studies of thickly topological sensitivity and multisensitivity of semigroups, one can refer to [24] in which all kinds of sensitivities are defined via a uniform structure of the involved space. In this paper, we introduce such sensitivities by an open cover of the phase space, so they are a little different.



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