The Topological Sensitivity with respect to Furstenberg Families

In this work, a dynamical system ( X, 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.


Introduction
For a compact system (X, 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, f)was firstly introduced by Ruelle [1] as if there exists δ > 0 such that for each x ∈ X and every open neighborhood V x of x, there is a nonnegative integer n such that sup d(f n (x), f n (y)): y ∈ V x > δ. One can write this in a slightly different way (see [2]) as follows. For a nonempty V ⊂ X and δ > 0, let where N 0 denotes the set of nonnegative integers. en, a compact system (X, f) is sensitive if and only if there exists δ > 0 such that S f (V, 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][9][10][11][12][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 pair (X, 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 ∈ N 0 such that 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 cover U satisfying that, for every opene set G in X, there are x, y ∈ G and n ∈ N 0 such that (x, y) ∉ f − n (U) × f − n (U) 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. e 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 of N 0 , 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.

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

Subsets of Nonnegative Integers.
roughout this paper, denote by N 0 the set of nonnegative integers, N the set of positive integers, Z the set of integers, and R the set of real numbers, respectively.
Let P be the collection of all subsets of N 0 . A subset F of P is called a Furstenberg family (family for short) of N 0 provided it is hereditary upward, that is, F 1 ⊂ F 2 and F 1 ∈ F imply F 2 ∈ F. A family F of N 0 is proper if it is a proper subset of P, namely, F ≠ ∅ and F ≠ P; translation invariant if for each n ∈ N 0 , F + n � i + n: i ∈ F { } ∈ F for each F ∈ F; and a filter if it is proper and closed under intersection, i.e., F 0 , is the smallest family generated by the family F.
A set F ⊂ N 0 is called thick if for each m ∈ N there exists t m ∈ N 0 such that t m , t m + 1, . . . , t m + m ⊂ F and syndetic if there exists m ∈ N such that, for every t ∈ N 0 , By their definitions, it is obvious that every thick subset of N 0 intersects each syndetic subset of N 0 .
A set F ⊂ N 0 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 s n 1 < s n 2 < · · · such that ∪ ∞ j�1 s n j , s n j + 1, . . . , s n j + n ⊂ F.
e upper density of a subset S of N 0 is defined as where |A| denotes the cardinality of the set A. Similarly one can define the lower density of S ⊂ N 0 . e upper Banach density of S ⊂ N 0 is defined as e supremun is taken over all segments of N 0 . One can see [16] for more details of families.
In general, we use F inf to denote the family consisting of all infinite subsets of N 0 and use F s , F t , F ts , F ps , F pud , F pld , and F pubd 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 N 0 . en, F s , F t , F ts , F ps , F pud , F pld , and F pubd 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]). A point x ∈ X is a transitive point of f if orb(x), 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 (orb(x), f | orb(x) ) is minimal. A subset M of X is called minimal if every point of M is minimal.
A point x ∈ X is called an almost periodic point of f if N(x, U) ∈ F s for each U ∈ U x . Denote by A(f) 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. Let F be a family of N 0 . A dynamical system (X, f) is F-transitive if for each pair of opene subsets U, V of X, N(U, V) ∈ F; F-central if for each opene subset V of X, N(V, V) ∈ F. Especially, a dynamical system (X, f) is transitive if for each pair of opene subsets U, V of X, N(U, V) ≠ ∅; topological ergodic if N(U, V) ∈ F s 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 2 Discrete Dynamics in Nature and Society properties for the hitting time set of any two opene subsets to compact systems.
A dynamical system (X, f) is called an M-system if it is transitive and the set of its minimal points is dense in X.
A dynamical system (X, 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 π(U) 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 Definitions 1 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 N 0 stated as below.

Definition 4.
Let (X, f) be a dynamical system, F be a family of N 0 , 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, Now, by using the notion of F-sensitivity covers, we introduce the topological version of sensitivity with respect to a Furstenberg family for dynamical systems.
Definition 5. Let F be a family of N 0 . A dynamical system (X, f) is called F-topologically sensitive if (X, f) admits an F-sensitivity cover.
Obviously, a dynamical system (X, 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. Let f be the self-map on R defined by f(x) � x + 1, and let us consider the following metric on R: en, 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 Take an open cover U � B(p, e/2): p ∈ R of R. For any opene subset G of R and x, y For any opene subset G of R and x, y ∈ G with x ≠ y and at is, f n i (x), f n i (y) ⊈B(p, e/2) for each p ∈ R and i ∈ N 0 . If not, there exist some opene subset G of R and x, y ∈ G with x ≠ y and q ∈ R and j ∈ N 0 such that ρ(f n j (x), q) < (e/2) and ρ(f n j (y), q) < (e/2); then, On the contrary, · e x − e y > e 1+j ≥ e, which is a contradiction. erefore, for every opene set G ⊂ R and every pair (

Topological Sensitivity with respect to Families of Dynamical Systems
In this section, we give some basic properties of F-topological sensitivity and prove that the F-topological sensitivity is invariant under semiconjugations and that a dynamical system Theorem 1. Suppose that (X, f) and (Y, g) are two dynamical systems and F is a family of N 0 . 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).
is an open cover of X. For each opene subset G of X, φ(G) has a nonempty interior since φ is semiopen. Take an opene set W ⊂ φ(G). By the F-topological sensitivity of (Y, g), us, for each n ∈ N g (W, V), there exist y n , y n ∈ W such that, for every

□
In the following, we will show that the F s -topological sensitivity as well as F t , F ps , F ts , F pud , F pld , and F pubd -topological sensitivity of a dynamical system is invariant under iterations.

Theorem 2.
Let (X, f) be a dynamical system. en, the following statements are equivalent: Discrete Dynamics in Nature and Society F ts -topologically sensitive) for some n ∈ N Proof. We only prove the case of F s -topological sensitivity since the proofs of other cases are similar: (1)⇒(2): assume that U is an F s -sensitivity cover for (X, f) and n ∈ N.
Let (X, f) be a dynamical system. en, the following statements are equivalent: F pubd -topologically sensitive) for some n ∈ N Proof. We only prove the case of F pud -topological sensitivity since the proofs of other cases are similar: Clearly, m � t m n + s m for some t m ∈ N 0 and 0 ≤ s m ≤ n − 1. Hence, (2)⇒(3) is obvious.
(3)⇒(1) is not difficult, so we leave it to the reader. □ Theorem 4. If a dynamical system (X, f) is topologically sensitive and there is no isolated points in X and f is sem- where U is a sensitivity cover for (X, f).
Proof. In fact, if there exists some opene set G ⊂ X such that Note f n+1 (x) ∈ X, and there exists some has a nonempty interior as f which is semiopen. So, there exists an opene set U ⊂ f n+1 (U x ) and N f (U, U) ≠ ∅ which gives that there are y 1 , y 2 ∈ U and m ∈ N f (U, U) such that Let x 1 , x 2 ∈ U x be such that y 1 � f n+1 (x 1 ) and y 2 � f n+1 (x 2 ); then, So, (14) which implies that m + n + 1 ∈ N f (G, U).
at is a contradiction. □ e following theorem reveals the relations between the topological sensitivity and the classical sensitivity and generalizes the result of eorem 2.3 of [15]. At first, we recall that τ d 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 N 0 . 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, Theorem 5. Let f be a continuous self-map on a metric space (X, d) and let us consider the following conditions: (1) implies (2). In addition, if X is compact, then (2) is equivalent to (1).
Proof. e proof of eorem 5 is similar to that of eorem 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.  Proof. Assume that V is an opene subset of X, by the given hypothesis, V∩BD * (f) ≠ ∅. Choose x ∈ V∩BD * (f); then, N(x, V) has positive upper Banach density. Take n 1 , n 2 ∈ N(x, V) with n 1 < n 2 ; then, f n 1 (x) ∈ V, f n 2 (x) ∈ V.
Set y � f n 1 (x) ∈ V; then, x ∈ f − n 1 ( y ). us, Namely, Note that F s 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 space X 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.
Let (X, f) be a dynamical system, where X is a T 3 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 F s -sensitivity cover with two elements. erefore, (X, f) is F s -topologically sensitive.
Proof. Since (X, f) is not minimal, take a ∈ X such that X − orb(a) ≠ ∅, pick b ∈ X − orb(a), then there exist U b ∈ U b and a neighborhood U of orb(a) such that N(G, U a ). Take N(G, U a ) there exists x ∈ G such that eorem 6 generalizes eorem 2.5 of [15] to a great extent since in [17] and there exists an example showing that the set A(f) 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. Proof. If the family [F] is a filter, then for any opene subsets U 1 , U 2 , V 1 , V 2 of X, Conversely, suppose (X, f) is weakly mixing and N(U 1 , V 1 ), N(U 2 , V 2 ) ∈ [F]. By the definition of weak mixing, there exists some m ∈ N(U 1 , For any k ∈ N(A, B), we have  Proof. Suppose U, V are opene subsets of X. By Lemma 3, for each n ∈ N, Suppose that (X, f) is a weakly mixing and topologically ergodic dynamical system, where X admits two opene subsets U and V such that U∩V � ∅. en, (X, f) has an F ps -sensitivity cover with two elements. erefore, (X, f) is F ps -topologically sensitive.  N(A, B) N(A, B) is syndetic which implies that N(A, B) is piecewise syndetic. Take arbitrarily n ∈ N(A, B) and note that N(A, B) ⊂ N(G, U)∩N(G, V), then G∩f − n (U) ≠ ∅ and G∩f − n (V) ≠ ∅. So, there exists x, y ∈ G such that f n (x) ∈ U and f n (y) ∈ V which yields that f n (x) ∉ X − U and f n (y) ∉ X − V. erefore, W is an F ps -sensitivity cover for (X, f) which drives that (X, f) is F ps -topologically sensitive.

N-Topological Sensitivity of Dynamical Systems
In this section, we introduce the notion of n-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.
Let (X, 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 ∈ N 0 and n different points x 1 , x 2 , . . . , x n ∈ G such that, for all x, y ∈ x 1 , x 2 , . . . , x n with x ≠ y, Definition 7. A dynamical system (X, f) (or simply the map f) is called n-topologically sensitive if (X, f) has an n-topological sensitivity cover.

Definition 8.
Let (X, f) be a dynamical system, F be a family of N 0 , 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, Now, by using the notion of n-F-sensitivity cover, we introduce the topological version of n-sensitivity with respect to a Furstenberg family.
Definition 9. Let F be a family of N 0 . 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 of n-topological sensitivity of dynamical systems.

Proof
(1) e proof is similar to that of (1) of eorem 1, so we omit it (2) If U is an n-topological sensitivity cover for (X, f m ) for some m ∈ N, then U is also an n-topological sensitivity cover for (X, f) Now, assume that f 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 U k � f −k (U): U ∈ U . Set V as the intersection of all the covers U 0 , . . . , U m−1 , i.e., V is the open cover of X consisting of all (nonempty) sets with the form We claim that V is an n-topological sensitivity cover for (X, f m ). In fact, let G be an opene subset of X, then there exist s ∈ N 0 and n different points x 1 , x 2 , . . . , x n in G such that, for any x i , x j ∈ x 1 , x 2 , . . . , x n with x i ≠ x j , Let q ∈ N 0 and 0 ≤ p < m such that s � qm + p. Now, for any x i , x j ∈ x 1 , x 2 , . . . , x n with x i ≠ x j , which implies that So, V is an n-topological sensitivity cover for (X, f m ) and (X, f m ) is n-topologically sensitive. 6 Discrete Dynamics in Nature and Society If (X, f m ) is n-topologically sensitive for each m ∈ N, then it is obvious that (X, f m ) is n-topologically sensitive for some m ∈ N. □ About n-topological sensitivity, we also have the following result that is similar to eorem 5.
Theorem 9. Let f be a continuous self-map on a metric space (X, d), and let us consider the following conditions: (2). In addition, if X is compact, then (2) is equivalent to (1).

Proof.
e proof is similar to that of eorem 5, so we omit it. □ Theorem 10. Assume that (X, f) is a weakly mixing dynamical system, where X is an infinite Uryshon space and there is no isolated points in X. en, (X, f) is n-thickly topologically sensitive for each n ∈ N.
Proof. Let n ∈ N. By the given conditions, there exist n different points x 1 , x 2 , . . . , x n ∈ X such that, for each i � 1, 2, . . . , n, x i admits a neighborhood U x i satisfying We claim that W 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 For each m ∈ H, there exist n distinct points y 1 , y 2 , . . . , y n ∈ G such that f m (y i ) ∈ U x i for each i ∈ A; then, for any x, y ∈ y 1 , y 2 , . . . , y n with x ≠ y, It follows that (X, f) is n-thickly topologically sensitive.

Remark 2
(1) eorem 10 generalizes Proposition 2.7 of [15] to a great extent. (2) Fix a family F of N 0 , 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 N 0 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, en, one can obtain the similar results for n-F-topological sensitivity as eorems 8-10.

Multitopological Sensitivity of Dynamical Systems
In this section, we introduce the notion of multisensitivity for dynamical systems. Review that a compact system (X, f) is multisensitive if there exists δ > 0 such that, for any finitely many opene subsets G 1 , . . . , G l of X, ∩ l i�1 S f (G i , 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.
Let (X, 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 G 1 , . . . , G n of X, Definition 11. A dynamical system (X, 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.
Let (X, f) and (Y, g) be two dynamical systems: (1) If (X, 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) (1) e proof is similar to that of (1) of eorem 1, so we omit it (2) If U is a multisensitivity cover for (X, f n ) for some n ∈ N, then it is clear that U is also a multisensitivity cover for (X, f) Now, assume that f is multisensitive and take a multisensitivity cover U for (X, f). Give arbitrarily n ∈ N. For every k ∈ 0, . . . , n − 1 Set V as the intersection of all the covers U 0 , · · · U n−1 , i.e., V is the open cover of X consisting of all (nonempty) sets with the form of U 0 ∩ U 1 ∩ · · · ∩U n−1 with U k ∈ U k for Discrete Dynamics in Nature and Society k � 0, 1, . . . , n − 1. We claim that V is a multisensitivity cover for (X, f n ). In fact let G 1 , . . . , G s be any finitely many opene subsets of X, by the multisensitivity of (X, f), there exist x i , x i ∈ G i and m ∈ N 0 such that for each 1 ≤ i ≤ s. Let q ∈ N 0 and 0 ≤ p < n such that m � qn + p. Now, we have for all 1 ≤ i ≤ s, which implies erefore, V is a multisensitivity cover for (X, f n ) and (X, f n ) is multisensitive.
If (X, f n ) is multisensitive for each n ∈ N, clearly (X, f n ) is multisensitive for some n ∈ N.
Summarize the above proof process, and we prove result (2). □ About multisensitivity, we have the following eorem 12 that is similar to eorems 5 and 9.
Theorem 12. Let f be a continuous self-map on a metric space (X, d), and let us consider the following conditions: (2). In addition, if X is compact, then (2) is equivalent to (1).

Proof.
e proof is similar to that of eorem 9, so we omit it. □ Theorem 13. Suppose that (X, f) is a weakly mixing dynamical system and X admits two opene subsets U and V of X such that U∩V � ∅. en, (X, f) is multisensitive.
Proof. By the conditions there exist two opene sets en, W is a multisensitivity cover of (X, f). In fact, for any finitely many opene sets U 1 , . . . , U n of X, since (X, f) is weakly mixing, by Lemma 3, N(U i , U) and N(U i , V) are thick for each 1 ≤ i ≤ n and A ≔ ∩ n i�1 (N(U i , U)∩N(U i , V)) ≠ ∅.
en, for each k ∈ A and each 1 ≤ i ≤ n, there exist erefore, W is a multisensitivity cover for (X, f) and (X, f) is multisensitive. Proof. We directly prove that W � X − U, X − V is an F ts -sensitivity cover for (X, f). Let W be any opene subset of X. For each k ∈ N, since f, f 2 , . . . , f k are continuous and A is 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 is also a minimal point of f, then there exists a syndetic set n j of N 0 such that f n j +m (x) ∈ U i A for all j ≥ 1 and each is shows that N(W, U) is a thickly syndetic set. By the same argument, N(W, V) is also thickly syndetic. us, N(W, U)∩N(W, V) is thickly syndetic, which implies that W is an F ts -sensitivity cover for (X, f) and (X, f) is F ts -topologically sensitive. □ Lemma 5. If (X, f) is multisensitive and there is no isolated points in X, then ∩ n i�1 N f (U i , U) is infinite for any finitely many opene sets U 1 , . . . , U n of X, where U is a multisensitivity cover for (X, f).
Proof. In fact, if there exist finitely many opene sets then m ∈ N f (G i , U) for all i � 1, . . . , k. us, there exist x i , y i ∈ G i such that for every 1 ≤ i ≤ k. Note that, for every 1 ≤ j ≤ m, there exists W i,j ∈ U such that f j (x i ) ∈ W i,j . By the continuities of f j , 1 ≤ j ≤ m, there is an open neighbourhood U i,j ⊂ G i of x i such that f j (U i,j ) ⊂ W i,j for each 1 ≤ j ≤ m and 1 ≤ i ≤ k. Now, for 1 ≤ i ≤ k, let G i � ∩ m j�1 U i,j ⊂ G i , and it is easy to check that f j (G i ) ⊂ W i,j ∈ U for every 1 ≤ j ≤ m and 1 ≤ i ≤ k. us, ∩ k i�1 N f (G i , U)} > m and ∩ k i�1 N f (G i , U) > m. at is a contradiction. Proof. Firstly assume that (X, f) is multisensitive with a multisensitivity cover U of X and k ∈ N. For any opene set G ⊂ X, we choose opene sets G i ⊂ f − i (G) for each i � 1, . . . , k. By Lemma 5, ∩ k i�1 N f (G i , U) is infinite, so we 8 Discrete Dynamics in Nature and Society can choose some n k ∈ ∩ k i�1 N f (G i , U) with n k > k. Note that n k ∈ N f (G i , U) for each i � 1, . . . , k, and there exist x i , y i ∈ G i which gives that f n k (x i ), f n k (y i ) ∈ f n k − i (G) such that en, for each i � 1, . . . , k, there exist x i , y i ∈ G such that f n k − i (x i ) � f n k (x i ), f n k − i (y i ) � f n k (y i ), and which implies n k − i ∈ N f (G, U) for all i � 1, . . . , k. erefore, (X, f) is thickly topologically sensitive. Now, we assume (X, 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 U 1 , . . . , U k be opene subsets of X. en, for each i � 1, . . . , k, there exists n i ∈ N such that f n i (x) ∈ U i . Hence, there exists V ∈ U x such that f n i (V) ∈ U i for each i � 1, . . . , k. By the given assumptions, there is s ∈ N 0 such that s, s + 1, . . . , s + n 1 + n 2 + · · · + n k ⊂ N f (V, U). (36) So, s + n i ∈ N f (V, U) for all i � 1, . . . , k which yields that there exist x i , y i ∈ V such that which implies that s ∈ N f (U i , U) for every i � 1, . . . , k, i.e., s ∈ ∩ k i�1 N f (U i , U). erefore, (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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Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.