A Note on Stronger Forms of Sensitivity for Non-Autonomous Dynamical Systems on Uniform Spaces

This paper introduces the notion of multi-sensitivity with respect to a vector within the context of non-autonomous dynamical systems on uniform spaces and provides insightful results regarding N-sensitivity and strongly multi-sensitivity, along with their behaviors under various conditions. The main results established are as follows: (1) For a k-periodic nonautonomous dynamical system on a Hausdorff uniform space (S,U), the system (S,fk∘⋯∘f1) exhibits N-sensitivity (or strongly multi-sensitivity) if and only if the system (S,f1,∞) displays N-sensitivity (or strongly multi-sensitivity). (2) Consider a finitely generated family of surjective maps on uniform space (S,U). If the system (S,f1,∞) is N-sensitive, then the system (S,fk,∞) is also N-sensitive. When the family f1,∞ is feebly open, the converse statement holds true as well. (3) Within a finitely generated family on uniform space (S,U), N-sensitivity (and strongly multi-sensitivity) persists under iteration. (4) We present a sufficient condition under which an nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates N-sensitivity.


Introduction
Chaos refers to the inherent unpredictability that arises in deterministic systems in the absence of stochastic variables.It is a fundamental area of study in nonlinear science, representing a universal dynamical behavior of nonlinear systems.Furthermore, chaos profoundly and globally influences the evolution of nonlinear dynamics.Sensitivity is a critical element of chaos, attracting significant attention from scholars for research purposes [1][2][3][4][5][6][7][8][9][10][11][12][13][14].
Ruelle and Takens provided the first definition of sensitivity in 1971 [1].It describes the unpredictable nature of chaotic processes and is essential to different kinds of chaos.Even a small change in a dynamical system's initial configuration might result in significantly different behavior later on.In 1980, Auslander and Yorke applied sensitivity to topological dynamical system [2].In 1989, the famous Devaney chaos was proposed [3].Since then, the study of sensitivity became popular.If a system is topologically transitive, contains a dense collection of periodic points, and is sensitive to initial conditions, it is said to be Devaney chaotic [3].Later, Banks et al. proved that the third condition (sensitivity) of the Devaney chaos is implied by the first two characteristics (transitivity and dense periodic points set) [4].Glasner and Weiss expanded it to demonstrate that a transitive non-minimal system with dense minimal points is sensitive [5].
One indicator of a system's sensitivity is the "largeness" of the time set where sensitivity occurs.From this perspective, Moothathu presented a number of stronger forms of sensitivity, namely, cofinite sensitivity, multisensitivity, and syndetic sensitivity [6].His work further deepened the study of sensitivity.Later, Li presented the concept of ergodic sensitivity [7], which is another stronger form of sensitivity.He also present some sufficient conditions for dynamical system (X, f ) to be ergodically sensitive.Liu et al. introduced thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, and strong sensitivity [8].
For more results about sensitivity, we refer to [9][10][11][12][13][14] Recently, there has been significant research interest in studying sensitivity and chaoticity on uniform spaces for dynamical systems.Ahmadi et al. studied the topological shadowing property, chain transitivity, total chain transitivity, and chain mixing property for dynamical systems on uniform spaces [15].Shah et al. presented and investigated the concept of distributional chaos on uniform spaces [16].The notions of weak uniformity, uniform rigidity, and multi-sensitivity for uniform spaces were initially proposed by Wu et al. in 2019 [17].The concepts of mean sensitivity and Banach mean sensitivity are expanded to uniform spaces by Wu et al. [18].
Consider a continuous map f : X → X operating on a compact metric space (X, d).A non-autonomous discrete system difference equation refers to where each f n is a continuous self-map on X.It should be noted that the autonomous dynamical system is a particular case of system (1).For other concepts and notations covered in this section, see Section 2.
The study of non-autonomous dynamical systems focuses on systems that vary with time, and compared to autonomous dynamical systems, the dynamics of non-autonomous dynamical systems are more complex.The applications of non-autonomous dynamical systems have been explored in various fields, including ecology, economics, climate science, biomedicine, and control engineering.These systems offer a valuable tool for understanding and predicting the behavior of dynamic systems under external influences and disturbances.By considering the time-varying nature of these systems, researchers can gain deeper insights into the intricacies of their dynamics and make more accurate predictions.The versatility and wide-ranging applications of non-autonomous dynamical systems make them an essential framework for studying and analyzing complex real-world phenomena.As a result, academics have been drawn to examine the complexity of such systems in recent years due to the rich dynamics [19][20][21][22][23][24][25][26][27].
Salman et al. introduced the notions of sensitivity, multi-sensitivity, cofinite sensitivity, and syndetic sensitivity within the realm of nonautonomous dynamical systems on uniform spaces.Additionally, they identified several adequate conditions wherein topological transitivity and the presence of densely distributed periodic points lead to sensitivity in nonautonomous systems residing on Hausdorff uniform spaces [28].
Inspired by their works, we present the notions of multi-sensitivity with respect to a vector, as well as N -sensitivity and strongly multi-sensitivity, within the context of nonautonomous dynamical systems on uniform spaces.Furthermore, we provide criteria under which a nonautonomous dynamical system on infinite Hausdorff uniform space demonstrates N -sensitivity.
The rest of the paper is organized as follows.In Section 2, some basic concepts are given.In Section 3, some stronger versions of sensitivities, namely, multi-sensitivity with respect to a vector, N -sensitive and strongly multi-sensitivity are introducd to nonautonomous dynamical systems on uniform spaces.In Section 4, it gives some sufficient conditions for an infinite Hausdorff uniform space to be N -sensitive.

Preliminaries
Throughout the paper, consider the symbols N = {1, 2, . ..} and Z + = {0, 1, . ..}.Consider a nonempty set S, the diagonal of S × S is denoted by ∆ S = {(x, x) : x ∈ S}.Suppose that A ⊂ S × S, A −1 which is defined by A −1 = {(y, x) : (x, y) ∈ A} is called the inverse of A. Specially, if A = A −1 , then A is said to be symmetric.Assume that A 1 , A 2 are subsets of S × S, the composite A 1 • A 2 refer to the collection {(x, y) ∈ S × S : there exits z ∈ S such that (x, z) ∈ A 1 and (z, y) ∈ A 2 }.Denote D S = {A ⊂ S × S : ∆ S ⊂ A and A = A −1 }.
The introduction of the notion of uniform space was attributed to Weil in [29].In this section, we provide a concise overview of uniform space, but for a more in-depth understanding, readers are encouraged to refer to ( [30], Chapter 8) for a comprehensive introduction to the subject.Definition 1 ([31]).Let U be a nonempty sets consist of the subsets of S × S, U is called a uniform structure on S, if the following conditions hold: For any A ∈ U , there exists B ∈ U such that B • B ⊂ A.
A uniform space is defined as a pair (S, U ), where S is a non-empty set and U is a uniform structure on it.Generally, we call U entourages.
In a uniform space denoted as (S, U ), a uniform topology can be established on S.This is characterized by a neighborhood base at each point s that belongs to S. This neighborhood base is composed of sets D[s] = {t ∈ S : (s, t) ∈ D}, with D representing all entourages of the uniform space (S, U ).For a map f : S → S on a uniform space (S, U ), if ( f × f ) −1 (U ) ⊂ U , then f is said to be uniformly continuous on (S, U ).
Next, we will introduce non-autonomous dynamical systems on uniform spaces.A nonautonomous discrete system is defined as (S, f 1,∞ ), (S, U ) is a nontrivial uniform space and for any n ∈ N, f n : S → S is uniformly continuous.This system consists of a sequence of uniformly continuous maps := id and the kth iterate For the periodic points for non-autonomous dynamical systems, scholars proposed two different definitions from different perspectives.Here, the two different periodic points is distinguish by P1 and P2.
Obviously, P2-periodic implies P1-periodic and the orbit of a periodic point in the sense of (P2) must be finite.However, by the Example 4.4 of [33], the orbit of a periodic point in sense of (P1) has the potential to be infinite.
Assume that D is a subset of S × S and V is a subset of S. Denote Assume that P is the collection of all subsets of Z + .F ⊂ P is called a Furstenberg family if it satisfies F 1 ⊂ F 2 and F 1 ∈ F imply F 2 ∈ F (for details see [34]).
The following definition about sensitivity were generated by Huang et al. [35] and Salman et al. [28].Let F be a Furstenberg family.A non-autonomous dynamical system (S, f 1,∞ ) on a uniform space (S, U ) is said to be sensitive, if there exists an entourage is finite, then the system becomes cofinitely sensitive.The entourage is called cofinitely sensitive entourage respectively.when N f 1,∞ (V, E) is syndetic set, i.e., there exists N ∈ N such that {i, i + 1, . . ., i for each j ∈ {1, 2, . . ., m} and for every m ∈ N.

Multi-Sensitivity with Respect to a Vector for Nonautonomous Dynamical System on Uniform Spaces
This section will introduce some stronger versions of sensitivity, namely, multisensitivity with respect to a vector, N -sensitive and strongly multi-sensitivity for nonautonomous dynamical systems on uniform spaces.Definition 4. Consider a nonautonomous dynamical system (S, f 1,∞ ), where (S, U ) is a uniform space.Let − → a = (a 1 , a 2 , . . .a r ).(S, U ) is said to be 1.
multi-sensitive with respect to − → a , if there exists an entourage D ∈ U such that for any

2.
N -sensitive, if there exists an entourage D ∈ U , for any strongly multi-sensitive, if there is an entourage D ∈ U , (S, f 1,∞ ) is multi-sensitive with respect to any vector in N n and any n ∈ N.
Theorem 2. A k-periodic nonautonomous dynamical system (S, f 1,∞ ) on Hausdorff uniform space is strongly multi-sensitive if and only if (S, For the necessity, assume that (S, g) is strongly multi-sensitive, where For any vector − → a = (a 1 , a 2 , . . ., a m ), m ∈ N and any non-empty open sets U 1 , U 2 , . . ., U m , as (S, g) is multi-sensitive with respect to − → a , that is, there exists an For the sufficiency, suppose that (S, f 1,∞ ) is strongly multi-sensitive.For any m ∈ N and any vector − → a = (a 1 , a 2 , . . . ,a m ), let U 1 , U 2 , . . ., U m be any nonempty open sets of S. By the hypothesis, (S, f 1,∞ ) is multi-sensitive with respect to vector (ka 1 , ka 2 , . . ., ka m ).That is, where F in f denote the collection of all infinite subsets of Z + .
Proof.Since sufficiency is obvious, we only need to prove necessity.Assume that (S, U ) is N -sensitive with sensitive entourage D ∈ U .We use the counterfactual.Suppose Given S is a uniform space, there exists an This contradicts with the notion that k is the maximum of the set Remark 1.By employing analogous reasoning, it can be confirmed that the aforementioned theorem holds true for cases of strong multi-sensitivity as well.
We subsequently demonstrate that for a modified nonautonomous dynamical system (S, f k,∞ ) with a feeble open family f 1,∞ , where (S, f k,∞ ) = { f n } ∞ n=k , N -sensitivity as well as strongly multi-sensitivity are preserved.Theorem 4. Assume that (S, U ) is a uniform space and f 1,∞ is finitely generated, where each f i be surjective.If the system (S, f 1,∞ ) exhibits N -sensitivity, it follows that the system (S, f k,∞ ) also displays N -sensitivity.Conversely, if the family f 1,∞ is feebly open, the converse holds true as well.
Proof.As (S, f 1,∞ ) is N -sensitive and by Theorem 3, there exists an entourage D ∈ U such that for any nonempty open sets Consider a set of generators g 1 , g 2 , • • • g s for the f 1,∞ , and let's fix an arbitrary integer k greater than or equal to 2. Define Γ := 1≤n≤2mk {1, 2, • • • , s} n , then take any element α from the set Γ, denote α = (α 1 , α 2 , • • • , α ℓ ) for 1 ≤ n ≤ 2mk.We define the composition of these generators as Clearly, each map g α , where α ∈ Γ, exhibits uniform continuity.This observation leads us to conclude that the set Note that Select a positive integer p and a non-negative integer r such that 0 ≤ r ≤ k − 1 and the equation M = pk + r holds.note that p ≥ 2.
Denote that α i = ((ip By the choice of E, this implies that ( f i(p−1)k k Conversely, assume that (S, f k,∞ ) is N -sensitive with sensitive entourage D ′ ∈ U .For nonempty open sets W 1 , W 2 , • • • , W m of S, due to the feeble openness exhibited by the family f 1,∞ , it can be inferred that W ′ i = int( f Denote that ( As each g β (β ∈ Γ) is uniform continuous, there exists U .This together with (3), implies that ( Evidently, the aforementioned outcome holds valid in the context of strong multisensitivity as well.
Theorem 5. Consider a finitely generated family f 1,∞ on a uniform space (S, U ).The pair (S, f 1,∞ ) possesses N -sensitivity if and only if, for any k ≥ 1, the pair (S, f Proof.Sufficiency: This is readily evident.
Necessity: Consider a generator set g 1 , g 2 , • • • , g s for f 1,∞ , and let k be any integer greater than or equal to 2. Suppose that f 1,∞ is N -sensitive with the N -sensitive entourage D ∈ U .For nonempty open sets U 1 , U 2 , . . ., U m ⊂ S, according to the Theorem 3, . Consequently, there exist elements x i and y i in U i , where ℓ > k, such that the pair ( f iℓ 1 (x i ), f iℓ 1 (y i )) lies outside the bounds of D, that is ( f iℓ 1 (x i ), f iℓ 1 (y i )) / ∈ D. Select a positive integer p and a non-negative integer r such that 0 ≤ r ≤ k − 1 and the equation M = pk + r holds.Denote Γ := 1≤n≤m(k+1) {1, 2, . . ., s} n , For any α ∈ Γ, where Evidently, the uniform continuity holds for every g α where α belongs to the set Γ.This leads to the implication that Example 2. Let I be the closed unit interval [0, 1] and f n (n ≥ 1) be defined by As each f n (x), n ≥ 3 is tent map, ([0, 1], f 3,∞ ) is cofinitely sensitive.Denote that f = f n , n ≥ 3. Actually, for any non-empty open set U ⊂ [0, 1].If 0 ∈ U, since there exists x ∈ U and m > 0 such that f m (x) = 1/2, and noting that f (0) = f (1) = 0, f (1/2) = 1, it follows that f n (U) = [0, 1] for n = m + 1.If 0 / ∈ U, let J be an open interval of U. Suppose that for each m ≥ 0, 1  2 / ∈ f m (J), then the length of f m (J) is 2 m times the length of J. Since m increases to infinity later, there must be some m > 0 such that the length of f m (J) greater than 1.This forms a contradiction.Therefore, there exists m 0 > 0 such that 1 2 ∈ f m 0 (J).Thus, as previously stated, there is also an n such that f n (U) = [0, 1].This implies that ([0, 1], f ) is sensitive.According to the equivalence between sensitivity and cofinite sensitivity on the interval ([6] Theorem 2), ([0, 1], f 3,∞ ) is cofinitely sensitive.

Conclusions
This paper introduces the concept of multi-sensitivity with respect to a vector in the context of non-autonomous dynamical systems on uniform spaces, providing an insights into N -sensitivity and strongly multi-sensitivity, as well as their behaviors under varying conditions.Compared to previous work, this article further extends the concept of strongly sensitivity and enriches the research on strongly sensitivity.Existing research work forms the basis for our work, and our work further extends and expands upon this existing research.We acknowledge the findings and methodologies established by previous studies and use them as a starting point.Our work builds upon the foundation laid by previous research and pushes the boundaries by delving deeper into the subject ) also exhibits N -sensitivity.