Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 52 https://internationalpubls.com Stability, and Almost Sensitivity of Induced Maps Amalraj. P1*, P.B.Vinod Kumar2 1*Research Scholar, APJ Abdul Kalam Technological University, (Department of Mathematics, Sanatana Dharma College, Alappuzha, Kerala, India) Email address: amalrp2929@gmail.com 2Research Supervisor, APJ Abdul Kalam Technological University, (Department of Mathematics, Rajagiri School of Engineering and Technology, Cochin, Kerala, India), Email address: vinodkumar.rajagiri@gmail.com Article History: Received: 22-07-2024 Revised: 09-09-2024 Accepted: 29-09-2024 Abstract: Suppose that X is a compact Hausdorff space and f : X → X is continuous. We consider the space K(X), the space of all compact subsets of X with Hausdorff metric H. Let f˜: K(X) → K(X) defined by f ̃(K)=f(K) . We discuss some interconnections between the orbit of f ̃and the orbit of f. By assuming the transitivity of f ̃, we conclude that X contains a cantor set C with (orb(f ̃,C)) ̅ = K(X). That is, orbit of a nowhere dense set is dense in the hyperspace. Along with this we introduce ”almost sensitivity” and ”stability” in K(X). We prove that f is stable in X if and only if f ̃ is stable in K(X). Again we prove that transitive maps are always ’almost sensitive’ in K(X) and hence the base map is ’almost sensitive’ in X. Keywords: Maps, continuous map, compact metric space. 1. Introduction Normally, for studying the dynamics of the hyperspace of a compact metric space, we consider a dynamical system defined by a continuous map f : X → X, describing the dynamics of points in the base space X and then we study the induced map 𝑓 : K(X) → K(X) defined by 𝑓(𝐾) = 𝑓(𝐾) for a compact set K ⊆ X as a form of collective dynamics. In this context, a very natural question arises: What is the connection between dynamical properties of the base map f and the induced map 𝑓 ? During the past years this question has attracted many researchers. (see[4],[5],[6],[7],[8], [9],[10] and [11]) In this paper, we prove that there exists a cantor set C in X, nowhere dense in X but its orbit is dense in K(X). At the same time we establish the fact that the set 𝐷 = {𝑥 ∈ 𝑋, 𝑜𝑟𝑏(𝑓, 𝑥)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑋}, 𝐷 ≠ 𝑋 is dense in X and so it is not closed and there fore 𝐷 ≠ 𝐾(𝑋). We also prove that f is ’stable’ if and only if 𝑓 is stable and 𝑓 is ’almost sensitive’ if 𝑓 is transitive and this implies the ’almost sensitivity’ of f. Through out this paper, X denotes a Hausdorff compact metric space without isolated points with metric d and f : X → X is continuous. K(X) denotes the space of all compact subsets of X with Hausdorff metric H induced by d . Let𝑓: K(X) → K(X) defined by 𝑓(𝐾) = 𝑓(𝐾). 2. Hyperspace and Induced Map 1.In this section we have study some properties of the induced map 𝑓 and the base map f. By assuming the transitivity of 𝑓, we show that the base space X contains a cantor set C with its orbit is dense in K(X). Along with this we also study the concept of ’almost sensitivity’ in X and K(X). Let us use the symbol 𝜙 for the induced map 𝑓 for the sake of convenience. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 53 https://internationalpubls.com Definition 2.1. If A = {Aµ}µ∈ℸ is a collection of non-empty subsections of X, then mesh(A ) = sup{diam(Aµ),µ ∈ ℸ}. In [2] it is proved that (K(X),H) is compact. Lemma 2.1 If 𝑜𝑟𝑏(𝜙, 𝐾) = 𝑋 then for individually A ∈ orb(ϕ,K) and for all x ∈ A, 𝑜𝑟𝑏(𝑓, 𝑥) = 𝑋 Proof. Let n ∈ ℕ and x ∈ f n(K) . We have to demonstrate that 𝑜𝑟𝑏(𝑓, 𝑥) = 𝑋. Take y ∈ X and ϵ > 0. Since 𝑜𝑟𝑏(𝜙, 𝐾) = 𝑋 we have 𝑜𝑟𝑏(𝜙, 𝑓𝑛(𝐾)) = 𝐾(𝑋) Let {𝑦} ∈ 𝑜𝑟𝑏(𝜙, 𝑓𝑛(𝐾)), then there exist j ∈ ℕ such that 𝐻 (𝑓𝑗(𝑓𝑛(𝐾), {𝑦})) < 𝜖. Hence 𝑓𝑗(𝑥) ∈ 𝑓𝑗(𝑓𝑛(𝐾)) ⊆ 𝐵𝜖(𝑦) So 𝐵𝜖(𝑦) ∩, 𝑜𝑟𝑏(𝑓, 𝑥) ≠ ∅ There fore , 𝑜𝑟𝑏(𝑓, 𝑥) = 𝑋. Lemma 2.2. If X is weekly mixing, then any power X × X × ··· × X is ergodic. Proof. see[1] Lemma 2.3. Let ϕ : K(X) → K(X) be transitive. Then there exist a cantor set C ⊆ X such that 𝑜𝑟𝑏(𝜙, 𝐶) = 𝐾(𝑋). Proof. Let δ₀ represent the diameter of the set X. For every positive integer n, we define a series of finite open covers of X, denoted by 𝑉�̃� = {𝑉𝑛,1, 𝑉𝑛,2, … . , 𝑉𝑛,𝑡𝑛 } where each 𝑉𝑛,𝑖 is a nonempty subset. These covers are constructed such that the size of each cover is smaller than δₙ. In essence, this arrangement ensures that each element of the set X is contained within at least one open set in the cover, and as n increases, the covers become increasingly finer, converging towards the set's diameter. Step 1 Let us assume W0 and W1 to be two non-empty disjoint open sets in X and mesh({W0∩W1} < δ1. Let 𝜆1 = {1,2, … . , 𝑡1} × {1,2, … . , 𝑡1} = {(𝑎, 𝑏)|𝑎, 𝑏 ∈ {1,2, … . , 𝑡1}}. Let us take into consideration the following 𝑡1 2+1 collection of open sets (W0,W1) and {(V1,a,V1,b) : (a,b) ∈ λ1} Then, by Lemma 1.4 [see 1], two closed subsets of X, 𝐶0 and 𝐶1, having the following characteristics, exist: • Each int(Ci) is nonempty and which is contained in Wi for i = 0,1. Hence C0 and C1 are disjoint. • for each A ∈ ⟨C0,C1⟩ = {𝐵 ∈ 𝐾(𝑋): 𝐵 ⊂ (𝐶0 ∪ 𝐶1)𝑎𝑛𝑑 𝐵 ∩ 𝐶𝑖 ≠ ∅ } and for each (a,b) ∈ λ1,there exist n ∈ N such that fn(A) ∈ ⟨U1,a,U1,b⟩. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 54 https://internationalpubls.com • Also, fn(A ∩ C0) ⊂ U1,a and fn(A ∩ C1) ⊂ U1,b. Let ℂ1= ⟨C0,C1⟩. Then diam(ℂ1) < δ1 for each A ∈ ℂ1, orb(ϕ,A) is δ1-close to F2(X),where F2(X) = {A ∈ K(X)\|A| ≤ 2}. For, given any {p,q} ∈ F2(X), there exist (a,b) ∈ λ1 such that p ∈ U1,a and q ∈ 𝑈1,𝑏. Since there exist n so that fn(A) ∈ ⟨U1,a,U1,b⟩. We conclude that H({p,q},fn(A)) < δ1 Step 2 Let W0,0,W1,0,W0,1.W1,1 are 4 non-empty open subsets of X with 𝑊0,0 ∩ 𝑊1,0 = ∅ and 𝑊0,1 ∩ 𝑊1,1 = ∅ 𝑊0,0 ∪ 𝑊1,0 ⊂ 𝐶0 and 𝑊0,1 ∪ 𝑊1,1 ⊂ 𝐶1 and 𝑚𝑒𝑠ℎ({𝑊0,0, 𝑊1,0, 𝑊0,1, 𝑊1,1 }) < 𝛿2. Let λ2 = {1,2,...t2} 4 = {(a1,a2,a3,a4)\ai ∈ {1,2,...t2}}. Consider the following 𝑡2 4 + 1 collection of open sets (𝑊0,0, 𝑊1,0, 𝑊0,1, 𝑊1,1),{(𝑈2, 𝑎1, 𝑈2, 𝑎2, 𝑈2, 𝑎3, 𝑈2, 𝑎4)}. Given four non-empty open subsets of X, 𝑊0,0, 𝑊1,0, 𝑊0,1, 𝑎𝑛𝑑 𝑊1,1, where 𝑊0,0 and 𝑊1,0 are disjoint, as are 𝑊0,1 and 𝑊1,1 . Each pair of subsets is contained within different closed sets, 𝐶0 and 𝐶1 , respectively. The mesh of these subsets is less than 𝛿2 . Then, 𝜆2 consists of all possible combinations of indices from 1 to 𝑡2. A collection of open sets is formed from the given subsets and 𝜆2. By the same result of Lemma 1.2[see 1] there exist 4 closed sets C0,0, C0,1,C1,0, C1,1 with the following properties: 1. Each int(Ci,j) is nonempty and contained in Wi,j for {i,j} ∈ {0,1} × {0,1} 2. For each A ∈ ⟨C0,0,C0,1,C1,0,C1,1⟩ and for each (a1,a2,a3,a4) ∈ λ2 there exist 𝑛 ∈ 𝑁such that fn(A) ∈ ⟨U2,a1,U2,a2,U2,a3,U2,a4⟩, subsets of ℂ2 namely ℂ2 𝑖 with 𝑓𝑛(𝐴 ∩ ℂ2 𝑖 ) ⊂ 𝑈2, 𝑎𝑖 for 1 ≤ 𝑖 ≤ 4 , where ℂ2 = ⟨𝐶0,0, 𝐶0,1, 𝐶1,0, 𝐶1,1⟩. Note that diam(ℂ2) < δ2 and ℂ2⊂ ℂ1 let A ∈ ℂ2 and {p1,p2,p3,p4} ∈ F4(X), then there exist (a1,a2,a3,a4) ∈ λ2 such that pi ∈ U2,ai Since there exist n so that fn(A) ∈ ⟨U2,a1,U2,a2,U2,a3,U2,a4⟩. We conclude that H({p1,a2,a3,a4},fn(A)) < δ2. So for every A ∈ℂ2 ,orb(ϕ ,A) is δ2-close to F4(X) . Step 3 Let's say that ℂ𝑟 has previously been defined and has the following attributes: ℂ𝑟 is defined as a collection of closed sets of X, represented as ⟨C0,0,0,...,...,C1,1,...⟩, where each closed set is indexed by a binary sequence :{Cj1,j2,...,jr\(j1,j2,...,jr) ∈ {0,1}r}. This indexing scheme corresponds to 2r possible combinations, denoting the presence or absence of each closed set in ℂ𝑟 . Each combination delineates the composition of ℂ𝑟 and its constituent closed sets. • (𝑗1, 𝑗2, … , 𝑗𝑟) ∈ {0,1}𝑟 and 𝑖𝑛𝑡(𝐶𝑗1𝑗2….𝑗𝑟 ) is non empty,𝐶𝑗1𝑗2….𝑗𝑟 ⊂ 𝐶𝑗2….𝑗𝑟 and 𝑑𝑖𝑎𝑚(𝐶𝑗1𝑗2….𝑗𝑟 ) < 𝛿𝑟 • diam(ℂ𝑟) is less than δr and ℂ𝑟 contained in ℂ𝑟−1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 55 https://internationalpubls.com • For each pair (𝑗1, 𝑗2, … 𝑗𝑟) ≠ (𝑙1, 𝑙2, … . , 𝑙𝑟) in {0,1}r; 𝐶𝑗1𝑗2….𝑗𝑟 and 𝐶𝑙1𝑙2….𝑙𝑟 are disjoint. • For each A ∈ ℂ𝑟 and each (𝑡1, 𝑡2, … , 𝑡2𝑟) in λr = {𝑡1, 𝑡2, … 𝑡𝑟}2𝑟 there exist 𝑛 ∈ 𝑁 , 𝑓𝑛(𝐴) ∈ 〈𝑈𝑟 , 𝑡1, 𝑈𝑟 , 𝑡2, … . , 𝑈𝑟 , 𝑡2𝑟〉, subsets of ℂ𝑟 namely ℂ𝑟 𝑖 with 𝑓𝑛(𝐴 ∩ ℂ𝑟 𝑖 ) ⊂ 𝑈𝑟 , 𝑡𝑖, for 1 ≤ 𝑖 ≤ 2𝑟. Then for if 𝐴 ∈ ℂ𝑟, then 𝑜𝑟𝑏(𝜙, 𝐴) is 𝛿𝑟 close to 𝐹2𝑟(X). Similarly, for ℂ𝑟+1. Step 4 So we get a declining order of compact subsets of 𝑋, {ℂ𝑟}1 ∞ Let {𝐶𝑙1𝑙2….𝑙𝑟:(𝑙1, 𝑙2, … , 𝑙𝑟) ∈ {0,1}𝑟} be the 2𝑟 compact subset of X that define ℂ𝑟 . ∞ Let 𝐶 = ⋂ (∪ {𝐶𝑙1,𝑙2,…𝑙𝑟 , (𝑙1, 𝑙2, … , 𝑙𝑟) ∈ {0,1}𝑟})∞ 𝑟=1 . Then C is a cantor set in X . for all 𝑟, 𝐶 ∈ ℂ𝑟, so 𝑜𝑟𝑏(𝜙, 𝐶) = 𝐾(𝑋). Theorem 2.1. There exist a cantor set C ⊆ X such that 𝑜𝑟𝑏(𝑓, 𝑥)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑋 for every x ∈ 𝑓𝑛(𝐶) for all n 𝑛 ∈ℕ . Proof. clear from Lemma 2.3 Theorem 2.2. Let 𝑓: K(X) → K(X) be transitive. Then there exist a cantor set C ⊆ X such that 𝑜𝑟𝑏(𝑓, 𝑥)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑋 for every x ∈ C Proof. clear from lemma 2.1 and theorem 2.2□ Theorem 2.3. Let 𝐷 = {𝑥 ∈ 𝑋, 𝑜𝑟𝑏(𝑓, 𝑥)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑋} and 𝑓 : K(X) → K(X) is transitive. Then C ⊆ D, that is D nonempty. Also D is a dense subset of X fully invariant under f, that is f(D) ⊆ D Proof. From theorem 2.3, we can find a cantor set C in X with C ⊆ D. That is D is nonempty. for the proof of the remaining part, see[3] and[4]□ Theorem 2.4. Let ϕ : K(X) → K(X) is transitive and 𝐷 ≠ 𝑋. Then 𝐷 ∉ 𝐾(𝑋). Proof. from theorem 2.4, D is dense in X. So D is not closed and so it is not compact. □ In the theorem we have a nowhere dense set C in X with its orbit orb(ϕ ,C) is dense in K(X). But at the same time we have a dense set D in X with even 𝐷 ∉ 𝐾(𝑋). Theorem 2.5. Let f˜: K(X) → K(X) is transitive and let C be the cantor set in X. Then for any U ⊆ X and for any integer m ,lim 𝑠𝑢𝑝 𝑛→∞ 𝐻 (𝑓𝑛(𝐶), 𝑓𝑛+𝑚(𝐶)) ≥ 𝐻(𝑈, 𝑓𝑚(𝑈)) Proof. Let U and m be given and let be the sequence. since f˜ is continuous and 𝑜𝑟𝑏(𝑓, 𝐶)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝐾(𝑋) there exist for every 1 2𝑛, a positive integer 𝑠𝑛 such that 𝑓𝑠𝑛(𝐶) so close as to U such that 𝐻(𝑓𝑠𝑛(𝐶), 𝑈) < 1 2𝑛+1 and 𝐻(𝑓𝑚 (𝑓𝑠𝑛(𝐶), 𝑓𝑚(𝑈)) < 1 2𝑛+1 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 56 https://internationalpubls.com This implies 𝐻 (𝑓𝑠𝑛(𝐶), 𝑓𝑠𝑛+𝑚(𝐶)) > 𝐻 (𝑈, 𝑓𝑚(𝑈)) − 1 2𝑛 [by using Triangle Inequality] Hence the result. Definition 2.3. Let X be a metric space with metric d and f : X → X be a continuous map. f is said to be almost sensitive if we can find an x ∈ X and 𝑚 ∈ ℕ such that for any lim 𝑠𝑢𝑝 𝑛→∞ 𝑑(𝑓𝑛(𝑥), 𝑓𝑛+𝑚(𝑦)) ≥ 𝑑(𝑦, 𝑓𝑚(𝑦)). Theorem 2.6. In K(X), transitive maps are almost sensitive. Proof. clear from Lemma 2.3 and Theorem 2.6 □ Theorem 2.7. 𝑓is almost sensitive in K(X) implies f is almost sensitive in X Proof. clear from Theorem 2.7 3.Stability of Induced Maps In this section we study the stability of the induced map 𝑓 : K(X) → K(X) and its connection with the stability of the continuous map f : X → X. In this section C(X) denotes the space of all continuous functions on X and C(K(X)) denotes the space of all continuous functions on K(X). Definition 3.1. A point x ∈ X is said to be stable if for all ϵ > 0 there is a δ > 0 such that if d(y ,x) < δ then d(f n(y),f n(x)) < ϵ for every n. A point x is said to be unstable if it is stable for 𝑓−1. Definition 3.2. 𝑓∈ K(X) is stable if given ϵ > 0, there exists a δ > 0 such that for each �̃�∈ C(K(X)) with dH(𝑓, �̃�) < δ, there exist a continuous map h ∈ C(X) such that 𝑓 ∘ �̃� = �̃� ∘ 𝑓 and dH(ℎ̃, 𝑖̃) < ϵ where 𝑖̃ : K(X) → K(X) is the identity map and dH(𝑓, �̃�)=sup {𝑑𝐻 (𝑓(𝐴), �̃�(𝐴)) , 𝐴 ∈ 𝐾(𝑋)}. Theorem 3.1. f : X → X is stable in C(X) if and only if 𝑓: K(X) → K(X) is stable in C(K(X)). Proof. To prove the theorem, first we prove the following lemmas Lemma 3.1. The map 𝜓 : C(X) → C(K(X)) given by 𝜓(𝑓) = 𝑓is an embedding of C(X) in to K(C(X)) Proof. 𝜓 is well defined since the induced map 𝑓 given by a continuous map is continuous. Next we have to show that 𝜓 is injective. Suppose 𝜓(𝑓) = 𝜓(𝑔). Let A = {x} for each x ∈ X We have,{f(x)} = 𝑓(A) = �̃�(A) = {g(x)} for each x ∈ X. Then, 𝜓 is injective. Next we have to show that 𝜓 and 𝜓−1 are continuous. Let {𝑓𝑛} be a sequence in sequence in C(X) which converges to f . Then {𝑓𝑛} converges to 𝑓 ̃in C(K(X)) Assume that for any ϵ > 0 and each A ∈ K(X) , there exist an N ∈ ℤ+ such that d(𝑓𝑛, f) < ϵ for every n ⩾ N. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 57 https://internationalpubls.com For all a ∈ A and n ⩾ N, we have d(f n(a),f(a)) < ϵ So ,f n(A) ⊆ B(fn(A),ϵ),n ⩾ N For all A ∈ K(X), we get f(A) ⊆ B(fn (A),ϵ) for every n ⩾ N Then there exist an N ∈ ℤ+ such that 𝑑(𝑓𝑛(𝐴), 𝑓(𝐴))< ϵ for every n ⩾ N and each A ∈ K(X). Similarly we show that if 𝑓�̃� → 𝑓 in C(K(X)) then 𝑓𝑛 → 𝑓 in C(X). There fore, 𝜓 is an embedding of C(X) in to K(C(X)).]□ Lemma 3.2. 𝜓(𝐶(𝑋))is closed in C(K(X)) Proof. We prove that there exist a sequence {𝑓�̃�} in 𝜓(𝐶(𝑋))which converges to F such that there is f ∈ C(X) satisfies 𝑓 = 𝐹. Consider 𝜓−1(𝑓𝑛) = 𝑓𝑛. For every ϵ > 0, take N > 0 such that 𝑑(𝑓𝑛, 𝑓𝑚) < ϵ, for every n,m ⩾ N. This means that any compact set A satisfies 𝑑(𝑓𝑛(𝐴), 𝑓𝑚(𝐴)) < ϵ for every n,m ⩾ N. For A ∈ K(X) ,choose Ax = {x} for every x ∈ X. Then d(fn, fm) < ϵ for every n,m ⩾ N and all x ∈ X. Since {fn} is a cauchy in C(X), which is a complete metric space, there exist f ∈ C(X) such that {fn} converges to f. Since 𝜓 is continuous {𝜓(𝑓𝑛)} = {𝑓𝑛}Since the limit is unique, F = 𝑓 There fore F ∈ 𝜓(C(X)). There fore 𝜓(C(X)) is closed in C(K(X)). Proof of The Theorem 3.1 Suppose f : X → X is stable and ϵ > 0 is given .Then there are δ > 0 and h ∈ C(X) as is the definition of stability for f. Since 𝜓−1 is continuous at 𝑓 ∈ 𝜓(C(X)),then for δ > 0, there exist a δ1 > 0 such that if �̃� ∈ 𝜓(C(X)) with 𝑑(𝑓, �̃�) < δ, then we have, d(𝜓−1(𝑓),𝜓−1(�̃�) = d(f ,g) < δ. From f ◦ h = h ◦ g, we have 𝜓(f ◦ h)(A) = (𝑓 ∘ �̃�)(A) = (f ◦ h)(A) = 𝑓(h(A)) =𝑓(ℎ̃(A)) = (𝑓 ∘ �̃�)(A),A ∈ K(X) and 𝜓(h ◦ g)(A) = (ℎ̃◦ �̃�)(A),A ∈ K(X) Thus 𝑓 ∘ ℎ̃ = ℎ̃ ∘ �̃�. Let A ∈ K(X).From the continuity of h and compactness of A, we can see that ℎ(𝐴) ⊆ ⋃ 𝐵(𝑎, 𝜖) 𝑎∈𝐴 and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 58 https://internationalpubls.com 𝐴 ⊆ ⋃ 𝐵(ℎ(𝑎), 𝜖) 𝑎∈𝐴 if and only if ℎ(𝐴) ⊆ 𝐵(𝐴, 𝜖) and 𝐴 ⊆ 𝐵(ℎ(𝐴), 𝜖) that is, h(A) ⊆ B(A,ϵ) and A ⊆ B(h(A),ϵ) for any A ∈ K(X). This implies 𝑑(ℎ̃, 𝑖̃)< ϵ There fore, 𝑓 is stable in 𝜓(C(X)). Conversely, Suppose 𝑓 ∈ 𝜓(𝐶(𝑋)) is stable. for any ϵ > 0, there exist a δ > 0 and ℎ̃ ∈ 𝜓(𝐶(𝑋)) satisfying topological stability of 𝑓. Since 𝜓 is continuous at f ∈ C(X), there is a δ2 > 0 such that for any g ∈ C(X) with d(f ,g) < δ2,we have 𝑑(𝜓(𝑓), 𝜓(𝑔)) = 𝑑(𝑓, �̃�)< δ Take A = {x} for each x ∈ X, then 𝑓 ∘ �̃� = �̃� ∘ 𝑓 that (𝑓 ∘ �̃� )(A) = {f(h(x))} = {h(g(x))} =(�̃� ∘ 𝑓 )(A), x ∈ X This implies f ◦ h = h ◦ g. From 𝑑(ℎ̃, 𝑖̃)<𝜖, we have 𝑓(𝐴) ⊆ 𝐵(𝐴, 𝜖) and 𝐴 ⊆ 𝐵(ℎ̃(𝐴), 𝜖) for 𝐴 ∈ 𝐾(𝑋). Taking A = {a}, for each a ∈ X, implies {h(a)} ⊆ B(a,ϵ) and {a} ⊆ B(h(a),ϵ) Since a ∈ X is arbitrary, we obtain d(h, i) < ϵ Hence f ∈ C(X) is stable. □ The authors would like to express their sincere gratitude to APJ Abdul Kalam Technological University, Kerala, India; the management and Staff of Sanatana Dharma College, Alappuzha, Kerala, India and the management and Staff of Rajagiri School of Engineering and Technology, Kerala, India for their support. References [1] Harry Frustenberg, Disjointness In Ergodic Theory,Minimal Sets,and A Problem In Diophantine Approximation, Mathematical System Theory, 1 (1967), 1-49. [2] Jack T.Goodykoontz,JR.and Choon Jai Rhee Local Properties of Hyperspaces, Topology Proceedings, 23(1998),183– 200.. [3] Paul S.Bourdon, Second Iterate of a Map with Dense Orbit, Proceedings of American Mathematical Society , 124,(1996),1577-1581. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 59 https://internationalpubls.com [4] Amalraj.P And P.B.Vinod Kumar, nth Iterate of A map with Dense Orbit, Topological Dynamics And Topological Data Analysis ,Springer ,(2018),177-180. [5] Hiroshi Hosokava, Induced Mappings On Hyperspace,Tsukuba J.Math , 21,1,(1997),239-250. [6] Amalraj.P and P.B. Vinod Kumar, Some properties of the Cantor Set in the Hyperspace K(X), Nanotechnology Perceptions,20 No S8(2024),1267-1274.