e Chaotic Generalized Shift Dynamical Systems

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1 Punjab University Journal of Mathematics (ISS ) Vol. 47(1)(2015) pp e Chaotic Generalized Shift Dynamical Systems Fatemah Ayatollah Zadeh Shirazi Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran fatemah@khayam.ut.ac.ir Hooman Zabeti Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran zabeti.hooman@ut.ac.ir Received: 10 April, 2015 / Accepted: 16 June, 2015 / Published online 17 June, 2015 Abstract. In the following text we prove that for bijection ϕ : and discrete set {1,..., k} with k 2, the generalized shift dynamical system ({1,..., k}, σ ϕ ) is e chaotic, (expansive, positively expansive) if and only if {{ϕ i (n) : i Z} : n } is a finite partition of (or equivalently there exists such that = {ϕ i ({1,..., }) : i Z}). AMS (MOS) Subject Classification Codes: 37B99, 54H20 Key Words: e-chaos, Expansive, Generalized shift dynamical system, Positively expansive 1. PRELIMIARES By a topological dynamical system (or briefly dynamical system) (X, f) we mean a topological space X (phase space) and continuous map f : X X. For a nonempty set X consider two maps one-sided shift σ 1 : X X and twosided shift σ 2 : X Z X Z with σ 1 ((x n ) n ) = (x n+1 ) n (for (x n ) n X ) and σ 2 ((y n ) n Z ) = (y n+1 ) n Z (for (y n ) n Z X Z ), where = {1, 2,...} is the set of all natural numbers and Z = {0, ±1, ±2,...} is the set of all integers. One-sided and twosided shifts are well-known and studied in several branches of mathematics, like ergodic theory and dynamical systems. For arbitrary nonempty set Γ, map ϕ : Γ Γ, nonempty set X with at least two elements, we call σ ϕ : X Γ X Γ with σ ϕ ((x α ) α Γ ) = (x ϕ(α) ) (for all (x α ) α Γ X Γ ) the generalized shift. If X is a topological space, consider X Γ under product (pointwise convergence) topology, so σ ϕ : X Γ X Γ is continuous. Generalized shifts in the above form has been introduced in [3], and their dynamical (and non-dynamical) properties has been studied in several texts, like [1], [2] and [6]. REMARK 1. Suppose X is a topological space with at least two elements and Γ is a nonempty set, equip X Γ with product topology. It is well-known that [7]: 135

2 136 Fatemah Ayatollah Zadeh Shirazi and Hooman Zabeti X Γ is compact if and only if X is compact; X Γ is metrizable if and only if X is metrizable and Γ is countable. REMARK 2. Suppose X is a nonempty set with at least two elements and Γ is a nonempty set, then the following statements are equivalent [3]: σ ϕ : X Γ X Γ is bijective; ϕ : Γ Γ is bijective. Hence if X has a topological structure, then σ ϕ : X Γ X Γ is a homeomorphism if and only if ϕ : Γ Γ is bijective. REMARK 3. If Γ is a nonempty set and ϕ : Γ Γ is arbitrary, for α, β Γ let α β if and only if there exists n, m with ϕ n (α) = ϕ m (β). Then is an equivalence relation on X. If ϕ : Γ Γ is bijective and α Γ, then the equivalence class of α under ϕ is α = {ϕ n (α) : n Z}, so α has exactly one of the following forms: there exists m with α = {α n : 0 n < m} where α i s are distinct ϕ(α i ) = α i+1 for i = 0,..., m 1 and ϕ(α m 1 ) = α 0 ; α = {α n : n Z}, α n s are distinct and ϕ(α n ) = α n+1 for n Z. In addition for bijective ϕ : Γ Γ and α, β Γ we have α β if and only if there exists n Z with ϕ n (α) = β. In the following text suppose X is a discrete finite set with at least two elements and Γ is a countable infinite set. So we may suppose X = {1,..., k} with discrete topology, k 2, and Γ =, also suppose ϕ : Γ Γ is bijective (note to Remarks 1 and 2). The main aim of this text is to study e chaotic generalized shift dynamical system ({1,..., k}, σ ϕ ). 2. WHE IS ({1,..., k}, σ ϕ ) EXPASIVE? We call the dynamical system (Y, f) (or briefly f : Y Y ) with compact metric space (Y, ρ) and homeomorphism f : Y Y, expansive if there exists µ > 0 such that for all distinct x, y Y there exists n Z with ρ(f n (x), f n (y)) > µ. REMARK 4. For arbitrary set Y we call the collection F of subsets of Y Y a uniform structure in Y if (let Y = {(x, x) : x Y }) [5]: α F ( Y α); α, β F (α β F); α F β F (β β 1 α); α F β Y Y (α β β F). Moreover for all α F and x Y let α[x] = {y Y : (x, y) α}. If F is a uniform structure in Y, then {U Y : x Y α F (α[x] U)} is a topology on Y, we call it uniform topology induced from F. We call the topological space uniformazable if there exists a uniform structure F in Y such that uniform topology induced from F coincides with original topology on Y, and in this case we call F compatible uniform structure in Y. Every compact Hausdorff (resp. compact metric) space is uniformazable and has a unique compatible uniform structure. If Y is a compact metric space, for ε > 0 let α ε = {(x, y) Y Y : ρ(x, y) ε}, and G = {D Y Y : δ > 0(α δ D)}, then G is a compatible uniform structure in Y. It s evident that homeomorphism f : Y Y is expansive if and only if there exists β G such that for all distinct x, y Y there exists n Z with (f n (x), f n (y)) / β. Since G is the unique compatible uniform structure in Y, expansivity of homeomorphism f : Y Y does not depends on ρ and we may choose any compatible metric on Y.

3 e Chaotic Generalized Shift Dynamical Systems 137 Consider the equivalence relation on as in Remark 3. We prove σ ϕ : X X is expansive if and only if = { α : α } is finite. Also using Remark 4 equip {1,..., k} with metric d((x n ) n, (y n ) n ) = n for (x n ) n, (y n ) n {1,..., k}, where: { 0 z = w, δ(z, w) = 1 z w. δ(x n, y n ) 2 n (*) So ({1,..., k}, d) is a compact metric space (metric topology on {1,..., k} induced from d, coincides with product topology on {1,..., k} (see [7])). LEMMA 2.1. If = { α 1,..., α s }, then for all distinct x, y {1,..., k} there exists n such that d(f n (x), f n 1 (y)) (consider metric d on {1,..., k} as in 2 max(α 1,...,αs) (*)). Proof. Consider distinct x = (x n ) n, y = (y n ) n {1,..., k}. There exists m with x m y m, there exists r {1,..., s} with m αr, so there exists k Z with ϕ k (α r ) = m. Suppose (z n ) n := σ k ϕ(x) = (x ϕ k (n)) n and (w n ) n := σ k ϕ(y) = (y ϕ k (n)) n. Thus which completes the proof. d(σ k ϕ(x), σ k ϕ(y)) = d((z n ) n, (w n ) n ) δ(z α r, w αr ) = δ(xϕk (α r ), y ϕk (α r )) 2 αr 2 αr = δ(x m, y m ) 2 α r = 1 2 α r 1 2 max(α 1,...,α s ) COROLLARY 2.1. If is finite, then σ ϕ : {1,..., k} {1,..., k} is expansive. Proof. If = { α 1,..., α s 1 } choose µ (0, ). By Lemma 2.1 for all 2 max(α 1,...,αs) distinct x, y {1,..., k} there exists n Z with d(σϕ(x), n σϕ(y)) n 1 > µ 2 max(α 1,...,α s) which leads to the desired result by Remark 4. LEMMA 2.2. If is infinite, then σ ϕ : {1,..., k} {1,..., k} is not expansive. 1 Proof. Consider µ > 0, then there exists such that < µ. Since 2n m is infinite, there exists m such that k for all k {1,..., }, i.e. m \ ( 1 ), and m \ ( 1 ) \ {1,..., } let x n = y n = 1 for all n \ {m}, x m = 1 and y m = 2. For x = (x n ) n, y = (y n ) n for all r Z we have: d(σϕ(x), r σϕ(y)) r = d((x ϕ r (n)) n, (y ϕ r (n)) n ) 1 2 n = 1 2 n x ϕ r (n) y ϕ r (n) Hence we have: ϕ r (n)=m n> nm µ > 0 x y r Z (d(f r (x), f r (y)) < µ). 1 2 n n> 1 2 n < µ.

4 138 Fatemah Ayatollah Zadeh Shirazi and Hooman Zabeti Using Remark 4, σ ϕ : {1,..., k} {1,..., k} is not expansive. THEOREM 2.1. For bijection ϕ : and discrete set {1,..., k} with k 2, the generalized shift dynamical system ({1,..., k}, σ ϕ ) is expansive if and only if is finite (i.e., there exists n 1,..., n s with = {ϕ i (n j ) : j {1,..., s}, i Z}). Proof. Use Corollary 2.1 and Lemma 2.2. EXAMPLE 1. Define ϕ 1, ϕ 2 : with: n + 2 n is odd { n + 1 n is odd ϕ 1 (n) = n 2 n > 2 is even and ϕ 2 (n) = n 1 n is even 1 n = 2 then ({1,..., k}, σ ϕ1 ) is expansive, and ({1,..., k}, σ ϕ2 ) is not expansive, since {} and 2 = {{2n 1, 2n} : n }. 1 = 3. e CHAOTIC GEERALIZED SHIFT DYAMICAL SYSTEM ({1,..., k}, σ ϕ ) We call the dynamical system (Y, f), e chaotic, if it is expansive and the set of all periodic points (of f : Y Y ) is dense in Y [9], we recall that a Y is a periodic point of f : Y Y if there exists n 1 with f n (a) = a. REMARK 5. If Y is a discrete topological space with at least two elements, Λ is a nonempty set and η : Λ Λ is arbitrary, then the set of periodic points of σ η : Y Λ Y Λ (σ η ((x α ) α Λ ) = (x η(α) ) α Λ ) is dense in Y Λ if and only if η : Λ Λ is one to one [4, Theorem 2.6]. THEOREM 3.1 (main). For bijection ϕ : discrete set {1,..., k} with k 2, in the generalized shift dynamical system ({1,..., k}, σ ϕ ), the following statements are equivalent: ({1,..., k}, σ ϕ ) is e chaotic; ({1,..., k}, σ ϕ ) is expansive; is finite (i.e., there exists n 1,..., n s with = {ϕ i (n j ) : j {1,..., s}, i Z}, or equivalently {{ϕ i (n) : i Z} : n } is a finite partition of ). Proof. Use Remark 5 and Theorem 2.1. EXAMPLE 2. Using [4, Theorem 2.13], for discrete topological space Y with at least two elements and η :, the generalized shift dynamical system (Y, σ η ) is Devaney chaotic if and only if η : is one to one without any periodic point. Let: L = the class of all generalized shift dynamical systems (Y, σ η ), where η : is bijective. L 1 = the class of all Devaney chaotic generalized shift dynamical systems (Y, σ η ), where η : is bijective. L 2 = the class of all e chaotic generalized shift dynamical systems (y, σ η ), where η : is bijective. We have the following diagram:

5 e Chaotic Generalized Shift Dynamical Systems 139 L 1 L 2 L (E2) (E4) (E1) (E3) where: E1 is ({1,..., k}, σ ϕ1 ) as in Example 1; E2 is ({1,..., k}, σ ϕ2 ) as in Example 1; E3 is ({1,..., k}, σ ϕ3 ) for ϕ 3 : with (k 2): { 1 n = 1, ϕ 3 (n) = ϕ 1 (n 1) + 1 n > 1 ; E4 is ({1,..., k}, σ ϕ4 ) for ϕ 4 : with (k 2)(suppose p m is the mth prime number and \ {p k m : m, k 1} = {w 1, w 2,...} for w 1 < w 2 < ): { ϕ p 1(k) m n = p ϕ 4 (n) = k m, w ϕ1 (k) n = w k. 4. MORE DETAILS O EXPASIVE GEERALIZED SHIFT DYAMICAL SYSTEMS Regarding previous sections, let s call the dynamical system ((Z, F), f) with uniform phase space (Z, F) and homeomorphism f : Z Z expansive if there exists µ F such that for all distinct x, y Z there exists n Z with (f n (x), f n (y)) / µ. In this section suppose (Y, K) is a uniform Hausdorff space with at least two elements, Λ is a nonempty set and λ : Λ Λ is an arbitrary map. It is well-known that product and subspaces of uniform spaces are uniformzable. In this section we prove that if the generalized shift dynamical system (Y Λ, σ λ ) with bijection λ : Λ Λ is expansive (with any compatible uniformity on Y Λ, where Y Λ equipped with product topology), then Λ is countable and {{λ n (α) : n Z} : α Λ} is a finite partition of Λ. COROLLARY 4.1. Using Theorem 3.1 if M is countable, W is finite discrete with at least two elements and ψ : M M is bijective, then the following statements are equivalent (note that W M with product topology is a compact metrizable space): (W M, σ ψ ) is e chaotic, (W M, σ ψ ) is expansive, M ψ is finite. THEOREM 4.1. For bijection λ : Λ Λ if the generalized shift dynamical system (Y Λ, σ λ ) is expansive, then Λ is countable and and Λ is finite. α Proof. First of all note that for all α Λ, = {λ n (α) : n Z} is countable. Suppose {α n } n 1 is a sequence in Λ and p, q are two distinct elements of Y. Let M = { αn : n 1}. Since (Y Λ, σ λ ) is expansive, ({p, q} M, σ λ M ) is expansive too. Since {p, q} is a discrete set with two elements and M is countable, using Corollary 4.1, for ψ = λ M M the set ψ (= { αn : n 1}) is finite. Thus we don t have any infinite sequence in Λ and Λ is finite. Since Λ α is finite and all ( Λ ) is countable, the set Λ = Λ is countable too.

6 140 Fatemah Ayatollah Zadeh Shirazi and Hooman Zabeti Positively expansive dynamical system. We call the dynamical system (Y, f) (or briefly f : Y Y ) with compact metric space (Y, ρ), positively expansive if there exists µ > 0 such that for all distinct x, y Y there exists n 0 with ρ(f n (x), f n (y)) > µ [8]. It s evident that for homeomorphism f : Y Y, if (Y, f) is positively expansive, then it is expansive. Using the same method described in Remark 4 positively expansivity of continuous map f : Y Y does not depends on ρ and we may choose any compatible metric on Y. Using the same proof as in Lemma 2.2, for arbitrary self-map µ :, if the generalized shift dynamical system ({1,..., k}, σ µ ) is positively expansive, then µ is finite. However for constant map µ : with µ(n) = 1, the dynamical system ({1,..., k}, σ µ ) is not positively expansive, although µ is finite. Moreover using Lemma 2.1 and Theorem 3.1 we have the following corollary. COROLLARY 4.2. For bijection ϕ : and discrete set {1,..., k} with k 2, the generalized shift dynamical system ({1,..., k}, σ ϕ ) is expansive if and only if it is positively expansive. 5. ACKOWLEDGMET The authors are grateful to the research division of the University of Tehran, for the grant which supported this research under the ref. no /1/07. This paper is a part of 2nd author s MSc thesis under the supervision of the first author, in the summer of 2014, when he was a student in the University of Tehran. Moreover the primary form of this article has been presented in a talk with the same abstract in The 11th Seminar of Differential Equations and Dynamical systems June 2014, Damghan University, Damghan, Iran. REFERECES [1] M. Akhavin, A. Giordano Bruno, D. Dikranjan, A. Hosseini and F. Ayatollah Zadeh Shirazi, Algebraic entropy of shift endomorphisms on abelian groups, Quaestiones Mathematicae 32, (2009) [2] F. Ayatollah Zadeh Shirazi and D. Dikranjan, Set-theoretical entropy: A tool to compute topological entropy, Proceedings ICTA2011, Islamabad, Pakistan, Cambridge Scientific Publishers (2012) [3] F. Ayatollah Zadeh Shirazi,. Karami Kabir and F. Heydari Ardi, A note on shift theory, Mathematica Pannonica, Proceedings of ITES , o. 2 (2008) [4] F. Ayatollah Zadeh Shirazi, J. azarian Sarkooh and B. Taherkhani, On Devaney chaotic generalized shift dynamical systems, Studia Scientiarum Mathematicarum Hungarica 50, o. 4 (2013) [5] J. Dugundji, Topology, Allyn and Bacon, Inc. Boston, [6] A. Giodano Bruno, Algebraic entropy of generalized shifts on direct products, Communications in Algebra 38, (2010) [7] J. R. Munkres, Topology: a first course, Prentice Hall, Inc [8] D. Richeson and J. Wiseman, Positively expansive dynamical systems, Topology and its Applications 154, (2007) [9] S. Shah and T. Das, On e chaos, International Journal of Mathematical Analysis 7, o. 12 (2013),