Weak stability of non-autonomous discrete dynamical systems

Accepted Manuscript Weak stability of non-autonomous discrete dynamical systems

Yaoyao Lan, Alfred Peris

PII: DOI: Reference:

S0166-8641(18)30411-5 https://doi.org/10.1016/j.topol.2018.10.006 TOPOL 6550

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Topology and its Applications

Received date: Revised date: Accepted date:

25 May 2018 16 October 2018 16 October 2018

Please cite this article in press as: Y. Lan, A. Peris, Weak stability of non-autonomous discrete dynamical systems, Topol. Appl. (2018), https://doi.org/10.1016/j.topol.2018.10.006

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Weak stability of non-autonomous discrete dynamical systems Yaoyao Lan1,, Alfred Peris2 a

Department of Mathematics Chongqing University of Arts and Sciences, 402160 Yongchuan, China b Institut Universitari de Matem` atica Pura i Aplicada Universitat Polit`ecnica de Val`encia, Edifici 8E, Acces F, 4a planta, 46022 Val`encia, Spain

Abstract In this paper we introduce a concept of weak stability for non-autonomous dynamical systems. We characterize the set of weak stable points and show that the set of weak stable points is residual, and investigate the relation between weak stability and shadowing property. We also discuss the relation between weak stability of a non-autonomous dynamical system and its induced set-valued system. Keywords: weak stability, non-autonomous dynamical system, set-valued system, shadowing property 2000 MSC: 54H20, 37B55

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1. Introduction Let F = {fn }∞ n=1 be a sequence of continuous maps fn : X → X acting on a compact metric space (X, d). A non-autonomous discrete system is a pair (X, F). The orbit of any point x ∈ X is the solution of the following non-autonomous difference equation  xn+1 = fn (xn ), (1) x0 = x. Email addresses: [email protected] (Yaoyao Lan), [email protected] (Alfred Peris2 )

Preprint submitted to Elsevier

October 19, 2018

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Note that the autonomous dynamical system is a special case of system (1) when fn = f for all n ≥ 1. We refer to Section 2 for other notions and notation needed in this paper. Non-autonomous discrete dynamical systems naturally appear as a suitable model to describe real processes like, e.g., when the map is perturbed in each iteration due to external forces or other reasons. For instance, they model very well the evolution of populations. The rich dynamics of nonautonomous discrete systems attracted the interest of several researchers, obtaining results on entropy [19], mixing properties [1, 26], Li-Yorke chaos [8], sensitivity [17, 25], and other properties [9, 31, 33]. For a survey on the dynamics of non-autonomous discrete dynamical systems we refer the reader to [2]. The basic task of the theory of non-autonomous dynamical systems is to understand the nature of all orbits. The dynamics in non-autonomous case can be vary complicated, and it is natural to study the pseudo-orbits for a better understanding of true orbits. Along this line, the study of shadowing property is an active topic of research (see, e.g., [31, 18, 29, 30, 32, 13, 36, 27, 11]). In [16], a concept of weak stability was studied, and it was shown that orbital shadowing property is generic in the set of weak stable homeomorphisms. Motivated by this idea, we discuss weak stability in nonautonomous dynamical systems. On the other hand, a discrete dynamical system uniquely induces its setvalued system on the space of compact subsets. It is natural to ask the following question: What is the relation between dynamical properties of the original and set-valued systems? The study of the interplay between dynamics and the corresponding induced set-valued systems has been extensively studied by several authors [3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 22, 23, 24, 21, 28, 34, 35]. In this paper, a concept of weak stability is studied, obtaining a characterization that is new even in the context of autonomous systems. The relation between weak stability of the non-autonomous discrete dynamical system and its set-valued system is also investigated.

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2. Preliminaries and notation

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We recall that F = {fn }∞ n=1 is a sequence of continuous maps defined on a compact metric space (X, d), and we denote, for each i, n ∈ N with i ≤ n, fin = fn ◦ fn−1 · · · ◦ fi , 2

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Note that fi0 is the identity map idX on X. An orbit of a point x ∈ X, denoted by o(x, F) is defined as o(x, F) = {x, f1 (x), f12 (x), . . . , f1n (x), . . . }

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ik+k ∞ We will also consider the associated non-autonomous systems Fk = {fik+1 }i=0 , k ≥ 1. For ε > 0 and x ∈ X we denote the ε-ball in (X, d) centered at x by

B(x, ε) = {y ∈ X ; d(x, y) < ε}. 45 46 47 48 49 50 51 52 53 54

For δ > 0, a δ-pseudo-orbit for F is a sequence {xi }∞ i=0 in X such that d(fi+1 (xi ), xi+1 ) < δ for i ∈ N0 , where N0 denotes the set of all nonnegative integers. A finite δ-pseudo-orbit {xi }bi=0 is called a δ-chain from x0 to xb with length b + 1. For ε > 0, a sequence {xi }∞ i=0 in X is said to be ε-shadowed by a true orbit starting at a point y ∈ X if d(f1i (y), xi ) < ε for all i ∈ N0 . F has shadowing property if, for any ε > 0, there is δ > 0 such that every δ-pseudo-orbit {xi }∞ i=0 of F can be ε-shadowed by an orbit starting in y ∈ X, i that is, d(f1 (y), xi ) < ε for all i ∈ N0 . A non-autonomous system (X, F) is called

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• topologically transitive if for each  pair of open sets U, V ∈ X, there exists n ∈ N0 such that f1n (U) V = ∅. In such a case there is a dense Gδ subset Y ⊂ X such that o(y, F) is dense in X. Every such point y with dense orbit is called a transitive point for F, and we denote by trans(F) the corresponding set of transitive points.

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• chain transitive if for any x, y ∈ X there is a δ-chain of F from x to y;

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• equicontinuous if for every ε > 0 and x ∈ X there exists δ > 0 such that d(x, y) < δ implies d(f1i (x), f1i (y)) < ε for all i ∈ N0 . By compactness, equicontinuity is the same as uniform equicontinuity, that is, we can choose δ > 0 independent of x ∈ X. Definition 2.1. Given a non-autonomous system (X, F), where (X, d) is a general metric space, we call x ∈ X a weak stable point of F (denoted by x ∈ ws(F)), or F is weak stable at x, if for every ε > 0 there exist δ > 0 and an integer Tx such that o(z, F) ⊂ Nε ({f1i (z) : i = 0, . . . , Tx }) for any z ∈ X with d(z, x) < δ. We say that F is weak stable if it is weak stable at every 3

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point x ∈ X. A weak stable system so that the integer Tx (given the arbitrary ε > 0) can be selected independent of each x ∈ X is called uniformly weak stable. Let K(X) be the collection of all non-empty compact subsets of X. Define the ε-ball around a nonempty subset A in X to be the set Nε (A) = {x ∈ X ; d(x, A) < ε},

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where d(x, A) = inf a∈A d(x, a). The Hausdorff separation ρ(A, B) of A, B ∈ K(X) is defined by ρ(A, B) = inf{ε > 0 ; A ⊆ Nε (B)},

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The Hausdorff metric on K(X) is defined by letting Hd (A, B) = max{ρ(A, B), ρ(B, A)}.

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We observe that KF (X), the set of all finite subsets of X, is dense in K(X). Also, (K(X), Hd ) is compact if and only if (X, d) is compact. Let f : X → X be continuous. By letting f¯(A) = {f (a) : a ∈ A} for every A ∈ K(X) one defines a continuous mapping on K(X). We call the pair (K(X), f¯) set-valued discrete dynamical system associated to f . A subset A ⊆ X is said to be residual if A contains a countable intersection of open and dense subsets of X.

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3. Weak stable systems and set-valued dynamics

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Let F = {fn }∞ n=1 be a sequence of continuous maps acting on a compact metric space (X, d). In this section we will study weak stability in a non-autonomous discrete dynamical system (X, F). We first obtain a useful characterization of weak stable points. Theorem 3.1. Let (X, F) be a non-autonomous dynamical system on a compact metric space X. Then x ∈ X is a weak stable point for F if, and only if, ∞  (wsp) o(x, F) = f1i (B(x, δ)). δ>0 i=0

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Proof. Let x ∈ X such that the equality (wsp) holds. Given ε > 0, by (wsp) we obtain δ1 > 0 such that ∞ 

f1i (B(x, δ1 )) ⊂ Nε/3 (o(x, F)).

i=0 95 96 97 98 99 100 101 102 103

Let {Uj }m j=1 be a finite cover of X by open balls of diameter ε/3. Let Lx ⊂ {1, . . . , m} be such that  o(x, F) ⊂ i∈Lx Ui , o(x, F) ∩ Uj = ∅ for every j ∈ Lx . For each i ∈ Lx there is ni ∈ N0 with f1ni (x) ∈ Ui . We set Tx = max{ni ; i ∈ Lx } and 0 < δ < δ1 such that d(f1j (z), f1j (x)) < ε/3 if z ∈ B(x, δ) and j ∈ {0, . . . , Tx }. Given any z ∈ B(x, δ) and k ∈ N there is l ∈ N0 such that d(f1k (z), f1l (x)) < ε/3. We select i ∈ Lx with f1l (x) ∈ Ui and j ∈ {0, . . . , Tx } with f1j (x) ∈ Ui . We obtain ε ε ε d(f1k (z), f1j (z)) < d(f1k (z), f1l (x))+d(f1l (x), f1j (x))+d(f1j (x), f1j (z)) < + + = ε, 3 3 3

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that is, o(z, F) ⊂ Nε ({f1j (z) ; j = 0, . . . , Tx }). Since z ∈ B(x, δ) was arbitrary, we get the weak stability of x. Conversely, if x ∈ ws(F) and we fix any y∈

∞ 

f1i (B(x, δ)),

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and ε1 > 0. We can find δx > 0 and Tx ∈ N associated to the weak stability of x for ε = ε1 /3. Also, by hypothesis we can obtain a sequence {zi }∞ i=1 in B(x, δx ) converging to x such that y ∈ Nε (o(zi , F)) for every i ∈ N. Thus, y ∈ Nε (o(zi , F)) ⊂ N2ε ({f1j (zi ) ; j = 0, . . . , Tx }).

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By taking limits when i → ∞, and by continuity, we obtain y ∈ N2ε ({f1j (x) ; j = 0, . . . , Tx }) ⊂ Nε1 (o(x, F)).

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Since ε1 was arbitrary, we conclude that y ∈ o(x, F), as desired. Some basic results are also established. 5

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Proposition 3.2. Let (X, F) be a non-autonomous dynamical system on a compact metric space X. (1) If (X, F) is equicontinuous then it is weak stable. (2) If (X, F) is weak stable then it is uniformly weak stable. (3) If (X, F) is topologically transitive then trans(F) = ws(F). Proof. Statement (1): By Theorem 3.1, given any x ∈ X we need to show that o(x, F) =

∞ 

f1i (B(x, δ)).

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The equicontinuity of the system yields ∞ 

f1i (B(x, δ))

⊂

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B(f1i (x), ε) = o(x, F),

ε>0 i=0

δ>0 i=0 121

∞ 

so (X, F) is weak stable. Statement (2): We fix ε > 0 and, given x ∈ X, let δx > 0 and Tx ∈ N such that o(z, F) ⊂ Nε ({f1i (z) ; i = 0, . . . , Tx }) for any z ∈ X with d(z, x) < δx . We find a finite collection {xi ∈ X ; i = 1, . . . , m} such that X⊂

m 

B(xi , δxi ).

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If we set T = max{Txi ; i = 1, . . . , m} we obtain the uniform weak stability of the system, since ε was arbitrary. Statement (3): Topological transitivity of the system means that ∞ 

f1i (B(x, δ)) = X

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for any x ∈ X. Therefore the set of weak stable points coincides with the set of transitive points by Theorem 3.1.

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The proof of the following result is inspired in the corresponding Theorem for genericity of the set of weak stable points for autonomous systems of homeomorphisms given in [16]. We include its proof for the sake of completeness. Theorem 3.3. Let F = {fn }∞ n=1 be a sequence of continuous maps on a compact metric space X. Then the set of weak stable points of F is residual in X. Proof. Let ε > 0 and let U = {Ui ; i = 1, 2, . . . , k} be an open finite cover of X with diam(Ui ) < 2ε . Set K = {1, 2, . . . , k}. As before, for each x ∈ X we select Lx ⊂ K satisfying  o(x, F) ⊂ i∈Lx Ui , o(x, F) ∩ Ui = ∅ for i ∈ Lx . Let Wε be the set of all a ∈ X such that for ε > 0, there exist δa > 0 and positive integer Ta such that o(x, F) ⊂ Nε ({f1i (x) i = 0, . . . , Ta }) for any x ∈ X with d(a, x) < δa . Obviously, Wε is open. To prove that Wε is dense in X, fix any x ∈ X and δ > 0 arbitrary. We take T ∈ N big enough so that {f1i (x) ; i = 0, . . . , T } ∩ Uj = ∅ for any j ∈ Lx .

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Since the diameter of each Ui is less than ε/2, we have  o(x, F) ⊂ Ui ⊂ Nε/2 ({f1i (x) ; i = 0, . . . , T }). i∈Lx

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Now we take δ > λ1 > 0 such that, for every z ∈ B(x, λ1 ), ε d(f1i (x), f1i (z)) < , 2

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where i = 0, . . . , T . Assume that x ∈ / Wε . For 0 < δ1 < λ1 there exists x1 ∈ B(x, δ1 ) such that we find m1 > T with d(f1m1 (x1 ), f1i (x1 )) ≥ ε,

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for any i = 0, . . . , T . We also have, for i = 0, . . . , T , ε d(f1m1 (x1 ), f1i (x)) ≥ . 2 7

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Indeed, if d(f1m1 (x1 ), f1i (x)) < 2ε , then d(f1m1 (x1 ), f1i (x1 )) ≤ d(f1m1 (x1 ), f1i (x)) + d(f1i (x), f1i (x1 )) <

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ε ε + = ε, 2 2

which is a contradiction. Consequently, a f1m1 (x1 ) ∈ X \ N 2ε ({f1i (a)}Ti=0 )⊂X\



Ui .

i∈Lx 156 157 158 159 160 161

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Then there exists j ∈ K−Lx such that f1m1 (x1 ) ∈ Uj . Thus Lx  Lx1 . In case that x1 ∈ Wε , we are done. Otherwise we repeat the process to find x2 ∈ X as close to x1 as we want (in particular, x2 ∈ B(x, λ1 )) so that Lx1  Lx2 . The process will finish when we arrive to certain xl ∈ B(x, λ1 ) with Lxl = K, since in this case xl ∈ Wε , which completes the proof of density of the set Wε . Set W = ∩∞ n=1 W 1 to show its residuality in X. n

In the sequel, the relation between weak stability of (X, F) and its induced set-valued system (K(X), F) has been discussed. Theorem 3.4. Let (X, F) be a non-autonomous dynamical system and let (K(X), F) be its induced set-valued system. Then F is weak stable if and only if F is weak stable. Proof. Assume that (X, F) is weak stable. Then, by Proposition 3.2, it is uniformly weak stable. Given ε1 > 0, let T ∈ N associated with ε = ε1 /2, due to the uniform weak stability of (X, F). For any K ∈ K(X) we have that   o(K, F) = o(x, F) ⊂ Nε ({f1i (x) ; i = 0, . . . , T }) x∈K

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x∈K

= Nε ({f1i (K) ; i = 0, . . . , T }), and (K(X), F) is weak stable. Conversely, fix any x ∈ X. Then {x} ∈ KF (X). To prove F is weak stable, it is sufficient to observe that d(x, y) = Hd ({x}, {y})

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and

n

n

Hd (f 1 ({y}), f 1 ({y})) = d(f1n (y), f1i (y)) 176

for every x, y ∈ X. This completes the proof. 8

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The relation between shadowing property and weak stability of F and Fk has been studied as follows: Lemma 3.5. If F has the shadowing property, then Fk has shadowing property. Proof. Let ε > 0 and δ > 0. Assume {yi }∞ i=0 is a δ-pseudo-orbit of Fk . Thus the following sequence k+2 2k−1 k−1 {xi }∞ (y0 ), y1 , fk+1 (y1 ), fk+1 (y1 ), . . . , fk+1 (y1 ), i=0 = {y0 , f1 (y0 ), . . . , f1 2k+2 3k−1 y2 , f2k+1 (y2 ), f2k+1 (y2 ), . . . , f2k+1 (y1 ), y3 , . . . }

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is a δ-pseudo-orbit of F. Since F has the shadowing property, there exists z ∈ X such that d(f1i (z), xi ) < ε for i ≥ 0. Consequently, we have k(i+1) d(f1 (z), yi+1 ) < ε, which leads to the shadowing property of Fk . Lemma 3.6. Let Fk be chain transitive for k ∈ N0 . If F has the shadowing property, then Fk is topological transitive. Proof. By Lemma 3.5, Fk has the shadowing property. Let B(x, r1 ) and B(y, r2 ) be balls of x, y ∈ X, respectively. For 0 < ε < min{r1 , r2 }, there exists δ > 0 such that every δ-pseudo-orbit of Fk can be ε-shadowed by some point of X. Since Fk is chain transitive, there exists a δ-chain {x = x0 , . . . , xn = y} from x to y. Thus there is z ∈ X such that d(z, x) < ε and d(f1kn (z), y) < ε. Consequently, f1kn (B(x, r1 )) ∩ B(y, r2 ) = ∅. It follows that Fk is topological transitive.

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Theorem 3.7. Let Fk be chain transitive for k ∈ N0 . If F has the shadowing property, then trans(F) = ws(F).

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Proof. It follows from Lemma 3.6 and Proposition 3.2 (3).

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Acknowledgements

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The first author was supported by the National Natural Science Foundation of China (NO. 11601051), China Scholarship Council Contract (NO. 201608505146), and Natural Science Foundation Project of Chongqing CSTC (No. cstc2014jcyjA00054). The second author was supported by MINECO, Project MTM2016-75963-P, and by Generalitat Valenciana, Project PROMETEO/2017/102.

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