Chaos in hyperspaces of nonautonomous discrete systems

Chaos, Solitons and Fractals 94 (2017) 68–74

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Chaos in hyperspaces of nonautonomous discrete systems Iván Sánchez a,1,∗, Manuel Sanchis a, Hugo Villanueva b,2 a

Institut de Matemàtiques i Aplicacions de Castelló (IMAC), Universitat Jaume I, Spain Facultad de Ciencias en Física y Matemáticas, Universidad Autónoma de Chiapas, Carretera Emiliano Zapata km 8.5, Rancho San Francisco, Ciudad Universitaria, Terán, C.P.29050, Tuxtla Gutiérrez, Chiapas, México

b

a r t i c l e

i n f o

Article history: Received 19 May 2016 Revised 17 November 2016 Accepted 21 November 2016 Available online 2 December 2016 Keywords: Nonautonomous dynamical systems Hyperspaces Devaney’s chaos Transitivity Weakly mixing

a b s t r a c t We study the interaction of some dynamical properties of a nonautonomous discrete dynamical system (X, f∞ ) and its induced nonautonomous discrete dynamical system (K (X ), f∞ ), where K (X ) is the hyperspace of non-empty compact sets in X, endowed with the Vietoris topology. We consider properties like transitivity, weakly mixing, points with dense orbit, density of periodic points, among others. We also present examples of nonautonomous discrete dynamical systems showing that transitivity, density of periodic points and sensitive dependence on initial conditions are independents on the unit interval, i.e., unlike autonomous discrete dynamical systems, in definition of Devaney chaotic there are not redundant conditions for NDS on the interval. Actually, our examples give an even more precise conclusion: the classical result stating that transitivity is a sufficient condition for an autonomous discrete dynamical system on the interval to be Devaney chaotic fails to be true for nonautonomous dynamical systems. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Let X be a topological space, fn : X → X a continuous function for each positive integer n, and f∞ = ( f1 , f2 , . . . , fn , . . . ). The pair (X, f∞ ) denotes the nonautonomous discrete dynamical system (NDS, for short) in which the orbit of a point x ∈ X under f∞ is defined as the set

orb(x, f∞ ) = {x, f1 (x ), f12 (x ), . . . , f1n (x ), . . .}, where

f1n := fn ◦ fn−1 ◦ · · · ◦ f2 ◦ f1 , for each positive integer n. In particular, when f∞ is the constant sequence ( f , f , . . . , f , . . . ), the pair (X, f∞ ) is the usual (autonomous) discrete dynamical system given by the continuous function f on X and it will be denoted by (X, f). Autonomous dynamical systems have been studied by many authors obtaining interesting and useful results. NDS were introduced in [16], and are related to nonautonomous difference equations. Indeed, a general form of a nonautonomous difference equation is the following:

∗

Corresponding author. E-mail addresses: [email protected] (I. Sánchez), [email protected] (M. Sanchis), [email protected] (H. Villanueva). 1 The author was supported by CONACyT of Mexico, grant number 259783. 2 The author was partially supported by Apoyo a la Incorporación de Nuevos PTC, PRODEP-SEP, DSA/103.5/14/10906. http://dx.doi.org/10.1016/j.chaos.2016.11.009 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

Given a compact metric space (X, d) and a sequence of continuous function ( fn : X → X )n∈N , for each x ∈ X we set



x0 = x, xn+1 = fn (xn ).

This kind of nonautonomous difference equations has been considered by several mathematicians (see for instance, among others, [22,26]). The most classical examples are when X = [0, 1] is the unit interval, and d is the usual euclidean metric. Observe that the orbit of a point forms a solution of a nonautonomous difference equation. Given a NDS (X, f∞ ), it induces a NDS (K (X ), f∞ ), where K (X ) is the hyperspace of all non-empty compact subsets of X endowed with the Vietoris topology and fn : K (X ) → K (X ) is the continuous function induced by fn . Here, fn (A ) = fn (A ) for each A ∈ K (X ). n Thus, f 1 = fn ◦ · · · ◦ f2 ◦ f1 . Usually, an autonomous discrete dynamical system can be regarded as describing dynamics of individuals (points) in the state space X and its induced continuous function on the hyperspace as a form of collective behavior. This interpretation raises a natural question: does individual chaos imply collective chaos? and conversely? For autonomous systems it is known that (K (X ), f ∞ ) is weakly mixing if and only if (X, f) is weakly mixing, and the latter condition is equivalent to the transitivity of (K (X ), f∞ ) (see [19, Theorem 2.1]). In [14], A. Khan and P. Kumar studied chaotic properties in the sense of Devaney for NDS, by considering the hyper-

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space K (X ) with the we -topology. Some results concerning chaotic properties in NDS were obtained in [2,7,12,17,18,24]. In Section 3 we study dynamical properties related to transitivity in a NDS (X, f∞ ) and the induced NDS (K (X ), f∞ ). In particular, we give some examples to show that [19, Theorem 2.1] cannot be extend to NDS and, however, that some implications remain true. We proved that if (K (X ), f∞ ) is weakly mixing, then (X, f∞ ) is weakly mixing (Corollary 3.9), and that the transitivity of (K (X ), f∞ ) implies that (X, f∞ ) satisfies Banks’s condition and is transitive (Propositions 3.5–3.6). The property which is preserved in both directions is to be weakly mixing of all orders (Proposition 3.10). In Section 4 we study several properties related to chaos in the sense of Devaney. Given a metric space X and a continuous function f: X → X, an autonomous discrete system (X, f) is said to be chaotic in the sense of Devaney ([6]) if it is transitive, has dense set of periodic points and is sensitive (dependence on initial conditions). It is a well-known fact that transitivity and density of periodic points imply the sensitivity of f (see [4] and [11, Theorem 9.20]) so that it is natural to ask if this result remains true for NDS (some conditions under this result is also valid for NDS are given in [27]). We address this question for NDS on the interval. Example 4.4 shows that the answer is negative for this relevant class of NDS. We also present an example that proves that the classic result that transitivity implies chaos in the sense of Devaney for autonomous dynamical systems on the interval ([1,21,25]) fails to be true for NDS. As a connection between Section 3 and Section 4, we can say that for an autonomous discrete system (X, f), there is no relation between Devaney chaos for (X, f) and its associated hyperspace (K (X ), f ) (see [10]): (X, f) is Devaney chaotic  (K (X ), f ) is Devaney chaotic (K (X ), f ) is Devaney chaotic  (X, f) is Devaney chaotic 2. Preliminaries Given a subset A of a topological space X, ClX (A) and IntX (A) denote the closure and the interior of A in X, respectively. A NDS (X, f∞ ) is point transitive if there exists x ∈ X with dense orbit in X, i.e. ClX (orb(x, f∞ )) = X. In this case, we say that x is a transitive point of (X, f∞ ). Also, (X, f∞ ) is topologically transitive if for any two non-empty open sets U and V in X, there exists a positive integer k such that f1k (U ) ∩ V = ∅. A NDS (X, f∞ ) is said to satisfy Banks’s condition if for any three non-empty open sets U, V, W in X, there exists a positive integer k such that f1k (U ) ∩ V = ∅ and f1k (U ) ∩ W = ∅. We say that (X, f∞ ) is weakly mixing if for any four non-empty open sets U1 , U2 , V1 , V2 in X, there exists a positive integer k such that f1k (Ui ) ∩ Vi = ∅, for each i ∈ {1, 2}. It is clear that if (X, f∞ ) is weakly mixing, then it has Banks’s condition and, this implies that it is transitive. For a NDS (X, f∞ ) we put X 2 = X × X and ( f∞ )2 = (g1 , g2 , . . . , gn , . . .), where gn = fn × fn for each positive integer n. Thus, (X2 , (f∞ )2 ) is a NDS. Note that

gn1 = gn ◦ gn−1 ◦ · · · ◦ g2 ◦ g1 = ( fn × fn ) ◦ ( fn−1 × fn−1 ) ◦ · · · ◦ ( f2 × f2 ) ◦ ( f1 × f1 ) = f1n × f1n . In general, for a positive integer m, we define the nonautonomous discrete dynamical system (Xm , (f∞ )m ), where

X

m

= X × ··· × X





m−times



and ( f∞ )m = (g1 , . . . , gn , . . . ), where

gn = f n × · · · × f n ,





m−times



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for each positive integer n. We say that (X, f∞ ) is weakly mixing of order m (m ≥ 2) if (Xm , (f∞ )m ) is transitive, i.e. for any non-empty open sets U1 , U2 , . . . , Um , V1 , V2 , . . . , Vm there is an integer n > 0 such that f1n (Ui ) ∩ Vi = ∅ for each 1 ≤ i ≤ m. Let X be a topological space. The symbol K (X ) will denote the hyperspace of all non-empty compact subsets of X endowed with the Vietoris topology. Let us recall that the following sets constitute a base of open sets for Vietoris topology:

U1 , . . . , Uk : = {K ∈ K (X ) : K ⊂

k 

Ui and

i=1

K ∩ Ui = ∅ for each i ∈ {1, . . . , k}}, where U1 , . . . , Uk are non-empty open subsets of X. Given a metric space (X, d), a point x ∈ X and A ∈ K (X ), let d (x, A ) = inf{d (x, a ) : a ∈ A}. For every  > 0, we define the open dball in X about A and radius  by Nd ( , A ) = {x ∈ X : d (x, A ) < } =  a∈A B ( , a ), where B( , a) denotes the open ball in X centred at a and radius  . We define in K (X ) the Hausdorff metric induced by d, denoted by Hd , as follows

Hd (A, B ) = inf{ > 0 : A ⊂ Nd ( , B ) and B ⊂ Nd ( , A )}, where A, B ∈ K (X ). In [13, Theorem 2.2] it is proved that, indeed, Hd is a metric on K (X ). Moreover, it is known [13, Theorem 3.1] that the topology induced by the Hausdorff metric coincides with the Vietoris topology. We denote by N( , A) (respectively, by H) the generalized open d-ball Nd ( , A) (respectively, the metric Hd ) when it is clear for the metric d that is used. If X is a compact metric space and A, B ∈ K (X ), it follows that H(A, B) <  if and only if A ⊂ N( , B) and B ⊂ N( , A). Given a continuous function f: X → X, it induces a continuous function on K (X ), f : K (X ) → K (X ) defined by f (K ) = f (K ) for every K ∈ K (X ). It is known that the continuity of f implies the continuity of f (see [13, Lemma 13.3]). Let (X, f∞ ) be a NDS and fn the induced continuous function of fn on K (X ), for each positive integer n. Then, the sequence f∞ = ( f1 , f2 , . . . , fn , . . . ) induces a nonautonomous discrete dynamn ical system (K (X ), f∞ ). In this case, f 1 = fn ◦ · · · ◦ f2 ◦ f1 . Note that n

f 1 = f1n . As usual, I denotes the unit interval and R the real numbers equipped with the usual topology. 3. Transitivity and related properties In the following theorem, the equivalence of (1) and (3) was independently showed by Banks [3] and Peris [19]. Moreover, in [5] was already shown that (2) implies (1). The results in this section are motivated by the following theorem of autonomous discrete dynamical systems. Our aim is to study which implications remain valid for the case of a NDS. We construct some examples to show that some implications can not be extended to nonautonomous discrete dynamical systems. Theorem 3.1. [19, Theorem 2.1]Let f: X → X be a continuous function on a topological space X. Then the following conditions are equivalent: (1) (X, f) is weakly mixing. (2) (K (X ), f ) is weakly mixing. (3) (K (X ), f ) is transitive. Clearly, (2) implies (3) even in nonautonomous discrete dynamical systems. The next example shows that (1) ⇒ (3) and (1) ⇒ (2) are false for a NDS. Given two points (a, b), (c, d ) ∈ R2 , [(a, b), (c, d )] stands for the segment whose endpoints are (a, b) and (c, d), respectively. Example 3.2. There is a NDS (I, f∞ ) which is weakly mixing, but (K (I ), f∞ ) is not transitive.

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Fig. 1. f is increasing.

Proof. Let F be the set of elements (a, b, c, d) such that a, b, c, d ∈ Q ∩ (0, 1 ), a < b, a = c, b = d and c = d. Clearly, F is countable. We will assign a homeomorphism f: I → I to every element (a, b, c, d) in F as follows Case 1. If c < d, then f is the function whose graphic is determined by the segments [(0, 0 ), (a, c )], [(a, c ), (b, d )] and [(b, d ), (1, 1 )]. Case 2. If c > d, then f is the function whose graphic is determined by the segments [(0, 1 ), (a, c )], [(a, c ), (b, d )] and [(b, d ), (1, 0 )]. Noting that in both cases f (a ) = c and f (b) = d. Let { fn : n ∈ N} be an enumeration of the functions induced by the elements of F. Consider f∞ = { f1 , f1−1 , f2 , f2−1 , . . . , fn , fn−1 , . . .}. Let us show that (I, f∞ ) is weakly mixing. Let U1 , U2 , V1 , V2 be non-empty open sets in I. By the density of the rational numbers in I, there exist rational numbers a ∈ U1 , b ∈ U2 , c ∈ V1 and d ∈ V2 with a = c, b = d and c = d. Without loss of generality, we can assume that a < b. Then, there is n ∈ N such that f n (a ) = c and fn (b) = d. We have thus proved that fn (U1 ) ∩ V1 = ∅ and fn (U2 ) ∩ V2 = ∅. Therefore, (I, f∞ ) is weakly mixing. We claim that (K (I ), f∞ ) is not transitive. Put 1 = {(x, x ) : x ∈ I} and 2 = {(x, 1 − x ) : x ∈ I}. Also, put U1 = (0.1, 0.2 ), U2 = (0.4, 0.6 ) and U3 = (0.8, 0.9 ). There is an open interval V1 ⊆ (0, 0.25) such that U1 × V1 is above 1 and U3 × V1 is above 2 . We can find an open interval V2 ⊆ (0.25, 0.5) such that neither 1 nor 2 intersects U2 × V2 . Also, there exists an open interval V3 ⊆ (0.75, 1) such that U1 × V3 is in the half-plane y + x − 1 > 0 (that is, above 2 ) and U3 × V3 is in the half-plane y − x > 0 (that is, above 1 ). Put U = U1 , U2 , U3 and V = V1 , V2 , V3 . We claim that there is n

no n > 1 such that f 1 (U ) ∩ V = ∅. Suppose the contrary. Then n must be an odd number. Put f1n = f . If f is increasing, then f(U1 ), f(U2 ) and f(U3 ) meet V1 , V2 and V3 , respectively. Therefore, the graphic of f meets 1 at least four times. This is a contradiction since f is as in Case 1 (see Fig. 1). If f is decreasing, then f(U1 ), f(U2 ) and f(U3 ) meet V3 , V2 and V1 , respectively. Therefore, the graphic of f meets 2 at least four times. This is a contradiction since f is as in Case 2 (see Fig. 2). We have thus proved that (K (I ), f∞ ) is not transitive.  The following observation is useful in the proof of Example 4.7.

Fig. 2. f is decreasing.

Remark 3.3. Note that in the previous example, each rational point of (0, 1) has dense orbit in (I, f∞ ). Indeed, if p is a rational number of (0, 1) and V is an open set, take a rational number q ∈ V. Thus, there exists a positive number n such that fn ( p) = q ∈ V . Let us prove that the implication (3) ⇒ (1) in Theorem 3.1 is false for NDS. Example 3.4. There is a NDS (I, f∞ ) such that (K (I ), f∞ ) is transitive, but (I, f∞ ) is not weakly mixing. Proof. For each k ≥ 1 take {ai : 1 ≤ i ≤ k}, {bi : 1 ≤ i ≤ k} ⊆ Q ∩ (0, 1 ) such that a1 < a2 <  < ak and b1 < b2 <  < bk and let f be an increasing homeomorphism from I onto itself such that f (ai ) = bi for each 1 ≤ i ≤ k. Let { fn : n ∈ N} be an enumeration of such functions. Put f∞ = { f1 , f1−1 , f2 , f2−1 , . . . , fn , fn−1 , . . .}.

Let us show that (K (I ), f∞ ) is transitive. Pick U1 , . . . , Uk , V1 , . . . , Vk non-empty open sets of I. Put U = U1 , . . . , Uk and V = V1 , . . . , Vk . Take ai ∈ Ui ∩ Q and bi ∈ Vi ∩ Q. Without loss of generality we can assume that a1 < a2 <  < ak and b1 < b2 <  < bk . Hence there exists n ∈ N such that fn (ai ) = bi for each 1 ≤ i ≤ k. Put K = {ai : 1 ≤ i ≤ k}. Then K ∈ U and fn (K ) = fn (K ) ∈ V . Therefore, fn (U ) ∩ V = ∅. This shows that (K (I ), f∞ ) is transitive. 1

To prove that (I, f∞ ) is not weakly mixing put U1 = 0, , 2 1

1

1

U2 = , 1 , V1 = , 1 and V2 = 0, . There is no n ∈ N such 2 2 2 n n that f1 (Ui ) ∩ Vi = ∅ since every f1 is increasing. This finishes the proof.  In [3, Theorem 1], Banks proved that in autonomous discrete dynamical systems the property of being weakly mixing is equivalent to satisfy Banks’s condition. So Theorem 3.1 and [3, Theorem 1] imply that an autonomous discrete dynamical system (X, f) satisfies Banks’s condition if and only if (K (X ), f ) is transitive. Example 3.2 satisfies Banks’s condition, but its induced NDS is not transitive. On the other hand, we have the following. Proposition 3.5. If (K (X ), f∞ ) is transitive, then (X, f∞ ) satisfies Banks’s condition. Proof. Let U, V1 , V2 be three non-empty open sets of X. Let U = U and V = V1 , V2 . By the transitivity of (K (X ), f∞ ), there exists a n

positive integer n such that f 1 (U ) ∩ V = ∅. Thus, there exists K ∈ U

I. Sánchez et al. / Chaos, Solitons and Fractals 94 (2017) 68–74 n

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such that f 1 (K ) ∈ V. Hence there are x, y ∈ K ⊂ U such that f1n (x ) ∈ V1 and f1n (y ) ∈ V2 . 

Corollary 3.12. If (I, f∞ ) is weakly mixing of order 3, then (K (I ), f∞ ) is weakly mixing of all orders.

Note that, by Proposition 3.5, Example 3.4 satisfies Banks’s condition, however, it is not weakly mixing. The following result is a consequence of previous proposition, since every NDS satisfying Banks’s condition is transitive.

Corollary 3.13. (I, f∞ ) is weakly mixing of order 3 if and only if (K (I ), f∞ ) is weakly mixing of order 3.

Proposition 3.6. If (K (X ), f∞ ) is transitive, then so is (X, f∞ ).

4. Chaos in the sense of Devaney

Corollary 3.8. Suppose that (K (X ), f∞ ) is weakly mixing of order m. Then so is (X, f∞ ).

Let X be a metric space and f: X → X a continuous function. The autonomous discrete dynamical system (X, f) is Devaney chaotic [6] if it is transitive, the set of periodic points of f is dense and has sensitive dependence on initial conditions. Surprisingly, to have sensitive dependence on initial conditions is “redundant” in the mentioned definition (see [4] and [11, Theorem 9.20]). Other result states that for autonomous dynamical systems on the unit interval I to be Devaney chaotic is equivalent to be transitive [1,21,25]. In this section we study, among other things, if both results can be extended to nonautonomous discrete dynamical systems on the interval. Given a NDS (X, f∞ ), a point x ∈ X is periodic if f1n (x ) = x for some positive integer n. Let us denote by Per(f∞ ) the set of periodic points of f∞ . For autonomous dynamical systems it is known that Per ( f∞ ) is dense in K (X ) if Per(f∞ ) is dense in X ([3, Lemma 1]). However, in the case of NDS, this is not necessarily true.

The next result shows that (2) ⇒ (1) of Theorem 3.1 remains valid for nonautonomous discrete dynamical systems.

Example 4.1. There is a NDS (I, f∞ ) which has dense set of periodic points but (K (I ), f∞ ) has no.

Corollary 3.9. If (K (X ), f∞ ) is weakly mixing, then so is (X, f∞ ).

Proof. Let {qn : n ∈ N} be an enumeration of the rational numbers in I. For each positive integer n, let fn : I → I be the continuous function given by fn (x ) = qn for every x ∈ I. Let f∞ = { f1 , f2 , . . . , fn , . . .}. It is clear that each qn is a periodic point of (I, f∞ ), thus Per ( f∞ ) = I. On the other hand, note that for each A ∈ K (I ) and n ∈ N we have that fn (A ) is a one-point subset of I. Thus, Per ( f∞ ) is not dense in K (I ). 

Lemma 3.7. A NDS (X, f∞ ) is weakly mixing of order m if and only if (Xm , (f∞ )m ) is transitive. Proof. Suppose that (X, f∞ ) is weakly mixing of order m. Let U1 × U2 ×  × Um and V1 × V2 ×  × Vm be two non-empty basic open sets in Xm . Then, there exists a positive integer k such that f1k (Ui ) ∩ Vi = ∅ for i ∈ {1, 2, . . . , m}. Then

( f1k × f1k × · · · × f1k )(U1 × U2 × · · · × Um ) ∩ (V1 × V2 × · · · × Vm ) = ∅. It follows that (Xm , (f∞ )m ) is transitive. A similar argument holds for the converse.  Proposition 3.6 and Lemma 3.7 imply the following.

The previous corollary permits us to conclude that in Example 3.4, (K (I ), f∞ ) is transitive and is not weakly mixing. Proposition 3.10. A NDS (X, f∞ ) is weakly mixing of all orders if and only if (K (X ), f∞ ) is weakly mixing of all orders. Proof. Assume that (X, f∞ ) is weakly mixing of all orders. Take U1 , U2 , . . . , Uk , V1 , V2 , . . . , Vk basic open sets in K (X ). We can suppose that there exists r ∈ N such that Ui = Ui,1 , . . . , Ui,r and Vi = Vi,1 , . . . , Vi,r for each 1 ≤ i ≤ k. Since (X, f∞ ) is weakly mixing of all orders, there exists n such that f1n (Ui, j ) ∩ Vi, j = ∅ for each 1 ≤ i ≤ k and 1 ≤ j ≤ r. Choose xi, j ∈ Ui, j such that yi, j = f (xi, j ) ∈ Vi, j . Put Ki = {xi, j : 1 ≤ j ≤ r} for each i. Then Ki ∈ Ui and f1n (Ki ) ∈ Vi for every i. Therefore, (K (X ), f∞ ) is weakly mixing of all orders. The converse is a consequence of Corollary 3.8  The following lemma shows that to be weakly mixing of order m in the sense of [2] is equivalent to our definition for Hausdorff spaces. Lemma 3.11. Let X be a Hausdorff space. Suppose that (X, f∞ ) is weakly mixing of order m. Then for any non-empty open sets U1 , . . . , Um , V1 , . . . , Vm and any N > 0 there is k > N such that f1k (Ui ) ∩ Vi = ∅ for i = 1, . . . , m. Proof. If X is an one-point set, then we are done. Now assume that X has infinitely many points. Since X is Hausdorff and (X, f∞ ) is weakly mixing of order m, X has no isolated points. Take U1 , . . . , Um , V1 , . . . , Vm and N > 0 as in hypothesis. Let Fn ⊆ X2 be the graphic of f1n . Then every Fn is a closed subset in X2 . Since X has no isolated points, the interior of Fn in X2 is empty. So  2 F= N n=0 Fn is a closed subset in X with empty interior. Then there exist non-empty open sets O1 , . . . , Om , W1 , . . . , Wm such that Oi × Wi ⊆ (Ui × Vi )F for each i. Since (X, f∞ ) is weakly mixing of order m, there is k such that f1k (Oi ) ∩ Wi = ∅ for i = 1, . . . , m, equivalently, Fk meets every open set Oi × Wi . Therefore, k > N.  The following two corollaries are consequences of Corollary 3.8, Proposition 3.10, Lemma 3.11 and [2, Theorem 11].

Now, we recall what means to have sensitive dependence on initial conditions for nonautonomous discrete dynamical systems. Definition 4.2. Let (X, d) be a metric space. We say that (X, f∞ ) has sensitive dependence on initial conditions if there exists δ > 0 such that for every point x and every open neighborhood U of x, there exist y ∈ U and n ∈ N such that d ( f1n (x ), f1n (y )) ≥ δ . We generalize [9, Theorem 3.2] for nonautonomous discrete dynamical systems. Proposition 4.3. Let (X, d) be a compact metric space. If (K (X ), f ∞ ) has sensitive dependence on initial conditions, then (X, f∞ ) does. Proof. Let δ > 0 such that for every A ∈ K (X ) and every open set U of K (X ) with A ∈ U, there exist B ∈ U and a positive integer n n n satisfying H ( f1 (A ), f1 (B )) ≥ δ . Take x ∈ X and let U be an open set of X such that x ∈ U. Let V = U . Then V is an open set in K (X ) such that {x} ∈ V. It follows that there exist L ∈ V and n ∈ N such that H ( f1n ({x} ), f1n (L )) ≥ δ . Note that f1n ({x} ) = { f1n (x )}. Suppose that d ( f1n (x ), f1n (y )) < δ for each y ∈ L. Hence f1n (L ) ⊂ B( f1n (x ), δ ) = N ({ f1n (x )}, δ ) and f1n (x ) ∈ N ( f1n (L ), δ ). Thus, H ( f1n ({x} ), f1n (L )) < δ which is a contradiction. Therefore, there exists y ∈ L ⊆ U such that d ( f1n (x ), f1n (y )) ≥ δ .  The next example shows that [11, Theorem 9.20] (see also [4]) does not remain valid for a NDS on the interval. Example 4.4. There is a NDS (I, f∞ ) which is transitive and has dense set of periodic points, but it does not have sensitive dependence on initial conditions.

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Consider f∞ = { f1 , f1−1 , f2 , f2−1 , . . . , fn , fn−1 , . . .}. The density of {rn : n ∈ N} in I × I implies that (I, f∞ ) is transitive. For every even number m, the function f1m is the identity in I. Thus, every point of I is periodic. Let us show that f∞ does not have sensitive dependence on initial conditions. Suppose the contrary. Then there is δ > 0 satisfy ing Definition 4.2. Put x = 12 ,  = min{ 14 , δ} and U = 12 −  , 12 +  . m m Then there exist y ∈ U and m ∈ N such that | f1 (x ) − f1 (y )| ≥ δ . Therefore, |x − y| <  ≤ δ, whence

| f1m (x ) − f1m (y )| > 1. |x − y| It follows that m must be odd. So f1m is as in Case 1 or Case 2. However, since x, y ∈ ( 41 , 34 ), the slope of the line determined by (x, f1m (x )) and (y, f1m (y )) is less than 1. This contradiction shows that f∞ does not have sensitive dependence on initial conditions.  Let us recall that a NDS (X, f∞ ) is point transitive if there exists x ∈ X with dense orbit in X. Proposition 4.5. If (K (X ), f∞ ) is point transitive, then so is (X, f∞ ). Proof. If (K (X ), f∞ ) is point transitive, then there exists K ∈ K (X ) such that Fig. 3. Case 1.

ClK(X ) (orb(K, f∞ )) = K (X ). Fix x0 ∈ K and take a non-empty open set U in X. Then U =

U is an open set in K (X ). Thus, there exists a positive integer n such that f1n (K ) ∈ U. Hence f1n (K ) ⊂ U. It follows that f1n (x0 ) ∈ U. We have showed that ClX (orb(x0 , f∞ )) = X.  It is known that point transitivity is equivalent to transitivity for autonomous discrete dynamical systems on complete separable metric spaces without isolated points. Every completely metrizable space has Baire property. Let us recall that a space have Baire property if the intersection of a countable family of dense open sets is non-empty. We borrow from [23] the technique to show the following result for nonautonomous discrete dynamical systems. Proposition 4.6. Suppose that X is a second-countable space with the Baire property. If (X, f∞ ) is transitive, then it is point transitive. Proof. Let {Un : n ∈ N} be a countable base for X. Put On =  m −1 m∈N ( f 1 ) (Un ) for each n ∈ N. Clearly, every On is open in X. Since (X, f∞ ) is transitive, each On is dense in X. By hypothesis,

n∈N On = ∅. Take x0 ∈ n∈N On . We claim that the orbit of x0 is dense in X. Indeed, pick Un . We have that x0 ∈ On . So there exists m ∈ N such that x0 ∈ ( f1m )−1 (Un ), i.e. f1m (x0 ) ∈ Un . 

Fig. 4. Case 2.

In contrast to the case of autonomous dynamical systems, the following example shows that the converse of the previous result fails to hold for NDS. Example 4.7. There is a NDS (I, g∞ ) which is point transitive but it is not transitive.

Proof. Let rn be an enumeration of elements in (0, 1) × (0, 1) with rational different coordinates, i.e. rn = ( p, q ) with p, q ∈ (0, 1 ) ∩ Q and p = q. For each n ∈ N we will construct a homeomorphism fn : I → I as follows: Case 1. p > q. Put s = max{ p, 34 } and let fn be the function whose graphic is the union of the segments [(0, 0 ), (s, qp s )] and

[(s, qp s ), (1, 1 )] (see Fig. 3). Note that in this case the slope of [(0, 0 ), (s, qp s )] is

q p

< 1. 1 4}

Case 2. p < q. Put s = min{ p, and let M be the intersection point of the lines x = s and the line containing (p, q) and (1, 1), say ln . Define fn as the function whose graphic is the union of the segments [(0, 0 ), M] and [M, (1, 1 )] (see Fig. 4). It is easy to see −q that the slope of ln is 11−p < 1.

Proof. Consider the continuous functions fn given in Example 3.2. Let g: I → I be the continuous function given by

g( x ) =

⎧ ⎨

0,

⎩ 2x − 1,

if 0 ≤ x ≤ if

1 ; 2

1 ≤ x ≤ 1. 2

Let g∞ = (g, f1 , f1−1 , f2 , f2−1 , . . . , fn , fn−1 , . . . ). By Remark 3.3, it is  not difficult to see that 34 ∈ 12 , 1 is a transitive point of (I, g∞ ). that  In order to show  that (I, g∞ ) is not transitive, note g 0, 12 = {0}. Put U = 0, 12 . Then, gm (U ) = g(U ) = {0} or gm (U ) = 1 1 fn1(g( U )) ⊆ {0, 1}. So no iteration of U meets the open set V = 2,1 . 

I. Sánchez et al. / Chaos, Solitons and Fractals 94 (2017) 68–74

As we say at the beginning of this section, for autonomous discrete dynamical systems on the unit interval I to be Devaney chaotic is equivalent to be transitive [25]. The next example shows that this is not necessarily true for NDS, even if we assume sensitive dependence on initial conditions. Example 4.8. There is a transitive NDS (I, g∞ ) with sensitive dependence on initial conditions such that the set of periodic points is not dense in I. Proof. Let g: I → I be the function whose graph is the union of the segments [(0, 0 ), ( 12 , π6 )] and [( 21 , π6 ), (1, 1 )]. Let rn be an enumeration of elements in (0, 1) × (0, 1) with rational different coordinates, i.e. rn = ( p, q ) with p, q ∈ (0, 1 ) ∩ Q and p = q. For each n ∈ N we will construct a countable family Fn of homeomorphisms from I onto itself. Case 1. If rn = ( p, q ) is below the graph of g, p ≥ 14 and q < 14 , then Fn is the family of functions fn, m whose graph is determined 1 1 by the segments [(0, 0 ), rn ], [rn , ( p + m ,1 − m )] and [( p + m1 , 1 − 1 1 1 ) , ( 1 , 1 ) ] for each m ∈ N such that p + < 1 and 1− m > q. Note m m that g(x) = fn, m (x) for every x ∈ (0, 14 ). Case 2. If rn = ( p, q ) is above the graph of g, p < 14 and q < 14 , then Fn is the family of functions fn, m whose graphic is determined 1 1 by the segments [(0, 0 ), rn ], [rn , ( p + m ,1 − m )] and [( p + m1 , 1 − 1 ) , ( 1 , 1 ) ] for each m > 4. Again, in this case g(x) = fn, m (x) for m every x ∈ (0, 14 ). Case 3. If rn is neither as in Case 1, nor as in Case 2, then Fn =  { fn }, where fn is the function with graph [(0, 0 ), rn ] [rn , (1, 1 )]. Since π is irrational and the quadrilateral with vertices at (0, 0), (1, 0), (1, 1) and ( 21 , π6 ) is convex, we have that g(x) = fn (x) for every x ∈ (0, 14 ).  Put F = n∈N Fn . Let gn be an enumeration of the members n −1 or gn = of F. Put g∞ = {g−1 , g1 , g−1 , . . . , gn , g−1 n , . . .}. Then g1 = g 1 1 gm ◦ g−1 for some m ∈ N. The density of {rn } implies that (I, g∞ ) is transitive. Let us show that g∞ has no periodic points in (0, 14 ). Suppose the contrary. Since g−1 has no fixed points in (0, 14 ), there exist x ∈ (0, 14 ) and m ∈ N such that gm (g−1 (x )) = x, i.e., gm (x ) = g(x ) which contradicts our construction of F. Now, let us prove that (I, g∞ ) has sensitive dependence on initial conditions. We propose δ = 14 . Pick x ∈ I and  > 0. There exists an interval [a, b] ⊆ (x −  , x +  ) ∩ [0, 1] such that b > a and x ∈ [a, b]. We can find rn = ( p, q ) and m > 4 such that q < 14 and 1 g−1 (a ) < p < p + m < g−1 (b). We take fn, m as in Case 1 if p ≥ 14 or take fn, m as in Case 2 if p < 14 . Since fn, m is increasing, we have

fn,m (g−1 (b)) − fn,m (g−1 (a )) > fn,m ( p + =1−

1 ) − fn,m ( p) m

1 1 1 1 −q>1− − = . m 4 4 2

Then

fn,m (g−1 (b)) − fn,m (g−1 (a )) = [ fn,m (g−1 (b)) − fn,m (g−1 (x ))] + [ fn,m (g−1 (x )) − fn,m (g−1 (a ))] >

1 . 2

It follows that fn,m (g−1 (b)) − fn,m (g−1 (x )) > 14 or fn,m (g−1 (x )) − fn,m (g−1 (a )) > 14 so that we obtain the required conclusion because a, b ∈ (x −  , x +  ).  5. Conclusions Let (X, f∞ ) be a NDS and (K (X ), f∞ ) its induced NDS to the hyperspace K (X ) of non-empty compact subsets of X. As a natural question we have: if (X, f∞ ) satisfies a chaotic property P, does

73

(K (X ), f∞ ) have P? (and conversely). This question has been studied for many authors in the case of autonomous discrete systems. In the case of NDS, we prove that if (K (X ), f∞ ) is weakly mixing, then (X, f∞ ) is weakly mixing. However, we prove that the converse is not true and we provide an example in which (X, f∞ ) is weakly mixing and (K (X ), f∞ ) is not transitive. We show that the transitivity of (K (X ), f∞ ) does not imply that (X, f∞ ) is weakly mixing and that the transitivity of (K (X ), f∞ ) implies that (X, f∞ ) has Banks’s condition and is transitive. The property which is preserved in both directions is to be weakly mixing of all orders. We can summarize our results in the following table: 

K ( X ), f ∞



(X, f∞ )

weakly mixing of all orders

⇔

weakly mixing of all orders

weakly mixing

⇒

weakly mixing

weakly mixing



weakly mixing

transitivity



weakly mixing

transitivity

⇒

Banks’s condition

Unlike autonomous discrete dynamical systems, we show that in NDS transitivity together density of periodic points does not imply the sensitivity of a NDS even in the case of the interval and that there exists a NDS on the interval without density of periodic points. We also prove that point transitivity is not equivalent to transitivity, but in second-countable spaces with the Baire property, one implication is true. Taking into account the nature of NDS, our results might lead to new applications to population biology, life sciences, engineering, etc. (see for instance [8,15,20]). Acknowledgements The authors wish to thank the referees for their valuable comments and suggestions for the improvement of this paper. References [1] Alsedà L, Kolyada S, Llibre J, Snoha L. Entropy and periodic points for transitive maps. Trans Amer Math Soc 1999;351(4):1551–73. [2] Balibrea F, Oprocha P. Weak mixing and chaos in nonautonomous discrete systems. Appl Math Lett 2012;25:1135–41. [3] Banks J. Chaos for induced hyperspace maps. Chaos Solitons Fractals 2005;25:1581–3. [4] Banks J, Brooks J, Cairns G, Davis G, Stacey P. On Devaney’s definition of chaos. Am Math Mon 1992;99(4):332–4. [5] Bauer W, Sigmund K. Topological dynamics of transformations induced on the space of probability measures. Monatsh Math 1975;79:81–92. [6] Devaney RL. An introduction to chaotic dynamical systems. Addison-Wesley; 1989. [7] Dvoráková J. Chaos in nonautonomous discrete dynamical systems. Commun Nonlinear Sci Numer Simul 2012;17:4649–52. [8] Elaydi S, Sacker R. Global stability of periodic orbits of nonautonomous difference equations and population biology. J Differ Equ 2005;208:258–73. [9] Gu R. Kato’S chaos in set valued discrete systems. Chaos Solitons Fractals 2007;31:765–71. [10] Guirao JL, Kwietniak D, Lampart M, Oprocha P, Peris A. Chaos on hyperspaces. Nonlinear Anal 2009;71:1–8. [11] Holmgren RA. A first course in discrete dynamical systems. New York: Springer-Verlag; 1994. [12] Huang Q, Shi Y, Zhang L. Sensitivity of non-autonomous discrete dynamical systems. Appl Math Lett 2015;39:31–4. [13] Illanes A, Nadler Jr SB. Hyperspaces of sets, fundamental and recent advances 1999;216. [14] Khan A, Kumar P. Chaotic properties on time varying map and its set valued extension. Adv Pure Math 2013;3:359–64. [15] Kloeden P, Pötzsche C. Nonautonomous dynamical systems in the life sciences. Mathematical Biosciences Subseries. Springer; 2013. [16] Kolyada S, Snoha L. Topological entropy of nonautonomous dynamical systems. Random Comput Dynam 1996;4:205–23. [17] Liu L, Sun Y. Weakly mixing sets and transitive sets for non-autonomous discrete systems. Adv Differ Equ 2014;2014(217):9. [18] Murillo-Arcila M, Peris A. Mixing properties for nonautonomous linear dynamics and invariant sets. Appl Math Lett 2013;26:215–18. [19] Peris A. Set-valued discrete chaos. Chaos Solitons Fractals 2005;26:19–23.

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