Chaotic sets of shift and weighted shift maps

Nonlinear Analysis 71 (2009) 2141–2152

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Chaotic sets of shift and weighted shift maps Xinchu Fu a,∗ , Yuncheng You b a

Department of Mathematics, Shanghai University, Shanghai 200444, PR China

b

Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA

article

info

Article history: Received 9 May 2008 Accepted 12 January 2009 Keywords: Li–Yorke chaotic set Devaney chaotic set Scrambled set Shift map Weighted shift map

a b s t r a c t In this paper Li–Yorke chaotic sets generated by shift and weighted shift maps are studied. The characterization of Li–Yorke chaotic sets by p-scrambled sets, maximal scrambled sets and orbit invariants are proved for the general shift maps. For weighted shift maps on infinite-dimensional spaces, the necessary and sufficient conditions for Li–Yorke chaotic are proved both in the abstract sequence setting and in the eigenfunction setting. Besides, a constructive proof is provided for the Devaney chaos of weighted shift maps on the Schwartz space. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Chaotic behavior is a complexity phenomenon of nonlinear dynamical systems. There exist different definitions of chaos, each characterizing some aspects of the complexity of dynamics. Two well-known definitions of chaos are given by Li and Yorke in [1] and by Devaney in [2]. The Li–Yorke chaos characterizes the existence of uncountable scrambled sets. The Devaney chaos describes sensitive dependence of longtime dynamics on initial conditions with transitivity and density of periodic points. Earlier results on infinite-dimensional linear chaotic systems constructed by shift maps appeared in 1990–1993, see [3– 5]. Godefroy and Shapiro [6] showed that a class of linear operators can be chaotic, such as the backward shift map in a separable Hilbert space and the weighted shift map with a weight greater than 1, which are studied also in Protopopescu [7] and MacCluer [8]. In MacCluer [8] it was further showed that a strongly continuous semigroup etB and some other operators which are infinite series in terms of B, with B being the backward shift map in a separable Hilbert space, are chaotic. Godefroy and Shapiro [6] showed that the translation operator in the Fréchet space of complex entire functions is chaotic, and this generalized Birkhoff’s result [9]. Chan and Shapiro [10] generalized this work to the space of slowly growing entire functions. Infinite-dimensional linear chaotic systems can be used as meaningful mathematical models for physical problems. For example, Protopopescu and Azmy [11] considered an infinite-dimensional system of linear differential equations in modeling of motions of particles with internal energy and showed that it is chaotic. Gulisashvili and MacCluer [12] showed that a linear quantum harmonic oscillator is chaotic by using a result of Godefroy and Shapiro [6]. In these cited papers, the chaos is in the sense of Wiggins [13] or in the sense of Devaney [2]. In [14], we considered the backward shift map in the space of infinite sequences of elements from a general Fréchet space, and showed that this backward shift map itself is chaotic in both senses of Devaney and of Li–Yorke in terms of orbit separation of nonwandering nonperiodic points. We also showed that the translation map, which can be treated as a continuous shift system, in the space of real continuous functions is chaotic in both senses of Devaney and of Li–Yorke. This

∗

Corresponding author. Tel.: +86 21 66132664; fax: +86 21 66133292. E-mail address: [email protected] (X. Fu).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.049

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space provides more natural framework for considering or modeling physical problems by linear or semilinear differential equations than the space of complex entire functions as used by Godefroy and Shapiro [6]. In [15,16] we studied shift and weighted shift maps related to linear harmonic oscillators and the ladder operators in quantum mechanics and proved that these maps are chaotic in the Li–Yorke sense. We also showed that the weighted backward shift map in a separable Hilbert space is chaotic in the Li–Yorke sense as well as in the Devaney sense. The idea of a scrambled set was introduced by Smítal [17] to characterize Li–Yorke chaotic sets in a certain way. There have been extensive studies on scrambled sets of an iterated continuous map on a compact interval. Some results have also been achieved on scrambled sets of continuous or discontinuous self-maps on higher-dimensional cube I n , n ≥ 2 or more general spaces [18–22]. In this paper we shall further investigate into the structures of the Li–Yorke chaotic sets and the Devaney chaotic sets generated by shift and weighted shift maps. In Section 2, preliminaries are given. In Section 3, we show the characterization of Li–Yorke chaotic sets by p-scrambled sets, maximal scrambled sets, and orbit invariants. In Section 4, we prove that the weighted shift maps are Li–Yorke chaotic for any nonzero weight and the weighted shift maps on eigenfunctions of separable Hilbert spaces are Li–Yorke chaotic for weight with norm no less than 1, and we give a constructive proof of the Devaney chaos for the weighted shift maps on the Schwartz class. Finally, in Section 5, we give some remarks as well as open problems. 2. Preliminaries For some integer N ≥ 2, let A = {0, 1, . . . , N − 1} and equipped with the discrete metric d. Denote by Σ (N ) the space consisting of unilateral sequences with entries in A. Any x ∈ Σ (N ) can be expressed as x = (x0 , x1 , . . . , xi , . . .), xi ∈ A, i ≥ 0. Let Σ (N ) be endowed with the product topology. Then Σ (N ) is metrizable, and the metric on Σ (N ) is chosen to be

ρ(x, y) =

∞ X d(xi , yi ) i =0

2i

,

x = (x0 , x1 , . . .), y = (y0 , y1 , . . .) ∈ Σ (N ).

Defined the shift map σ : Σ (N ) → Σ (N ) by (σ (x))i = xi+1 , i = 0, 1, . . . . A bilateral shift map is defined similarly, acting by the same formula but on bilaterally infinite sequences of elements of A. The shift map σ : Σ (N) → Σ (N) over a countable alphabet N can be defined similarly by letting A = N = {0, 1, . . . , k, . . .}. Here in this paper we shall consider only the unilateral shift maps. Let (X , d) be a Fréchet space over the complex field C (that is a complete linear metric space over C). We denote by Σ (X ) the space of functions from the nonnegative integers to X , c : Z+ → X , where c (k) is denoted by ck and c = (c0 , c1 , . . .). Similarly we define the backward shift map σ : Σ (X ) → Σ (X ) by (σ (c ))k = ck+1 and a weighted backward shift map b : Σ (X ) → Σ (X ) as (b(c ))k = µck+1 , where µ ∈ C is a constant or depending on k. Let Σ (X ) be endowed with the metric of the product topology,

ρ(x, y) =

∞ X 1

d(xi , yi )

2i 1 + d(xi , yi ) i =0

,

x = (x0 , x1 , . . .), y = (y0 , y1 , . . .) ∈ Σ (X ).

These shift maps on the sequence space associated with (X , d) generalize the usual shift maps on finite alphabets. Note that for a Fréchet space (X , d), the metric d is translation-invariant. So the metric ρ on Σ (X ) is regarded as translation-invariant. When X is nontrivial, Σ (X ) is infinite-dimensional. Define the addition ‘‘⊕’’ and the scalar multiplication ‘‘·’’ in Σ (X ) as follows:

(x ⊕ y)i = xi + yi , (α · x)i = α xi ,

x, y ∈ Σ (X ), α ∈ C, i = 0, 1, . . . ,

then it is easy to verify that (Σ (X ), ρ) is a Fréchet space over C. From ρ(σ (x), σ (y)) ≤ 2ρ(x, y), the backward shift map σ is continuous. (Σ (X ), σ ) is a general symbolic dynamical system, cf. [23, Chapter 13] or [24]. It can be verified that σ (α · x ⊕ β · y) = α · σ (x) ⊕ β · σ (y), ∀ x, y ∈ Σ (X ), ∀ α, β ∈ C , that is, the shift map σ : Σ (X ) → Σ (X ) is linear and (Σ (X ), σ ) a linear dynamical system. For the orbit {σ n (x), n ≥ 0} from x and the orbit {σ n (y), n ≥ 0} from y, consider their linear combinations. We have α · σ n (x) ⊕ β · σ n (y) = σ n (α · x ⊕ β · y), in other words, {α · σ n (x) ⊕ β · σ n (y), n ≥ 0} is the orbit from α · x ⊕ β · y. Therefore, the shift map σ satisfies the orbit superposition principle. We denote by x = (x0 , x1 , . . .)T ∈ Σ (X ) as an infinite-dimensional column vector. Define the infinite matrix A = (aij )i,j=0,1,...,+∞ ,

 where aij =

1, 0,

j=i+1 otherwise,

then the shift map σ has a matrix representation σ (x) = Ax. For any eigenvalue λ ∈ C of σ , there are infinitely many eigenvectors associated with it, i.e. x = (x0 , λx0 , λ2 x0 , . . . , λk x0 , . . .),

∀x0 6= 0.

It is obvious that for any eigenvalue λ ∈ C of σ , the corresponding eigenspace is isomorphic to X , and the spectrum of σ consists of eigenvalues only [14].

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Many familiar operators have finite or infinite countable number of eigenvalues in discrete distribution. But the shift map

σ here has uncountable eigenvalues in continuum distribution. This is an indication that the system (Σ (X ), σ ) may have complicated properties. The following theorem substantiates this expectation. It shows that the subshifts of (Σ (X ), σ ) can be used as models of other continuous maps on infinite-dimensional metric spaces.

Theorem 2.1 ([3,25]). Let (X , d) be a metric space. For any continuous map τ : X → X , there exists a subshift (Στ , στ ) of (Σ (X ), σ ), such that τ is topologically conjugate to στ , where στ is σ restricted to Στ . The above theorem shows the importance of subshifts of (Σ (X ), σ ). Moreover, it reveals the extraordinarily plentiful structure of subshifts in (Σ (X ), σ ). As mentioned before, there are a few mathematical definitions of chaos. Although the definition in Devaney [2] and in Wiggins [13] seems a popular one, other definitions in Li and Yorke [1] and Kirchgraber and Stoffer [26], which capture or describe various different dynamical behavior of a system, are proposed and used in nonlinear sciences. Li–Yorke’s definition of chaos does characterize the complexity of dynamical systems. For example, for one-dimensional dynamical systems such as the iteration of continuous maps on intervals, Li–Yorke’s chaos is equivalent to topological chaos, i.e., the topological entropy of the system is positive [27]. The same conclusion holds for subshifts of finite type. Recently it was further shown that in a compact metric space, a continuous map is topologically chaotic implies it is Li–Yorke chaotic [21]. This motivates us to study Li–Yorke chaos of the shift and weighted shift maps. Let f : X → X be a continuous map of a compact metric space X with metric d. We classify pairs of points in X into distal and proximal, with a subdivision of the proximal group into asymptotic and scrambled. Definition 2.1. A pair (a1 , a2 ) is called distal for the map f , if lim inf d(f n (a1 ), f n (a2 )) > 0; n→∞

called proximal if lim inf d(f n (a1 ), f n (a2 )) = 0; n→∞

called asymptotic if lim sup d(f n (a1 ), f n (a2 )) = 0, n→∞

i.e. lim d(f n (a1 ), f n (a2 )) = 0. n→∞

A pair (a1 , a2 ) is called scrambled for the map f if lim sup d(f n (a1 ), f n (a2 )) > 0,

and

n→∞

lim inf d(f n (a1 ), f n (a2 )) = 0. n→∞

If a scrambled set S of f is also uncountable, and S ⊆ Ω (f ) − P (f ), where Ω (f ) is the nonwandering set for f and P (f ) is the periodic set for f , then we call S a chaotic set for f , and we say that f is chaotic in the Li–Yorke sense. For an integer p > 0, x ∈ X is called p-distal, if for the orbit of x under f , orb(x) = {x0 = x, x1 = f (x), . . . , xn = f n (x), . . .}, lim infn→∞ d(xn , xn+p ) > 0; or x ∈ X is called p-scrambled, if lim supn→∞ d(xn , xn+p ) > 0 and lim infn→∞ d(xn , xn+p ) = 0. From this definition we see that a nondistal pair is proximal, and a proximal pair is either asymptotic or scrambled. A subset S ⊆ X containing at least two points is a scrambled set of f , if for any a1 , a2 ∈ S, a1 6= a2 , (a1 , a2 ) is a scrambled pair for f , cf. [28]. Thus a scrambled set of f is a subset S ⊆ X such that whenever a1 and a2 are two distinct points of S one has simultaneously lim sup d(f n (a1 ), f n (a2 )) > 0 n→∞

and

lim inf d(f n (a1 ), f n (a2 )) = 0. n→∞

Definition 2.2. A discrete topological dynamical system (X , f ) is said to be Li–Yorke chaotic if there exists a chaotic set for f in X . Remark 2.1. The original characterization of chaos in the Li–Yorke’s theorem is based on three conditions. The third condition is lim sup d(f n (x), f n (a)) > 0, n→∞

for any x ∈ S and for any periodic point a ∈ P (f ). But it has been shown [14] that this condition is not essential and is removable. For a continuous self-map f on a compact interval, the existence of a scrambled pair is sufficient for f to be Li–Yorke chaotic [29]. For subshifts of finite type the same result is valid [27].

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Remark 2.2. The concepts of scrambled pair and scrambled set are used for better understanding of Li–Yorke type chaos. In [30] scrambled pair (a1 , a2 ) is further required to be nonwandering. With this additional assumption on nonwandering, Li–Yorke chaos for one-dimensional maps on intervals is equivalent to topological chaos. The same conclusion holds for subshifts of finite type. Also note that without this additional assumption some maps may be Li–Yorke chaotic but have zero topological entropy [31]. We now give the definition for Devaney chaos according to Devaney [2] and Wiggins [13]. Definition 2.3. (X , f ) is said to be Devaney chaotic, if there is a compact invariant subset S ⊆ X , such that (1) f has sensitive dependence on initial conditions on S: there exists δ > 0 such that for any x in S and any neighborhood U of x, there exists y ∈ U and n ≥ 0 such that ρ(f n (x), f n (y)) > δ; (2) f is topologically transitive on S: for any two open subsets U and V in S, there exists k > 0, such that f k (U ) ∩ V 6= ∅. The following condition is given in [2] by Devaney, but not included in our definition. (3) The periodic points of f are dense in S. Remark 2.3. Note that the conditions (1), (2) and (3) are not independent, since conditions (2) and (3) imply condition (1), as shown in [32]. There exist systems whose dynamical behavior is chaotic but there are only few periodic points. So condition (3) may be too strong and restrictive. Therefore we usually use conditions (1) and (2) to define Devaney chaos. However, the importance and relevance to chaos of condition (3) should not be neglected, see [13]. Remark 2.4. Huang and Ye proved in [33] that Devaney chaos under the three conditions implies Li–Yorke chaos. 3. Chaotic sets of shift maps In [14] it was shown that the general backward shift map σ : Σ (X ) → Σ (X ) is both Li–Yorke and Devaney chaotic. Precisely, suppose (X , d) is a nontrivial separable Fréchet space. Then the infinite-dimensional linear dynamical system (Σ (X ), σ ) is chaotic in the senses of both Li–Yorke and Devaney. In this section we further study the structure of Li–Yorke chaotic sets of shift maps. Note that the existence of the Li–Yorke chaotic sets for shift maps is already known, our new contribution here is the characterization of the Li–Yorke chaotic sets. We shall characterize Li–Yorke chaotic sets by p-scrambled sets, maximal scrambled sets, and orbit invariants. Note also that for a shift map the Li–Yorke chaotic set exists but it is usually not the whole space (see Corollary 3.10 below), and thus it is worthwhile to locate it. n Definition 3.1. We call γ an orbit invariant on S ⊆ Σ (X ) for σ , if (i) γ : n=0 σ (S ) → (0, 1) is a function; (ii) γ |S is injective; (iii) γ (σ n (x)) = γ (x), ∀x ∈ S , ∀n ≥ 0. Thus γ has the same value on an orbit, and different values on different orbits.

S∞

Lemma 3.1. Let (X , d) be a metric space and σ : Σ (X ) → Σ (X ) be the shift map. If S ⊆ Σ (X ) is a scrambled set of σ , and σ m (S ) ∩ σ n (S ) ⊆ P (σ ), for any m, n ≥ 0, m 6= n, then there exists a point a ∈ P (σ ), such that for any m, n ≥ 0, m 6= n, σ m (S ) ∩ σ n (S ) ⊆ orb(a), where P (σ ) is the set of periodic points under σ and orb(a) = {σ n (a), n ≥ 0} is the orbit of a under the shift map σ . Proof. Note that S contains at most one asymptotically periodic point [14]. Note that those eventually periodic points must also be asymptotically periodic points. Hence S contains at most one eventually periodic point. Suppose there exists an eventually periodic point a ∈ S, namely, there is e ∈ S , k ≥ 0, a ∈ P (σ ), such that σ k (e) = a. Suppose σ m (a1 ) = σ n (a2 ) = a3 ∈ P (σ ), a1 , a2 ∈ S , m, n ≥ 0, m 6= n. Then a1 , a2 are eventually periodic points. Therefore a1 = a2 = e, and a3 ∈ orb(a). So σ m (S ) ∩ σ n (S ) ⊆ orb(a). The conclusion holds.  3.1. Characterization of Li–Yorke chaotic sets In this first part we shall study the p-scrambled set in detail and use it to characterize Li–Yorke chaotic sets. Lemma 3.2. If for any integer p > 0, x ∈ Σ (X ) is p-scrambled, then the orbit orb(x) from x under the shift map σ is a scrambled set. Proof. Let x ∈ Σ (X ), consider the orbit from x under σ : orb(x) = {x(0) = x, x(1) = σ (x), . . . , x(n) = σ n (x), . . .}. If x(0) is a p-scrambled point, then (x(0) , x(p) ) is a scrambled pair. For any k ≥ 0, we have lim sup ρ(σ n−k (x(k) ), σ n−k (x(k+p) )) > 0 n→∞

and

lim inf ρ(σ n−k (x(k) ), σ n−k (x(k+p) )) = 0. n→∞

That is, (x(k) , x(k+p) ), for any k ≥ 0 are all scrambled pairs. If for any integer p > 0, x is p-scrambled, then (x(k) , x(k+p) ) (k ≥ 0, p > 0) are all scrambled pairs, so that the orbit from x under the shift map σ is a scrambled set.  It is immediate to show that the result in the above lemma is also valid for general continuous maps.

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For the shift map, by using Lemma 3.2 and the continuity of σ , we have the following result. Corollary 3.3. If x ∈ Σ (X ) is p-scrambled for all integers p > 0, and orb(x) is dense in a closed set S ⊆ Σ (X ), then S is a scrambled set of the shift map σ .  We know that no periodic point is p-scrambled for any integer p > 0, and every p-periodic point is q-distal for q 6= kp, where k > 0 is an integer. The converse is almost true, as stated in the next proposition. Proposition 3.4. If for all integers p > 0, x ∈ Σ (X ) is not p-scrambled and not p-distal, then x is a nonperiodic and asymptotically periodic point of the shift map σ . Proof. By the assumption on x, for an integer q > 0, we have lim inf ρ(σ n (x), σ n+q (x)) = 0

and

n→∞

lim sup ρ(σ n (x), σ n+q (x)) = 0. n→∞

If for every periodic point y ∈ P (σ ), lim supn→∞ ρ(σ n (x), σ n (y)) > 0, suppose the period of y is p, then

ρ(σ n (x), σ n+p (x)) ≥ ρ(σ n (x), σ n+p (y)) − ρ(σ n+p (y), σ n+p (x)) = ρ(σ n (x), σ n (y)) − ρ(σ n+p (x), σ n+p (y)). Therefore, lim sup ρ(σ n (x), σ n+p (x)) ≥ lim sup(ρ(σ n (x), σ n (y)) − ρ(σ n+p (x), σ n+p (y))) > 0, n→∞

n→∞

which is a contradiction. So there must exist an y ∈ P (σ ), such that lim supn→∞ ρ(σ n (x), σ n (y)) = 0, i.e., x is asymptotically periodic. Because x is not p-distal for all integers p > 0, x is nonperiodic. So x is a nonperiodic and asymptotically periodic point of the shift map σ .  An interesting topic is how to characterize a scrambled set of shift maps by finding the necessary and sufficient conditions. For the shift map σ : Σ (N ) → Σ (N ), N ≥ 2, we show in the next theorem that together with the condition η(x, y) = 1, x, y ∈ S, the existence of an orbit invariant γ on S is a sufficient condition for S ⊆ Σ (N ) to be a scrambled set of σ . Here η : S × S → [0, 1] is defined as:

η(x, y) = lim sup

N (x, y, n)

n→∞

n

,

where N (x, y, n) = card {k : xk = yk , 0 ≤ k ≤ n}. Theorem 3.5. Let S ⊆ Σ (N ), cardS ≥ 2. If there exists an orbit invariant γ on S for σ , and η(x, y) = 1 for all x, y ∈ S, then S is a scrambled set of σ , and ∃a ∈ P (σ ), ∀m, n ≥ 0, m 6= n, σ m (S ) ∩ σ n (S ) ⊆ orb(a). Proof. ∀x, y ∈ S , x 6= y, we claim that there exists a subsequence {nk , k = 0, 1, . . .} such that xnk 6= ynk . Otherwise, ∃M > 0, ∀k ≥ M , xk = yk . That is, σ k (x) = σ k (y), k ≥ M. So

γ (x) = γ (σ k (x)) = γ (σ k (y)) = γ (y), which is a contradiction to η(x, y) = 1. Hence it holds that lim sup ρ(σ n (x), σ n (y)) ≥ lim ρ(σ nk (x), σ nk (y)) > 0. k→+∞

n→+∞

Define β :

S+∞ n =0

σ n (S ) → {0, 1, . . . , N − 1} as:

β(x) = x0 , ∀x = (x0 , x1 , . . .) ∈

+∞ [

σ n (S ),

n =0

then we can declare that for any x, y ∈ S, and for any K > 0, there exists n(K ) ≥ 0 such that β(σ i (x)) = β(σ i (y)), n(K ) ≤ i ≤ n(K ) + K . Otherwise, there is an M0 > 0, for all m ≥ 0, there exists k satisfying m ≤ k ≤ m + M0 , such that xk 6= yk , then N (x, y, n) ≤ n −





n M0 + 1

,

this implies η(x, y) ≤

M0 M0 + 1

that is a contradiction. It follows that lim inf ρ(σ n (x), σ n (y)) ≤ lim ρ(σ n(K ) (x), σ n(K ) (y)) = 0. n→+∞

K →+∞

Therefore, S is a scrambled set of σ .

< 1,

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If there exist x, y ∈ S, such that σ m (x) = σ n (y), where m > n ≥ 0, then γ (σ m (x)) = γ (σ n (y)), so x = y, that is σ (σ n (y)) = σ n (y), i.e., σ n (y) ∈ P (σ ). Therefore, for any m, n ≥ 0, m 6= n, we have σ m (S ) ∩ σ n (S ) ⊆ P (σ ). By Lemma 3.1, we can conclude that there exists a ∈ P (σ ), such that σ m (S ) ∩ σ n (S ) ⊆ orb(a). The proof is completed.  m−n

We conjecture that the condition η(x, y) = 1, for x, y ∈ S in the above theorem may be replaced by a weaker condition η(x, y) ≥ δ, x, y ∈ S for some (or any given) small positive constant δ < 1. Corollary 3.6. Let S ⊆ Σ (N ). If the following conditions are satisfied, (i) there exists a surjective map γ : any n ≥ 0; (ii) η(x, y) > δ > 0, for all x, y ∈ S,

S∞

n =0

σ n (S ) → (0, 1) such that γ |S is injective and γ (σ n (x)) = γ (x), for all x ∈ S and for

then S is a Li–Yorke chaotic set for σ and there exists a ∈ P (σ ) such that for any m, n ≥ 0, m 6= n, it holds that σ m (S ) ∩ σ n (S ) ⊆ orb(a). Proof. We only need to check that S is uncountable. In fact, this is true because

ℵ = card γ

∞ [

! σ (S ) = card γ (S ) = card(S ).  n

n =0

3.2. Maximal scrambled sets of shift maps In this section we deal with some problems related to maximal scrambled sets of shift maps. We call a scrambled set S ⊆ X a maximal scrambled set, if there exists no scrambled set which contains S as its proper subset. We call a continuous map f : X → X totally chaotic if the whole space X is a scrambled set of f , cf. [20]. Let (X , d) be a compact metric space and a1 , a2 ∈ X . We say that a1 is equivalent to a2 [34], denoted by a1 ≈ a2 , if limn→∞ d(f n (a1 ), f n (a2 )) = 0. Two sets S1 , S2 ⊆ X are equivalent, denoted by S1 ≈ S2 , if there is a bijection map ϕ : S1 → S2 such that x ≈ ϕ(x), for any x ∈ S1 . Two sets S , R ⊆ X are called separated, if there exists δ > 0 such that lim infn→∞ d(f n (x), f n (y)) ≥ δ for any x ∈ S and y ∈ R. We remark that ≈ is used as an equivalence relation in X and 2X , respectively. We view a scrambled pair (x, y) for a general backward shift map σ : Σ (X ) → Σ (X ) as a point of the product space Σ (X ) × Σ (X ), and denote by B the set of all scrambled pairs in Σ (X ) × Σ (X ), where X P is a countable set. Consider the (p0 , p1 , . . . , pk , . . .)-shift map ([35]) σ : (Σ (X ), B , m) → (Σ (X ), B , m) with pi > 0, and pi = 1. Here (Σ (X ), B , m) is a measure space whose measure is denoted by m. Let f : Y → Y be continuous, and suppose ν be a nontrivial weakly mixing measure with support Y × Y . Then ν × ν is ergodic, so the set of generic points (x, y) for f × f has full measure for ν × ν . Note that the orbit of (x, y) under f × f is dense in Y × Y , so (x, y) is a scrambled pair. In particular, we have the following result. Proposition 3.7. The set B ⊂ Σ (X ) × Σ (X ) of all scrambled pairs has full measure under the product measure m × m. In particular, for the (p0 , p1 , . . . , pN −1 )-shift map σ : (Σ (N ), B , m) → (Σ (N ), B , m) with pi = 1/N , i = 0, 1, . . . , N − 1, the set of all scrambled pairs is a full measure subset in Σ (N ) × Σ (N ). Proposition 3.7 shows the set B of all scrambled pairs is a quite big subset in Σ (X ) × Σ (X ). Let PB be the projection of B to Σ (X ). Then for any maximal scrambled set M of σ , we have M ⊆ PB . But PB may not be a scrambled set of σ , simply because (a1 , a2 ) and (a1 , a3 ) being scrambled pairs do not always imply that (a2 , a3 ) is a scrambled pair. Now we address the maximal scrambled sets of general shift maps σ . If M is a maximal scrambled set of σ : Σ (X ) → e = M ∪ {a2 } \ {a1 } is also a maximal Σ (X ), it is easy to find two points a1 , a2 , such that a1 ∈ M , a1 ≈ a2 , so that M scrambled set of σ . This implies that the maximal scrambled set of σ is not unique and two maximal scrambled sets may have nonempty intersection. For the general shift maps, the following result further shows that the maximal scrambled set of σ is not unique even in the sense of equivalence. Theorem 3.8. Suppose X is a compact metric space. Then there exists no scrambled set of σ : Σ (X ) → Σ (X ) such that, for any scrambled set S of σ , either it contains S or it has a subset which is equivalent to S. Proof. We give a constructive proof for this theorem to show the structure of maximal scrambled sets of shift maps. Let e (X ) = {˜x : x ∈ Σ (X )}. Define ρ(˜ a˜ = {x ∈ Σ (X ) : x ≈ a}, Σ ˜ x, y˜ ) = lim supn→∞ ρ(σ n (x), σ n (y)), σ˜ (˜x) = σ] (x). Then e (X ), ρ) e (X ), σ˜ ) a dynamical system. (Σ ˜ is a compact metric space, and (Σ Since σ : Σ (X ) → Σ (X ) is topologically transitive [23], there exists a point a ∈ Σ (X ) (a is a nonperiodic recurrent point) such that Σ (X ) = cl(orb(a)). Since P (σ ) 6= ∅, from [30, Theorem 5.1], σ has a scrambled set S which contains a. e (X ) = cl(orb(˜a)) and e Then Σ S := {˜x : x ∈ S } is a scrambled set of σ˜ , a˜ ∈ e S. We only need to prove that there exists no scrambled set of σ˜ such that it contains any scrambled sets of σ˜ as its subsets. e of σ˜ and it contains any scrambled sets of σ˜ as its subsets, then e e Otherwise, suppose there exists a scrambled set M S ⊆ M.

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e When x is a fixed point of σ , x˜ is a fixed point of σ˜ . Since N ≥ 2, there exists a fixed point, say p˜ , such that p˜ 6∈ M. e Otherwise, if there exist two distinct fixed points p˜1 , p˜2 ∈ M, then 

lim inf ρ( ˜ σ˜ n (p˜1 ), σ˜ n (p˜2 )) = lim inf lim sup ρ(σ m+n (p1 ), σ m+n (p2 )) n→∞

n→∞

m→∞



= ρ(p1 , p2 ) > 0,

which is a contradiction. There exists a subsequence {ni } such that σ˜ ni (˜a) → p˜ as i → ∞. So we have lim infn→∞ ρ( ˜ σ˜ n (˜a), σ˜ n (˜p)) = 0. e But this is a contradiction. Therefore, it must be If lim supn→∞ ρ( ˜ σ˜ n (˜a), σ˜ n (˜p)) > 0, then {˜a, p˜ } ⊆ M. limn→∞ ρ( ˜ σ˜ n (˜a), σ˜ n (˜p)) = 0. That is,

 lim

n→∞

lim sup ρ(σ m+n (a), σ m+n (p)) m→∞



= 0.

It follows that liml→∞ ρ(σ l (a), σ l (p)) = 0, i.e. a ≈ p, again a contradiction. The proof is hence complete.



We have the following corollaries. Corollary 3.9. If (Σ , σ ) is a subshift of (Σ (X ), σ ), which is scrambled, topologically transitive, with card P (σ |Σ ) ≥ 2, then the maximal scrambled sets of σ |Σ are not unique.  Corollary 3.10. The shift map σ : Σ (X ) → Σ (X ) is not totally chaotic.



4. Chaotic sets of weighted shift maps It is well known that the global attractor of the shift map (l2 , σ ) discussed in [36] is a one-point set {θ }. This shows that not every shift map is chaotic. The chaotic property of a shift map closely relates to the structure of the phase space of the system. Gulisashvili and MacCluer [12] showed that a weighted shift map related to the annihilation operator for a linear quantum harmonic oscillator is Devaney chaotic. It is proved in [15] by using the result in [14] that this weighted shift map is also Li–Yorke chaotic. 4.1. Chaotic sets of a constantly weighted shift map The weighted shift map b : Σ (X ) → Σ (X ) is defined by b = µσ , µ ∈ C, where X is a complex Banach space equipped with the norm k · k, Σ (X ) = {(x0 , x1 , . . . , xk , . . .) : xk ∈ X , k ≥ 0}, and the metric in Σ (X ) is defined by m(x, y) =

∞ X 1 n=0

2n

kxn − yn k . 1 + kxn − yn k

Theorem 4.1. When µ 6= 0, the weighted shift map b : Σ (X ) → Σ (X ) is Li–Yorke chaotic. Proof. When µ 6= 0, construct the set S = {xθ = (xθ0 , xθ1 , . . . , xθn , . . .) : θ ∈ (0, 1)}, where xθ0 = 0, and

  1 e, xθn = µn 0,

if n = k2 , [kθ ] − [(k − 1)θ] = 1 otherwise,

where e is a unit element in X . Take x(N ) = (xθ0 , xθ1 , . . . , xθN , xθ0 · µ−(N +1) , xθ1 · µ−(N +1) , . . . , xθN · µ−(N +1) , . . . , xθn · µ−(N +1) , . . .). Then we have m(x(N ) , xθ ) ≤

∞ X 1 n =N +1

2n

→ 0,

as N → ∞.

There exists a sufficiently large N = Nε , such that x(Nε ) ∈ V (xθ , ε), where V (xθ , ε) is the ε -neighborhood of xθ . Note that bNε +1 x(Nε ) = xθ , which implies that (bNε +1 V (xθ , ε)) ∩ V (xθ , ε) 6= ∅. Hence S ⊆ Ω (b). It is obvious that S ∩ P (b) = ∅. Thus we have S ⊆ Ω (b) − P (b), and it is not difficult to show that S is an uncountable set. θ θ For any θ1 6= θ2 , there exist infinitely many kn , such that x 12 6= x 22 . Then we have kn



1 2 θ1 θ2

kxk2 − xk2 k = 2 e = |µ|−kn ,

k n n µn

kn

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X. Fu, Y. You / Nonlinear Analysis 71 (2009) 2141–2152

and θ1

k2n

θ2

k2n

m(b (x ), b (x )) =

l=0

θ

θ

2

|µ|kn kxl+1 k2 − xl+2 k2 k

∞ X 1

n

n

θ

2l 1 + |µ|k2n kxθ1

|µ| kxk2 − xk2 k n

n

θ1

2

n

θ2

θ1

k2n

≥

− xl+2 k2 k

l+k2n

1 + |µ|kn kx

k2n

=

θ

− xk22 k

1

> 0.

2

n

It follows that lim sup m(bk (xθ1 ), bk (xθ2 )) > 0,

for any θ1 6= θ2 .

k→∞

On the other hand, we have 2 2 m(bk +1 (xθ1 ), bk +1 (xθ2 )) =

2 +1

|µ|k

∞ X 1 l=0

2l 1 + |µ|k2 +1 kxθ1 2 − xθ2 2 k l+k +1 l +k +1

∞ X

≤

2 +1

|µ|k

1

N =k+1

1

N =k+1

2 2 2N −k −1

≤2

∞  X i=1

θ

θ

kxN12 − xN22 k

2 2 2N −k −1 1 + |µ|k2 +1 kxθ12 − xθ22 k N N

∞ X

≤

θ

θ

kxl+1 k2 +1 − xl+2 k2 +1 k

1

=

∞ X

1

i=1

2 2i +2ki−1

i

22k

→ 0 as k → ∞,

so that lim infk→∞ m(bk (xθ1 ), bk (xθ2 )) = 0. Thus we have proved that the weighted shift map b is Li–Yorke chaotic.



4.2. Chaotic sets of a nonconstantly weighted shift map Now we introduce a weighted shift map with nonconstant weight arising in a quantum harmonic oscillator without forcing, which is modeled by the Schrödinger equation [37] as follows, ih¯ ψt = −

h¯ 2

k

ψxx + x2 ψ,

2m

2

with wave function ψ(x, t ), displacement x, mass m, stiffness k and Planck number h¯ . The nondimensionalized steady states in terms of the eigenfunctions in the separable Hilbert space X = L2 (−∞, ∞) form an orthonormal basis e−x /2 Hn (x) ψn (x) = p√ , π 2n n! 2

n = 0, 1, . . . ,

2

where Hn (x) = (−1)n ex

d n −x 2 e is the nth Hermite polynomial. dxn

The natural phase space for the quantum harmonic oscillator is the Schwartz class or called Schwartz space F of rapidly decreasing functions in X = L2 (−∞, ∞) as defined below,

( F =

φ ∈ L2 (−∞, ∞) : φ =

∞ X

cn ψn ,

n =0

∞ X

) |cn |2 (n + 1)r < ∞, ∀r ≥ 0 .

n =0

F is an infinite-dimensional Fréchet space with topology defined by the system of seminorms pr (·) of the form [38] pr (φ) = pr

∞ X

! cn ψn

=

n =0

∞ X

!1/2 |cn | (n + 1) 2

r

,

r ≥ 0.

n =0

This topology on F is also given by the metric ρ

ρ(φ, ψ) =

∞ X

2−m pm (φ − ψ) · (1 + pm (φ − ψ))−1 .

m=0

Gulisashvili and MacCluer [12] defined an unbounded, closed, linear operator which is a weighted shift map B on F , B : F → F, 1 Bψn ≡ √ 2

 x+

d dx



ψn =

√

nψn−1 .

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By using a result of Godefroy and Shapiro [6], Gulisashvili and MacCluer [12] showed that the weighted shift map B is Devaney chaotic [2], namely, it has topological transitivity (dense orbits), sensitivity to initial conditions (orbit divergence), and density of periodic points. In [15] it has been shown furthermore that the weighted shift map B : F → F is also Li–Yorke chaotic. The chaotic set was constructed P∞ asθ follows. Fix θ ∈ (0, 1) and define φ θ = n=0 cn ψn by

 θ c0 = 0  √ 1/ n!, θ cn = 0,

if n = k2 , [kθ] − [(k − 1)θ] = 1, otherwise,

where k = 1, 2, . . ., and [·] denotes the integer part of a real number. Let S = {φ θ : θ ∈ (0, 1)}. By using the idea and result in [14], it can be shown that all points in S are nonwandering and nonperiodic, and S is a chaotic set for B, i.e., the weighted shift map B is Li–Yorke chaotic. Godefroy and Shapiro [6] showed that the following weighted shift map β defined on a separable Hilbert space H with a complete orthonormal basis {φn }, β : H → H , β φn = µ φn−1 , is Devaney chaotic, whenever |µ| > 1. Similar to the proof of Theorem 4.1, we can also discuss the chaos of this weighted shift map β in the sense of Li–Yorke. When |µ| > 1, the chaotic set S of β can be constructed as follows:

(

∞ X

θ

φ =

S=

) θ

cn φn : θ ∈ (0, 1) ,

n =0

where

 θ 0 c0 =  cnθ =

1/µn 0

if n = k2 , [kθ ] − [(k − 1)θ] = 1, k ≥ 1 otherwise.

Hence β is also Li–Yorke chaotic whenever |µ| > 1. When µ = 1, β is similar to the shift map σ discussed in Section 3, so β is also Li–Yorke chaotic when |µ| = 1. However, when |µ| < 1, because the phase space here is different from that in Theorem 4.1, the weighted shift map is not chaotic in any sense, in fact, the global attractor of β is a single-point set containing only the zero vector. Thus β is not chaotic in the sense of Li–Yorke or Devaney. In comparison with the result in Theorem 4.1, where the weighted shift map b is Li–Yorke chaotic for all µ 6= 0, we have the following result on the weighted shift map β . Theorem 4.2. The weighted shift map β : φn → φn−1 defined on a separable Hilbert space H with a complete orthonormal basis φn is Li–Yorke chaotic if and only if |µ| ≥ 1. As mentioned above, B is Devaney chaotic [12]. For better understanding of Devaney chaos of the weighted shift map B, now we give a constructive proof of this result. Theorem 4.3. The weighted shift map B : F → F is Devaney chaotic. Proof. According to Definition 2.3, we need to show that B is topologically transitive and B has sensitive dependence on initial conditions. Note that F ⊆ L2 (−∞, ∞) is a second category space, hence F is separable. So there exists a countable subset F , such that E = F . E = {φl }∞ l=0 ⊆P ∞ l l l l l l Let φl = n=0 an ψn , l ≥ 0, Ak = (a0 a1 · · · ak ), l, k ≥ 0. Rearrange the countable set {Ak }l,k≥0 by a certain way k k k to a one-index sequence and denote it by {Dk }k≥0 . Let Dk = (d0 d1 · · · dNk −1 ), where Nk is the length of Dk , and denote s0 = 0, sk =

Pk−1 i =0

φ (0) =

Ni , k ≥ 1. Let

N 0 −1

X

d0n ψn ,

n=0

φ (k) =

sk+1 −1

X

1

dkn−sk (n(n − 1) · · · (n − sk + 1))− 2 ψn ,

k ≥ 1.

n=sk

Take e φ=

P∞

k=0

φ (k) , then

sk+1 −1

Bsk e φ =

X

1

dkn−sk (n(n − 1) · · · (n − sk + 1))− 2

n=sk sk+1 −sk −1

=

X n =0

N k −1

dkn ψn + · · · =

X n =0

dkn ψn + · · · .

√ √ n n − 1···

p

n − sk + 1ψn−sk + · · ·

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X. Fu, Y. You / Nonlinear Analysis 71 (2009) 2141–2152

For any given φ ∈ F , φ =

P∞

n=0

cn ψn , and for any ε > 0, there exists φl ∈ E, such that ρ(φl , φ) < 2ε , and there exists k,

such that the first Nk coefficients in Bsk e φ and φl agree. We can choose k such that

ρ(Bsk e φ, φl ) <

ε 2

.

Therefore ρ(Bsk e φ, φ) < ε. Thus {Bke φ :kP = 0, 1, . . .} is a dense orbit in F . This confirms the topological transitivity of B on F . ∞ Take 0 < δ < 1. For any φ ∈ F , φ = n=0 an ψn , and for every neighborhood U of φ , take

φ(N ) =

N X

an ψn + (aN +1 + 1)ψN +1 +

n=0

∞ X

an ψn ,

N +2

then φ(N ) ∈ F , since N X

|an |2 (n + 1)r + |aN +1 + 1|2 (N + 2)r +

n =0

≤

∞ X

|an |2 (n + 1)r

N +2

∞ X

|an |2 (n + 1)r + (|aN +1 |2 + 2)(N + 2)r < ∞,

∀r ≥ 0 .

n=0

When N is big enough, φ(N ) ∈ U, while

ρ(BN +1 φ(N ) , BN +1 φ) =

∞ X

2−m pm (BN +1 (φ(N ) − φ))(1 + pm (BN +1 (φ(N ) − φ)))−1

m=0

=

∞ X

2−m pm (BN +1 ψN +1 )(1 + pm (BN +1 ψN +1 ))−1

m=0

=

∞ X

√

2

m=0

=

pm ( (N + 1)!ψ0 )

−m

√

1 + pm ( (N + 1)!ψ0 )

√ ∞ X (N + 1)! 1 = 1 > δ. ≥ √ m m+1 2 2 1 + ( N + 1 )! m=0 m=0 ∞ X 1

So B has sensitive dependence on initial conditions. This proved that B is Devaney chaotic.



5. Some remarks and open problems In this section we give some remarks and present some open problems on scrambled sets of shift and weighted shift maps. We remark that some results in this paper, such as Lemmas 3.1 and 3.2, Corollary 3.3, and Proposition 3.4, are also valid for a general continuous map f : X → X on a compact metric space X [22]. In Theorem 3.5 and Corollary 3.6 we give the sufficient conditions for S to be a scrambled or Li–Yorke chaotic set of σ . We may further ask that if it is possible to formulate necessary and sufficient conditions for S to be a scrambled or Li–Yorke chaotic set of σ ? A possible route to find the necessary and sufficient conditions is to re-define the function η(·, ·) so that it can better characterize symbolic sequences in S. To study a chaotic system, it is essential to confirm that there exists a scrambled set with a ‘‘big’’ enough size in a certain scale. In a measure space, measure is a good scale for the size of a set. We can say the size of a scrambled set is big if it has positive (or full) measure. For the one-sided (p0 , p1 , . . . , pN −1 )-shift σ : (Σ (N ), B , m) → (Σ (N ), B , m) with pi > 0, i = 0, 1, . . . , N − 1, we conjecture that every maximal scrambled set of σ possesses positive m-measure. (We remark here that the one-dimensional Hausdorff measure H 1 (S ) of a maximal scrambled set S ⊆ Σ (N ) is zero, if S is H 1 -measurable [39].) It can be shown that a totally chaotic subshift must be a subshift of infinite type [22]. The question is open whether it is possible to give an example of a totally scrambled subshift σ on Σ ⊆ Σ (N ), i.e., with the whole space Σ being a scrambled set of σ |Σ . Topological entropy and scrambled sets are closely related [40,34,41,42]. For subshifts of finite type, we know that their topological entropies can be simply characterized by scrambled sets. So we guess under certain conditions that for subshifts of infinite type their topological entropies can also be similarly characterized via their maximal scrambled sets. Though there are some weakly mixing symbolic systems with entropy 0, since they are weakly mixing, by the main result of [43] they have scrambled Mycielski sets (i.e., countable unions of Cantor sets). Examples available in the literature are the Chacon system and related subshifts, and the Weiss example [44], but there are many others. On the other hand, it is possible that positive topological entropy cannot go along with really big scrambled sets. A totally scrambled system always has zero entropy [45].

X. Fu, Y. You / Nonlinear Analysis 71 (2009) 2141–2152

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The weighted shift map B discussed in Section 4.2 has complex periodic motion. We discuss this below. P∞ √ fn , n ≥ 0. The fixed points of B: From Bψ = ψ , we have ψ = n=0 fn ψn , where f0 is arbitrary, and fn+1 = The k-periodic points of B: From Bk ψ = ψ , we have ψ = g gn+k = √(n+1)(n+n2)···(n+k) , n ≥ 0. Density of the periodic points P (B) of B: for any φ ∈ F , φ =

  h n = an ,

hn

,  hn +N = √ (n + 1)(n + 2) · · · (n + N )

P∞

n =0

P∞

n=0

n+1

gn ψn , where g0 , g1 , . . . , gk−1 are arbitrary, and

an ψn , for N > 0, take φ (N ) =

P∞

n=0

hn ψn , such that

n = 0, 1, . . . , N − 1, n ≥ 0.

Then φ (N ) is an N-periodic point of B, and φ (N ) can enter an arbitrary neighborhood of φ when N is big enough. So P (B) is dense in F . From the remarks above, B is also Devaney chaotic under the three conditions. Duan, Fu and Lawson [16] discussed the chaotic trajectory in the quantum state space of an arbitrary solution to the Schrödinger equation for Hydrogen atom under the repeated application of ladder operators, which shift the eigenfunctions un of the Schrödinger equation into un−1 and un+1 . The properties of this weighted bilateral shift map are in some sense quite close to that of weighted shift maps. In [15] we have shown that every point in the constructed chaotic set S in the space F is nonwandering. P Actually, we can ∞ further show that every point in the whole space F is nonwandering, i.e., Ω (B) = F . In fact, for any φ = n=0 cn ψn ∈ F , take

φN =

N X

cn ψn +

n=0

∞ X

1

cn−(N +1) (n(n − 1) · · · (n − N ))− 2 ψn ,

n =N +1

then pr (φN − φ) → 0 as N → ∞, and BN +1 φN = φ . There are various open questions and challenging issues about the Li–Yorke chaotic sets and the Li–Yorke chaos, such as possible measurements of their sizes (see [46] for recent progresses), characterization of a Li–Yorke chaotic sets by a weighted shift maps, and other properties. It would be interesting to further consider some general weighted shifts with the form b = λn σ and their dynamical properties. Acknowledgements This research was supported jointly by NSFC grant 10672146 and Shanghai Leading Academic Discipline Project, Project Number: J50101. XCF was also grateful to University of South Florida for support via a visiting professorship. References [1] T.-Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975) 985–992. [2] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, 1989. [3] X.C. Fu, The general symbolic dynamics and its applications II, in: Nonlinear Oscillations, Bifurcations and Chaos, Tianjin Univ. Press, Tianjin, China, 1992 (in Chinese). [4] X.C. Fu, The general symbolic dynamics and its applications I, in: The National Conference on Nonlinear Dynamics, Univ. of Science and Technology of China, 1990. [5] X.C. Fu, Linearity, nonlinearity and chaoticity, in: The National Conference on Nonlinear Science, Shanghai, July, 1991 (Invited talk); also see Research on New Subjects, Beijing, China Science and Technology Press, 1993. [6] G. Godefroy, J.H. Shapiro, Operators with dense invariant cyclic vector manifolds, J. Funct. Anal. 98 (1991) 229–269. [7] V. Protopopescu, Linear vs. nonlinear infinite vs. finite: An interpretation of chaos, Oak Ridge National Laboratory Report, TM-11667, Oak Ridge, Tennessee, USA, 1990. [8] C.R. MacCluer, Chaos in linear distributed systems, J. Dyn. Syst. Meas. Control 114 (1992) 322–324. [9] G.D. Birkhoff, Démonstration d’un théoreme elementaire sur les fonctions entiéres, C. R. Acad. Sci. Paris 189 (1929) 473–475. [10] K.C. Chan, J.H. Shapiro, The cyclic behavior of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J. 40 (1991) 1421–1449. [11] V. Protopopescu, Y.Y. Azmy, Topological chaos for a class of linear models, Math. Models Methods Appl. Sci. 2 (1992) 79–90. [12] A. Gulisashvili, C.R. MacCluer, Linear chaos in the unforced quantum harmonic oscillator, J. Dyn. Syst. Meas. Control 118 (1996) 337–338. [13] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 1990. [14] X.C. Fu, J. Duan, Infinite-dimensional linear dynamical systems with chaoticity, J. Nonlinear Sci. 9 (1999) 197–211. [15] J. Duan, X.C. Fu, P.D. Liu, A.K. Manning, On a linear chaotic quantum harmonic oscillator, Appl. Math. Lett. 12 (1999) 15–19. [16] J. Duan, X.C. Fu, J. Lawson, Chaotic operators from quantum mechanics, Manuscript. [17] J. Smítal, A chaotic function with some extremal properties, Proc. Amer. Math. Soc. 87 (1983) 54–56. [18] R. Wolfe, H.C. Morris, Chaotic solutions of systems of first order partial differential equations, SIAM J. Math. Anal. 18 (1987) 1040–1063. [19] X.C. Fu, Criteria for a general continuous self-map to have chaotic sets, Nonlinear Anal. TMA 26 (1996) 329–334. [20] J. Mai, Continuous maps with the whole space being a scrambled set, Chinese Sci. Bull. 42 (1997) 1603–1606. [21] F. Blanchard, E. Glasner, S. Kolyada, A. Maass, On Li–Yorke pairs, Z. Reine Angew. Math. 547 (2002) 51–68. [22] X.C. Fu, Z.R. Liu, H.J. Gao, Chaotic sets of continuous and discontinuous maps, 2008 (submitted for publication). [23] X.C. Fu, H.W. Chou, K.H. Xu, Bifurcations, Chaos, and Symbolic Dynamics, Wuhan Univ. Press, Wuhan, China, 1993 (in Chinese). [24] X.C. Fu, H.W. Chou, Chaotic behaviour of the general symbolic dynamics, Appl. Math. Mech. 13 (1992) 117–123. [25] X.C. Fu, W. Lu, P. Ashwin, J. Duan, Symbolic representations of iterated maps, Topol. Methods Nonlinear Anal. 18 (2001) 119–148. [26] U. Kirchgraber, D. Stoffer, On the definition of chaos, Z. Angew. Math. Mech. 69 (1989) 175–185. [27] Z.L. Zhou, Chaos and topological entropy, Acta Math. Sinica 31 (1988) 83–87.

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