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A-coupled-expanding and distributional chaos
Chaos, Solitons and Fractals 77 (2015) 291–295
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
A-coupled-expanding and distributional chaosR Cholsan Kim a,b, Hyonhui Ju b, Minghao Chen a,∗, Peter Raith c a b c
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria
a r t i c l e
i n f o
Article history: Received 10 February 2015 Accepted 13 June 2015
Keywords: Chaos Coupled-expanding map Distributional chaos
a b s t r a c t The concept of A-coupled-expanding maps is one of the more natural and useful ideas generalized from the horseshoe map which is commonly known as a criterion of chaos. It is well known that distributional chaos is one of the concepts which reflect strong chaotic behavior. In this paper, we focus on the relationship between A-coupled-expanding and distributional chaos. We prove two theorems which give sufficient conditions for a strictly A-coupledexpanding map to be distributionally chaotic in the senses of two kinds, where A is an m × m irreducible transition matrix.
1. Introduction The concept of A-coupled-expanding maps which has been recognized as one of the criteria of chaos defined in [16]. But it should be noted that this notion goes back to the notion of turbulence introduced by Block and Coppel in [2]. A continuous map f: I → I, where I ⊂ R is the unit interval, is said to be turbulent if there exist two closed nondegenerate subintervals (i.e. subintervals which are not singletons) J and K with pairwise disjoint interiors such that f(J) ⊃ J ∪ K and f(K) ⊃ J ∪ K. Furthermore, it is said to be strictly turbulent if the subintervals J and K can be chosen disjoint. Actually, the same concept was studied in one-dimensional dynamical systems earlier by Misiurewicz in [6] and [7]. This property was called “horseshoe”, because it is similar to the Smale’s horseshoe effect. Let f: I → I be an interval map (i.e. f is continuous) and J1 , . . ., Jn be nondegenerate subintervals with pairwise disjoint interiors such that J1 ∪ . . . ∪ Jn ⊂ f ( Ji )
R This work was supported by the National Natural Science Foundation of China (Grant Nos.11271099 and 11471088) ∗ Corresponding author: Tel: +86 451 8641 2607. E-mail addresses: [email protected] (C. Kim), [email protected], [email protected] (M. Chen).
http://dx.doi.org/10.1016/j.chaos.2015.06.010 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
© 2015 Elsevier Ltd. All rights reserved.
for i ∈ {1, . . ., n}. Then ( J1 , . . ., Jn ) is called an n-horseshoe, or simply a horseshoe if n ≥ 2. This concept was generalized to general metric spaces. The term “turbulence” was changed to “coupled-expansion”, because it was well-established in fluid mechanics (see [15,18]). There were some results on chaos of coupledexpanding maps (see [2,14,15,17,18,24]). Later, the notion was extended to coupled-expanding maps for a transitive matrix A (simply called A-coupled-expanding map) in [16], which is the same as the concept of coupled-expanding maps if each entry of the matrix A equals to 1. Recently, by applying symbolic dynamical system theory, many important results about criteria of chaos using the Acoupled-expanding map have been established. For instance, it has been verified that under certain conditions, a strictly A-coupled-expanding map is chaotic in the sense of Li-Yorke or Devaney or Wiggins (see [4,14–16,22]). It was also proved that some A-coupled-expanding maps have positive topological entropy in [11,23]. On the other hand, since the concept of distributional chaos has been introduced in [13], a large number of papers have been devoted to the study on distributional chaos, including researches on the relations between the distributional chaos and many other definitions of chaos such as Li-Yorke chaos, Devaney chaos and so on ([1,3,8–10,19–21]).
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In [4], it was shown that if an A-coupled-expanding map f satisfies some expanding conditions, then there exists an finvariant set such that f | is conjugate to a subshift of finite type. Consequently, the map f is distributionally chaotic of type 1 (Remark 3.3 of [4]). As mentioned above, the concept of A-coupled-expanding maps is one of the more natural and useful ideas generalized from the horseshoe map, which is a well known criterion of chaos recently, but except [4], there is no result on the relationship between Acoupled-expanding and distributional chaos yet. Moreover, the concept of distributional chaos is recognized as one of the concepts reflecting strong chaotic behavior. This implies that it is natural to study the relationship between A-coupledexpanding and distributional chaos, or complement conditions for one to be another. The rest of this paper is organized as follows. In Section 2, some basic concepts and notations used in this paper are introduced. In Section 3, a sufficient condition for a strictly A-coupled-expanding map on a compact metric space to be distributionally chaotic in a sequence is established (Theorem 3.1) and an example is presented to illustrate that our results are remarkable. Finally it is proved that the map is distributionally chaotic of type 1 under stronger conditions (Theorem 3.2). 2. Preliminaries Let N be the set of all positive integers and N0 = N ∪ {0}. For m ≥ 2, let A = (ai j ) be an m × m matrix. The definitions of transition and irreducible matrices follow [22]. Set A = {1, . . . , m} and the element of A is called alphabet. It is well known that the set m = {(a0 a1 . . . ) | ai ∈ A, i ∈ N0 } is a compact metric space with the metric
0 2−(k+1)
ρ(α , β) =
if α = β , if α = β and k = min {i | ai = bi },
where α = (a0 a1 . . . ), β = (b0 b1 . . . ) ∈ m . For α = (a0 a1 . . . ai . . . ), the subscript i ∈ N0 of ai is called the index of ai in the α . Define the shift σ : m → m as σ (α) = (a1 a2 . . . ), where α = (a0 a1 a2 . . . ) ∈ m . It is well known that for an m × m transition matrix A = (ai j ), A = {(b0 b1 . . . ) ∈ m | abi bi+1 = 1, i ∈ N0 } is a compact subset of m , which is σ −invariant (i.e. σ ( A ) ⊂ A ). The map σA = σ |A is called the subshift of finite type for matrix A (for more details, see [12]). For n ∈ N, let bi ∈ A for i ∈ {1, . . . , n}, then the word (b1 b2 . . . bn ) is called admissible for the matrix A if abi bi+1 = 1 for all i ∈ {1, . . . , n − 1}. Use |(b1 b2 . . . bn )| = n to denote the length of the admissible word (b1 b2 . . . bn ). For s, t ∈ N, let u = (u0 u1 . . . us ), v = (v0 v1 . . . vt ), where ui , vi ∈ A. Let uv = (u0 u1 . . . us v0 v1 . . . vt ) and call it the combination of u and v. For n ∈ N, denote un = uu . . . u (n times repeated). The following definitions can be found in [5,10,16].
for all 1 ≤ i ≤ m, then the map f is said to be A-coupledexpanding in i (1 ≤ i ≤ m). Moreover, the map f is said to be strictly A-coupled-expanding in i (1 ≤ i ≤ m) if d(i , j ) > 0 for all 1 ≤ i = j ≤ m, where d(i , j ) denotes the distance between two sets i and j . Definition 2.2 ([10]). Let (X, d) be a compact metric space and f: X → X be a continuous map. For x, y ∈ X and n ∈ N, set
Fxy(n) (t ) =
1 #{i : d( f i (x), f i (y)) < t, 0 ≤ i < n} , n
where #A denotes the cardinal number of the set A. Furthermore set
Fxy (t ) = lim inf Fxy(n) (t ), n→∞
Fxy∗
(t ) = lim sup Fxy(n) (t ) . n→∞
∗ (t ) are called lower or upper disThe functions Fxy (t) and Fxy tribution functions of x and y respectively. We consider the following three conditions: ∗ (t ) = 1 for all t > 0 and Fxy (s) = 0 for some D(1) Fxy s > 0, ∗ (t ) = 1 for all t > 0 and Fxy (s) < 1 for some D(2) Fxy s > 0, D(3) Fxy (s) < 1 for some s > 0. If a pair of points (x, y) satisfies the condition D(k) (k = 1, 2, 3), then the pair (x, y) is called distributionally chaotic of type k. A subset of X containing at least two points is called a distributionally scrambled set of type k if any pair of its distinct points is distributionally chaotic of type k. Finally, f is said to be distributionally chaotic of type k if there exists an uncountable distributionally scrambled set of type k.
Definition 2.3 ([5]). Let (X, d) be a compact metric space and f: X → X be a continuous map. Suppose that (pi ) is an increasing sequence in N, x, y ∈ X and t > 0. Set
#{i : d( f pk (x), f pk (y)) < t, 1 ≤ k ≤ n} , n #{i : d( f pk (x), f pk (y)) < t, 1 ≤ k ≤ n} Fxy∗ (t, ( pi )) = lim sup . n n→∞
Fxy (t, ( pi )) = lim inf n→∞
A subset D ⊂ X is called distributionally chaotic in a sequence (or in the sequence (pi )) if for any x, y ∈ D with x = y, (1) there is an ε > 0 such that Fxy (ε , ( pk )) = 0 and ∗ (t, ( p )) = 1 for every t > 0. (2) Fxy k If there exists an uncountable distributionally chaotic set in a sequence, the map f is said to be distributionally chaotic in a sequence. Obviously, distributional chaos of type 1 implies distributional chaos in a sequence. 3. Main results
Definition 2.1 ([16]). Let (X, d) be a metric space and f: D ⊂ X → X. Suppose that A = (ai j ) is an m × m transition matrix for some m ≥ 2. If there exist m nonempty subsets i (1 ≤ i ≤ m) of D with pairwise disjoint interiors such that
Assume that (X, d) is a compact metric space, m ≥ 2 and the sets V1 , V2 , . . . , Vm are compact subsets of X with pairwise disjoint interiors. Let f : m i=1 Vi → X be a continuous map, A = (ai j ) be an irreducible m × m transition matrix which satisfies the assumption that
f (i ) ⊃
there exists an i0 such that
j
ai j =1
j
m j=1
ai0 j ≥ 2.
(∗)
C. Kim et al. / Chaos, Solitons and Fractals 77 (2015) 291–295
Suppose that f is strictly A-coupled-expanding in Vi (1 ≤ i ≤ m). Let c = (c0 c1 . . . cn ) be an admissible word for the matrix A,
Vc = Vc0 c1 ...cn =
n
f −i (Vci ).
i=0
Then Vc is a nonempty compact subset. Moreover
Vc0 c1 ...cn−1 ⊃ Vc0 c1 ...cn−1 cn and
f (Vc0 c1 ...cn ) = Vc1 ...cn . If (c0 c1 . . . cn ) and (d0 d1 . . . dn ) are two different admissible words for the matrix A, then
Vc0 c1 ...cn ∩ Vd0 d1 ...dn = ∅. For β = (b0 b1 . . . ) ∈ A , set
Vβ =
∞
f −n (Vbn ),
n=0
where f0 is the identity map. Then
Vβ =
∞
f (Vβ ) = VσA (β) . If α , β ∈ A are different, then
Vα ∩ Vβ = ∅ (see [23]). Theorem 3.1. Let (X, d) be a compact metric space. Suppose that f: X → X is a continuous map, A is an irreducible m × m transition matrix (m ≥ 2) and satisfies the assumption(∗). Let f be strictly A-coupled-expanding in compact sets Vi (1 ≤ i ≤ m). If there exists an α = (a0 a1 . . . ) ∈ A such that Vα = ∞ −n (V ) is a singleton, then there exists a sequence (p ) an k n=0 f such that f is distributionally chaotic in the sequence (pk ). Proof. From the assumptions of this theorem and the facts presented earlier in this section, we get that for any n ∈ N0 ,
(Vα ) = VσAn (α)
(1)
and fn (Vα ) is also a singleton. There exists at least one alphabet appearing infinitely in α = (a0 a1 . . . ). Without loss of generality we may assume that a0 appears infinitely in α . It means that there exists a strictly increasing sequence (ν k ) in N such that aνk = a0 for all k ∈ N. Now for k ∈ N, set uk = (a0 a1 . . . aνk −1 ). Obviously, for any n ∈ N, unk is admissible for the matrix A. By the definition of irreducible matrix, the assumption (∗) and Lemma 2.4 in [22], there is an a ∈ A such that aa a0 = 1. Moreover, there are two different admissible words for the matrix A, v1 = (a0 . . . a ) and v2 = (a0 . . . a ), such that |v1 | = |v2 |. Obviously, any combination of uk , v1 and v2 is admissible for matrix A. Let ϕ be a map 2 → A given by
ϕ(c) = vsc10 us12 vsc31 vsc42 us15 us26 vsc73 vsc84 vsc95 us110 us211 us312 vsc136 . . .
s1 = 2 0 , s2 = 21 |vsc10 |, s3 = 22 |vsc10 us12 |, s4 = 23 |vsc10 us12 vsc31 |, s5 = 24 |vsc10 us12 vsc31 vsc42 |, s6 = 25 |vsc10 us12 vsc31 vsc42 us15 |, ......... The map ϕ is well defined and obviously bijective. By the definition of the map ϕ , any two elements in the ϕ ( 2 ) coincide in the parts where combinations of ui appear. For any two elements of ϕ ( 2 ), the first alphabets a0 of admissible words ui or vj appear in the same places. Now, for any α ∈ ϕ ( 2 ), rearranging the indexes of the first alphabets a0 of ui or vj in the α , we make a strictly increasing sequence ( pk )∞ in N0 . As ( 2 , σ ) is chaotic in the sense of Li-Yorke, k=1 there is an uncountable scrambled set S in 2 . Put D0 = ϕ(S), then D0 is also uncountable. By the facts presented earlier in this section, it follows that for any αˆ = (aˆ0 aˆ1 . . . ) ∈ D0 , Vαˆ is nonempty and αˆ = βˆ ∈ D0 implies Vαˆ ∩ Vβˆ = ∅. For each
Now we prove that the set G is uncountable distri. It is obvibutionally scrambled in the sequence ( pk )∞ k=1 ous that G is uncountable. By assumptions of this theorem and the facts presented earlier in this section, we obtain that (diam(Va0 a1 ...an ))∞ n=0 is a nonincreasing sequence and limn→∞ diam(Va0 a1 ...an ) = 0, where diam(V) denotes diameter of V. That is, for any ε > 0 there exists an n0 ∈ N such that
and Vβ is nonempty and compact. Also, we have
f
for c = (c0 c1 c2 c3 c4 . . . ) ∈ 2 , where si are defined as follows:
αˆ ∈ D0 , we choose only one element from Vαˆ and denote it by xαˆ . Set G = {xαˆ | αˆ ∈ D0 }.
Vb0 b1 ...bn
n=0
n
293
(2)
diam(Va0 a1 ...an ) < ε
(3)
for all n ∈ N with n ≥ n0 . For any αˆ = (aˆ0 aˆ1 . . . ) ∈ D0 we can choose a subsequence ( pk j ) of (pk ) such that ( pk j )-th alphas
bet is the first alphabet (that is, a0 ) of a combination ui k . By (2), there is a pk j such that
(aˆ pk j aˆ pk j +1 aˆ pk j +2 . . . aˆ pk j +1 −1 ) = (a0 a1 . . . a pk j +1 −pk j −1 ) = u pk +1 −pk j
j
and
|u pk j +1 −pk j | = pk j +1 − pk j > n0 .
(4)
From (2) there is a subsequence (sk ) of (si ) such that αˆ conj
sk
tains combination of admissible words, u p j
k j +1 −pk j
. Let j ∈ N.
By (3) and (4), for any i with k j ≤ i ≤ k j + sk − 1, it follows that f pi (xαˆ ), f pi (xβˆ ) ∈ Vu p
diam(Vu p
k j +1 −pk j
j
, for any xαˆ , xβˆ ∈ G. Since ) < ε, we have d( f pi (xαˆ ), f pi (xβˆ )) < ε. Thus, k j +1 −pk j
from the definition of si , we can obtain
#{i : d( f pi (xαˆ ), f pi (xβˆ )) < ε , 1 ≤ i ≤ k j + skj − 1} k j + skj − 1 ≥
skj k j + skj − 1
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C. Kim et al. / Chaos, Solitons and Fractals 77 (2015) 291–295
skj
≥
butionally scrambled set in a sequence. However, f does not satisfy the ε − δ condition of Theorem 1 in [4], that is, let B = {2.5}, then diam(B) = 0 and diam( f −1 (B) ∩ V1 ) = 13 .
pk j + skj − 1 skj
=
2
−kj +1
skj + skj − 1
→ 1 ( j → ∞).
By the above equation and the definition of superior limit of sequences, we have
lim sup
#{i : d( f pi (xαˆ ), f pi (xβˆ )) < ε , 1 ≤ i ≤ n} n
n→∞
= 1.
(5)
Next set d0 = d(Vv1 , Vv2 ) > 0. Since any two different elements of the set S ⊂ 2 is Li-Yorke pair, the set {i: ci = di } is infinite for (c0 c1 . . . ) = (d0 d1 . . . ) ∈ S. Without loss of generality, we may assume that for αˆ = βˆ ∈ D0 , there exists a subof the sequence (pk ) and a subsequence sequence ( pq j )∞ j=1
(sq )∞ j j=1
of the sequence (si ) such that sq
σ pq j (α) ˆ = (v1 j . . . )
sq
ˆ = (v j . . . ) σ pq j (β) 2
and
hold. For j ∈ N, i with q j ≤ i ≤ q j + sq − 1, it folj
lows that f pi (xαˆ ) ∈ Vv1 and f pi (xβˆ ) ∈ Vv2 . It means that
d( f pi (xαˆ ), f pi (xβˆ )) ≥ d0 . Thus
#{ i : d ( f
pi
qj
q j + sqj − 1 qj − 1 q j + sqj − 1
≤
2−q j +1 sqj − 1
≤
2−q j +1 sqj + sqj − 1
→ 0 ( j → ∞),
#{i : d( f pi (xαˆ ), f pi (xβˆ )) < d0 , 1 ≤ i ≤ n} n
n→∞
= 0.
(6)
From (5) and (6), the set G is uncountable distributionally scrambled in the sequence (pk ), therefore, we can conclude that the map f is distributionally chaotic in the sequence (pk ). Example 3.1. Define f: [0, 3] → [0, 3] as follows:
f (x) =
⎧ ⎪ ⎪1.5x + 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2.5,
⎪ 1.5x + 1.5, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3, ⎪ ⎪ ⎩ −3x + 9,
Set
A=
0 1
1 , 3 1 2 , , if x ∈ 3 3 2 if x ∈ ,1 , 3 if x ∈ (1, 2], if x ∈
A
n ∈ N0 . Furthermore, for any n ∈ {0, 1, . . . , T − 1}, the sequence (diam(Van ...an+i ))∞ is nonincreasing and satisi=0 fies limi→∞ diam(Van ...an+i ) = 0. Set dk = max{diam(Van ...an+k ) | 0 ≤ n ≤ T − 1} for k ∈ N, then
lim dk = 0.
k→∞
(7)
(aq+1 aq+2 . . . aT ). Define a map ψ : {1, 2} → {u, v} as ψ(1) = u, ψ(2) = v. Construct a map ϕ : 2 → A as follows:
hence
lim inf
Proof. Denote the period of α by T, that is, σAT (α) = α , then #{σAn (α) | n ∈ N0 } = T . From the assumptions of this the orem, f n (Vα ) = Vσ n (α) = ∞ i=0 Van an+1 ...an+i is a singleton for
By the definition of irreducible matrix and the assumption (∗), there exists an a ∈ A such that aa a0 = 1. There are two different admissible words for the matrix A, u = (a0 . . . a ), v = (a0 . . . a ), such that |u| = |v| = l. Since the matrix A is irreducible, there exist i, j ∈ A such that aa0 i = a ja0 = 1 and an admissible word for the matrix A, C = (i . . . j). For p ∈ N, set B p = (a0 a1 . . . a p−1 ) and define B¯ p as follows. If a p−1 = a0 we assume that B¯ p vanishes. Otherwise, there is a q ∈ {1, 2, . . . , T − 1} such that a p−1 = aq . In this case set B¯ p =
(xαˆ ), f (xβˆ )) < d0 , 1 ≤ i ≤ q j + s − 1} pi
Theorem 3.2. Let (X, d) be a compact metric space. Suppose that f: X → X is a continuous map and A is an irreducible m × m transition matrix (m ≥ 2) and satisfies the assumption(∗). Let f be strictly A-coupled-expanding in the compact sets Vi (1 ≤ i ≤ m). If there exists a periodic point α = (a0 a1 . . . ) ∈ A such −n (V ) is a singleton, then f is distributionally that Vα = ∞ an n=0 f chaotic of type 1.
0,
if x ∈ (2, 3].
1 , 1
V1 = [0, 1] and V2 = [2, 3]. This matrix A is an irreducible transition matrix and satisfies the assumption (∗) and f is strictly A-coupled-expanding in V1 and V2 . Furthermore, the map f satisfies the remaining conditions of Theorem 3.1. Therefore, there exists an uncountable distri-
ϕ(a0 a1 a2 . . . ) = a0C ψ(a0 )m1 B p2 B¯ p2 C ψ(a1 )m3 ψ(a2 )m4 B p5 B¯ p5 C ψ(a3 )m6 ψ(a4 )m7 ψ(a5 )m8 B p9 B¯ p9 C ψ(a6 )m10 ψ(a7 )m11 ψ(a8 )m12 ψ(a9 )m13 . . . , (8) where mi is 2i th power of the length of the subword to the left of ψ(a j )mi in (8) and pi is 2i th power of the length of the subword to the left of B pi in (8). Obviously, any subword in the sequence defined in (8) is admissible for the matrix A, that is, the map ϕ is well defined. By the construction of (8), the map ϕ is bijective. Since the shift ( 2 , σ ) is chaotic in sense of Li-Yorke, there exists an uncountable scrambled set D in 2 . Let D0 = ϕ(D). For any αˆ ∈ D0 , we choose only one element from Vαˆ and denote it by xαˆ . Denote the set consist ing of these elements by Dx (⊂ m i=1 Vi ). Now we prove that this set Dx is uncountable distributionally scrambled of type 1. It is obvious that Dx is uncountable. The equation (7) implies that for every t > 0 there is a k ∈ N with dk < t. Let xαˆ , xβˆ ∈ Dx be different, then αˆ = βˆ ∈ D0 . Obviously, for any i, the end alphabet of the combination B pi B¯ pi is a0 . Denote the index of the a0 in αˆ as ni . By definition of ϕ , ni is also the index of the end a0 of B pi B¯ pi in βˆ . Note that |B¯ pi | ≤ T, then it is obvious that ni ≤ pi (1 + 2−i ) + T . The choice of pi guarantees that |B pi | < |B p j | for i < j, and there exists an i0 such that |B pi | > k for i ≥ i0 .
C. Kim et al. / Chaos, Solitons and Fractals 77 (2015) 291–295
Therefore, for i ≥ i0 ,
assumptions is topologically conjugate to a subshift of finite type in a compact f-invariant subset. Moreover, in the Remark 3.3 of [4], they said that if assumptions of Theorem 2 are satisfied, then f is distributionally chaotic. However, in Example 3.1 f does not satisfy the assumptions of Theorem 2 in [4].
#{s : d( f (xαˆ ), f (xβˆ )) < t, 0 ≤ s < ni } ni #{s : d( f s (xαˆ ), f s (xβˆ )) < dk , 0 ≤ s < ni } ≥ ni #{s : (as . . . as+k ) lies in B pi } ≥ ni pi − k = ni pi − k ≥ → 1 (i → ∞), pi (1 + 2−i ) + T s
295
s
Acknowledgments The authors are grateful to the Editor-in-Chief, Prof. S. Boccaletti, and the referees for their helpful comments and valuable suggestions. References
hence
lim sup
#{s : d( f s (xαˆ ), f s (xβˆ )) < t, 0 ≤ s < n}
n→∞
n
= 1.
(9)
Denote the set of all admissible words for the matrix A with the length l as W. Choose d0 such that
0 < d0 < min{d(Vc , Vd ) | c = d ∈ W }. Assume that αˆ = (aˆ0 aˆ1 . . . ) = βˆ = (bˆ 0 bˆ 1 . . . ) ∈ D0 . From the definition of ϕ and the construction of D0 we may assume without loss of generality that there is an increasing semr mr quence (rj ) in N such that combinations u j and v j appear ˆ in the same place of αˆ and β , respectively. Denote ν j as the original index of a in the sequence αˆ (or βˆ ) which is the last mr mr a in u j (or v j ). Since ν j = mr j l + 2−r j mr j we obtain
#{s : d( f s (xαˆ ), f s (xβˆ )) < d0 , 0 ≤ s < ν j }
νj ν j − #{s : (es . . . el+s−1 ) ∈ umr j (or vmr j ), s ≥ 0} ≤ νj ν j − (mr j l − l + 1) = νj =
2−r j mr j + l − 1 mr j l + 2−r j mr j
→ 0 ( j → ∞).
From the definition of inferior limit of sequences, we have
lim inf n→∞
#{s : d( f s (xαˆ ), f s (xβˆ )) < d0 , 0 ≤ s < n} n
= 0.
(10)
From (9) and (10) we can see that any pair of two different elements in Dx is distributionally chaotic of type 1. Therefore the map f is distributionally chaotic of type 1. Remark 3.1. Consider the map f of Example 3.1. For the periodic point α = (121212 . . . ), Vα is a singleton. Thus the map f is also distributionally chaotic of type 1. Remark 3.2. In Theorem 2 of [4], it was proved that a strictly A-coupled-expanding map f satisfying two additional
[1] Balibrea F, Smítal J, Štefánková M. The three versions of distributional chaos. Chaos Solitons Fractals 2005;23:1581–3. [2] Block L, Coppel W. Dynamics in one dimension, lecture notes in mathematics. Berlin: Springer; 1992. [3] Forti GL. Various notions of chaos for discrete dynamical systems, a brief survey. Aequationes Math 2005;70:1–13. [4] Kulczycki M, Oprocha P. Coupled-expanding maps and matrix shifts. Internat J Bifur Chaos 2013;23:1–6. [5] Liu H, Wang L, Chu Z. Devaney’s chaos implies distributional chaos in a sequence. Nonlinear Anal 2009;71:6144–7. [6] Misiurewicz M. Horseshoes for mappings of the interval. Bull Acad Polon Sci Sér Sci Math 1979;27:167–9. [7] Misiurewicz M. Horseshoes for continuous mappings of an interval. In: Dynamical systems (Bressanone, 1978), Liguori, Naples; 1980. [8] Oprocha P. Relations between distributional and Devaney chaos. Chaos 2006;16:1–5. [9] Oprocha P. Distributional chaos via semiconjugacy. Nonlinearity 2007;20:2661–79. [10] Oprocha P. Distributional chaos revisited. Trans Amer Math Soc 2009;361:4901–25. [11] Ri C., Ju H. Entropy for A-coupled expanding map and chaos. Preprint. [12] Robinson C. Dynamical system - stability, symbolic dynamics and chaos. Boca Raton: CRC Press; 1999. [13] Schweizer B, Smítal J. Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans Amer Math Soc 1994;344:737–54. [14] Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos Solitons Fractals 2004;22:555–71. [15] Shi Y., Chen G. Some new criteria of chaos induced by coupledexpanding maps. In: Proc. 1st IFAC Conf. Analysis and Control of Chaotic systems, Reims, France, pp. 157–162. [16] Shi Y, Ju H, Chen G. Coupled-expanding maps and one-sided symbolic dynamical systems. Chaos Solitons Fractals 2009;39:2138–49. [17] Shi Y, Yu P. Study on chaos induced by turbulent maps in noncompact sets. Chaos Solitons Fractals 2006;28:1165–80. [18] Shi Y, Yu P. Chaos induced by regular snap-back repellers. J Math Anal Appl 2008;337:1480–94. [19] Smítal J. Various notions of chaos, recent results, open problems, Report on the Summer Symposium in Real Analysis XXVI. Real Anal Exchange 2002:81–5 26th Summer Symposium Conference, Suppl. [20] Wang H, Xiang J. Chaos for subshifts of finite type. Acta Math Sinica 2005;21:1407–14. [21] Wang L, Huang G, Huan S. Distributional chaos in a sequence. Nonlinear Anal 2007;67:2131–6. [22] Zhang X, Shi Y. Coupled-expanding maps for irreducible transition matrices. Internat J Bifur Chaos 2010;20:3769–83. [23] Zhang X., Shi Y., Chen G. Some properties of coupled-expanding maps in compact sets. Preprint. [24] Zhang ZS. On the shift invariant sets of self-maps. Acta Math Sinica 1984;27:564–76.
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
A-coupled-expanding and distributional chaosR Cholsan Kim a,b, Hyonhui Ju b, Minghao Chen a,∗, Peter Raith c a b c
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria
a r t i c l e
i n f o
Article history: Received 10 February 2015 Accepted 13 June 2015
Keywords: Chaos Coupled-expanding map Distributional chaos
a b s t r a c t The concept of A-coupled-expanding maps is one of the more natural and useful ideas generalized from the horseshoe map which is commonly known as a criterion of chaos. It is well known that distributional chaos is one of the concepts which reflect strong chaotic behavior. In this paper, we focus on the relationship between A-coupled-expanding and distributional chaos. We prove two theorems which give sufficient conditions for a strictly A-coupledexpanding map to be distributionally chaotic in the senses of two kinds, where A is an m × m irreducible transition matrix.
1. Introduction The concept of A-coupled-expanding maps which has been recognized as one of the criteria of chaos defined in [16]. But it should be noted that this notion goes back to the notion of turbulence introduced by Block and Coppel in [2]. A continuous map f: I → I, where I ⊂ R is the unit interval, is said to be turbulent if there exist two closed nondegenerate subintervals (i.e. subintervals which are not singletons) J and K with pairwise disjoint interiors such that f(J) ⊃ J ∪ K and f(K) ⊃ J ∪ K. Furthermore, it is said to be strictly turbulent if the subintervals J and K can be chosen disjoint. Actually, the same concept was studied in one-dimensional dynamical systems earlier by Misiurewicz in [6] and [7]. This property was called “horseshoe”, because it is similar to the Smale’s horseshoe effect. Let f: I → I be an interval map (i.e. f is continuous) and J1 , . . ., Jn be nondegenerate subintervals with pairwise disjoint interiors such that J1 ∪ . . . ∪ Jn ⊂ f ( Ji )
R This work was supported by the National Natural Science Foundation of China (Grant Nos.11271099 and 11471088) ∗ Corresponding author: Tel: +86 451 8641 2607. E-mail addresses: [email protected] (C. Kim), [email protected], [email protected] (M. Chen).
http://dx.doi.org/10.1016/j.chaos.2015.06.010 0960-0779/© 2015 Elsevier Ltd. All rights reserved.
© 2015 Elsevier Ltd. All rights reserved.
for i ∈ {1, . . ., n}. Then ( J1 , . . ., Jn ) is called an n-horseshoe, or simply a horseshoe if n ≥ 2. This concept was generalized to general metric spaces. The term “turbulence” was changed to “coupled-expansion”, because it was well-established in fluid mechanics (see [15,18]). There were some results on chaos of coupledexpanding maps (see [2,14,15,17,18,24]). Later, the notion was extended to coupled-expanding maps for a transitive matrix A (simply called A-coupled-expanding map) in [16], which is the same as the concept of coupled-expanding maps if each entry of the matrix A equals to 1. Recently, by applying symbolic dynamical system theory, many important results about criteria of chaos using the Acoupled-expanding map have been established. For instance, it has been verified that under certain conditions, a strictly A-coupled-expanding map is chaotic in the sense of Li-Yorke or Devaney or Wiggins (see [4,14–16,22]). It was also proved that some A-coupled-expanding maps have positive topological entropy in [11,23]. On the other hand, since the concept of distributional chaos has been introduced in [13], a large number of papers have been devoted to the study on distributional chaos, including researches on the relations between the distributional chaos and many other definitions of chaos such as Li-Yorke chaos, Devaney chaos and so on ([1,3,8–10,19–21]).
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In [4], it was shown that if an A-coupled-expanding map f satisfies some expanding conditions, then there exists an finvariant set such that f | is conjugate to a subshift of finite type. Consequently, the map f is distributionally chaotic of type 1 (Remark 3.3 of [4]). As mentioned above, the concept of A-coupled-expanding maps is one of the more natural and useful ideas generalized from the horseshoe map, which is a well known criterion of chaos recently, but except [4], there is no result on the relationship between Acoupled-expanding and distributional chaos yet. Moreover, the concept of distributional chaos is recognized as one of the concepts reflecting strong chaotic behavior. This implies that it is natural to study the relationship between A-coupledexpanding and distributional chaos, or complement conditions for one to be another. The rest of this paper is organized as follows. In Section 2, some basic concepts and notations used in this paper are introduced. In Section 3, a sufficient condition for a strictly A-coupled-expanding map on a compact metric space to be distributionally chaotic in a sequence is established (Theorem 3.1) and an example is presented to illustrate that our results are remarkable. Finally it is proved that the map is distributionally chaotic of type 1 under stronger conditions (Theorem 3.2). 2. Preliminaries Let N be the set of all positive integers and N0 = N ∪ {0}. For m ≥ 2, let A = (ai j ) be an m × m matrix. The definitions of transition and irreducible matrices follow [22]. Set A = {1, . . . , m} and the element of A is called alphabet. It is well known that the set m = {(a0 a1 . . . ) | ai ∈ A, i ∈ N0 } is a compact metric space with the metric
0 2−(k+1)
ρ(α , β) =
if α = β , if α = β and k = min {i | ai = bi },
where α = (a0 a1 . . . ), β = (b0 b1 . . . ) ∈ m . For α = (a0 a1 . . . ai . . . ), the subscript i ∈ N0 of ai is called the index of ai in the α . Define the shift σ : m → m as σ (α) = (a1 a2 . . . ), where α = (a0 a1 a2 . . . ) ∈ m . It is well known that for an m × m transition matrix A = (ai j ), A = {(b0 b1 . . . ) ∈ m | abi bi+1 = 1, i ∈ N0 } is a compact subset of m , which is σ −invariant (i.e. σ ( A ) ⊂ A ). The map σA = σ |A is called the subshift of finite type for matrix A (for more details, see [12]). For n ∈ N, let bi ∈ A for i ∈ {1, . . . , n}, then the word (b1 b2 . . . bn ) is called admissible for the matrix A if abi bi+1 = 1 for all i ∈ {1, . . . , n − 1}. Use |(b1 b2 . . . bn )| = n to denote the length of the admissible word (b1 b2 . . . bn ). For s, t ∈ N, let u = (u0 u1 . . . us ), v = (v0 v1 . . . vt ), where ui , vi ∈ A. Let uv = (u0 u1 . . . us v0 v1 . . . vt ) and call it the combination of u and v. For n ∈ N, denote un = uu . . . u (n times repeated). The following definitions can be found in [5,10,16].
for all 1 ≤ i ≤ m, then the map f is said to be A-coupledexpanding in i (1 ≤ i ≤ m). Moreover, the map f is said to be strictly A-coupled-expanding in i (1 ≤ i ≤ m) if d(i , j ) > 0 for all 1 ≤ i = j ≤ m, where d(i , j ) denotes the distance between two sets i and j . Definition 2.2 ([10]). Let (X, d) be a compact metric space and f: X → X be a continuous map. For x, y ∈ X and n ∈ N, set
Fxy(n) (t ) =
1 #{i : d( f i (x), f i (y)) < t, 0 ≤ i < n} , n
where #A denotes the cardinal number of the set A. Furthermore set
Fxy (t ) = lim inf Fxy(n) (t ), n→∞
Fxy∗
(t ) = lim sup Fxy(n) (t ) . n→∞
∗ (t ) are called lower or upper disThe functions Fxy (t) and Fxy tribution functions of x and y respectively. We consider the following three conditions: ∗ (t ) = 1 for all t > 0 and Fxy (s) = 0 for some D(1) Fxy s > 0, ∗ (t ) = 1 for all t > 0 and Fxy (s) < 1 for some D(2) Fxy s > 0, D(3) Fxy (s) < 1 for some s > 0. If a pair of points (x, y) satisfies the condition D(k) (k = 1, 2, 3), then the pair (x, y) is called distributionally chaotic of type k. A subset of X containing at least two points is called a distributionally scrambled set of type k if any pair of its distinct points is distributionally chaotic of type k. Finally, f is said to be distributionally chaotic of type k if there exists an uncountable distributionally scrambled set of type k.
Definition 2.3 ([5]). Let (X, d) be a compact metric space and f: X → X be a continuous map. Suppose that (pi ) is an increasing sequence in N, x, y ∈ X and t > 0. Set
#{i : d( f pk (x), f pk (y)) < t, 1 ≤ k ≤ n} , n #{i : d( f pk (x), f pk (y)) < t, 1 ≤ k ≤ n} Fxy∗ (t, ( pi )) = lim sup . n n→∞
Fxy (t, ( pi )) = lim inf n→∞
A subset D ⊂ X is called distributionally chaotic in a sequence (or in the sequence (pi )) if for any x, y ∈ D with x = y, (1) there is an ε > 0 such that Fxy (ε , ( pk )) = 0 and ∗ (t, ( p )) = 1 for every t > 0. (2) Fxy k If there exists an uncountable distributionally chaotic set in a sequence, the map f is said to be distributionally chaotic in a sequence. Obviously, distributional chaos of type 1 implies distributional chaos in a sequence. 3. Main results
Definition 2.1 ([16]). Let (X, d) be a metric space and f: D ⊂ X → X. Suppose that A = (ai j ) is an m × m transition matrix for some m ≥ 2. If there exist m nonempty subsets i (1 ≤ i ≤ m) of D with pairwise disjoint interiors such that
Assume that (X, d) is a compact metric space, m ≥ 2 and the sets V1 , V2 , . . . , Vm are compact subsets of X with pairwise disjoint interiors. Let f : m i=1 Vi → X be a continuous map, A = (ai j ) be an irreducible m × m transition matrix which satisfies the assumption that
f (i ) ⊃
there exists an i0 such that
j
ai j =1
j
m j=1
ai0 j ≥ 2.
(∗)
C. Kim et al. / Chaos, Solitons and Fractals 77 (2015) 291–295
Suppose that f is strictly A-coupled-expanding in Vi (1 ≤ i ≤ m). Let c = (c0 c1 . . . cn ) be an admissible word for the matrix A,
Vc = Vc0 c1 ...cn =
n
f −i (Vci ).
i=0
Then Vc is a nonempty compact subset. Moreover
Vc0 c1 ...cn−1 ⊃ Vc0 c1 ...cn−1 cn and
f (Vc0 c1 ...cn ) = Vc1 ...cn . If (c0 c1 . . . cn ) and (d0 d1 . . . dn ) are two different admissible words for the matrix A, then
Vc0 c1 ...cn ∩ Vd0 d1 ...dn = ∅. For β = (b0 b1 . . . ) ∈ A , set
Vβ =
∞
f −n (Vbn ),
n=0
where f0 is the identity map. Then
Vβ =
∞
f (Vβ ) = VσA (β) . If α , β ∈ A are different, then
Vα ∩ Vβ = ∅ (see [23]). Theorem 3.1. Let (X, d) be a compact metric space. Suppose that f: X → X is a continuous map, A is an irreducible m × m transition matrix (m ≥ 2) and satisfies the assumption(∗). Let f be strictly A-coupled-expanding in compact sets Vi (1 ≤ i ≤ m). If there exists an α = (a0 a1 . . . ) ∈ A such that Vα = ∞ −n (V ) is a singleton, then there exists a sequence (p ) an k n=0 f such that f is distributionally chaotic in the sequence (pk ). Proof. From the assumptions of this theorem and the facts presented earlier in this section, we get that for any n ∈ N0 ,
(Vα ) = VσAn (α)
(1)
and fn (Vα ) is also a singleton. There exists at least one alphabet appearing infinitely in α = (a0 a1 . . . ). Without loss of generality we may assume that a0 appears infinitely in α . It means that there exists a strictly increasing sequence (ν k ) in N such that aνk = a0 for all k ∈ N. Now for k ∈ N, set uk = (a0 a1 . . . aνk −1 ). Obviously, for any n ∈ N, unk is admissible for the matrix A. By the definition of irreducible matrix, the assumption (∗) and Lemma 2.4 in [22], there is an a ∈ A such that aa a0 = 1. Moreover, there are two different admissible words for the matrix A, v1 = (a0 . . . a ) and v2 = (a0 . . . a ), such that |v1 | = |v2 |. Obviously, any combination of uk , v1 and v2 is admissible for matrix A. Let ϕ be a map 2 → A given by
ϕ(c) = vsc10 us12 vsc31 vsc42 us15 us26 vsc73 vsc84 vsc95 us110 us211 us312 vsc136 . . .
s1 = 2 0 , s2 = 21 |vsc10 |, s3 = 22 |vsc10 us12 |, s4 = 23 |vsc10 us12 vsc31 |, s5 = 24 |vsc10 us12 vsc31 vsc42 |, s6 = 25 |vsc10 us12 vsc31 vsc42 us15 |, ......... The map ϕ is well defined and obviously bijective. By the definition of the map ϕ , any two elements in the ϕ ( 2 ) coincide in the parts where combinations of ui appear. For any two elements of ϕ ( 2 ), the first alphabets a0 of admissible words ui or vj appear in the same places. Now, for any α ∈ ϕ ( 2 ), rearranging the indexes of the first alphabets a0 of ui or vj in the α , we make a strictly increasing sequence ( pk )∞ in N0 . As ( 2 , σ ) is chaotic in the sense of Li-Yorke, k=1 there is an uncountable scrambled set S in 2 . Put D0 = ϕ(S), then D0 is also uncountable. By the facts presented earlier in this section, it follows that for any αˆ = (aˆ0 aˆ1 . . . ) ∈ D0 , Vαˆ is nonempty and αˆ = βˆ ∈ D0 implies Vαˆ ∩ Vβˆ = ∅. For each
Now we prove that the set G is uncountable distri. It is obvibutionally scrambled in the sequence ( pk )∞ k=1 ous that G is uncountable. By assumptions of this theorem and the facts presented earlier in this section, we obtain that (diam(Va0 a1 ...an ))∞ n=0 is a nonincreasing sequence and limn→∞ diam(Va0 a1 ...an ) = 0, where diam(V) denotes diameter of V. That is, for any ε > 0 there exists an n0 ∈ N such that
and Vβ is nonempty and compact. Also, we have
f
for c = (c0 c1 c2 c3 c4 . . . ) ∈ 2 , where si are defined as follows:
αˆ ∈ D0 , we choose only one element from Vαˆ and denote it by xαˆ . Set G = {xαˆ | αˆ ∈ D0 }.
Vb0 b1 ...bn
n=0
n
293
(2)
diam(Va0 a1 ...an ) < ε
(3)
for all n ∈ N with n ≥ n0 . For any αˆ = (aˆ0 aˆ1 . . . ) ∈ D0 we can choose a subsequence ( pk j ) of (pk ) such that ( pk j )-th alphas
bet is the first alphabet (that is, a0 ) of a combination ui k . By (2), there is a pk j such that
(aˆ pk j aˆ pk j +1 aˆ pk j +2 . . . aˆ pk j +1 −1 ) = (a0 a1 . . . a pk j +1 −pk j −1 ) = u pk +1 −pk j
j
and
|u pk j +1 −pk j | = pk j +1 − pk j > n0 .
(4)
From (2) there is a subsequence (sk ) of (si ) such that αˆ conj
sk
tains combination of admissible words, u p j
k j +1 −pk j
. Let j ∈ N.
By (3) and (4), for any i with k j ≤ i ≤ k j + sk − 1, it follows that f pi (xαˆ ), f pi (xβˆ ) ∈ Vu p
diam(Vu p
k j +1 −pk j
j
, for any xαˆ , xβˆ ∈ G. Since ) < ε, we have d( f pi (xαˆ ), f pi (xβˆ )) < ε. Thus, k j +1 −pk j
from the definition of si , we can obtain
#{i : d( f pi (xαˆ ), f pi (xβˆ )) < ε , 1 ≤ i ≤ k j + skj − 1} k j + skj − 1 ≥
skj k j + skj − 1
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C. Kim et al. / Chaos, Solitons and Fractals 77 (2015) 291–295
skj
≥
butionally scrambled set in a sequence. However, f does not satisfy the ε − δ condition of Theorem 1 in [4], that is, let B = {2.5}, then diam(B) = 0 and diam( f −1 (B) ∩ V1 ) = 13 .
pk j + skj − 1 skj
=
2
−kj +1
skj + skj − 1
→ 1 ( j → ∞).
By the above equation and the definition of superior limit of sequences, we have
lim sup
#{i : d( f pi (xαˆ ), f pi (xβˆ )) < ε , 1 ≤ i ≤ n} n
n→∞
= 1.
(5)
Next set d0 = d(Vv1 , Vv2 ) > 0. Since any two different elements of the set S ⊂ 2 is Li-Yorke pair, the set {i: ci = di } is infinite for (c0 c1 . . . ) = (d0 d1 . . . ) ∈ S. Without loss of generality, we may assume that for αˆ = βˆ ∈ D0 , there exists a subof the sequence (pk ) and a subsequence sequence ( pq j )∞ j=1
(sq )∞ j j=1
of the sequence (si ) such that sq
σ pq j (α) ˆ = (v1 j . . . )
sq
ˆ = (v j . . . ) σ pq j (β) 2
and
hold. For j ∈ N, i with q j ≤ i ≤ q j + sq − 1, it folj
lows that f pi (xαˆ ) ∈ Vv1 and f pi (xβˆ ) ∈ Vv2 . It means that
d( f pi (xαˆ ), f pi (xβˆ )) ≥ d0 . Thus
#{ i : d ( f
pi
qj
q j + sqj − 1 qj − 1 q j + sqj − 1
≤
2−q j +1 sqj − 1
≤
2−q j +1 sqj + sqj − 1
→ 0 ( j → ∞),
#{i : d( f pi (xαˆ ), f pi (xβˆ )) < d0 , 1 ≤ i ≤ n} n
n→∞
= 0.
(6)
From (5) and (6), the set G is uncountable distributionally scrambled in the sequence (pk ), therefore, we can conclude that the map f is distributionally chaotic in the sequence (pk ). Example 3.1. Define f: [0, 3] → [0, 3] as follows:
f (x) =
⎧ ⎪ ⎪1.5x + 2, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2.5,
⎪ 1.5x + 1.5, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3, ⎪ ⎪ ⎩ −3x + 9,
Set
A=
0 1
1 , 3 1 2 , , if x ∈ 3 3 2 if x ∈ ,1 , 3 if x ∈ (1, 2], if x ∈
A
n ∈ N0 . Furthermore, for any n ∈ {0, 1, . . . , T − 1}, the sequence (diam(Van ...an+i ))∞ is nonincreasing and satisi=0 fies limi→∞ diam(Van ...an+i ) = 0. Set dk = max{diam(Van ...an+k ) | 0 ≤ n ≤ T − 1} for k ∈ N, then
lim dk = 0.
k→∞
(7)
(aq+1 aq+2 . . . aT ). Define a map ψ : {1, 2} → {u, v} as ψ(1) = u, ψ(2) = v. Construct a map ϕ : 2 → A as follows:
hence
lim inf
Proof. Denote the period of α by T, that is, σAT (α) = α , then #{σAn (α) | n ∈ N0 } = T . From the assumptions of this the orem, f n (Vα ) = Vσ n (α) = ∞ i=0 Van an+1 ...an+i is a singleton for
By the definition of irreducible matrix and the assumption (∗), there exists an a ∈ A such that aa a0 = 1. There are two different admissible words for the matrix A, u = (a0 . . . a ), v = (a0 . . . a ), such that |u| = |v| = l. Since the matrix A is irreducible, there exist i, j ∈ A such that aa0 i = a ja0 = 1 and an admissible word for the matrix A, C = (i . . . j). For p ∈ N, set B p = (a0 a1 . . . a p−1 ) and define B¯ p as follows. If a p−1 = a0 we assume that B¯ p vanishes. Otherwise, there is a q ∈ {1, 2, . . . , T − 1} such that a p−1 = aq . In this case set B¯ p =
(xαˆ ), f (xβˆ )) < d0 , 1 ≤ i ≤ q j + s − 1} pi
Theorem 3.2. Let (X, d) be a compact metric space. Suppose that f: X → X is a continuous map and A is an irreducible m × m transition matrix (m ≥ 2) and satisfies the assumption(∗). Let f be strictly A-coupled-expanding in the compact sets Vi (1 ≤ i ≤ m). If there exists a periodic point α = (a0 a1 . . . ) ∈ A such −n (V ) is a singleton, then f is distributionally that Vα = ∞ an n=0 f chaotic of type 1.
0,
if x ∈ (2, 3].
1 , 1
V1 = [0, 1] and V2 = [2, 3]. This matrix A is an irreducible transition matrix and satisfies the assumption (∗) and f is strictly A-coupled-expanding in V1 and V2 . Furthermore, the map f satisfies the remaining conditions of Theorem 3.1. Therefore, there exists an uncountable distri-
ϕ(a0 a1 a2 . . . ) = a0C ψ(a0 )m1 B p2 B¯ p2 C ψ(a1 )m3 ψ(a2 )m4 B p5 B¯ p5 C ψ(a3 )m6 ψ(a4 )m7 ψ(a5 )m8 B p9 B¯ p9 C ψ(a6 )m10 ψ(a7 )m11 ψ(a8 )m12 ψ(a9 )m13 . . . , (8) where mi is 2i th power of the length of the subword to the left of ψ(a j )mi in (8) and pi is 2i th power of the length of the subword to the left of B pi in (8). Obviously, any subword in the sequence defined in (8) is admissible for the matrix A, that is, the map ϕ is well defined. By the construction of (8), the map ϕ is bijective. Since the shift ( 2 , σ ) is chaotic in sense of Li-Yorke, there exists an uncountable scrambled set D in 2 . Let D0 = ϕ(D). For any αˆ ∈ D0 , we choose only one element from Vαˆ and denote it by xαˆ . Denote the set consist ing of these elements by Dx (⊂ m i=1 Vi ). Now we prove that this set Dx is uncountable distributionally scrambled of type 1. It is obvious that Dx is uncountable. The equation (7) implies that for every t > 0 there is a k ∈ N with dk < t. Let xαˆ , xβˆ ∈ Dx be different, then αˆ = βˆ ∈ D0 . Obviously, for any i, the end alphabet of the combination B pi B¯ pi is a0 . Denote the index of the a0 in αˆ as ni . By definition of ϕ , ni is also the index of the end a0 of B pi B¯ pi in βˆ . Note that |B¯ pi | ≤ T, then it is obvious that ni ≤ pi (1 + 2−i ) + T . The choice of pi guarantees that |B pi | < |B p j | for i < j, and there exists an i0 such that |B pi | > k for i ≥ i0 .
C. Kim et al. / Chaos, Solitons and Fractals 77 (2015) 291–295
Therefore, for i ≥ i0 ,
assumptions is topologically conjugate to a subshift of finite type in a compact f-invariant subset. Moreover, in the Remark 3.3 of [4], they said that if assumptions of Theorem 2 are satisfied, then f is distributionally chaotic. However, in Example 3.1 f does not satisfy the assumptions of Theorem 2 in [4].
#{s : d( f (xαˆ ), f (xβˆ )) < t, 0 ≤ s < ni } ni #{s : d( f s (xαˆ ), f s (xβˆ )) < dk , 0 ≤ s < ni } ≥ ni #{s : (as . . . as+k ) lies in B pi } ≥ ni pi − k = ni pi − k ≥ → 1 (i → ∞), pi (1 + 2−i ) + T s
295
s
Acknowledgments The authors are grateful to the Editor-in-Chief, Prof. S. Boccaletti, and the referees for their helpful comments and valuable suggestions. References
hence
lim sup
#{s : d( f s (xαˆ ), f s (xβˆ )) < t, 0 ≤ s < n}
n→∞
n
= 1.
(9)
Denote the set of all admissible words for the matrix A with the length l as W. Choose d0 such that
0 < d0 < min{d(Vc , Vd ) | c = d ∈ W }. Assume that αˆ = (aˆ0 aˆ1 . . . ) = βˆ = (bˆ 0 bˆ 1 . . . ) ∈ D0 . From the definition of ϕ and the construction of D0 we may assume without loss of generality that there is an increasing semr mr quence (rj ) in N such that combinations u j and v j appear ˆ in the same place of αˆ and β , respectively. Denote ν j as the original index of a in the sequence αˆ (or βˆ ) which is the last mr mr a in u j (or v j ). Since ν j = mr j l + 2−r j mr j we obtain
#{s : d( f s (xαˆ ), f s (xβˆ )) < d0 , 0 ≤ s < ν j }
νj ν j − #{s : (es . . . el+s−1 ) ∈ umr j (or vmr j ), s ≥ 0} ≤ νj ν j − (mr j l − l + 1) = νj =
2−r j mr j + l − 1 mr j l + 2−r j mr j
→ 0 ( j → ∞).
From the definition of inferior limit of sequences, we have
lim inf n→∞
#{s : d( f s (xαˆ ), f s (xβˆ )) < d0 , 0 ≤ s < n} n
= 0.
(10)
From (9) and (10) we can see that any pair of two different elements in Dx is distributionally chaotic of type 1. Therefore the map f is distributionally chaotic of type 1. Remark 3.1. Consider the map f of Example 3.1. For the periodic point α = (121212 . . . ), Vα is a singleton. Thus the map f is also distributionally chaotic of type 1. Remark 3.2. In Theorem 2 of [4], it was proved that a strictly A-coupled-expanding map f satisfying two additional
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