aa r X i v : . [ m a t h . D S ] N ov SCRAMBLED AND DISTRIBUTIONALLY SCRAMBLED n -TUPLES JANA DOLEˇZELOV ´A*
Abstract.
This article investigates the relation between the distributional chaos and the existence ofa scrambled triple. We show that for a continuous mapping f acting on a compact metric space ( X, d ),the possession of an infinite extremal distributionally scrambled set is not sufficient for the existenceof a scrambled triple. We also construct an invariant Mycielski set with an uncountable extremaldistributionally scrambled set without any scrambled triple.
Mathematics Subject Classification.
Primary 37D45; 37B10.
Key words:
Distributional chaos; scrambled tuples; Morse minimal set. Introduction
The first definition of chaotic pairs appeared in the paper [4] by Li and Yorke in 1975. One of themost important extensions of the concept of Li-Yorke chaos is distributional chaos introduced in [1]. Thisextended definition is much stronger - there are many mappings which are chaotic in the sense of Li-Yorkebut not distributionally chaotic. Another way how to extend the Li-Yorke chaos is looking on dynamics oftuples instead dynamics of pairs. Since Xiong [5] and Sm´ıtal [7] constructed some interval maps with zerotopological entropy which are Li-Yorke chaotic, the Li-Yorke chaos is not sufficient condition for positivetopological entropy. But interval maps with zero topological entropy never contain scramled triples [6]and hence existence of scrambled triple implies positive topological entropy. Consequently we can find adynamical system which is Li-Yorke chaotic but contains no scrambled triple. The natural question wasif there is a dynamical system which is distributionally chaotic but contains no distributionally scrambledtriple. Example in [8] contains no distributionally scrambled triple but still there were some scrambledtriples (in the sense of Li-Yorke) and therefore another open problem appeared - is there a distributionallychaotic dynamical system without any scrambled triple? In this paper, we construct a dynamical systemwhich possess an infinite extremal distributionally scrambled set but without any scrambled triple. Weshow the existence of an invariant Mycielski set which possess an uncountable extremal distributionallyscrambled set and has no scrambled triple. Because some scrambled triples occur in the closure of thisinvariant set, the following question remains open:
Does the existence of uncountable distributionallyscrambled set imply the existence of a scrambled triple? Terminology
Let (
X, d ) be a non-empty compact metric space. Let us denote by (
X, f ) the topological dynamicalsystem , where f is a continous self-map acting on X . We define the forward orbit of x , denoted by Orb + f ( x )as the set { f n ( x ) : n ≥ } . A non-empty closed invariant subset Y ⊂ X defines naturally a subsystem( Y, f ) of (
X, f ). For n ≥
2, we denote by ( X n , f ( n ) ) the product system ( X × X × ... × X, f × f × ... × f )and put ∆ ( n ) = { ( x , x , ..., x n ) ∈ X n : x i = x j for some i = j } . By a perfect set we mean a nonemptycompact set without isolated points. A Cantor set is a nonempty, perfect and totally disconnected set.A set D ⊂ X is invariant if f ( D ) ⊂ D . A Mycielski set is defined as a countable union of Cantor sets.
Definition 1.
A tuple ( x , x , ..., x n ) ∈ X n is called n -scrambled iflim inf k →∞ max ≤ i
For an n -tuple ( x , x , ..., x n ) of points in X , define the lower distribution function gener-ated by f as Φ ( x ,x ,...,x n ) ( δ ) = lim inf m →∞ m { < k < m ; min ≤ i A subset S of X is called extremal distributionally n -scrambled if every n -tuple ( x , x , ..., x n ) ∈ S n \ ∆ ( n ) is distributionally n -scrambled with Φ ( x ,x ,...,x n ) ( δ ) = 0, for any δ < diam X. Let A = { , , ..., n − } , n ≥ 2, be a finite alphabet and Σ n a set of all infinite sequences on A, thatis, for u ∈ Σ n , u = u u u . . . , where u i ∈ A for all i ≥ 1. We define a metric on Σ n by d ( u, v ) = ∞ X i =1 δ ( u i , v i )2 i , where δ ( u i , v i ) = (cid:26) , u i = v i , u i = v i . The shift transformation is a continuous map σ : Σ n → Σ n given by σ ( u ) i = u i +1 .The dynamical system(Σ n , σ ) is called the one-sided shift on n symbols. Any closed subset X ⊂ Σ n invariant for σ is called asubshift of (Σ n , σ ).Any finite string B of some u ∈ Σ n is called a word (or a block) and the length of B is denoted by | B | .Let B = b . . . b n and G = g . . . g n be words. Denote by BG = b . . . b n g . . . g n and for the case of Σ denote by ¯ B the binary complement of B .The Morse block M i is defined inductively such that M = 0, and M i = M i − M i − , for all i > Morse sequence m ∈ Σ is the limit of the Morse blocks, i.e. m = lim i →∞ M i . This sequence m generates the infinite Morse minimal set M = cl { m, σ ( m ) , σ ( m ) , . . . } and it is known that, for all words B ⊂ m , the sequence m contains no block BBb , where b is the first element of block B (cf. [3]). Denotethis property by P . 3. Scrambled and distributionally scrambled n -tuples We will show that the existence of an infinite extremal distributionally scrambled set doesn’t implythe existence of a scrambled triple. Then we construct an invariant Mycielski set with an uncountableextremal distributionally scrambled set without any scrambled triple. crambled and distributionally scrambled n -tuples 3 Lemma 1. Let { a i } ∞ i =1 be a strictly increasing sequence of positive integers such that, for every n ≥ , a n and n have the same parity. Then the point x = M a M a M a . . . is contained in the Morse minimalset.Proof. Since a n and a n +1 have different parity, for every n ≥ 1, the Morse block M a n +1 ends with ¯ M a n M a n +1 = M a n ¯ M a n ¯ M a n M a n . . . ¯ M a n . By the construction of the Morse sequence m and the previous property, we can observe σ · a − a ( m ) = M a M a ¯ M a M a M a ¯ M a . . .σ · a − a − a ( m ) = M a M a M a ¯ M a M a M a ¯ M a . . . ...Since σ r n ( m ) starts with blocks M a M a M a . . . M a n , where r n = 3 · a n − P n − i =1 a i , for all n > 1, thepoint x = lim n →∞ σ r n ( m ) is contained in the Morse minimal set. (cid:3) Theorem 1. There exists a dynamical system X with an infinite extremal distributionally scrambled setbut without any scrambled triple.Proof. Let W , W , W , . . . be the following infinite decomposition of even numbers into infinite sets: W = { n · , n ≥ } ,W = { n · , n ≥ } ,W = { n · , n ≥ } , ...Let { a n } ∞ n =1 be an increasing sequence of positive integers with lim n →∞ a n /a n +1 = 0 and, for every n ≥ a n and n have the same parity. Thenlim n →∞ P n − i =1 a i a n = 0 . (3)We construct the point x i as a sequence of blocks M a M ia M a M ia M a M ia M a M ia . . . where M ia j = ( M a j , if j / ∈ W i ¯ M a j , if j ∈ W i . Remark 1 Let i be a fixed positive integer. Then the first complementary block ¯ M a j appears in theconstruction of x i for j = 2 · (2 i − ,x i = M a M a M a M a M a . . . M a · (2 i − − ¯ M a · (2 i − . . . . Hence the sequence { x i } ∞ i =1 converges and lim i →∞ x i = x , where x is the point constructed in Lemma 1.Let D = { x i } ∞ i =1 . We claim D is distributionally 2-scrambled set and X = cl ( ∪ ∞ i =0 σ i ( D )) is the wanteddynamical system. I. D is extremal distributionally 2-scrambled set Let ( x i , x j ) ∈ D be a pair of distinct points. For simplicity denote s k = P kn =1 a n . Let l be afixed positive integer and ǫ = l . Since ( x i ) n = ( x j ) n if s k < n ≤ s k +1 , for any k > 0, we have d ( σ n ( x i ) , σ n ( x j )) < ǫ for all s k < n < s k +1 − l . By (3),lim k →∞ a k +1 − l a k +1 + s k = 1 , so it is easy to see that Φ ∗ x i ,x j ( ǫ ) = 1, for arbitrary small ǫ , and hence Φ ∗ x i ,x j ≡ { l k } ∞ k =1 ⊂ W i ∪ W j such that ( x i ) n = ( x j ) n if s l k − < n ≤ s l k , forany integer k . Since ( x i ) m = ( x j ) m , for m = 1 , , . . . r , implies d ( x i , x j ) ≥ P rm =1 12 m and ( σ n ( x i )) m = Jana Doleˇzelov´a ( σ n ( x j )) m , for all s l k − < n < s l k − r and m = 1 , , . . . r , it follows d ( σ n ( x i ) , σ n ( x j )) ≥ P rm =1 12 m for all s l k − < n < s l k − r . Because lim k →∞ s l k − s l k − + 2 a lk − r = 0 , it is easy to see that Φ x i ,x j ( P rm =1 12 m ) = 0, for arbitrary large r , and hence Φ x i ,x j ( δ ) = 0, for any0 < δ < . II. S ∞ i =0 σ i ( D ) has no scrambled triples Let ( x i , x j , x k ) ∈ D \ ∆ (3) . Since M i n = M j n or M i n = M k n or M j n = M k n , and M n − is the common block for all x i , x j , x k and every n > 1, we can assumelim n →∞ min { d ( σ n ( x i ) , σ n ( x j )) , d ( σ n ( x i ) , σ n ( x k )) , d ( σ n ( x k ) , σ n ( x j )) } = 0and consequently, condition (2) is not satisfied and D has no scrambled triples. For the same reason σ n ( D ) has no scrambled triples, for any n > 0. It follows that any potential scrambled triple in X mustcontain some pair σ p ( u ) , σ q ( v ), where p < q and u, v ∈ D . To prove that for such tuple the condition (1)is not fulfilled, it is sufficient to show thatlim inf k →∞ d ( σ k ( σ p ( u )) , σ k ( σ q ( v ))) > , where p < q and u, v ∈ D. Assume the contrary - let lim inf k →∞ d ( σ k ( σ p ( u )) , σ k ( σ q ( v ))) = 0 and denote r = q − p > 0. Then we can find an infinite subsequence { k n } ∞ n =1 such that both σ k n ( σ p ( u )) and σ k n ( σ q ( v )) begin with the same block G n of length 14 r and obviously these blocks can be found also inthe sequence u and, shifted by r , in v . For sufficiently large n , G n is in the sequence u either containedin some Morse block or is on the edge of two following Morse blocks, but at least the first 7 r digits orthe last 7 r digits of block G n are contained in a single Morse block. Denote this block M ( u ) a j , where M ( u ) a j is either M a j or M a j ), depending on u and j , and these 7 r consecutive digits of G n by G = g u g u ...g u r .This G appears in v shifted by r , so we can conclude that the first 6 r digits g v g v ...g v r of G in v arein M ( v ) a j . The block M ( v ) a j is either the same Morse block as M ( u ) a j or its binary complement. In thefirst case, g u = g v = g ur +1 = g vr +1 = g u r +1 = ... = g u r +1 and similarly for g u , ..., g ur and thereforewe obtained a block BBBBBB which is impossible since M ( u ) a j is a Morse block. In the second case, g u = g v = g ur +1 = g vr +1 = g u r +1 = g u r +1 = ... = g u r +1 and similarly for g u , ..., g u r and therefore weobtained a block BBB which is a contradiction with M ( u ) a j is a Morse block. Remark 2 By the same argument, ( x i , x j , x ) is never scrambled triple, where x = lim i →∞ x i . III. If y ∈ X \ S ∞ i =0 σ i ( D ) , then there exists a nonnegative integer n such that σ n ( y ) is contained inthe Morse minimal set M . Suppose the contrary. By [3], the points of Morse minimal sets are characterised by the property P andtherefore we can find two distinct blocks B B b and B B b which appears in y , where b denotes thefirst element of B and b denotes the first element of B . Suppose that the last element of B B b is onthe k -th position in y , the last element of B B b is on the k -th position in y and k > k + 2 · | B | .Since y is contained in the closure of ∪ ∞ i =0 σ i ( D ), there are sequences { x m i } ∞ i =1 and { n i } ∞ i =1 such that y = lim i →∞ σ n i ( x m i ) , and suppose all sequences σ n i ( x m i ) have the same first k symbols. Let j be an integer such that2 a j > k . We can observe that n i is bounded by P j − l =1 l for all i > n i the block of thefirst k symbols would be part of a single Morse block, but by assumption there is B B b inside of theblock of first k symbols. Hence there is a nonnegative integer N and a subsequence { x m ik } ∞ k =1 such that y = lim i →∞ σ n i ( x m i ) = lim i →∞ σ N ( x m ik ) = σ N ( x ), where the last identity follows from the Remark 1.By Lemma 1, y = σ N ( x ) ∈ M and this is a contradiction with assumptions. IV. X has no scrambled triples By [9] the Morse minimal set is a distal system, i.e. all pairs inthis set are either distal or asymptotic. Hence the only potential scrambled triples are ( x i , x j , y ), where y ∈ X \ ∪ ∞ i =0 σ i ( D ) and x i , x j ∈ D . By the previous step, it is sufficient to show that ( x i , x j , y ), where y ∈ M , is not a scrambled triple. Since y ∈ M and x ∈ M by Lemma 1, the pair ( x, y ) is either distal or crambled and distributionally scrambled n -tuples 5 asymptotic:a) ( x, y ) is distal pairSequences x, x i , x j are exactly the same except of blocks M ia l and M ja l where l ∈ W i ∪ W j and it holds M ia l = M ja l , for l ∈ W i ∪ W j . Therefore ( x i , x j , y ) is not proximal.b) ( x, y ) is asymptotic pairThe triple ( x i , x j , x ) is not scrambled by Remark 2, therefore ( x i , x j , y ) is not scrambled. (cid:3) To prove Theorem 2, we need the next lemma: Lemma 2. There is a Cantor set B ⊂ { , } N such that, for any distinct α = { α ( i ) } ∞ i =1 and β = { β ( i ) } ∞ i =1 in B , the set { j ∈ N ; α ( j ) = β ( j ) } is infinite. (4) Proof. By Lemma 5.4 in [1] there is an uncountable Borel set B ⊂ { , } N satisfying (4). The resultfollows from Alexandrov-Hausdorff Theorem [2]. (cid:3) Theorem 2. There exists an invariant Mycielski set X ⊂ Σ with an uncountable extremal distribution-ally 2-scrambled set but without any 3-scrambled tuple.Proof. We will denote by M i the Morse block M i and by M i the binary complement of M i . Let α = { α ( i ) } ∞ i =1 be a point of B where B is the set from Lemma 2. Let { a n } ∞ n =1 be an increasing sequenceof positive integers with lim n →∞ a n /a n +1 = 0. Thenlim n →∞ P n − i =1 a i a n = 0 . (5)We construct a point x α as a sequence of blocks M α a M a M α a M a M α a M a M α a . . . . Let D = { x α ; α ∈ B } . We claim D is extremal distributionally 2-scrambled set and X = ∪ ∞ i =0 σ i ( D ) isthe wanted Mycielski set. I. D is extremal distributionally 2-scrambled set Let ( u, v ) ∈ D be a pair of distinct points. For simplicity denote s i = P ij =1 a j . Let l be a fixedinteger and ǫ = l . Since u i = v i if s k − < i ≤ s k , for any k > 0, we have d ( σ i ( u ) , σ i ( v )) < ǫ for all s k − < i < s k − l . By (4), lim k →∞ a k − l a k + s k − = 1 , so it is easy to see that Φ ∗ u,v ( ǫ ) = 1, for arbitrary small ǫ , and hence Φ ∗ u,v ≡ { l k } ∞ k =1 such that u i = ¯ v i if s l k − < i ≤ s l k , for any integer k .Since u m = ¯ v m , for m = 1 , , . . . r , implies d ( u, v ) ≥ P rm =1 12 r and it follows d ( σ i ( u ) , σ i ( v )) ≥ P rm =1 12 r for all s l k − < i < s l k − r and m = 1 , , . . . r . Becauselim k →∞ s l k − s l k − + 2 a lk − r = 0 , it is easy to see Φ u,v ( P rm =1 12 m ) = 0, for arbitrary large r , and hence Φ u,v ( δ ) = 0, for any 0 < δ < . II. X has no scrambled triples Let ( x α , x β , x γ ) ∈ D \ ∆ (3) . Since α i , β i , γ i ∈ { , } , for any integer i , M α i i − = M β i i − or M α i i − = M γ i i − or M β i i − = M γ i i − , and M i is the common block for all x α , x β , x γ , we can assumelim k →∞ min { d ( σ k ( x α ) , σ k ( x β )) , d ( σ k ( x α ) , σ k ( x γ )) , d ( σ k ( x γ ) , σ k ( x β )) } = 0and consequently, condition (2) is not satisfied and D has no scrambled triples. For the same reason σ i ( D ) has no scrambled triples, for any i > 0. It follows that any potential scrambled triple in X must Jana Doleˇzelov´a contain some pair σ p ( u ) , σ q ( v ), where p < q and u, v ∈ D . The fact that for such tuple the condition(1) is not fulfilled, can be proven in the similar way to the second step of the proof of the previous theorem. III. X is a Mycielski set Let h : B → D be a bijection such that, for all α ∈ B , h ( α ) = x ( α ) . To prove that h is homeomorphism, it is sufficient to show that h is continuous. Let { α m } ∞ m =1 be aconverging sequence in B , i.e. lim m →∞ α m = α . Then for an arbitrary i > m suchthat, for all m > m , the first i members of the sequences α m and α are equal. Therefore also the first2 a +2 a +2 a + . . . +2 a (2 i − members of x α m and x α are equal and this exactly means lim m →∞ x α m = x α ,hence h is homeomorphism and D is a Cantor set. Since D is distributionally 2-scrambled, the mapping σ i | D : D → σ i ( D ) is one-to-one and σ i | D is homeomorphism for every i ≥ 1. Thus X is the union ofCantor sets. (cid:3) Remark 3 There are some scrambled triples ( x α , x β , x ) in the closure of X , where x ∈ cl ( X ) \ X anddepends on the parity of { a n } ∞ n =1 . If a n and n have the same parity, then there exist α and β such that( x α , x β , x ) is scrambled, where x = M a M a M a M a . . . . Acknowledgment. I sincerely thank my supervisor, Professor Jaroslav Sm´ıtal, for valuable guidance.I am grateful for his constant support and help. References [1] Schweizer B., Sm´ıtal J. , Measures of chaos and a spectral decomposition of dynamical systems on the interval , Trans.Amer. Math. Soc. , (1994), 737 – 754.[2] Kuratowski K. , Topologie , Vol. II., Academic Press and Polisch Scientific Publischers, (1968).[3] Gottschalk W. H., Hedlund G. A. , A characterization of the Morse minimal set , Proc. Amer. Math. Soc. ,(1964), 70–74.[4] Li T., Yorke J. , Period three implies chaos , Amer. Math. Monthly , (1975), 985–992.[5] Xiong J. , A chaotic map with topological entropy zero , Acta Math. Sci. , (1986), 439–443.[6] Li J. , Chaos and entropy for interval maps , J. Dynam. Differential Eq. , (2011), 333–352.[7] Sm´ıtal J. , Chaotic functions with zero topological entropy , Trans. Amer. Math. Soc. , (1986), 269 – 282.[8] Oprocha P., Li J. , On n -scrambled tuples and distributional chaos in a sequence , J. Difference Eq. Appl. , (2013),927–941[9] Blanchard F., Glasner E., Kolyada S., Maas A. , On Li-Yorke pairs , J. reine angew. Math. , (2002), 51–68 J . Doleˇzelov´a, Mathematical Institute, Silesian University, CZ-746 01 Opava, Czech Republic E-mail address ::