Distributional chaos and Li-Yorke chaos in metric spaces
aa r X i v : . [ m a t h . F A ] J a n DISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRICSPACES
MARKO KOSTI´C
Abstract.
In this paper, we introduce several new types and generalizationsof the concepts distributional chaos and Li-Yorke chaos. We consider the gen-eral sequences of binary relations acting between metric spaces, while in aseparate section we focus our attention to some special features of distribu-tionally chaotic and Li-Yorke chaotic multivalued linear operators in Fr´echetspaces. Introduction and Preliminaries
Let X be a separable Fr´echet space. A linear operator T on X is said to behypercyclic iff there exists an element x ∈ D ∞ ( T ) ≡ T n ∈ N D ( T n ) whose orbit { T n x : n ∈ N } is dense in X ; T is said to be topologically transitive, resp. topo-logically mixing, iff for every pair of open non-empty subsets U, V of X, thereexists n ∈ N such that T n ( U ) ∩ V = ∅ , resp. there exists n ∈ N such that, forevery n ∈ N with n ≥ n , T n ( U ) ∩ V = ∅ . A linear operator T on X is said to bechaotic iff it is topologically transitive and the set of periodic points of T, definedby { x ∈ D ∞ ( T ) : ( ∃ n ∈ N ) T n x = x } , is dense in X. The basic facts about topological dynamics of linear continuous operators inBanach and Fr´echet spaces can be obtained by consulting the monographs [3] byF. Bayart, E. Matheron and [14] by K.-G. Grosse-Erdmann, A. Peris. In a jointresearch study with C.-C. Chen, J. A. Conejero and M. Murillo-Arcila [11], theauthor has recently introduced and analyzed a great deal of topologically dynamicalproperties for multivalued linear operators (cf. also the article [1] by E. Abakumov,M. Boudabbous and M. Mnif). The notion has been extended in [12] and [20] forgeneral sequences of binary relations over topological spaces.Distributional chaos for interval maps was introduced by B. Schweizer and J.Sm´ıtal in [26] (this type of chaos was called strong chaos there, 1994). For linearcontinuous operators, distributional chaos was firstly investigated in the researchstudies of quantum harmonic oscillator, by J. Duan et al [13] (1999) and P. Oprocha[24] (2006). Distributional chaos for linear continuous operators in Fr´echet spaceswas analyzed by N. C. Bernardes Jr. et al [6] (2013), while distributional chaos forclosed linear operators in Fr´echet spaces was investigated by J. A. Conejero et al[10] (2016).On the other hand, the notion of Li-Yorke chaos received enormous attentionafter the foundational paper of T. Y. Li and J. A. Yorke [22] (1975). Li-Yorke chaoticlinear continuous operators on Banach and Fr´echet spaces have been systematically
Mathematics Subject Classification.
Key words and phrases.
Distributional chaos; Li-Yorke chaos; binary relations; metric spaces;multivalued linear operators.The author is partially supported by grant 174024 of Ministry of Science and TechnologicalDevelopment, Republic of Serbia. analyzed in [5] and [8]. For more details about Li-Yorke chaos and distributionalchaos in metric and Fr´echet spaces, we refer the reader to [2], [4], [16]-[17], [21],[23]-[25], [28] and references cited therein.The main aim of this paper is to introduce various notions of distributional chaosand Li-Yorke chaos for binary relations and their sequences, working in the setting ofmetric spaces. In particular, we analyze the notions of reiteratively ˜ X -distributionalchaos of types 1 and 2, ( ˜ X, i )-mixed chaos, where i = 1 , , , , and ˜ X -Li-Yorkechaos; here, ˜ X is a non-empty subset of metric space X under our consideration.In [7], N. C. Bernardes Jr. et al have considered the notion of distributional chaosof type s for linear continuous operators acting on Banach spaces ( s ∈ { , , , } ).In this paper, we extend the notion from this paper for general sequences of binaryrelations, as well (up to now, the notion from [7] has not been introduced for linearunbounded operators on Banach spaces and their sequences). Finally, we analyzedistributionally chaotic and Li-Yorke chaotic multivalued linear operators in Fr´echetspaces by enquiring into the basic properties of associated distributionally chaoticand Li-Yorke chaotic irregular vectors and their submanifolds. Plenty of usefulcomments, observations and open problems enriches our study.Albeit defined in this general framework, we feel duty bound to say that thenotion of distributional chaos, its specifications and various generalizations are themost intriguing for orbits of continuous linear operators in Banach spaces. Thisfollows from a series of simple counterexamples presented in this paper, which showin particular that distributional chaos and Li-Yorke chaos occur even for the generalsequences of continuous linear operators on finite-dimensional spaces. All aspectsand connections between the introduced concepts cannot be easily perceived withinjust one research paper and our intention here, actually, was to create a solid basefor further explorations of distributional chaos and Li-Yorke chaos. Because of that,we can freely say that this paper is heuristic to a large extent.The organization of material is briefly described as follows. In Subsection 1.1and Subsection 1.2, we recall the basic things about lower and upper densities aswell as binary relations and multivalued linear operators, respectively. In Section2 and Section 3, we analyze various types of distributional chaos, Li-Yorke chaosand distributional chaos of type s ( s ∈ { , , , } ) for binary relations over metricspaces. The fourth section of paper is reserved for the study of distributional chaosand Li-Yorke chaos for multivalued linear operators in Fr´echet spaces; in a separatesubsection, we investigate irregular vectors and irregular manifolds. In addition tothe above, we include the conclusion and remark section at the end of paper.Before proceeding further, we need to recall that for each set D = { d n : n ∈ N } ,where ( d n ) n ∈ N is a strictly increasing sequence of positive integers, we define itscomplement D c := N \ D and difference set { e n := d n +1 − d n | n ∈ N } . Let usrecall that an infinite subset A of N is said to be syndetic, or relatively dense, iffits difference set is bounded. The difference set of any finite subset of N , definedsimilarly as above, is finite. Set N n := { , ··· , n } ( n ∈ N ) and S := { z ∈ C ; | z | = 1 } . By P ( A ) we denote the power set of A .1.1. Lower and upper densities.
In this subsection, we recall the basic thingsabout lower and upper densities that will be necessary for our further work.
ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 3
Let A ⊆ N be non-empty. The lower density of A, denoted by d ( A ) , is definedby d ( A ) := lim inf n →∞ | A ∩ [1 , n ] | n , and the upper density of A, denoted by d ( A ) , is defined by d ( A ) := lim sup n →∞ | A ∩ [1 , n ] | n . Further on, the lower Banach density of A, denoted by Bd ( A ) , is defined by Bd ( A ) := lim s → + ∞ lim inf n →∞ | A ∩ [ n + 1 , n + s ] | s and the (upper) Banach density of A, denoted by Bd ( A ) , is defined by Bd ( A ) := lim s → + ∞ lim sup n →∞ | A ∩ [ n + 1 , n + s ] | s . It is well known that the limits appearing in definitions of Bd ( A ) and Bd ( A ) existas s tends to + ∞ , as well as that0 ≤ Bd ( A ) ≤ d ( A ) ≤ d ( A ) ≤ Bd ( A ) ≤ , (1.1) d ( A ) + d ( A c ) = 1(1.2)and Bd ( A ) + Bd ( A c ) = 1 . (1.3)1.2. Binary relations and multivalued linear operators.
Let
X, Y, Z and T be given non-empty sets. A binary relation between X into Y is any subset ρ ⊆ X × Y. If ρ ⊆ X × Y and σ ⊆ Z × T with Y ∩ Z = ∅ , then we define ρ − ⊆ Y × X and σ ◦ ρ ⊆ X × T by ρ − := { ( y, x ) ∈ Y × X : ( x, y ) ∈ ρ } and σ ◦ ρ := (cid:8) ( x, t ) ∈ X × T : ∃ y ∈ Y ∩ Z such that ( x, y ) ∈ ρ and ( y, t ) ∈ σ (cid:9) , respectively. Domain and range of ρ are introduced by D ( ρ ) := { x ∈ X : ∃ y ∈ Y such that ( x, y ) ∈ ρ } and R ( ρ ) := { y ∈ Y : ∃ x ∈ X such that ( x, y ) ∈ ρ } , respectively; ρ ( x ) := { y ∈ Y : ( x, y ) ∈ ρ } ( x ∈ X ), x ρ y ⇔ ( x, y ) ∈ ρ. If ρ isa binary relation on X and n ∈ N , then we define ρ n inductively; ρ − n := ( ρ n ) − and ρ := { ( x, x ) : x ∈ X } . Put D ∞ ( ρ ) := T n ∈ N D ( ρ n ) and ρ ( X ′ ) := { y : y ∈ ρ ( x ) for some x ∈ X ′ } ( X ′ ⊆ X ).In the remaining part of this subsection, we present a brief overview of the nec-essary definitions and properties of multivalued linear operators. For more detailsabout the subject, we refer the reader to the monographs [9] by R. Cross and [15]by A. Favini, A. Yagi (in [15], applications of multivalued linear operators to ab-stract degenerate differential equations have been thoroughly analyzed; for someother approaches, the reader may consult the monograph [27] by G. A. Sviridyukand V. E. Fedorov).Let X and Y be two Fr´echet spaces over the same field of scalars K . For anymapping A : X → P ( Y ) we define ˇ A := { ( x, y ) : x ∈ D ( A ) , y ∈ A x } . Then A is amultivalued linear operator (MLO) iff the associated binary relation ˇ A is a linearrelation in X × Y, i.e., iff ˇ A is a linear subspace of X × Y. In our work, we willidentify A and its associated linear relation ˇ A , so that the notion of D ( A ) , whichis a linear subspace of X, as well as the sets R ( A ) and D ∞ ( A ) are clear. The set MARKO KOSTI´C A − { x ∈ D ( A ) : 0 ∈ A x } is called the kernel of A and it is denoted henceforthby N ( A ) or Kern( A ) . The inverse A − and the power A n of a MLO, introduced inthe sense of corresponding definition for general binary relations, are MLOs ( n ∈ N ).If X = Y, then we say that A is an MLO in X. An almost immediate consequenceof definition is that, for every x, y ∈ D ( A ) and λ, η ∈ K with | λ | + | η | 6 = 0 , we have λ A x + η A y = A ( λx + ηy ) . If A is an MLO, then A Y and A x = f + A x ∈ D ( A ) and f ∈ A x. The sum A + B of MLOs A and B , defined by D ( A + B ) := D ( A ) ∩ D ( B ) and ( A + B ) x := A x + B x ( x ∈ D ( A + B )), islikewise an MLO. We write A ⊆ B iff D ( A ) ⊆ D ( B ) and A x ⊆ B x for all x ∈ D ( A ) . The scalar multiplication of an MLO A : X → P ( Y ) with the number z ∈ K , z A for short, is defined by D ( z A ) := D ( A ) and ( z A )( x ) := z A x, x ∈ D ( A ) . It is clearthat z A : X → P ( Y ) is an MLO and ( ωz ) A = ω ( z A ) = z ( ω A ) , z, ω ∈ K . By aperiodic point of A we mean any vector x ∈ D ∞ ( A ) such that there exists n ∈ N with x ∈ A n x. Suppose that A is an MLO in X. Then we say that a point λ ∈ K is an eigen-value of A iff there exists a vector x ∈ X \ { } such that λx ∈ A x ; we call x aneigenvector of operator A corresponding to the eigenvalue λ. Observe that, if A ispurely multivalued (i.e., A = 0), a vector x ∈ X \ { } can be an eigenvector ofoperator A corresponding to different values of scalars λ. The point spectrum of A ,σ p ( A ) for short, is defined as the union of all eigenvalues of A . If A : X → P ( Y ) is an MLO, then we define the adjoint A ∗ : Y ∗ → P ( X ∗ ) of A by its graph A ∗ := n(cid:0) y ∗ , x ∗ (cid:1) ∈ Y ∗ × X ∗ : (cid:10) y ∗ , y (cid:11) = (cid:10) x ∗ , x (cid:11) for all pairs ( x, y ) ∈ A o . Distributional chaos and Li-Yorke chaos for binary relations
In this section, it will be always assumed that (
X, d ) and (
Y, d Y ) are metricspaces. Suppose that σ > ǫ > x k ) k ∈ N , ( y k ) k ∈ N are two given sequencesin Y. Consider the following conditions: Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) = 0 ,Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 , (2.1) Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) = 0 ,d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 , (2.2) d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) = 0 ,Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0(2.3)and d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) = 0 ,d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 . (2.4)In the following definition, we introduce the notion of reiterative distributionalchaos (reiterative distributional chaos of type 1 or 2): ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 5
Definition 2.1.
Suppose that, for every k ∈ N , ρ k ⊆ X × Y is a binary relation and˜ X is a non-empty subset of X. If there exist an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X and σ > ǫ > x, y ∈ S of distinct pointswe have that for each k ∈ N there exist elements x k ∈ ρ k x and y k ∈ ρ k y suchthat (2.4) holds, resp. (2.1) [(2.2)/(2.3)] holds, then we say that the sequence( ρ k ) k ∈ N is ˜ X -distributionally chaotic, resp. ˜ X -reiteratively distributionally chaotic[ ˜ X -reiteratively distributionally chaotic of type 1/ ˜ X -reiteratively distributionallychaotic of type 2].The sequence ( ρ k ) k ∈ N is said to be densely ˜ X -distributionally chaotic, resp. ˜ X -reiteratively distributionally chaotic [ ˜ X -reiteratively distributionally chaotic of type1/ ˜ X -reiteratively distributionally chaotic of type 2], iff S can be chosen to be densein ˜ X. A binary relation ρ ⊆ X × X is said to be (densely) ˜ X -distributionallychaotic, resp. ˜ X -reiteratively distributionally chaotic [ ˜ X -reiteratively distribution-ally chaotic of type 1/ ˜ X -reiteratively distributionally chaotic of type 2], iff thesequence ( ρ k ≡ ρ k ) k ∈ N is. The set S is said to be σ ˜ X -scrambled set, resp. σ ˜ X -reiteratively scrambled set [ σ ˜ X -reiteratively scrambled set of type 1/ σ ˜ X -reiterativelyscrambled set of type 2] ( σ -scrambled set, resp. σ -reiteratively scrambled set [ σ -reiteratively scrambled set of type 1/ σ -reiteratively scrambled set of type 2], inthe case that ˜ X = X ) of the sequence ( ρ k ) k ∈ N (the binary relation ρ ); in the casethat ˜ X = X, then we also say that the sequence ( ρ k ) k ∈ N (the binary relation ρ )is distributionally chaotic, resp. reiteratively distributionally chaotic [reiterativelydistributionally chaotic of type 1/reiteratively distributionally chaotic of type 2].It is well known that, for any infinite set A ⊆ N , being syndetic and having a pos-itive Banach density is the same thing. Therefore, if the sets { k ∈ N : d Y ( x k , y k ) <σ } and { k ∈ N : d Y ( x k , y k ) ≥ ǫ } are infinite, then they have unbounded differencesets iff (2.1) holds. If one of these sets is finite, say the first one, then there exists k = k ( σ ) ∈ N such that [ k , ∞ ) ⊆ { k ∈ N : d Y ( x k , y k ) ≥ ǫ } and the secondequality in (2.1) cannot be satisfied. Therefore, in definition of ˜ X -reiterative distri-butional chaos, we can equivalently replace the equation (2.1) with the statementsthat the difference sets of { k ∈ N : d Y ( x k , y k ) < σ } and { k ∈ N : d Y ( x k , y k ) ≥ ǫ } are unbounded.The following definition seems to be new even for the sequences of linear notcontinuous operators on Banach and Fr´echet spaces as well as for the sequences oflinear continuous operators that are not orbits of one single operator: Definition 2.2.
Suppose that, for every k ∈ N , ρ k ⊆ X × Y is a binary relation and˜ X is a non-empty subset of X. If there exist an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X such that for each pair x, y ∈ S of distinct points and for each k ∈ N there existelements x k ∈ ρ k x and y k ∈ ρ k y such thatlim inf k →∞ d Y (cid:0) x k , y k (cid:1) = 0 and lim sup k →∞ d Y (cid:0) x k , y k (cid:1) > , (2.5)then we say that the sequence ( ρ k ) k ∈ N is ˜ X -Li-Yorke chaotic.The sequence ( ρ k ) k ∈ N is said to be densely ˜ X -Li-Yorke chaotic iff S can be chosento be dense in ˜ X. A binary relation ρ ⊆ X × X is said to be (densely) ˜ X -Li-Yorkechaotic iff the sequence ( ρ k ≡ ρ k ) k ∈ N is. The set S is said to be ˜ X -scrambled Li-Yorke set (scrambled Li-Yorke set, in the case that ˜ X = X ) of the sequence ( ρ k ) k ∈ N MARKO KOSTI´C (the binary relation ρ ); in the case that ˜ X = X, then we also say that the sequence( ρ k ) k ∈ N (the binary relation ρ ) is (densely) Li-Yorke chaotic.Besides the conditions introduced so far, we can also examine the following ones: Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) = 0 and lim inf k →∞ d Y (cid:0) x k , y k (cid:1) = 0 , (2.6) d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) = 0 and lim inf k →∞ d Y (cid:0) x k , y k (cid:1) = 0 , (2.7) lim sup k →∞ d Y (cid:0) x k , y k (cid:1) > Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 , (2.8) lim sup k →∞ d Y (cid:0) x k , y k (cid:1) > d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 . (2.9)The following definition is meaningful, as well: Definition 2.3.
Suppose that, for every k ∈ N , ρ k ⊆ X × Y is a binary relation and˜ X is a non-empty subset of X. If there exist an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X and σ > ǫ > x, y ∈ S of distinctpoints we have that for each k ∈ N there exist elements x k ∈ ρ k x and y k ∈ ρ k y such that (2.6)/(2.9) holds, then we say that the sequence ( ρ k ) k ∈ N is ( ˜ X, X, X, i )-mixed chaotic sequence ( ρ k ) k ∈ N (the binary rela-tion ρ ), where i ∈ N , the corresponding ( σ ˜ X , i )-mixed scrambled set (( σ, i )-mixedscrambled set, in the case that ˜ X = X ), where i ∈ N , the corresponding ( ˜ X, i )-mixed scrambled set ( i -mixed scrambled set, in the case that ˜ X = X ), where i ∈ N \ N , of the sequence ( ρ k ) k ∈ N (the binary relation ρ ) is introduced as above;in the case that ˜ X = X and i ∈ N , then we also say that the sequence ( ρ k ) k ∈ N (the binary relation ρ ) is i -mixed chaotic.Keeping in mind (1.1) and (1.3), an elementary line of reasoning shows thatany ˜ X -distributionally chaotic sequence (binary relation) is already ˜ X -reiterativelydistributionally chaotic as well as that any ˜ X -reiteratively distributionally chaoticsequence (binary relation) is both ˜ X -reiteratively distributionally chaotic of type1 and ˜ X -reiteratively distributionally chaotic of type 2. It is also predictablethat any ˜ X -reiteratively distributionally chaotic sequence of binary relations (any˜ X -reiteratively distributionally chaotic binary relation) needs to be ˜ X -Li-Yorkechaotic. To see this, suppose that the sequence ( ρ k ) k ∈ N is ˜ X -reiteratively distri-butionally chaotic with given σ ˜ X -scrambled set S. Let x = y − z for some twodifferent elements y, z ∈ S. Due to our assumption, for each number k ∈ N it isvery simple to construct two strictly increasing sequences ( s n,k ) n ∈ N and ( l n,k ) n ∈ N ofpairwise disjoint positive integers such that d Y ( x s n,k , ≥ σ and d Y ( x l n,k , ≤ /k for all n ∈ N as well as l k +1 ,k +1 > l k,k + 2 k , n k +1 ,k +1 > n k,k + 2 k for all k ∈ N and s , < l , < s , < l , < · · · ; here, x s n,k ∈ ρ s n,k x and x l n,k ∈ ρ l n,k x. If n / ∈ S k ∈ N { s k,k , l k,k } , then we take any vector x n ∈ ρ n x, which clearly exists becausethe set T ∞ k =1 D ( ρ k ) ∩ ˜ X is at least countable. If n = s k,k ( n = l k,k ) for some k ∈ N , then we set x n := x s k,k ( x n := x l k,k ). Then it is clear that lim inf n →∞ d Y ( x n ,
0) = 0and lim sup n →∞ d Y ( x n , > n →∞ d Y ( x l n,n ,
0) = 0 and the subse-quence ( d Y ( x s n,n , d Y ( x n , ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 7 hand, it is clear that ( ˜
X, i )-mixed chaos implies ˜ X -Li-Yorke chaos for any i ∈ N ;the above conclusions also hold for any kind of dense ˜ X -chaos considered above.Therefore, we have proved the following proposition: Proposition 2.4.
Let for each k ∈ N we have that ρ k ⊆ X × Y is a binary relation,and let ˜ X be a non-empty subset of X. Consider the following statements: (i) ( ρ k ) k ∈ N is (densely) ˜ X -distributionally chaotic. (ii) ( ρ k ) k ∈ N is (densely) reiteratively ˜ X -distributionally chaotic of type . (iii) ( ρ k ) k ∈ N is (densely) reiteratively ˜ X -distributionally chaotic of type . (iv) ( ρ k ) k ∈ N is (densely) reiteratively ˜ X -distributionally chaotic. (v) ( ρ k ) k ∈ N is (densely) ( ˜ X, -mixed chaotic. (vi) ( ρ k ) k ∈ N is (densely) ( ˜ X, -mixed chaotic. (vii) ( ρ k ) k ∈ N is (densely) ( ˜ X, -mixed chaotic. (viii) ( ρ k ) k ∈ N is (densely) ( ˜ X, -mixed chaotic. (ix) ( ρ k ) k ∈ N is (densely) ˜ X -Li-Yorke chaotic.Then (i) implies (ii) - (ix) ; (ii) implies (iv) - (v) , (vii) - (viii) and (ix) ; (iii) implies (iv) - (vii) and (ix) ; (iv) implies (v) , (vii) and (ix) ; (v) , (vii) or (viii) implies (ix) ; (vi) implies (v) and (ix) . It is worth noting that any two different types of chaos considered above, as wellas in Definition 3.1 and Definition 4.1 below, do not coincide even for the sequencesof continuous linear operators on finite-dimensional spaces. This can be inspectedas in Example 4.4 below.3.
Distributional chaos of type s for binary relations ( s ∈ { , , , } ) As in the previous one, in this section we assume that (
X, d ) and (
Y, d Y ) aremetric spaces. Suppose that σ, σ ′ > ǫ > x k ) k ∈ N , ( y k ) k ∈ N are two givensequences in Y. Consider the following conditions: Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ σ (cid:9)(cid:17) > ,Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0;(3.1) Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ σ (cid:9)(cid:17) > ,d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0;(3.2) d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ σ (cid:9)(cid:17) > ,Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0;(3.3) d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ σ (cid:9)(cid:17) > ,d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0;(3.4)there exist c > r > Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) < c < Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) (3.5) MARKO KOSTI´C for 0 < σ < r ;there exist c > r > Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) < c < d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) (3.6)for 0 < σ < r ;there exist c > r > d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) < c < Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) (3.7)for 0 < σ < r ;there exist c > r > d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) < c < d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) < σ (cid:9)(cid:17) (3.8)for 0 < σ < r ;there exist a, b, c > σ ∈ [ a, b ];(3.9) there exist a, b, c > σ ∈ [ a, b ];(3.10) there exist a, b, c > σ ∈ [ a, b ];(3.11) there exist a, b, c > σ ∈ [ a, b ] . (3.12)Let i ∈ { , } . For (reiterative) distributional chaos (reiterative distributionalchaos of type i ) it is also said that it is (reiterative) distributional chaos of type0; 1 (reiterative distributional chaos of type i ; 1). Now we would like to propose thefollowing notion: Definition 3.1.
Let i ∈ { , , } and s ∈ { , , , } . Suppose that, for every k ∈ N , ρ k ⊆ X × Y is a binary relation and ˜ X is a non-empty subset of X. (i) If there exist an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X and σ > ǫ > x, y ∈ S of distinct points wehave that for each k ∈ N there exist elements x k ∈ ρ k x and y k ∈ ρ k y such that (3.1) [(3.2)/(3.3)/(3.4)] holds, then we say that the sequence( ρ k ) k ∈ N is reiteratively ˜ X -distributionally chaotic of type 0; 2 [reiteratively˜ X -distributionally chaotic of type 1; 2/reiteratively ˜ X -distributionally chaoticof type 2; 2/ ˜ X -distributionally chaotic of type 0; 2].(ii) If there exist an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X and numbers σ, c, r > ǫ > x, y ∈ S of distinct pointswe have that for each k ∈ N there exist elements x k ∈ ρ k x and y k ∈ ρ k y such that (3.5) [(3.6)/(3.7)/(3.8)] holds for 0 < σ < r , then we saythat the sequence ( ρ k ) k ∈ N is reiteratively ˜ X -distributionally chaotic of type2 [reiteratively ˜ X -distributionally chaotic of type 1; 2 /reiteratively ˜ X -distributionally chaotic of type 2; 2 / ˜ X -distributionally chaotic of type 2 ].(iii) If there exist an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X and real numbers σ, a, b, c > ǫ > x, y ∈ S ofdistinct points we have that for each k ∈ N there exist elements x k ∈ ρ k x and y k ∈ ρ k y such that (3.9) [(3.10)/(3.11)/(3.12)] holds for a < σ < b , thenwe say that the sequence ( ρ k ) k ∈ N is reiteratively ˜ X -distributionally chaoticof type 3 [reiteratively ˜ X -distributionally chaotic of type 1; 3/reiteratively ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 9 ˜ X -distributionally chaotic of type 2; 3/ ˜ X -distributionally chaotic of type3].The sequence ( ρ k ) k ∈ N is said to be densely (reiteratively) ˜ X -distributionally chaoticof type i ; s iff S can be chosen to be dense in ˜ X. A binary relation ρ ⊆ X × X is said to be (densely) reiteratively ˜ X -distributionally chaotic of type i ; s iff thesequence ( ρ k ≡ ρ k ) k ∈ N is. The set S is said to be (reiteratively) ( σ ˜ X , s )-scrambledset ((reiteratively) ( σ, s )-scrambled set, in the case that ˜ X = X ) of the sequence( ρ k ) k ∈ N (the binary relation ρ ); in the case that ˜ X = X, then we also say that thesequence ( ρ k ) k ∈ N (the binary relation ρ ) is densely (reiteratively) distributionallychaotic of type i ; s .Keeping in mind the inequality (1.1), we are in a position to immediately clarifythe following: Proposition 3.2.
Suppose that i ∈ { , , } , s, s , s ∈ { , , , } , s ≤ s , ˜ X is a non-empty subset of X and ( ρ k ) k ∈ N is a given sequence of binary relations.Then, for ( ρ k ) k ∈ N , we have the following:(dense, reiterative) ˜ X -distributional chaos of type i ; s implies (dense, reiterative) ˜ X -distributional chaos of type i ; s ,(dense) ˜ X -distributional chaos of type s implies (dense) reiterative ˜ X -distributional chaos of types s and s and(dense) reiterative ˜ X -distributional chaos of type s or s implies (dense)reiterative ˜ X -distributional chaos of type s. As it is well known, ˜ X -distributional chaos of type 3 is a very weak form oflinear chaos: it is still unknown whether there exists a complex matrix that isdistributionally chaotic of type 3 (cf. [7, Problem 51]). The same problem can beposed for reiterative distributional chaos of type 3 . Concerning the relation of distributional chaos and distributional chaos of type2 , it is worth noting that these two notions are equivalent for linear continuousoperators on Banach spaces (see [7, Theorem 2]). As mentioned at the end ofprevious section, this is far from being true for general sequences of linear continuousoperators.A further analysis of distributional chaos of type s and their generalizations willbe carried out somewhere else.4. Distributional chaos and Li-Yorke chaos in Fr´echet spaces
In the remaining part of the paper, we assume that X is an infinite-dimensionalFr´echet space over the field K ∈ { R , C } and that the topology of X is induced bythe fundamental system ( p n ) n ∈ N of increasing seminorms (separability of X is notassumed a priori in future). Then the translation invariant metric d : X × X → [0 , ∞ ) , defined by(4.1) d ( x, y ) := ∞ X n =1 n p n ( x − y )1 + p n ( x − y ) , x, y ∈ X, enjoys the following properties: d ( x + u, y + v ) ≤ d ( x, y ) + d ( u, v ) , x, y, u, v ∈ X, d ( cx, cy ) ≤ ( | c | + 1) d ( x, y ) , c ∈ K , x, y ∈ X, and d ( αx, βx ) ≥ | α − β | | α − β | d (0 , x ) , x ∈ X, α, β ∈ K . By Y we denote another Fr´echet space over the same field of scalars as X ; thetopology of Y will be induced by the fundamental system ( p Yn ) n ∈ N of increasingseminorms. Define the translation invariant metric d Y : Y × Y → [0 , ∞ ) by replacing p n ( · ) with p Yn ( · ) in (4.1). If ( X, k ·k ) or ( Y, k ·k Y ) is a Banach space, then we assumethat the distance of two elements x, y ∈ X ( x, y ∈ Y ) is given by d ( x, y ) := k x − y k ( d Y ( x, y ) := k x − y k Y ). Keeping in mind this terminological change, our structuralresults clarified in Fr´echet spaces continue to hold in the case that X or Y is aBanach space. By L ( X, Y ) we denote the space consisting of all linear continuousmappings from X into Y ; L ( X ) ≡ L ( X, X ) . In Fr´echet spaces, of importance are the following conditions:the sequence (cid:0) x k − y k (cid:1) k ∈ N is unbounded and lim inf k →∞ d Y (cid:0) x k , y k (cid:1) = 0 , (4.2)the sequence (cid:0) x k − y k (cid:1) k ∈ N is unbounded and Bd (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 , (4.3)the sequence (cid:0) x k − y k (cid:1) k ∈ N is unbounded and d (cid:16)(cid:8) k ∈ N : d Y (cid:0) x k , y k (cid:1) ≥ ǫ (cid:9)(cid:17) = 0 . (4.4)Albeit the most intriguing for multivalued linear operators, the following notioncan be introduced for general binary relations, as well: Definition 4.1.
Suppose that, for every k ∈ N , ρ k ⊆ X × Y is a binary relation and˜ X is a non-empty subset of X. If there exists an uncountable set S ⊆ T ∞ k =1 D ( ρ k ) ∩ ˜ X such that for each pair x, y ∈ S of distinct points and for each ǫ > k ∈ N there exist elements x k ∈ ρ k x and y k ∈ ρ k y such that (4.2) holds, resp.(4.3) [(4.4)] holds, then we say that the sequence ( ρ k ) k ∈ N is strongly ˜ X -Li-Yorkechaotic, resp. h ˜ X, i -mixed chaotic [ h ˜ X, i -mixed chaotic].The notion of densely strong ˜ X -Li-Yorke chaotic sequence, resp. densely h ˜ X, i i -mixed chaotic sequence ( ρ k ) k ∈ N (the binary relation ρ ), where i ∈ N , the corre-sponding strong ˜ X -Li-Yorke scrambled set, resp. h ˜ X, i i -mixed scrambled set (strongLi-Yorke scrambled set, resp. h i i -mixed scrambled set, in the case that ˜ X = X ),where i ∈ N , of the sequence ( ρ k ) k ∈ N (the binary relation ρ ) is introduced as above;in the case that ˜ X = X and i ∈ N , then we also say that the sequence ( ρ k ) k ∈ N (the binary relation ρ ) is strong Li-Yorke chaotic, resp. i -mixed chaotic.With the exception of implications clarified in Proposition 2.4, we can only statethe following ones, in general:(A) strong ˜ X -Li-Yorke chaos implies ˜ X -Li-Yorke chaos;(B) h ˜ X, i -mixed chaos implies strong ˜ X -Li-Yorke chaos;(C) h ˜ X, i -mixed chaos implies h ˜ X, i -mixed chaos and strong ˜ X -Li-Yorke chaos. ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 11
We continue by observing the following: If X is a Banach space and T ∈ L ( X ) , then T is (densely) Li-Yorke chaotic iff T is (densely) reiteratively distributionallychaotic. To see this, let us recall that we have k T k > T is Li-Yorke chaotic. By [5, Theorem 5], we have the existence of a vector x ∈ X such that lim inf n →∞ k T n x k = 0 and lim sup n →∞ k T n x k = ∞ . Therefore, thereexist two strictly increasing sequences of positive integers ( n k ) and ( l k ) such thatmin( n k +1 − n k , l k +1 − l k ) > k , k T n k x k < − k and k T l k x k > k for all k ∈ N . Put A := S k ∈ N [ n k , n k + k ] and B := S k ∈ N [ l k , l k − k ] . Then Bd ( A ) = Bd ( B ) = 1 and foreach n ∈ A ( n ∈ B ) there exists k ∈ N such that n ∈ [ n k , n k + k ] ( n ∈ [ l k , l k − k ])and therefore k T n x k ≤ (1 + k T k ) k − k ( k T n x k ≥ k T k − k k ). This, in turn, impliesthat the notions of 1-mixed chaos, 3-mixed chaos, reiterative distributional chaosand Li-Yorke chaos coincide in this case (this also holds for dense analogues).On the other hand, as already mentioned, the situation is completely differentfor the sequences of continuous linear operators on finite-dimensional spaces. Take,for instance, T k = 0 if k is even and T k = I if k is odd. Then the sequence ( T k )is Li-Yorke chaotic on ony Fr´echet space X but it is not strongly Li-Yorke chaotic(this trivial counterexample also shows that the assertions of [5, Theorem 5] and[8, Theorem 9], where it has been proved that the notions of (dense) Li-Yorkechaos and (dense) strong Li-Yorke chaos coincide for the orbits of linear continuousoperators, do not hold for the sequences of continuous linear operators on Banachand Fr´echet function spaces).If ( ρ k ) k ∈ N and ˜ X are given in advance, then we define the binary relations ρ ′ k : D ( ρ ′ k ) ⊆ X → Y by D ( ρ ′ k ) := D ( ρ k ) ∩ ˜ X and ρ ′ k x := ρ k x, x ∈ D ( ρ ′ k ) ( k ∈ N ).We can simply prove the following proposition: Proposition 4.2.
Let i ∈ { , } . Then ( ρ k ) k ∈ N is (reiteratively) ˜ X -distributionallychaotic, resp. reiteratively ˜ X -distributionally chaotic of type i /(strong) ˜ X -Li-Yorkechaotic, iff ( ρ ′ k ) k ∈ N is (reiteratively) distributionally chaotic, resp. reiteratively dis-tributionally chaotic of type i /(strong) Li-Yorke chaotic. The same holds for h ˜ X, i i -mixed chaos and ( ˜ X, j ) -mixed chaos, where j ∈ N . In our further work, we will consider only the sequences ( A k ) k ∈ N of MLOs be-tween the spaces X and Y as well as the orbits of an MLO A in X. First of all, wewould like to observe the following:
Remark . Let S := T ∞ k =1 D ( A k ) = { } and let any operator A k be purelymultivalued ( k ∈ N ). Choosing numbers σ > , ǫ > x, y ∈ S of distinct points arbitrarily, we can always find appropriate elements x k ∈ A k x and y k ∈ A k x such that the set { k ∈ N : d Y ( x k , y k ) < σ } is finite and the firstequations in (2.4) and (2.1) automatically hold. Therefore, it is very importantto assume that the second parts in the equations, e.g. (2.4) and (2.1)-(2.3), holdwith the same elements x k ∈ A k x and y k ∈ A k x (not for some other elements x ′ k ∈ A k x and y ′ k ∈ A k x ). If we accept this weaker notion of distributional chaosand reiterative distributional chaos (of type 1 or 2), with different vectors x ′ k ∈ A k x and y ′ k ∈ A k x in the second equality of (2.4) and (2.1)-(2.3), then we will be in aposition to construct a great number of densely distributionally chaotic operatorsand sequences of MLOs. For example, suppose that A ∈ L ( X ) and the linearsubspace X := { x ∈ X : lim k →∞ A k x = 0 } is dense in X. Set A k x := A k x + W k ,k ∈ N , where W k = { } is a subspace of X ( k ∈ N ). Since the first equation in (2.4)holds, setting S := X and x ′ k := A k x, y ′ k := A k y ( k ∈ N , x, y ∈ X ), it readily follows that the sequence ( A k ) k ∈ N will be densely distributionally chaotic in thisweaker sense. In the sequel, we will follow solely the notion in which x ′ k = x k and y ′ k = y k ( k ∈ N ).We can simply verify that the notions of distributional chaos, reiterative distribu-tional chaos, reiterative distributional chaos of type 1 and reiterative distributionalchaos of type 2 do not coincide: Example 4.4.
It is well known that a subset A of N has the upper Banach density1 iff, for every integer d ∈ N , the set A contains infinitely many pairwise disjointintervals of d consecutive integers. Therefore, it is very simple to construct twodisjoint subsets A and B of N such that N = A ∪ B, d ( A ) < Bd ( A ) = Bd ( B ) = 1 . After that, set X := K , T k := kI ( k ∈ A ) and T k := 0 ( k ∈ B ). Thenit can be simply checked that the sequence ( T k ) k ∈ N is reiteratively distributionallychaotic but not reiteratively distributionally chaotic of type 2, as well as that thecorresponding reiteratively scrambled set S can be chosen to be the whole space X. Furthermore, there exist two possible subcases: d ( B ) = 1 or d ( B ) < . In thefirst one, the sequence ( T k ) k ∈ N is reiteratively distributionally chaotic of type 1,while in the second one the sequence ( T k ) k ∈ N is not reiteratively distributionallychaotic of type 1 . Keeping in mind the obvious symmetry between the reiterativedistributional chaos of type 1 and reiterative distributional chaos of type 2 , weobtain the claimed.4.1. Irregular vectors and irregular manifolds.
We start this section by in-troducing the following notion (cf. [6, Definition 18] and [10, Definition 3.4] forsingle-valued linear case):
Definition 4.5.
Suppose that for each k ∈ N , A k : D ( A k ) ⊆ X → Y is an MLO,˜ X is a closed linear subspace of X, x ∈ T ∞ k =1 D ( A k ) and m ∈ N . Then we say that:(i) x is (reiteratively) distributionally near to 0 for ( A k ) k ∈ N iff there exists A ⊆ N such that ( Bd ( A ) = 1) d ( A ) = 1 and for each k ∈ A there exists x k ∈ A k x such that lim k ∈ A,k →∞ x k = 0;(ii) x is (reiteratively) distributionally m -unbounded for ( A k ) k ∈ N iff there exists B ⊆ N such that ( Bd ( B ) = 1) d ( B ) = 1 and for each k ∈ B there exists x ′ k ∈ A k x such that lim k ∈ B,k →∞ p Ym ( x ′ k ) = ∞ ; x is said to be (reiteratively)distributionally unbounded for ( A k ) k ∈ N iff there exists q ∈ N such that x is(reiteratively) distributionally q -unbounded for ( A k ) k ∈ N (if Y is a Banachspace, this simply means that lim k ∈ B,k →∞ k x ′ k k Y = ∞ );(iii) x is a (reiteratively) ˜ X -distributionally irregular vector for ( A k ) k ∈ N iff x ∈ T ∞ k =1 D ( A k ) ∩ ˜ X, (i) holds with with some subset A of N satisfying d ( A ) = 1( Bd ( A ) = 1) and the sequence ( x k ), as well as the second part of (ii) holdswith some subset B of N satisfying d ( B ) = 1 ( Bd ( B ) = 1) and the samesequence ( x ′ k = x k ) as in (i) ( for the sake of brevity, we will assume in anypart (iv) - (xiii) below that x ′ k = x k , with the meaning clear );(iv) x is a reiteratively ˜ X -distributionally irregular vector of type 1 for ( A k ) k ∈ N iff x ∈ ˜ X is distributionally near to zero and x is reiterativelty distribution-ally chaotic for ( A k ) k ∈ N ;(v) x is a reiteratively ˜ X -distributionally irregular vector of type 2 for ( A k ) k ∈ N iff x ∈ ˜ X is reiteratively distributionally near to zero and x is distribution-ally chaotic for ( A k ) k ∈ N ; ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 13 (vi) x is a strong ˜ X -Li-Yorke irregular vector for ( A k ) k ∈ N iff x ∈ T ∞ k =1 D ( A k ) ∩ ˜ X and for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N is unboundedand has a subsequence converging to zero;(vii) x is a ˜ X -Li-Yorke irregular vector for ( A k ) k ∈ N iff x ∈ T ∞ k =1 D ( A k ) ∩ ˜ X andfor each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N does not convergeto zero but it has a subsequence converging to zero;(viii) x is a ( ˜ X, A k ) k ∈ N iff x is reiterativelydistributionally unbounded for ( A k ) k ∈ N and for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N has a subsequence converging to zero;(ix) x is a ( ˜ X, A k ) k ∈ N iff x ∈ ˜ X is distri-butionally unbounded for ( A k ) k ∈ N and for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N has a subsequence converging to zero;(x) x is a ( ˜ X, A k ) k ∈ N iff for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N does not converge to zero and x ∈ ˜ X is reiteratively distributionally near to 0 for ( A k ) k ∈ N ;(xi) x is a ( ˜ X, A k ) k ∈ N iff for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N does not converge to zero and x ∈ ˜ X is distributionally near to 0 for ( A k ) k ∈ N ;(xii) x is a h ˜ X, i -mixed chaotic irregular vector for ( A k ) k ∈ N iff x ∈ T ∞ k =1 D ( A k ) ∩ ˜ X and for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N is unboundedand x is reiteratively distributionally near to 0 for ( A k ) k ∈ N ;(xiii) x is a h ˜ X, i -mixed chaotic irregular vector for ( A k ) k ∈ N iff x ∈ T ∞ k =1 D ( A k ) ∩ ˜ X and for each k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N is unboundedand x is distributionally near to 0 for ( A k ) k ∈ N ;(xiv) x is ˜ X -Li-Yorke near to zero for ( A k ) k ∈ N iff x ∈ T ∞ k =1 D ( A k ) ∩ ˜ X and foreach k ∈ N there exists x k ∈ A k x such that ( x k ) k ∈ N has a subsequenceconverging to zero.If A : D ( A ) ⊆ X → X is an MLO, then x is a (reiteratively) ˜ X -distributionallyirregular vector for A iff x is a (reiteratively) ˜ X -distributionally irregular vector forthe sequence ( A k ≡ A k ) k ∈ N ; we accept this definition for all other parts (iii)-(xiv).Keeping in mind the inequality d ( A ) ≤ Bd ( A ), it readily follows that the state-ments (A)-(C) and all implications clarified in Proposition 2.4 can be formulatedfor irregular vectors introduced above. Further on, we would like to note there aresome important differences between Banach spaces and Fr´echet spaces with regardto the existence of (reiteratively) distributionally unbounded vectors for sequencesof MLOs: Example 4.6. (i) Suppose that the upper (Banach) density of set ˜ B := { k ∈ N : A k is purelly multivalued } is equal to 1 , and Y is a Banach space. Thenany vector x ∈ T ∞ k =1 D ( A k ) is (reiteratively) distributionally unbounded.To see this, observe that in the part (ii) of previous definition we can take B = ˜ B ; then for any k ∈ B, choosing arbitrary x ′ k ∈ A k x, we can always find y ′ k ∈ A k k x k k Y = k x ′ k + y ′ k k Y > k , with x k = x ′ k + y ′ k . (ii) The situation is quite different in the case that Y is a Fr´echet space, weagain assume that the set ˜ B defined above has the upper (Banach) den-sity equal to 1 : Then there need not exist a vector x ∈ T ∞ k =1 D ( A k ) thatis (reiteratively) distributionally m -unbounded for some m ∈ N . To illus-trate this, consider the case in which X := Y := C ( R ) , equipped with the usual topology, and the operator A k is defined by D ( A k ) := X and A k f := f + C [ k, ∞ ) ( R ) , k ∈ N , where C [ k, ∞ ) ( R ) := { f ∈ C ( R ) : supp( f ) ⊆ [ k, ∞ ) } . Then ˜ B = N but for any f ∈ X we have k f + g k Ym = k f k Ym ≡ sup x ∈ [ − m,m ] | f ( x ) | , g ∈ C [ k, ∞ ) ( R ) , m ≤ k. Despite of the above, it should be noted that the existence of a scalar λ ∈ σ p ( A )with | λ | > x ∈ X and any integer k ∈ N we have λ k x ∈ A k x, which in particular shows that x has distributionallyunbounded orbit under A . In [11, Theorem 3.5], we have proved that the hypercyclicity of an MLO A implies σ p ( A ∗ ) = ∅ . This is no longer true for dense Li-Yorke chaos, where we canstate the following (see [8, Proposition 11, Remark 12] for single-valued case):
Proposition 4.7.
Suppose that A is an MLO and λ ∈ σ p ( A ∗ ) satisfies | λ | ≥ . Then A cannot have a dense set of Li-Yorke near to zero vectors.Proof. Suppose the contrary, i.e., there exists a dense set S of Li-Yorke near to zerovectors. Let x ∗ ∈ X ∗ \ { } be such that λx ∗ ∈ A ∗ x ∗ . Then it can be simply shownthat for each x ∈ S and n ∈ N the supposition x n ∈ A n x implies (cid:10) x ∗ , x n (cid:11) = (cid:10) λ n x ∗ , x (cid:11) . (4.5)Take now any x ∈ S such that h x ∗ , x i 6 = 0 . Then there exists a sequence ( x n ) n ∈ N in X such that x n ∈ A n x for all n ∈ N and ( x n ) n ∈ N has a subsequence convergingto zero. By (4.5), it readily follows that | λ | < , which is a contradiction. (cid:3) The following result is a kind of Godefroy-Shapiro and Dech-Schappacher-WebbCriterion for multivalued linear operators:
Theorem 4.8. (cf. [10, Theorem 3.8] ) Suppose that Ω is an open connected subsetof K = C satisfying Ω ∩ S = ∅ . Let f : Ω → X \ { } be an analytic mapping suchthat λf ( λ ) ∈ A f ( λ ) for all λ ∈ Ω . Set ˜ X := span { f ( λ ) : λ ∈ Ω } . Then the operator A | ˜ X is topologically mixing in the space ˜ X and the set of periodic points of A | ˜ X isdense in ˜ X. Now we would like to propose the following problem:
Problem 1.
Suppose that the requirements of Theorem 4.8 hold true. Is it truethat the operator A | ˜ X is densely distributionally chaotic in the space ˜ X ?Assuming that the answer to Problem 1 is affirmative, we will be in a position toconstruct a substantially large class of densely distributionally chaotic MLOs (seee.g. [11, Example 3.10, Example 3.12, Example 3.13]).To state the next problem, let us assume that T ∈ L ( X ) and there exists a denselinear submanifold X of X such that for each x ∈ X one has lim n →∞ T n x = 0 . Then it is well known that the existence of a distributionally unbounded vector x for T (a bounded sequence ( x n ) in X such that the sequence ( T n x n ) is unbounded)implies that there exists a dense distributionally irregular manifold (dense Li-Yorkeirrregular manifold) for T ; see [6, Theorem 15] and [8, Theorem 20]. Now we wouldlike to raise the following issue: Problem 2.
Do there exist similar conditions ensuring dense distributional chaos(dense Li-Yorke chaos) for orbits of MLOs?We continue by introducing the following notion:
ISTRIBUTIONAL CHAOS AND LI-YORKE CHAOS IN METRIC SPACES 15
Definition 4.9.
Let { } 6 = X ′ ⊆ ˜ X be a linear manifold and let i ∈ { , } . Thenwe say that:(i) X ′ is (reiteratively) ˜ X -distributionally irregular manifold, resp. reitera-tively ˜ X -distributionally irregular manifold of type i /(strong) ˜ X -Li-Yorkeirregular manifold for ( A k ) k ∈ N ((reiteratively) distributionally irregular man-ifold, resp. reiteratively distributionally irregular manifold of type i /(strong)Li-Yorke irregular manifold in the case that ˜ X = X ) iff any element x ∈ ( X ′ ∩ T ∞ k =1 D ( A k )) \ { } is a (reiteratively) ˜ X -distributionally irreg-ular vector, resp. reiteratively ˜ X -distributionally irregular vector of type i /(strong) ˜ X -Li-Yorke irregular vector for ( A k ) k ∈ N ;(ii) X ′ is a uniformly (reiteratively) ˜ X -distributionally irregular manifold for( A k ) k ∈ N (uniformly (reiteratively) distributionally irregular manifold in thecase that ˜ X = X ) iff there exists m ∈ N such that any vector x ∈ ( X ′ ∩ T ∞ k =1 D ( A k )) \{ } is both (reiteratively) distributionally m -unbounded and(reiteratively) distributionally near to 0 for ( A k ) k ∈ N . The notions of a uniformly reiteratively ˜ X -distributionally irregular manifold oftype i and a uniformly ( ˜ X, i )-mixed irregular manifold for i ∈ N as well as ( ˜ X, i )-mixed irregular manifold for i ∈ N and h ˜ X, i i -mixed irregular manifold for i ∈ N are introduced analogically. The notion of any type of (uniformly) ˜ X -irregularmanifold for an MLO A : D ( A ) ⊆ X → X is defined as before, by using thesequence ( A k ≡ A k ) k ∈ N .Let i ∈ { , } . Using the elementary properties of metric, it can be simply ver-ified that X ′ is 2 − m ˜ X -(reiteratively) scrambled set for ( A k ) k ∈ N whenever X ′ is auniformly (reiteratively) ˜ X -distributionally irregular manifold for ( A k ) k ∈ N ; a sim-ilar notion holds for uniformly reiteratively X ′ -distributionally irregular manifoldsof type i and uniformly ( ˜ X, i )-mixed irregular manifold for i ∈ N . Clearly, if X ′ is a (strong) ˜ X -Li-Yorke irregular manifold for ( A k ) k ∈ N , then X ′ is a (strong)˜ X -scrambled Li-Yorke set for ( A k ) k ∈ N ; a similar statement holds for ( ˜ X, i )-mixedirregular chaos, where i ∈ N , and h ˜ X, i i -mixed chaos, where i ∈ N . Furthermore,it can be simply verified that, if 0 = x ∈ ˜ X ∩ T ∞ k =1 D ( A k ) is a (reiteratively) ˜ X -distributionally irregular vector, resp. reiteratively ˜ X -distributionally irregular vec-tor of type i /(strong) ˜ X -Li-Yorke irregular vector for ( A k ) k ∈ N , then X ′ ≡ span { x } is a uniformly (reiteratively) ˜ X -distributionally irregular manifold, resp. uniformlyreiteratively ˜ X -distributionally irregular manifold of type i /(strong) ˜ X -Li-Yorke ir-regular manifold) for ( A k ) k ∈ N ; a similar statement holds for ( ˜ X, i )-mixed irregularchaos, where i ∈ N , and h ˜ X, i i -mixed chaos, where i ∈ N . If X ′ is dense in ˜ X, then the notions of dense (reiteratively) ( ˜ X -)distributionallyirregular manifolds, dense uniformly (reiteratively) ( ˜ X -)distributionally irregularmanifolds, and so forth, are defined analogically. The same agreements are acceptedfor all other types of chaos considered above.If ( A k ) k ∈ N and ˜ X are given in advance, then we define the MLOs A k : D ( A k ) ⊆ X → Y by D ( A k ) := D ( A k ) ∩ ˜ X and A k x := A k x, x ∈ D ( A k ) ( k ∈ N ). Then thefollowing holds: Proposition 4.10.
Let i ∈ { , } . (i) A vector x is a (reiteratively) ˜ X -distributionally irregular vector, resp. re-iteratively ˜ X -distributionally irregular vector of type i /(strong) ˜ X -Li-Yorkeirregular vector for ( A k ) k ∈ N iff x is a (reiteratively) distributionally ir-regular vector, resp. reiteratively distributionally irregular vector of type i /(strong) Li-Yorke irregular vector for ( A k ) k ∈ N . The same holds for h ˜ X, i i -mixed chaos and ( ˜ X, j ) -mixed chaos, where j ∈ N . (ii) A linear manifold X ′ is a (uniformly, (reiteratively)) ˜ X -distributionally ir-regular manifold, resp. (uniformly) reiteratively ˜ X -distributionally irregu-lar manifold of type i /(strong) ˜ X -Li-Yorke irregular manifold for ( A k ) k ∈ N iff X ′ is a (uniformly, (reiteratively)) distributionally irregular manifold,resp. (uniformly) reiteratively distributionally irregular manifold of type i /(strong) Li-Yorke irregular manifold for the sequence ( A k ) k ∈ N . The sameholds for ( ˜
X, i ) -mixed chaos. The fundamental distributionally chaotic properties of linear, not necessarilycontinuous, operators have been clarified in [11, Corollary 3.12, Theorem 3.13].The proofs of these results, which are intended solely for the analysis of single-valued operators, lean heavily on the methods and ideas from the theory of C -regularized semigroups (see [18]-[19] and references cited therein for more detailson the subject). For the investigations of distributionally chaotic properties of pureMLOs, we do not have such a powerful technique by now.5. Conclusions and final remarks
In this paper, we have introduced a great number of distributionally chaotic andLi-Yorke chaotic properties for general sequences of binary relations acting betweenmetric spaces. We have carried out a special study of distributionally chaotic andLi-Yorke chaotic multivalued linear operators in Fr´echet spaces, as well, providing agreat number of illustrative examples and observations about problems considered.In a series of recent research studies, N. C. Bernardes Jr. et al and T. Berm´udezet al have analyzed the notions of mean Li-Yorke chaos, absolute Ces`aro bound-edness and Ces`aro hypercyclicity for linear continuous operators in Banach spaces.We close the paper with the observation that these concepts can be analyzed forgeneral sequences of binary relations over metric spaces.
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Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovi´ca 6, 21125Novi Sad, Serbia
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