CC BY
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Chelyabinsk Physical and Mathematical Journal. 2019. Vol. 4, iss. 1. P. 42-58.
DOI: 10.24411/2500-0101-2019-14104
DISTRIBUTIONAL CHAOS AND LI — YORKE CHAOS IN METRIC SPACES
M. Kostic
University of Novi Sad, Novi Sad, Serbia marco.s @verat.net
We introduce several new types and generalizations of the concepts distributional chaos and Li — Yorke chaos. We consider the general sequences of binary relations acting between metric spaces, while in a separate section we focus our attention to some special features of distributionally chaotic and Li — Yorke chaotic multivalued linear operators in Frechet spaces.
Ключевые слова: distributional chaos, Li — Yorke chaos, binary relation, metric space, multivalued linear operator.
1. Introduction and Preliminaries
Let X be a separable Frechet space. A linear operator T on X is said to be hypercyclic iff there exists an element x E D^(T) = ПneND(Tn) whose orbit {Tnx : n E N0} is dense in X; T is said to be topologically transitive, resp. topologically mixing, iff for every pair of open non-empty subsets U, V of X, there exists n0 E N such that Tn0 (U) П V = 0, resp. there exists n0 E N such that, for every n E N with n > n0, Tn(U) П V = 0. A linear operator T on X is said to be chaotic iff it is topologically transitive and the set of periodic points of T, defined by {x E D^(T) : (3n E N) Tnx = x}, is dense in X.
The basic facts about topological dynamics of linear continuous operators in Banach and Frechet spaces can be obtained by consulting the monographs [1] by F.Bayart, E. Matheron and [2] by K.-G. Grosse-Erdmann, A. Peris. In a joint research study with C.-C. Chen, J.A. Conejero and M. Murillo-Arcila [3], the author has recently introduced and analyzed a great deal of topologically dynamical properties for multivalued linear operators (cf. also the article [4] by E. Abakumov, M. Boudabbous and M. Mnif). The notion has been extended in [5] and [6] for general sequences of binary relations over topological spaces.
Distributional chaos for interval maps was introduced by B. Schweizer and J. Smital in [7] (this type of chaos was called strong chaos there, 1994). For linear continuous operators, distributional chaos was firstly investigated in the research studies of quantum harmonic oscillator, by J.Duan et al [8] (1999) and P. Oprocha [9] (2006). Distributional chaos for linear continuous operators in Frechet spaces was analyzed by N.C. Bernardes Jr. et al [10] (2013), while distributional chaos for closed linear operators in Frechet spaces was investigated by J.A. Conejero et al [11] (2016).
On the other hand, the notion of Li — Yorke chaos received enormous attention after the foundational paper of T.Y.Li and J.A. Yorke [12] (1975). Li — Yorke chaotic linear continuous operators on Banach and Frechet spaces have been systematically analyzed
The author is partially supported by grant no. 174024 of Ministry of Science and Technological Development, Republic of Serbia.
in [13] and [14]. For more details about Li — Yorke chaos and distributional chaos in metric and Frechet spaces, we refer the reader to [15-22] and references cited therein.
The main aim of this paper is to introduce various notions of distributional chaos and Li — Yorke chaos for binary relations and their sequences, working in the setting of metric spaces. In particular, we analyze the notions of reiteratively X-distributional chaos of types 1 and 2, (X, i)-mixed chaos, where i = 1, 2, 3, 4, and X-Li — Yorke chaos; here, X is a non-empty subset of metric space X under our consideration. In [23], N.C. Bernardes Jr. et al have considered the notion of distributional chaos of type s for linear continuous operators acting on Banach spaces (s E {1, 2, 22, 3}). In this paper, we extend the notion from this paper for general sequences of binary relations, as well (up to now, the notion from [23] has not been introduced for linear unbounded operators on Banach spaces and their sequences). Finally, we analyze distributionally chaotic and Li — Yorke chaotic multivalued linear operators in Frechet spaces by enquiring into the basic properties of associated distributionally chaotic and Li — Yorke chaotic irregular vectors and their submanifolds. Plenty of useful comments, observations and open problems enriches our study.
Albeit defined in this general framework, we feel duty bound to say that the notion of distributional chaos, its specifications and various generalizations are the most intriguing for orbits of continuous linear operators in Banach spaces. This follows from a series of simple counterexamples presented in this paper, which show in particular that distributional chaos and Li — Yorke chaos occur even for the general sequences of continuous linear operators on finite-dimensional spaces. All aspects and connections between the introduced concepts cannot be easily perceived within just one research paper and our intention here, actually, was to create a solid base for 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 1.1 and Subsection 1.2, we recall the basic things about lower and upper densities as well as binary relations and multivalued linear operators, respectively. In Section 2 and Section 3, we analyze various types of distributional chaos, Li — Yorke chaos and distributional chaos of type s (s E {1, 2, 21, 3}) for binary relations over metric spaces. The fourth section of paper is reserved for the study of distributional chaos and Li — Yorke chaos for multivalued linear operators in Frechet spaces; in a separate subsection, we investigate irregular vectors and irregular manifolds. In addition to the 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 = {dn : n E N}, where (dn)neN is a strictly increasing sequence of positive integers, we define its complement Dc := N \ D and difference set {en := dn+i — dn| n E N}. Let us recall that an infinite subset A of N is said to be syndetic, or relatively dense, iff its difference set is bounded. The difference set of any finite subset of N, defined similarly as above, is finite. Set Nn := {1, ■ ■ ■, n} (n E N) and Si := {z E 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 things about lower and upper densities that will be necessary for our further work.
Let A C N be non-empty. The lower density of A, denoted by d(A), is defined by
d(A) := lim inf |A n [1-n" ,
n—^^o n
and the upper density of A, denoted by d(A), is defined by
:= lim sup J^^. Further on, the lower Banach density of A, denoted by Bd(A), is defined by
Bd(A):= lim liminf |A ° [n + ^ + s]|
s—n—TO s
and the (upper) Banach density of A, denoted by Bd(A), is defined by
-TTu a\ i- V |A n [n +1,n + s]| Bd(A) := lim lim sup -------.
s—n—^^o s
It is well known that the limits appearing in definitions of Bd(A) and Bd(A) exist as s tends to as well as that
0 < Bd(A) < d(A) < d(A) < Bd(A) < 1, (1)
d(A) + d(Ac) = 1
and
Bd(A)+ Bd(Ac) = 1. (2)
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 p C X x Y. If p C X x Y and a C Z x T with Y n Z = 0, then we define p-1 C Y x X and a o p C X x T by p-1 := {(y, x) G Y x X : (x, y) G p} and
a o p := {(x, t) G X x T : 3y G Y n Z such that (x, y) G p and (y, t) G ^,
respectively. Domain and range of p are introduced by D(p) := {x G X : 3y G Y such that (x,y) G p} and R(p) := {y G Y : 3x G X such that (x,y) G p}, respectively; p(x) := {y G Y : (x,y) G p} (x G X), xpy ^ (x, y) G p. If p is a binary relation on X and n G N, then we define pn inductively; p-n := (pn)-1 and p0 := {(x, x) : x G X}. Put D^(p) := P|neND(pn) and p(X') := {y : y G p(x) for some x G X'} (X' C X).
In the remaining part of this subsection, we present a brief overview of the necessary definitions and properties of multivalued linear operators. For more details about the subject, we refer the reader to the monographs [24] by R.Cross and [25] by A. Favini, A.Yagi (in [25], applications of multivalued linear operators to abstract degenerate differential equations have been thoroughly analyzed; for some other approaches, the reader may consult the monograph [26] by G.A. Sviridyuk and V.E. Fedorov).
Let X and Y be two Frechet spaces over the same field of scalars K. For any mapping A : X ^ P(Y) we define A := {(x,y) : x G D(A),y G Ax}. Then A is a multivalued linear operator (MLO) iff the associated binary relation A is a linear relation in X x Y, i.e., iff A is a linear subspace of X x Y. In our work, we will identify A and its associated linear relation A, so that the notion of D(A), which is a linear subspace of X, as well as the sets R(A) and D^(A) are clear. The set A-10 = {x G D(A) : 0 G Ax} is called the kernel of A and it is denoted henceforth by N (A) or Kern(A). The inverse A-1 and the power An of a MLO, introduced in the sense of corresponding definition for general
binary relations, are MLOs (n G N). If X = Y, then we say that A is an MLO in X. An almost immediate consequence of definition is that, for every x, y G D(A) and À, n G K with |À| + |n| = 0, we have ÀAx + nAy = A(Àx + ny). If A is an MLO, then A0 is a linear manifold in Y and Ax = f + A0 for any x G D(A) and f G Ax. The sum A + B of MLOs A and B, defined by D(A + B) := D(A) n D(B) and (A + B)x := Ax + Bx (x G D(A + B)), is likewise an MLO. We write A Ç B iff D(A) Ç D(B) and Ax Ç Bx for all x G D(A). The scalar multiplication of an MLO A : X ^ P(Y) with the number z G K, zA for short, is defined by D(zA) := D(A) and (zA)(x) := zAx, x G D(A). It is clear that zA : X ^ P(Y) is an MLO and (wz)A = w(zA) = z(wA), z,w G K. By a periodic point of A we mean any vector x G Dœ(A) such that there exists n G N with x G Anx.
Suppose that A is an MLO in X. Then we say that a point À G K is an eigenvalue of A iff there exists a vector x G X \ {0} such that Àx G Ax; we call x an eigenvector of operator A corresponding to the eigenvalue À. Observe that, if A is purely multivalued (i.e. A0 = 0), a vector x G X \ {0} can be an eigenvector of operator A corresponding to different values of scalars À. The point spectrum of A, op (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* := j (y*,x*) G Y * x X * : (y*, y) = (x*,x) for all pairs (x,y) G Aj.
2. Distributional chaos and Li — Yorke chaos for binary relations
In this section, it will be always assumed that (X, d) and (Y, dy) are metric spaces. Suppose that o > 0, e > 0 and (xk)keN, (yk)keN are two given sequences in Y. Consider the following conditions:
Bd({k G N : dy (xfc,yfc) < o}) = 0,
V ' (3)
{k G N : dy (xfc,yfc) > e}) =0;
B^f{k G N : dy (xfc,yfc) < o}) =0,
(4)
^{k G N : dy (xfc,yfc) > ^ =0; ^{k G N : dy (xfc,yfc) < ^ = 0,
} (5)
Bd({k G N : dy (xfc,yfc) > e^ =0
and
d({k G N : dy (xfc,yfc) < ^ = 0, ; < (6) d({k G N : dy (xfc,yk) > ^ =0.
In the following definition, we introduce the notion of reiterative distributional chaos (reiterative distributional chaos of type 1 or 2):
Definition 1. Suppose that, for every k G N, pk Ç X x Y is a binary relation and XX is a non-empty subset of X. If there exist an uncountable set S Ç P|D(pk) n X and o > 0 such that for each e > 0 and for each pair x, y G S of distinct points we have that for each k G N there exist elements xk G pkx and yk G pky such that (6) holds, resp. (3) [(4)/(5)] holds, then we say that the sequence (pk)keN is X-distributionally chaotic,
resp. X-reiteratively distributionally chaotic [X-reiteratively distributionally chaotic of type 1/X-reiteratively distributionally chaotic of type 2].
The sequence (pk )keN is said to be densely X-distributionally chaotic, resp. X-reiteratively distributionally chaotic [X-reiteratively distributionally chaotic of type 1/X-reiteratively distributionally chaotic of type 2], iff S can be chosen to be dense in X. A binary relation p C X x X is said to be (densely) X-distributionally chaotic, resp. X-reiteratively distributionally chaotic [X-reiteratively distributionally chaotic of type 1/X-reiteratively distributionally chaotic of type 2], iff the sequence (pk = pk)keN is. The set S is said to be a-£-scrambled set, resp. aj-reiteratively scrambled set [aj-reiteratively scrambled set of type 1/aj-reiteratively scrambled set of type 2] (a-scrambled set, resp. a-reiteratively scrambled set [a-reiteratively scrambled set of type 1/a-reiteratively scrambled set of type 2], in the case that X = X) of the sequence (pk)keN (the binary relation p); in the case that X = X, then we also say that the sequence (pk)keN (the binary relation p) is distributionally chaotic, resp. reiteratively distributionally chaotic [reiteratively distributionally chaotic of type 1 /reiteratively distributionally chaotic of type 2].
It is well known that, for any infinite set A C N, being syndetic and having a positive Banach density is the same thing. Therefore, if the sets {k G N : dY(xk, yk) < a} and {k G N : dY(xk, yk) > e} are infinite, then they have unbounded difference sets iff (3) holds. If one of these sets is finite, say the first one, then there exists k0 = k0(a) G N such that [k0, ro) C {k G N : dY(xk,yk) > e} and the second equality in (3) cannot be satisfied. Therefore, in definition of X-reiterative distributional chaos, we can equivalently replace the equation (3) with the statements that the difference sets of {k G N : dY(xk, yk) < a} and {k G N : dY(xk, yk) > e} are unbounded.
The following definition seems to be new even for the sequences of linear not continuous operators on Banach and Frechet spaces as well as for the sequences of linear continuous operators that are not orbits of one single operator:
Definition 2. Suppose that, for every k G N, pk C X x Y is a binary relation and X is a non-empty subset of X. If there exist an uncountable set S C P|D(pk) n X such that for each pair x, y G S of distinct points and for each k G N there exist elements xk G pkx and yk G pky such that
liminf dY (xk , yk) = 0 and limsup dY (xk, yk) > 0,
k—TO k—>-(TO
then we say that the sequence (pk)keN is X-Li — Yorke chaotic.
The sequence (pk)keN is said to be densely X-Li — Yorke chaotic iff S can be chosen to be dense in X. A binary relation p C X x X is said to be (densely) X-Li — Yorke chaotic iff the sequence (pk = pk)keN 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 (pk)keN (the binary relation p); in the case that X = X, then we also say that the sequence (pk)keN (the binary relation p) is (densely) Li — Yorke chaotic.
Besides the conditions introduced so far, we can also examine the following ones:
Bd({k G N : (xfc,yfc) < a}j =0 and liminf (xfc,yfc) = 0, (7)
V / k—
G N : dy (xk< a} j =0 and liminf dy (xk, y^ =0, (8)
\ J k—^TO
limsupdY(xk> 0 and Bdi {k G N : dY(xk> ^ ) =0, (9)
k — TO ^ '
limsup dy (xfc,yk) > 0 and d( {k G N : dy (xfc,yfc) > e} =0. (10)
The following definition is meaningful, as well:
Definition 3. Suppose that, for every k G N, pk C X x Y is a binary relation and X is a non-empty subset of X. If there exist an uncountable set S C P|D(pk) n X and a > 0 such that for each e > 0 and for each pair x, y G S of distinct points we have that for each k G N there exist elements xk G pkx and yk G pky such that (7)/(10) holds, then we say that the sequence (pk)keN is (X, l)-mixed chaotic/(X, 4)-mixed chaotic.
The notion of densely (X,i)-mixed chaotic sequence (pk)keN (the binary relation p), where i G N4, the corresponding (a^,i)-mixed scrambled set ((a, i)-mixed scrambled set, in the case that X = X), where i G N2, the corresponding (X,i)-mixed scrambled set (i-mixed scrambled set, in the case that X = X), where i G N4 \ N2, of the sequence (pk)fceN (the binary relation p) is introduced as above; in the case that X = X and i G N4, then we also say that the sequence (pk)keN (the binary relation p) is i-mixed chaotic.
Keeping in mind (1) and (2), an elementary line of reasoning shows that any X-distributionally chaotic sequence (binary relation) is already X-reiteratively distributionally chaotic as well as that any X-reiteratively distributionally chaotic sequence (binary relation) is both X-reiteratively distributionally chaotic of type 1 and X-reiteratively distributionally chaotic of type 2. It is also predictable that any X-reiteratively distributionally chaotic sequence of binary relations (any X-reiteratively distributionally chaotic binary relation) needs to be X-Li — Yorke chaotic. To see this, suppose that the sequence (pk)keN is X-reiteratively distributionally chaotic with given ax-scrambled set S. Let x = y — z for some two different elements y, z G S. Due to our assumption, for each number k G N it is very simple to construct two strictly increasing sequences (sn,k)neN and (/n,k)neN of pairwise disjoint positive integers such that dY (xSn k, 0) > a and dY (x1n k, 0) < 1/k for all n G N as well as /k+1,k+1 > Zk,k + 2k, nfc+i,fc+i > nfc,fc + 2k for all k G N and sM < /1,1 < ^2,2 < ¿2,2 < ...; here, xSn,k G pSn,kx and xin,k G pin,kx. If n G UfceN{sfc,fc, lk,k}, then we take any vector x„ G p„x, which clearly exists because the set P|D(pk) nX is at least countable. If n = sk,k (n = /k,k) for some k G N, then we set
xn : - xS, J, (xn := x1k k). Then it is clear that liminfra^TO dY(xn, 0) = 0
and limsupn^TO dY(xn, 0) > 0 because dY (x1n,n, 0) = 0 and the subsequence
(dY(xSn,n, 0)) of (dY(xn, 0)) is bounded away from zero. On the other hand, it is clear that (X, i)-mixed chaos implies X-Li —Yorke chaos for any i G N4; the above conclusions also hold for any kind of dense X-chaos considered above.
Therefore, we have proved the following proposition:
Proposition 1. Let for each k G N we have that pk C X x Y is a binary relation, and let X be a non-empty subset of X. Consider the following statements:
(i) (pk)keN is (densely) X-distributionally chaotic;
(ii) (pk)keN is (densely) reiteratively X-distributionally chaotic of type 1;
(iii) (pk)keN is (densely) reiteratively X-distributionally chaotic of type 2;
(iv) (pk)keN is (densely) reiteratively X-distributionally chaotic;
(v) (pk)keN is (densely) (X, 1)-mixed chaotic;
(vi) (pk)keN is (densely) (X, 2)-mixed chaotic;
(vii) (pk)keN is (densely) (X, 3)-mixed chaotic;
(viii) (pk)keN is (densely) (X, 4)-mixed chaotic;
(ix) (pk)keN is (densely) X-Li — Yorke chaotic.
Then (i) implies (ii)-(ix); (ii) implies (iv),(v), (vii)-(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 well as in Definition 4 and Definition 3 below, do not coincide even for the sequences of continuous linear operators on finite-dimensional spaces. This can be inspected as in Example 1 below.
3. Distributional chaos of type s for binary relations
(s G {1, 2, 22,3})
As in the previous one, in this section we assume that (X, d) and (Y, dy) are metric spaces. Suppose that o, o' > 0, e > 0 and (xk)keN, (yk)fceN are two given sequences in Y. Consider the following conditions:
Bd({k G N : dy (xfc,yfc) > o}) > 0, Bd({k G N : dy (xfc,yfc) > e}) =0;
Bd({k G N : dy (xfc,yfc) > o}) > 0, {k G N : dy (xfc,yfc) > e}) =0; {k G N : dy (xfc,yfc) > o}) > 0, Bd({k G N : dy (xfc,yfc) > e}) =0; {k G N : dy (xfc,yfc) > o}) > 0,
{k G N : dy (xfc,yfc) > e}) =0; there exist c > 0 and r > 0 such that
Bd({k G N : dy(xk,yk) < o}j < c < Bd({k G N : dy (xk,yk) < o}) (15
for 0 < o < r;
there exist c > 0 and r > 0 such that
for 0 < o < r;
there exist c > 0 and r > 0 such that
for 0 < o < r;
there exist c > 0 and r > 0 such that
11)
12)
13)
14)
Bd({k G N : dy (xfc,yfc) < o}) < c < d({k G N : dy (xfc,yfc) < o}) (16)
d({k G N : dy (xfc,yfc) < o}) < c < Bd({k G N : dy (xfc,yfc) < o}) (17)
d({k G N : dy (xk,yk) < a}) < c< d({k G N : dy (xk,yk) < a}) (18)
for 0 < a < r;
there exist a, b, c > 0 such that (15) holds for all a G [a, b]; (19)
there exist a, b, c > 0 such that (16) holds for all a G [a, b]; (20)
there exist a, b, c > 0 such that (17) holds for all a G [a, b]; (21)
there exist a, b, c > 0 such that (18) holds for all a G [a, b]. (22)
Let i G {1, 2}. For (reiterative) distributional chaos (reiterative distributional chaos of type i) it is also said that it is (reiterative) distributional chaos of type 0; 1 (reiterative distributional chaos of type i; 1). Now we would like to propose the following notion:
Definition 4. Let i G {0,1, 2} and s G {1, 2, 22,3}. Suppose that, for every k G N, pk C X x Y is a binary relation and X is a non-empty subset of X.
(i) If there exist an uncountable set S C P|D(pk) n X and a > 0 such that for each e > 0 and for each pair x, y G S of distinct points we have that for each k G N there exist elements xk G pkx and yk G pky such that (11) [(12)/(13)/(14)] holds, then we say that the sequence (pk)keN is reiteratively X-distributionally chaotic of type 0; 2 [reiteratively X-distributionally chaotic of type 1; 2/reiteratively X-distributionally chaotic of type 2; 2/X-distributionally chaotic of type 0; 2].
(ii) If there exist an uncountable set S C P|D(pk) n X and numbers a, c, r > 0 such that for each e > 0 and for each pair x, y G S of distinct points we have that for each k G N there exist elements xk G pkx and yk G pky such that (15) [(16)/(17)/(18)] holds for 0 < a < r, then we say that the sequence (pk)keN is reiteratively X-distributionally chaotic of type 22 [reiteratively X-distributionally chaotic of type 1;22/reiteratively X-distributionally chaotic of type 2; 22/X-distributionally chaotic of type 21].
(iii) If there exist an uncountable set S C flTO=1 D(pk)nX and real numbers a, a, b, c > 0 such that for each e > 0 and for each pair x, y G S of distinct points we have that for each k G N there exist elements xk G pkx and yk G pky such that (19) [(20)/(21)/(22)] holds for a < a < b, then we say that the sequence (pk)keN is reiteratively X-distributionally chaotic of type 3 [reiteratively X-distributionally chaotic of type 1;3/reiteratively X-distributionally chaotic of type 2; 3/X-distributionally chaotic of type 3].
The sequence (pk)keN is said to be densely (reiteratively) X-distributionally chaotic of type i; s iff S can be chosen to be dense in X. A binary relation p C X x X is said to be (densely) reiteratively X-distributionally chaotic of type i; s iff the sequence (pk = pk)keN is. The set S is said to be (reiteratively) (aj, s)-scrambled set ((reiteratively) (a, s)-scrambled set, in the case that X = X) of the sequence (pk)keN (the binary relation p); in the case that X = X, then we also say that the sequence (pk)keN (the binary relation p) is densely (reiteratively) distributionally chaotic of type i; s.
Keeping in mind the inequality (1), we are in a position to immediately clarify the following:
Proposition 2. Suppose that i G {0,1, 2}, s,s1,s2 G {1, 2, 21, 3}, s1 < s2, X is a nonempty subset of X and (pk)keN is a given sequence of binary relations. Then, for (pk)keN, we have the following:
(dense, reiterative) X-distributional chaos of type i; s1 implies (dense, reiterative)
X-distributional chaos of type i; s2;
(dense) X-distributional chaos of type 0; s implies (dense) reiterative X-distributional
chaos of types 1; s and 2; s;
(dense) reiterative X-distributional chaos of type 1; s or 2; s implies (dense) reiterative
X-distributional chaos of type 0; s.
As it is well known, X-distributional chaos of type 3 is a very weak form of linear chaos: it is still unknown whether there exists a complex matrix that is distributionally chaotic of type 3 (cf. [23, Problem 51]). The same problem can be posed for reiterative distributional chaos of type 3.
Concerning the relation of distributional chaos and distributional chaos of type 2, it is worth noting that these two notions are equivalent for linear continuous operators on Banach spaces (see [23, Theorem 2]). As mentioned at the end of previous section, this is far from being true for general sequences of linear continuous operators.
A further analysis of distributional chaos of type s and their generalizations will be carried out somewhere else.
4. Distributional chaos and Li — Yorke chaos in Frechet spaces
In the remaining part of the paper, we assume that X is an infinite-dimensional Frechet space over the field K G {R, C} and that the topology of X is induced by the fundamental system (pn)raeN of increasing seminorms (separability of X is not assumed a priori in future). Then the translation invariant metric d : X x X ^ [0, ro), defined by
d(x, y) := £ 2n 1+n(x(-y) ), x,y G X, (23)
2n 1+ p„(x - y)
enjoys the following properties:
d(x + u, y + v) < d(x,y) + d(u, v), x,y,u, v G X, d(cx, cy) < (|c| + 1)d(x,y), c G K, x,y G X,
and
d(ax,^x) > ——d(0,x), x G X, a,^ G K. 1 + |a — p|
By Y we denote another Frechet space over the same field of scalars as X; the topology of Y will be induced by the fundamental system (p^)neN of increasing seminorms. Define the translation invariant metric dy : Y x Y ^ [0, ro) by replacing pn(-) with p^(') in (23). If (X, || ■ ||) or (Y, || ■ ||y) is a Banach space, then we assume that the distance of two elements x, y G X (x, y G Y) is given by d(x, y) := ||x — y|| (dy(x, y) := ||x — y||y). Keeping in mind this terminological change, our structural results clarified in Frechet spaces continue to hold in the case that X or Y is a Banach space. By L(X, Y) we denote the space consisting of all linear continuous mappings from X into Y; L(X) = L(X, X). In Frechet spaces, of importance are the following conditions:
the sequence (xk — yk)keN is unbounded and liminf dy (xk,yk) = 0, (24)
the sequence (xk — yk)keN is unbounded and G N : dy (xk,yk) > ^^ = 0, (25)
the sequence (xk — yk)keN is unbounded and G N : dy (xk,yk) > ^^ = 0. (26)
Albeit the most intriguing for multivalued linear operators, the following notion can be introduced for general binary relations, as well:
Definition 5. Suppose that, for every k G N, pk C X x Y is a binary relation and X is a non-empty subset of X. If there exists an uncountable set S C P|D(pk) if X such that for each pair x,y G S of distinct points and for each e > 0 we have that for each k G N there exist elements xk G pkx and yk G pky such that (24) holds, resp. (25) [(26)] holds, then we say that the sequence (pk)keN is strongly X-Li — Yorke chaotic, resp. (X, 1)-mixed chaotic [(X, 2)-mixed chaotic].
The notion of densely strong X-Li — Yorke chaotic sequence, resp. densely (X, i)-mixed chaotic sequence (pk)keN (the binary relation p), where i G N2, the corresponding strong X-Li — Yorke scrambled set, resp. (X,i)-mixed scrambled set (strong Li — Yorke scrambled set, resp. (i)-mixed scrambled set, in the case that X = X), where i G N2, of the sequence (pk)k^N (the binary relation p) is introduced as above; in the case that X = X and i G N2, then we also say that the sequence (pk)keN (the binary relation p) is strong Li — Yorke chaotic, resp. i-mixed chaotic.
With the exception of implications clarified in Proposition 1, we can only state the following ones, in general:
(A) strong X-Li — Yorke chaos implies X-Li — Yorke chaos;
(B) (X, 1)-mixed chaos implies strong X-Li — Yorke chaos;
(C) (X, 2)-mixed chaos implies (X, 1)-mixed chaos and strong X-Li — Yorke chaos.
We continue by observing the following: If X is a Banach space and T G L(X), then T is (densely) Li — Yorke chaotic iff T is (densely) reiteratively distributionally chaotic. To see this, let us recall that we have ||T|| > 1 due to the fact that T is Li — Yorke chaotic. By [13, Theorem 5], we have the existence of a vector x G X such that liminfra^^ ||Tnx|| = 0 and limsupn^^ ||Tnx|| = ro. Therefore, there exist two strictly increasing sequences of positive integers (nk) and (1k) such that min(nk+1 —nk, 1k+1—1k) > k2, ||Tnkx|| < 2-k2 and ||T1kx|| > 2k2 for all k G N. Put A := (JkeN[nk, nk + k] and B := UkeN[1k, 1k — k]. Then Bd(A) = Bd(B) = 1 and for each n G A (n G B) there exists k G N such that n G K, nk + k] (n G [Ik, Ik — k]) and therefore ||Tnx|| < (1 + ||T||)k2-k2 (||Tnx|| > ||T||-k2k ). This, in turn, implies that the notions of 1-mixed chaos, 3-mixed chaos, reiterative distributional chaos and Li — Yorke chaos coincide in this case (this also holds for dense analogues).
On the other hand, as already mentioned, the situation is completely different for the sequences of continuous linear operators on finite-dimensional spaces. Take, for instance, Tk = 0 if k is even and Tk = I if k is odd. Then the sequence (Tk) is Li — Yorke chaotic on ony Frechet space X but it is not strongly Li — Yorke chaotic (this trivial counterexample also shows that the assertions of [13, Theorem 5] and [14, Theorem 9], where it has been proved that the notions of (dense) Li — Yorke chaos and (dense) strong Li — Yorke chaos coincide for the orbits of linear continuous operators, do not hold for the sequences of continuous linear operators on Banach and Frechet function spaces).
If (pk)keN and X are given in advance, then we define the binary relations p'k : D(pk) C X ^ Y by D(pk) := D(pk) f X and p'kx := pkx, x G D(p'k) (k G N). We can simply prove the following proposition:
Proposition 3. Let i G {1, 2}. Then (pk)keN is (reiteratively) X-distributionally chaotic, resp. reiteratively X-distributionally chaotic of type i/(strong) X-Li — Yorke chaotic, iff (pk)keN is (reiteratively) distributionally chaotic, resp. reiteratively distributionally chaotic of type i/(strong) Li — Yorke chaotic. The same holds for (X,i)-mixed chaos and (X, j)-mixed chaos, where j G N4.
In our further work, we will consider only the sequences (Ak)k^N of MLOs between the spaces X and Y as well as the orbits of an MLO A in X. First of all, we would like to observe the following:
Remark 1. Let S := P|TO=1 D(Ak) = {0} and let any operator Ak be purely multivalued (k G N). Choosing numbers a > 0, e > 0 and each pair x,y G S of distinct points arbitrarily, we can always find appropriate elements xk G Akx and yk G Akx such that the set {k G N : dY(xk,yk) < a} is finite and the first equations in (6) and (3) automatically hold. Therefore, it is very important to assume that the second parts in the equations, e. g. (6) and (3)-(5), hold with the same elements xk G Akx and yk G Akx (not for some other elements x'k G Akx and yk G Akx). If we accept this weaker notion of distributional chaos and reiterative distributional chaos (of type 1 or 2), with different vectors x'k G Akx and yk G Akx in the second equality of (6) and (3)-(5), then we will be in a position to construct a great number of densely distributionally chaotic operators and sequences of MLOs. For example, suppose that A G L(X) and the linear subspace X0 := {x G X : limk—TO Akx = 0} is dense in X. Set Akx := Akx + Wk, k G N, where Wk = {0} is a subspace of X (k G N). Since the first equation in (6) holds, setting S := X0 and x'k := Akx, yk := Aky (k G N, x,y G X0), it readily follows that the sequence (Ak)k^N will be densely distributionally chaotic in this weaker sense. In the sequel, we will follow solely the notion in which x'k = xk and yk = yk (k G N).
We can simply verify that the notions of distributional chaos, reiterative distributional chaos, reiterative distributional chaos of type 1 and reiterative distributional chaos of type 2 do not coincide:
Example 1. It is well known that a subset A of N has the upper Banach density 1 iff, for every integer d G N, the set A contains infinitely many pairwise disjoint intervals of d consecutive integers. Therefore, it is very simple to construct two disjoint subsets A and B of N such that N = A U B, d(A) < 1 and Bd(A) = Bd(B) = 1. After that, set X := K, Tk := kI (k G A) and Tk := 0 (k G B). Then it can be simply checked that the sequence (Tk)keN is reiteratively distributionally chaotic but not reiteratively distributionally chaotic of type 2, as well as that the corresponding 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) < 1. In the first one, the sequence (Tk)keN is reiteratively distributionally chaotic of type 1, while in the second one the sequence (Tk)keN is not reiteratively distributionally chaotic of type 1. Keeping in mind the obvious symmetry between the reiterative distributional chaos of type 1 and reiterative distributional chaos of type 2, we obtain the claimed.
4.1. Irregular vectors and irregular manifolds
We start this section by introducing the following notion (cf. [10, Definition 18] and [11, Definition 3.4] for single-valued linear case):
Definition 6. Suppose that for each k G N, Ak : D(Ak) C X ^ Y is an MLO, X is a closed linear subspace of X, x G P|TO=1 D(Ak) and m G N. Then we say that:
(i) x is (reiteratively) distributionally near to 0 for (Ak)keN iff there exists A C N such that (Bd(A) = 1) d(A) = 1 and for each k G A there exists xk G Akx such that
limkeA,k—TO xk = 0;
(ii) x is (reiteratively) distributionally m-unbounded for (Ak)k^N iff there exists B C N such that (Bd(B) = 1) d(B) = 1 and for each k G B there exists x'k G Akx such that limkeB,k—TO ^(xk) = ro; x is said to be (reiteratively) distributionally
unbounded for (Ak)k^N iff there exists q G N such that x is (reiteratively) distributionally q-unbounded for (Ak)keN (if Y is a Banach space, this simply means that limkeB,k^^ ||x'k||Y = ro);
(iii) e is a (reiteratively) X-distributionally irregular vector for (Ak)k^N iff x G nr= 1 D(Ak) n X, (i) holds with with some subset A of N satisfying d(A) = 1 (Bd(A) = 1) and the sequence (xk), as well as the second part of (ii) holds with some subset B of N satisfying d(B) = 1 (Bd(B) = 1) and the same sequence (xk = xk) as in (i) (for the sake of brevity, we will assume in any part (iv)-(xiii) below that x'k = xk, with the meaning clear);
(iv) x is a reiteratively X-distributionally irregular vector of type 1 for (Ak)keN iff x G X is distributionally near to zero and x is reiterativelty distributionally chaotic for (Ak )keN;
(v) x is a reiteratively X-distributionally irregular vector of type 2 for (Ak)k^N iff x G X is reiteratively distributionally near to zero and x is distributionally chaotic for (Ak )keN;
(vi) x is a strong X-Li — Yorke irregular vector for (Ak)k^N iff x G P|D(Ak) n X and for each k G N there exists xk G Akx such that (xk)keN is unbounded and has a subsequence converging to zero;
(vii) x is a X-Li — Yorke irregular vector for (Ak)k^N iff x G P|D(Ak) n X and for each k G N there exists xk G Akx such that (xk)keN does not converge to zero but it has a subsequence converging to zero;
(viii) x is a (X, 1)-distributionally irregular vector for (Ak)keN iff x is reiteratively distributionally unbounded for (Ak)keN and for each k G N there exists xk G Akx such that (xk)keN has a subsequence converging to zero;
(ix) x is a (X, 2)-distributionally irregular vector for (Ak)keN iff x G XX is distributionally unbounded for (Ak)keN and for each k G N there exists xk G Akx such that (xk)keN has a subsequence converging to zero;
(x) x is a (XX, 3)-distributionally irregular vector for (Ak)k^N iff for each k G N there exists xk G Akx such that (xk)keN does not converge to zero and x G XX is reiteratively distributionally near to 0 for (Ak)keN;
(xi) x is a (XX, 4)-distributionally irregular vector for (Ak)k^N iff for each k G N there exists xk G Akx such that (xk)keN does not converge to zero and x G XX is distributionally near to 0 for (Ak)keN;
(xii) x is a (XX, 1)-mixed chaotic irregular vector for (Ak)k^N iff x G P|D(Ak) n X and for each k G N there exists xk G Akx such that (xk)keN is unbounded and x is reiteratively distributionally near to 0 for (Ak)keN;
(xiii) x is a (XX, 2)-mixed chaotic irregular vector for (Ak)keN iff x G P|D(Ak) n XX and for each k G N there exists xk G Akx such that (xk)keN is unbounded and x is distributionally near to 0 for (Ak)keN;
(xiv) x is X-Li — Yorke near to zero for (Ak)keN iff x G P|D(Ak) n XX and for each k G N there exists xk G Akx such that (xk)keN has a subsequence converging to zero.
If A : D(A) C X ^ X is an MLO, then x is a (reiteratively) X-distributionally irregular vector for A iff x is a (reiteratively) X-distributionally irregular vector for the sequence (Ak = Ak)keN; we accept this definition for all other parts (iii)-(xiv).
Keeping in mind the inequality d(A) < Bd(A), it readily follows that the statements (A)-(C) and all implications clarified in Proposition 1 can be formulated for irregular vectors introduced above. Further on, we would like to note there are some important
differences between Banach spaces and Frechet spaces with regard to the existence of (reiteratively) distributionally unbounded vectors for sequences of MLOs:
Example 2.
(i) Suppose that the upper (Banach) density of set B := {k G N : Ak is purelly multivalued} is equal to 1, and Y is a Banach space. Then any vector x G P|D(Ak) 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 G B, choosing arbitrary x'k G Akx, we can always find yk G Ak0 such that we have
||xkHy = ||xk + ykHY > 2k, with xk = xk + yk.
(ii) The situation is quite different in the case that Y is a Frechet space, we again assume that the set B defined above has the upper (Banach) density equal to 1: Then there need not exist a vector x G P|D(Ak) that is (reiteratively) distributionally m-unbounded for some m G N. To illustrate this, consider the case in which X := Y := C(R), equipped with the usual topology, and the operator Ak is defined by D(Ak) := X and Akf := f + C[k,^)(R), k G N, where C[k,<x>)(R) := {f G C(R) : supp(f) C [k, ro)}. Then B = N but for any f G X we have ||f + g^ = Hf H|m = SUPxe[-m,m] |f(x)1, g G C^)^ m < k.
Despite of the above, it should be noted that the existence of a scalar A G ap(A) with |A| > 1 implies that for any corresponding eigenvector x G X and any integer k G N we have Akx G Akx, which in particular shows that x has distributionally unbounded orbit under A.
In [3, Theorem 3.5], we have proved that the hypercyclicity of an MLO A implies ap(A*) = 0. This is no longer true for dense Li — Yorke chaos, where we can state the following (see [14, Proposition 11, Remark 12] for single-valued case):
Proposition 4. Suppose that A is an MLO and A G ap(A*) satisfies |A| > 1. 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 zero vectors. Let x* G X* \ {0} be such that Ax* G A*x*. Then it can be simply shown that for each x G S and n G N the supposition xn G Anx implies
<x*,x„> = <Anx*,x>. (27)
Take now any x G S such that (x*,x) = 0. Then there exists a sequence (xn)neN in X such that xn G Anx for all n G N and (xn)neN has a subsequence converging to zero. By (27), it readily follows that |A| < 1, which is a contradiction. □
The following result is a kind of Godefroy — Shapiro and Dech — Schappacher — Webb Criterion for multivalued linear operators:
Theorem 2. (cf. [11, Theorem 3.8]). Suppose that Q is an open connected subset of K = C satisfying Q f S1 = 0. Let f : Q ^ X \ {0} be an analytic mapping such that Af (A) G Af (A) for all A G Q. Set X := span{f (A) : A G Q}. Then the operator A|X is topologically mixing in the space X and the set of periodic points of A|X is dense in X.
Now we would like to propose the following problem:
Problem 1. Suppose that the requirements of Theorem 2 hold true. Is it true that 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 to construct a substantially large class of densely distributionally chaotic MLOs (see e. g. [3, Example 3.10, Example 3.12, Example 3.13]).
To state the next problem, let us assume that T G L(X) and there exists a dense linear submanifold Xo of X such that for each x G Xo one has Tnx = 0. Then
it is well known that the existence of a distributionally unbounded vector x for T (a bounded sequence (xn) in X such that the sequence (Tnxn) is unbounded) implies that there exists a dense distributionally irregular manifold (dense Li — Yorke irrregular manifold) for T; see [10, Theorem 15] and [14, Theorem 20]. Now we would like 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:
Definition 7. Let {0} = X' C X be a linear manifold and let i G {1, 2}. Then we say that:
(i) X' is (reiteratively) X-distributionally irregular manifold, resp. reiteratively X-distributionally irregular manifold of type i/(strong) X-Li — Yorke irregular manifold for (Ak)k^N ((reiteratively) distributionally irregular manifold, resp. reiteratively distributionally irregular manifold of type i/(strong) Li — Yorke irregular manifold in the case that X = X) iff any element x G (X'ifQD(Ak)) \ {0} is a (reiteratively) X-distributionally irregular vector, resp. reiteratively X-distributionally irregular vector of type i/(strong) X-Li — Yorke irregular vector for (Ak)keN;
(ii) X' is a uniformly (reiteratively) X-distributionally irregular manifold for (Ak)keN (uniformly (reiteratively) distributionally irregular manifold in the case that X = X) iff there exists m G N such that any vector x G (X'ifQD(Ak)) \{0} is both (reiteratively) distributionally m-unbounded and (reiteratively) distributionally near to 0 for (Ak)keN.
The notions of a uniformly reiteratively X-distributionally irregular manifold of type i and a uniformly (X,i)-mixed irregular manifold for i G N2 as well as (X,i)-mixed irregular manifold for i G N4 and (X,i)-mixed irregular manifold for i G N2 are introduced analogically. The notion of any type of (uniformly) X-irregular manifold for an MLO A : D(A) C X ^ X is defined as before, by using the sequence (Ak = Ak)keN.
Let i G {1, 2}. Using the elementary properties of metric, it can be simply verified that X' is 2xm-(reiteratively) scrambled set for (Ak)k^N whenever X' is a uniformly (reiteratively) X-distributionally irregular manifold for (Ak)keN; a similar notion holds for uniformly reiteratively X'-distributionally irregular manifolds of type i and uniformly (X,i)-mixed irregular manifold for i G N2. Clearly, if X' is a (strong) X-Li — Yorke irregular manifold for (Ak)keN, then X' is a (strong) X-scrambled Li — Yorke set for (Ak)keN; a similar statement holds for (X,i)-mixed irregular chaos, where i G N4, and (X, i)-mixed chaos, where i G N2. Furthermore, it can be simply verified that, if 0 = x G X if P|D(Ak) is a (reiteratively) X-distributionally irregular vector, resp. reiteratively X-distributionally irregular vector of type i/(strong) X-Li — Yorke irregular vector for (Ak)keN, then X' = span{x} is a uniformly (reiteratively) X-distributionally irregular manifold, resp. uniformly reiteratively X-distributionally irregular manifold of type i/(strong) X-Li — Yorke irregular manifold) for (Ak)keN; a similar statement holds for (X, i)-mixed irregular chaos, where i G N4, and (X, i)-mixed chaos, where i G N2.
If X' is dense in X, then the notions of dense (reiteratively) (X-)distributionally irregular manifolds, dense uniformly (reiteratively) (X-)distributionally irregular manifolds, and so forth, are defined analogically. The same agreements are accepted for all other types of chaos considered above.
If (Ak)keN and X are given in advance, then we define the MLOs Ak : D(Ak) C X ^ Y by D(Ak) := D(Ak) n X and Akx := Akx, x G D(Ak) (k G N). Then the following holds:
Proposition 5. Let i G {1, 2}.
(i) A vector x is a (reiteratively) X-distributionally irregular vector, resp. reiteratively X-distributionally irregular vector of type i/(strong) X-Li — Yorke irregular vector for (Ak)keN iff x is a (reiteratively) distributionally irregular vector, resp. reiteratively distributionally irregular vector of type i/(strong) Li — Yorke irregular vector for (Ak)keN. The same holds for (XX, i)-mixed chaos and (XX, j)-mixed chaos, where j G N4.
(ii) A linear manifold X' is a (uniformly, (reiteratively)) XX-distributionally irregular manifold, resp. (uniformly) reiteratively XX-distributionally irregular manifold of type i/(strong) X-Li — Yorke irregular manifold for (Ak)keN 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 (Ak)keN. The same holds for (X,i)-mixed chaos.
The fundamental distributionally chaotic properties of linear, not necessarily continuous, operators have been clarified in [3, 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 [27, 28] and references cited therein for more details on the subject). For the investigations of distributionally chaotic properties of pure MLOs, 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 and Li — Yorke chaotic properties for general sequences of binary relations acting between metric spaces. We have carried out a special study of distributionally chaotic and Li — Yorke chaotic multivalued linear operators in Frechet spaces, as well, providing a great number of illustrative examples and observations about problems considered.
In a series of recent research studies, N.C. Bernardes Jr. et al and T. Bermudez et al have analyzed the notions of mean Li — Yorke chaos, absolute Cesaro boundedness and Cesaro hypercyclicity for linear continuous operators in Banach spaces. We close the paper with the observation that these concepts can be analyzed for general sequences of binary relations over metric spaces.
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Accepted article received 29.12.2018 Corrections received 15.02.2019
УДК 517.987.5; 517.983.2 DOI: 10.24411/2500-0101-2019-14104
РАСПРЕДЕЛИТЕЛЬНЫЙ ХАОС И ХАОС ЛИ — ЙОРКА В МЕТРИЧЕСКИХ ПРОСТРАНСТВАХ1
М. Костич
Университет Нови-Сада, Нови-Сад, Сербия marco.s @verat.net
Мы вводим несколько новых типов и обобщений понятий распределительного хаоса и хаоса Ли — Йорка. Рассматриваются общие последовательности бинарных отношений, действующих между метрическими пространствами, а в отдельном параграфе мы фокусируем наше внимание на некоторых отличительных чертах распределительно-хаотических и хаотических в смысле Ли — Йорка многозначных линейных операторах в пространствах Фреше.
Ключевые слова: 'распределительный хаос, хаос Ли — Йорка, бинарное отношение, метрическое пространство, многозначный линейный оператор.
Поступила в редакцию 29.12.2018 После переработки 15.02.2019
Сведения об авторе
Костич Марко, профессор, факультет технических наук, Университет Нови-Сада, Нови-Сад, Сербия; e-mail: marco.s@verat.net.
1 Автор частично поддержан грантом № 174024 Министерства науки и технологического развития Республики Сербия.
