Distributional Chaoticity of C 0-Semigroup on a Frechet Space

This paper is mainly concerned with distributional chaos and the principal measure of C0-semigroups on a Frechet space. New definitions of strong irregular (semi-irregular) vectors are given. It is proved that if C0-semigroup T has strong irregular vectors, then T is distributional chaos in a sequence, and the principal measure μp(T ) is 1. Moreover, T is distributional chaos equivalent to that operator Tt is distributional chaos for every ∀t > 0.


Introduction
Chaotic properties of dynamical systems have been ardently studied since the term chaos (namely, Li-Yorke chaos) was defined in 1975 by Li and Yorke [1].To describe unpredictability in the evolution of dynamical systems, many properties related to chaos have been discussed (for example, References [2][3][4][5][6][7][8][9][10][11][12][13], where References [4][5][6][7] are some of our works done in recent years).In 1994, Schweizer and Smital in Reference [8] introduced a popular concept named distributional chaos for interval maps, by considering the dynamics of pairs with some statistical properties.The goal was to extend the definition of Li-Yorke chaos, and it was equivalent to positive topological entropy.Later, Reference [9] summarizes the connections between Li-Yorke, distributional, and ω-chaos.The notions of distributional chaos and principal measures were extended to general dynamical systems [10,11] and especially to the framework of linear dynamics in the last few years.It seems that the first example of a distributional chaotic operator on a Frechet space was given by Oprocha [14], whom investigated the annihilation operator of a quantum harmonic oscillator.Wu and Zhu [15] further proved that the principal measure of the annihilation operator studied in Reference [14] is 1.Since then, distributional chaos for linear operators has been studied by many authors, see for instance References [16][17][18][19][20][21].
The study of hypercyclicity and chaoticity for operators and C 0 -semigroups has became a hot and active research area in the past two decades (such as References [22,23]).In Reference [24], Devaney chaos for C 0 -semigroup of unbounded operators was discussed.The extension of distributional chaos to C 0 -semigroup on weighted spaces of integrable functions was done in Reference [25].Devaney chaos and distributional chaos are closely tied for the C 0 -semigroup.Distributionally chaotic C 0 -semigroups on Banach spaces were found in Reference [16].A systematic investigation of distributional chaos for linear operators on Frechet space was given by Bernardes [17].Recently, an extension of distributional chaos for a family of operators (including C 0 -semigroups) on Frechet spaces were proposed by Conejero [26].For other studies of C 0 -semigroups or Frechet spaces see References [27][28][29][30][31][32][33][34] and others.
In the present work, in Section 2 we deal with the notion of strong irregular (semi-irregular) vectors for C 0 -semigroups of operators on a Frechet spaces.It is proved that if a C 0 -semigroup T on a Frechet space admits a strong irregular vector, then T is distributionally chaotic in a sequence, and the principal measure µ p (T ) is 1.In Section 3, using the properties of the upper density and lower density, we point out that the distributional chaoticity of T is equivalent to the distributional chaoticity of T t (∀t > 0).
Throughout this paper, the set of natural numbers is denoted by N = {1, 2, 3, • • • } and the set of positive real numbers is denoted by R + = (0, +∞).

Preliminaries
The Frechet space in this paper is a vector space X, endowed with a separating increasing sequence ( • k ) k∈N of seminorms in the following metric.
Throughout this paper, the Frechet space is denote by (X, ( • k ) k∈N , ρ) (or simply X) without otherwise statements and we let L(X) be the space of continuous linear operators on X.
One parameter family T = {T t } t≥0 ⊆ L(X) is called a C 0 -semigroup of linear operators on X if: (i) T 0 = I (where I is the identity operator on X); In References [17,33], Peris et al. introduced the notions of an irregular vector and a distributional irregular vector for operators in order to characterize distributional chaos.Similarly, we give notions of a strong irregular vector and strong semi-irregular vector.
x ∈ X is called a strong irregular vector for a C 0 -semigroup T on a Frechet space X if limsup t→∞ T t x k = ∞ and liminf t→∞ T t x k = 0 for every k ∈ N.
x ∈ X is called a strong semi-irregular vector, if and there exists a sequence {T t i } i∈N such that liminf i→∞ T t i x k = 0 for every k ∈ N.

Distributional Chaos in a Sequence of C 0 -Semigroup
For any x, y ∈ X and a sequence {p i } i∈N ⊂ R + , we define the distributional function in a sequence of x and y with respect to T = {T t } t≥0 as: where card{M} denotes the cardinality of the set M (or denoted by |M|).
The upper and lower distributional functions of x and y are then defined by respectively for ∀ε > 0.
Definition 1.Let (X, ( • k ) k∈N , ρ) be a Frechet space.A C 0 -semigroup of operators T = {T t } t≥0 on X is said to be distributionally chaotic in a sequence if there exists a sequence {p i } i∈N , an uncountable subset of S ⊂ X and δ > 0 such that for ∀x, y ∈ S : x = y and ∀ε > 0, we have that: In this case, S is called a distributionally δ-scrambled set in a sequence, and (x, y) is called a distributionally chaotic pair in a sequence.
To measure the degree of chaos for a given dynamical system, the concept of principal measure was introduced for general dynamical systems accompanying the appearance of distributional chaos [8,11].For the study of principal measures of certain linear operators, we refer to References [14,15,35].Naturally, the concept for the case of C 0 -semigroup of operators on Frechet spaces can be extended.Definition 2. Let T = {T t } t≥0 be a C 0 -semigroup of operators on a Frechet space X.The principal measure µ p (T ) of T is defined as follows: where Φ * x,0 (s) and Φ x,0 (s) are the upper and lower distributional functions of x and 0, and diam(X) is the diameter of X.Now we shall establish the relationship between strong irregular vectors and the distributional chaos of the C 0 -semigroup of operators on Frechet spaces.Theorem 1.Let T is a C 0 -semigroup on a Frechet space X.If T admits a strong irregular vector, then T is distributionally chaotic in a sequence.
Proof.Let x ∈ X.Since T admits a strong irregular vector, there exists two increasing sequences {n j } j∈N and {m j } j∈N are, respectively, the subsequence of {n j } j∈N and {m j } j∈N such that m j < n j when j ≤ b 1 or l 2s < j < l 2s+1 , and n j < m j when l 2s−1 < j < l 2s for any s ∈ N. Let The following prove that Γ is a distributional δ-scrambled set of T in {p j } j∈N for some δ > 0.
In fact, for any pair x, y ∈ Γ with x = y, it is clear that there exists α ∈ (0, 1) such that x − y = αx.
This completes the proof.
As an important class of operators in linear dynamics, the backward shift [35,36] admits principal measure 1 if it is distributionally chaotic.In addition, it is easy to see that every distributionally chaotic operator on a Banach space (as a special Frechet space) has a principal measure of 1.So we wonder whether the C 0 -semigroup on the Frechet space above with a principal measure of 1 is distributionally chaotic.The answer is positive.Theorem 2. Let T be a C 0 -semigroup of operators on a Frechet space X.Assume that T admits a strong irregular vector X 0 , then the principal measure µ p (T ) = 1.

Distributionally Chaotic C 0 -Semigroup
For any x, y ∈ X and any t > 0, the distributional function of x and y with respect to T = {T t } t≥0 is defined as follows: x,y (ε) = 1 t µ({0 ≤ i ≤ t : ρ(T s (x), T s (y)) < ε}).∀ε > 0 where µ denotes the Lebesgue measure on R. The upper and lower distributional functions of x and y are then defined by: on X is said to be distributionally chaotic if one can find an uncountable subset S ∈ X and δ > 0 such that, for ∀x, y ∈ S : x = y and for ∀ε > 0, we have: In this case, S is called a distributionally δ-scrambled set and (x, y) a distributionally chaotic pair.Let E ∈ R + be a Lebesgue measurable set; the upper density and lower density of E are defined as: respectively.Then, the conditions Φ * x,y (ε) = 1, Φ x,y (δ) = 0 in Definition 3 are equivalent to: Dens({t ≥ 0 : ρ(T t (x), T t (y)) < ε}) = 1 and Dens({t ≥ 0 : ρ(T t (x), T t (y)) < δ}) = 0 respectively.
Given M ⊂ N + , the upper density and lower density of M are defined as respectively.The conditions in the definition of the distributional chaos for operator T are equivalent to Then for every ε, δ > 0 and all N > 0: and because t * − (s − 1)t 0 ≤ t 0 , then That is, Then, for every t ∈ [(s − 1)t 0 , st 0 ], we have that (The last inequality is right for the reason that st 0 − t ≤ t 0 ).Hence, Thus, (iii) and (iv) can be obtained with analogous considerations.This completes the proof.
Theorem 5. Let T = {T t } t≥0 be a C 0 -semigroup of operators on a Frechet space X.Then the following properties are equivalent.
(ii) implies (iii).It is trivial.(iii) implies (i).The proof is analogous to the first implication.This completes the proof.

Discussion
Inspired by the definition of an irregular vector given by N.C.Bernardes Jr in Reference [17], this paper defines the strong irregular vector.In particular, it is proved that a C 0 -semigroup on a Frechet space is distributionally chaotic in a sequence if it admits a strong irregular vector.In addition, the principal measure µ p (T ) = 1.These results extend the corresponding results in References [16,17,31,35].In Section 4, using upper density and lower density, it is showed that the distributional chaoticity of µ p (T ) = {T t } t≥0 is equivalent to the distributional chaoticity of some T t 0 (t 0 > 0).This result is consistent with the similar conclusion in Banach space or other Frechet spaces (see References [17,27,29,31,33] and others).Then, some further results regarding C 0 -semigroups or Frechet spaces may be obtained in the future.Since Li-Yorke chaos is a special case of distributional chaos, therefore, the conclusions of this paper are also correct for Li-Yorke chaos.