Distributional Chaoticity of C₀-Semigroup on a Frechet Space

Abstract

This paper is mainly concerned with distributional chaos and the principal measure of $C_0$-semigroups on a Frechet space. New definitions of strong irregular (semi-irregular) vectors are given. It is proved that if $C_0$-semigroup $\mathcal{T}$ has strong irregular vectors, then $\mathcal{T}$ is distributional chaos in a sequence, and the principal measure ${\mu }_p(\mathcal{T})$ is 1. Moreover, $\mathcal{T}$ is distributional chaos equivalent to that operator $T_t\hspace{0.17em}$ is distributional chaos for every $\forall t>0$.

1. 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 $\omega$-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 $\mathcal{T}$ on a Frechet space admits a strong irregular vector, then $\mathcal{T}$ is distributionally chaotic in a sequence, and the principal measure ${\mu }_p(\mathcal{T})$ is 1. In Section 3, using the properties of the upper density and lower density, we point out that the distributional chaoticity of $\mathcal{T}$ is equivalent to the distributional chaoticity of $T_t\hspace{0.17em}(\forall t>0)$.

Throughout this paper, the set of natural numbers is denoted by $\mathbb{N}=\{1,2,3,\cdots \}$ and the set of positive real numbers is denoted by ${ℝ}^{+}=(0,+\infty )$.

2. Preliminaries

The Frechet space in this paper is a vector space $X$, endowed with a separating increasing sequence ${({\Vert \hspace{0.17em}·\hspace{0.17em}\Vert }_k)}_{k\in \mathbb{N}}$ of seminorms in the following metric.

$$\rho (x,y)={\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}}·\frac{{\Vert x-y\Vert }_k}{1+{\Vert x-y\Vert }_k},\hspace{0.17em}\forall x,y\in X.$$

Throughout this paper, the Frechet space is denote by $(X,\hspace{0.17em}{({\Vert \hspace{0.17em}·\hspace{0.17em}\Vert }_k)}_{k\in \mathbb{N}},\hspace{0.17em}\rho )$ (or simply $X$) without otherwise statements and we let $ℒ(X)$ be the space of continuous linear operators on $X$.

One parameter family $\mathcal{T}={\{T_t\}}_{t\ge 0}\subseteq ℒ(X)$ is called a $C_0$-semigroup of linear operators on $X$ if:

(i)

$T_0=I$ (where $I$ is the identity operator on $X$);

(ii)

$T_t{T}_s=T_{t+s},\hspace{0.17em}\forall s,t\ge 0$;

(iii)

$\underset{s\to t}{\lim }{T}_s(x)=T_t(x),\hspace{0.17em}\forall x\in X,\hspace{0.17em}\forall s,t\ge 0$.

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.

$x\in X$ is called a strong irregular vector for a $C_0$-semigroup $\mathcal{T}$ on a Frechet space $X$ if

$$\underset{t\to \infty }{\lim \sup }{\Vert T_{t}x\Vert }_k=\infty and\underset{t\to \infty }{\lim \inf }{\Vert T_{t}x\Vert }_k=0$$

for every $k\in \mathbb{N}$.

$x\in X$ is called a strong semi-irregular vector, if

$$\underset{t\to \infty }{\lim \sup }{\Vert T_{t}x\Vert }_k=\infty but\underset{t\to \infty }{\lim \inf }{\Vert T_{t}x\Vert }_k\ne 0for\hspace{0.17em}some\hspace{0.17em}k\in \mathbb{N}$$

and there exists a sequence ${\{T_{t_i}\}}_{i\in \mathbb{N}}$ such that

$$\underset{i\to \infty }{\lim \inf }{\Vert T_{t_i}x\Vert }_k=0$$

for every $k\in \mathbb{N}$.

3. Distributional Chaos in a Sequence of C₀-Semigroup

For any $x,y\in X$ and a sequence ${\{p_i\}}_{i\in \mathbb{N}}\subset {ℝ}^{+}$, we define the distributional function in a sequence of $x$ and $y$ with respect to $\mathcal{T}={\{T_t\}}_{t\ge 0}$ as:

$$\begin{gathered} {\mathsf{\Phi }}_{xy}^n:{ℝ}^{+}\to [0,1] \\ {\mathsf{\Phi }}_{xy}^n(\epsilon )=\frac{1}{n}card\{1\le i\le n:\rho (T_{p_i}(x),T_{p_i}(y))<\epsilon \}\hspace{0.17em}(\forall \epsilon \in {ℝ}^{+}) \end{gathered}$$

where $card\{M\}$ denotes the cardinality of the set $M$ (or denoted by $\left|M\right|$).

The upper and lower distributional functions of $x$ and $y$ are then defined by

$${\mathsf{\Phi }}_{xy}^{\ast }(\epsilon ,{\{p_i\}}_{i\in \mathbb{N}})=\underset{n\to \infty }{\lim \sup }{\mathsf{\Phi }}_{xy}^n(\epsilon )and{\mathsf{\Phi }}_{xy}^{}(\epsilon ,{\{p_i\}}_{i\in \mathbb{N}})=\underset{n\to \infty }{\lim \inf }{\mathsf{\Phi }}_{xy}^n(\epsilon )$$

respectively for $\forall \epsilon >0$.

Definition 1.

Let$(X,{({\Vert \hspace{0.17em}·\hspace{0.17em}\Vert }_k)}_{k\in \mathbb{N}},\rho )$be a Frechet space. A$C_0$-semigroup of operators$\mathcal{T}={\{T_t\}}_{t\ge 0}$on$X$is said to be distributionally chaotic in a sequence if there exists a sequence${\{p_i\}}_{i\in \mathbb{N}}$, an uncountable subset of$S\subset X$and$\delta >0$such that for$\forall x,y\in S:x\ne y$and$\forall \epsilon >0$, we have that:

$${\mathsf{\Phi }}_{xy}^{\ast }(\epsilon ,{\{p_i\}}_{i\in \mathbb{N}})=1and{\mathsf{\Phi }}_{xy}^{}(\delta ,{\{p_i\}}_{i\in \mathbb{N}})=0.$$

In this case, $S$ is called a distributionally $\delta$-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$\mathcal{T}={\{T_t\}}_{t\ge 0}$be a$C_0$-semigroup of operators on a Frechet space$X$. The principal measure${\mu }_p(\mathcal{T})$of$\mathcal{T}$is defined as follows:

$${\mu }_p(\mathcal{T})=\underset{x\in X}{\sup }\frac{1}{diam(X)}{\displaystyle {\int }_0^{\infty }({\mathsf{\Phi }}_{x,0}^{\ast }(s)-{\mathsf{\Phi }}_{x,0}(s))ds\hspace{0.17em}},$$

where ${\mathsf{\Phi }}_{x,0}^{\ast }(s)$ and ${\mathsf{\Phi }}_{x,0}(s)$ 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$\mathcal{T}$is a$C_0$-semigroup on a Frechet space$X$. If$\mathcal{T}$admits a strong irregular vector, then$\mathcal{T}$is distributionally chaotic in a sequence.

Proof. 

Let $x\in X$. Since $\mathcal{T}$ admits a strong irregular vector, there exists two increasing sequences ${\{n_j\}}_{j\in \mathbb{N}}\subset {ℝ}^{+}$ and ${\{m_j\}}_{j\in \mathbb{N}}\subset {ℝ}^{+}$ such that

$$\underset{j\to \infty }{\lim }{\Vert T_{n_j}(x)\Vert }_k=\infty and\underset{j\to \infty }{\lim }{\Vert T_{m_j}(x)\Vert }_k=0$$

for every $k\in \mathbb{N}$.

Let

$b_1=l_1=2,\hspace{0.17em}{b}_2=2^{b_1},\hspace{0.17em}{b}_3=2^{b_1+b_2},\cdots ,b_i=2^{b_1+\cdots +b_{i-1}}=2^{{\displaystyle {\sum }_{k=1}^{i-1}{b}_k}}$ for all $\hspace{0.17em}i>1$;

$l_2=b_1+b_2,l_3=b_1+b_2+b_3,\cdots ,l_i={\displaystyle {\sum }_{h=1}^i{b}_h}$ for all $\hspace{0.17em}i>1$.

${\{n_j{}^{\prime }\}}_{j\in \mathbb{N}}$ and ${\{m_j{}^{\prime }\}}_{j\in \mathbb{N}}$ are, respectively, the subsequence of ${\{n_j\}}_{j\in \mathbb{N}}$ and ${\{m_j\}}_{j\in \mathbb{N}}$ such that $m_j{}^{\prime }<n_j{}^{\prime }$ when $j\le b_1$ or $l_{2s}<j<l_{2s+1}$, and $n_j{}^{\prime }<m_j{}^{\prime }$ when $l_{2s-1}<j<l_{2s}$ for any $s\in \mathbb{N}$.

Let

$$p_j=\{\begin{array}{cc}{n}_j{}^{\prime }j\le b_1\hspace{0.17em}or\hspace{0.17em}{l}_{2s}<j<l_{2s+1},\hfill & s\in \mathbb{N}\hfill \\ m_j{}^{\prime }{l}_{2s-1}<j<l_{2s},\hfill & s\in \mathbb{N}\hfill \end{array}$$

then, ${\{p_j\}}_{j\in \mathbb{N}}\subset {ℝ}^{+}$ is an increasing sequence.

Denote $\mathsf{\Gamma }=\{\alpha x:\alpha \in (0,1)\}$. The following prove that $\mathsf{\Gamma }$ is a distributional $\delta$-scrambled set of $\mathcal{T}$ in ${\{p_j\}}_{j\in \mathbb{N}}$ for some $\delta >0$.

In fact, for any pair $x,y\in \mathsf{\Gamma }$ with $x\ne y$, it is clear that there exists $\alpha \in (0,1)$ such that $x-y=\alpha x$.

Since $\underset{j\to \infty }{\lim }{\Vert T_{m_j}(x)\Vert }_k=0\hspace{0.17em}\hspace{0.17em}(\forall k\in \mathbb{N})$, then, for $\forall \epsilon >0$, there exists $N\in \mathbb{N}$ such that ${\Vert T_{m_j{}^{\prime }}(\alpha x)\Vert }_k<\epsilon$ for each $j\ge N$. Then, $\forall k\in \mathbb{N}$,

$$\frac{{\Vert T_{p_j}(x)-T_{p_j}(y)\Vert }_k}{1+{\Vert T_{p_j}(x)-T_{p_j}(y)\Vert }_k}=\frac{{\Vert T_{p_j}(x-y)\Vert }_k}{1+{\Vert T_{p_j}(x-y)\Vert }_k}=\frac{\alpha {\Vert T_{p_j}\Vert }_k}{1+\alpha {\Vert T_{p_j}\Vert }_k}<\frac{\alpha \epsilon }{1+\alpha \epsilon }$$

So,

$$\begin{array}{ll}{\mathsf{\Phi }}_{xy}^{\ast }(\epsilon ,{\{p_j\}}_{j\in \mathbb{N}})\hfill & =\underset{n\to \infty }{\lim \sup }\frac{1}{n}card\{1\le j\le n:\rho (T_{p_j}(x),T_{p_j}(y))<\epsilon \}\hfill \\ & =\underset{n\to \infty }{\lim \sup }\frac{1}{n}card\{1\le j\le n:{\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}·\frac{{\Vert T_{p_j}(x)-T_{p_j}(y)\Vert }_k}{1+{\Vert T_{p_j}(x)-T_{p_j}(y)\Vert }_k}}<\epsilon \}\hfill \\ & \ge \underset{n\to \infty }{\lim \sup }\frac{1}{n}card\{1\le j\le n:{\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}·\frac{\alpha x}{1+\alpha x}}<\epsilon \}\hfill \\ & =\underset{s\to \infty }{\lim \sup }\frac{1}{l_{2s}}card\{1\le j\le l_{2s}:{\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}·\frac{\alpha x}{1+\alpha x}}<\epsilon \}\hfill \\ & \ge \underset{s\to \infty }{\lim \sup }\frac{b_{2s}}{l_{2s}}\hfill \\ & =\underset{s\to \infty }{\lim \sup }\frac{2^{b_1+b_2+\cdots +b_{2s-1}}}{{\displaystyle {\sum }_{h=1}^{2s-1}{b}_h+2^{b_1+b_2+\cdots +b_{2s-1}}}}\hfill \\ & =1.\hfill \end{array}$$

(1)

Let $\delta =1$.

Since $\underset{j\to \infty }{\lim }{\Vert T_{n_j}(x)\Vert }_k=\infty \hspace{0.17em}\hspace{0.17em}(\forall k\in \mathbb{N})$, there exists $M\in \mathbb{N}$ such that ${\Vert T_{n_j{}^{\prime }}(\alpha x)\Vert }_k>\delta (\forall k\in \mathbb{N})$ for each $j\ge M$.

Thus,

$$\begin{array}{cc}{\mathsf{\Phi }}_{xy}(\delta ,{\{p_j\}}_{j\in \mathbb{N}})\hfill & =\underset{n\to \infty }{\lim \inf }\frac{1}{n}card\{1\le j\le n:\rho (T_{p_j}(x),T_{p_j}(y))<\delta \}\hfill \\ & =\underset{s\to \infty }{\lim \inf }\frac{1}{l_{2s+1}}card\{1\le j\le l_{2s+1}:{\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}·\frac{\alpha \delta }{1+\alpha \delta }}<\delta \}\hfill \\ & \le \underset{s\to \infty }{\lim \inf }\frac{l_{2s}}{l_{2s+1}}\hfill \\ & =\underset{s\to \infty }{\lim \inf }\frac{{\displaystyle {\sum }_{h=1}^{2s}{b}_h}}{{\displaystyle {\sum }_{h=1}^{2s}{b}_h+2^{b_1+b_2+\cdots +b_{2s}}}}\hfill \\ & =0.\hfill \end{array}$$

(2)

By (1) and (2), $\mathsf{\Gamma }=\{\alpha x:\alpha \in (0,1)\}$ is a distributionally $\delta$-scrambled set of $\mathsf{\Gamma }$ in ${\{p_j\}}_{j\in \mathbb{N}}$. So, $\mathsf{\Gamma }$ is distributionally chaotic in a sequence as $\mathsf{\Gamma }$ is uncountable.

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$\mathcal{T}$be a$C_0$-semigroup of operators on a Frechet space$X$. Assume that$\mathcal{T}$admits a strong irregular vector$X_0$, then the principal measure${\mu }_p(\mathcal{T})=1$.

Proof. 

From the definition of a strong irregular vector, for every $k\in \mathbb{N}$, one has:

$$\underset{t\to \infty }{\lim \inf }{\Vert T_t(x_0)\Vert }_k=0\mathrm{and}\underset{t\to \infty }{\lim \sup }{\Vert T_t(x_0)\Vert }_k=\infty .$$

Given arbitrary $\epsilon \in (0,1)$, one can find a sequence ${\{t_i^{\epsilon }\}}_{i\in \mathbb{N}}\in {ℝ}^{+}$ and a positive number $N_1$ such that $\rho (T_{t_i^{\epsilon }}(x_0),0)<\epsilon$ for all $\{t_i^{\epsilon }:t_i^{\epsilon }\in \{t_1^{\epsilon },t_2^{\epsilon },\cdots \},t_i^{\epsilon }>N_1\}$.

$$\begin{array}{cc}{\mathsf{\Phi }}_{x_0,0}^{\ast }(\epsilon ,{\{t_1^{\epsilon }\}}_{i\in \mathbb{N}})\hfill & =\underset{n\to \infty }{\lim \sup }\frac{1}{n}card\{1\le i\le n:\rho (T_{t_i^{\epsilon }}(x_0),0)<\epsilon \}\hfill \\ & \ge \underset{n\to \infty }{\lim \sup }\frac{1}{n}(n-N_1)\hfill \\ & =1.\hfill \end{array}$$

On the other hand, we show that $\forall k\in \mathbb{N}$, ${\mathsf{\Phi }}_{x_0,0}(\epsilon )=0$ for every $\epsilon \in (0,diam(X))$.

In fact, given $\forall \epsilon \in (0,diam(X))$. Since $\underset{t\to \infty }{\lim \sup }{\Vert T_t(x_0)\Vert }_k=\infty$ for every $k\in \mathbb{N}$, then for any sequence ${\{t_i\}}_{i\in \mathbb{N}}\in {\mathbb{N}}^{+}$, there exists a positive number $N_2$ such that $\rho (T_{t_i}(x_0),0)>\epsilon$ for $\{t_i:t_i\in {\{t_i\}}_{i\in \mathbb{N}},t_i>N_2\}$. So

$${\mathsf{\Phi }}_{x_0,0}(\epsilon ,{\{t_i^{\epsilon }\}}_{i\in \mathbb{N}})=\underset{n\to \infty }{\lim \inf }\frac{1}{n}card\{1\le i\le n:\rho (T_{t_i}(x_0),0)<\epsilon \}\le \underset{n\to \infty }{\lim \inf }\frac{N_2}{n}=0.$$

Hence,

$$\begin{array}{cc}{\mu }_n(\mathcal{T})\hfill & =\underset{x\in X}{\sup }\frac{1}{diam(X)}{\displaystyle {\int }_0^{\infty }({\mathsf{\Phi }}_{x,0}^{\ast }(\epsilon )-{\mathsf{\Phi }}_{x,0}^{}(\epsilon ))d_{\epsilon }}\hfill \\ & =\underset{x\in X}{\sup }\frac{1}{diam(X)}{\displaystyle {\int }_0^{diam(X)}({\mathsf{\Phi }}_{x,0}^{\ast }(\epsilon )-{\mathsf{\Phi }}_{x,0}^{}(\epsilon ))d_{\epsilon }}\hfill \\ & \ge \frac{1}{diam(X)}{\displaystyle {\int }_0^{diam(X)}({\mathsf{\Phi }}_{x_0,0}^{\ast }(\epsilon ,{\{t_i^{\epsilon }\}}_{i\in \mathbb{N}})-{\mathsf{\Phi }}_{x_0,0}^{}(\epsilon ,{\{t_i^{\epsilon }\}}_{i\in \mathbb{N}}))d_{\epsilon }}\hfill \\ & =1.\hfill \end{array}$$

This completes the proof.  □

4. Distributionally Chaotic C₀-Semigroup

For any $x,y\in X$ and any $t>0$, the distributional function of $x$ and $y$ with respect to $\mathcal{T}={\{T_t\}}_{t\ge 0}$ is defined as follows:

• ${\mathsf{\Phi }}_{x,y}^t:{ℝ}^{+}\to [0,1]$
• ${\mathsf{\Phi }}_{x,y}^t(\epsilon )=\frac{1}{t}\mu (\{0\le i\le t:\rho (T_s(x),T_s(y))<\epsilon \}).\forall \epsilon >0$

where $\mu$ denotes the Lebesgue measure on $ℝ$.

The upper and lower distributional functions of $x$ and $y$ are then defined by:

$${\mathsf{\Phi }}_{x,y}^{\ast }(\epsilon )=\underset{t\to \infty }{\lim \sup }{\mathsf{\Phi }}_{x,y}^t(\epsilon )\mathrm{and}{\mathsf{\Phi }}_{x,y}^{}(\epsilon )=\underset{t\to \infty }{\lim \inf }{\mathsf{\Phi }}_{x,y}^t(\epsilon ),\forall \epsilon >0$$

respectively.

Definition 3.

Let$(X,{({\Vert \hspace{0.17em}·\hspace{0.17em}\Vert }_k)}_{k\in \mathbb{N}},\rho )$be a Frechet space. A$C_0$-semigroup of operators$\mathcal{T}={\{T_t\}}_{t\ge 0}$on$X$is said to be distributionally chaotic if one can find an uncountable subset$S\in X$and$\delta >0$such that, for$\forall x,y\in S:x\ne y$and for$\forall \epsilon >0$, we have:

$${\mathsf{\Phi }}_{x,y}^{\ast }(\epsilon )=1\mathrm{and}{\mathsf{\Phi }}_{x,y}(\delta )=0.$$

In this case,$S$is called a distributionally$\delta$-scrambled set and$(x,y)$a distributionally chaotic pair.

Let$E\in {ℝ}^{+}$be a Lebesgue measurable set; the upper density and lower density of$E$are defined as:

$$\overline{Dens}(E)=\underset{t\to \infty }{\lim \sup }\frac{\mu (E\cap [0,t])}{t}\mathrm{and}\underset{\_}{Dens}(E)=\underset{t\to \infty }{\lim \inf }\frac{\mu (E\cap [0,t])}{t}$$

respectively. Then, the conditions${\mathsf{\Phi }}_{x,y}^{\ast }(\epsilon )=1$,${\mathsf{\Phi }}_{x,y}(\delta )=0$in Definition 3 are equivalent to:

$$\overline{Dens}(\{t\ge 0:\rho (T_t(x),T_t(y))<\epsilon \})=1\mathrm{and}\underset{\_}{Dens}(\{t\ge 0:\rho (T_t(x),T_t(y))<\delta \})=0$$

respectively.

Given$M\subset {\mathbb{N}}^{+}$, the upper density and lower density of$M$are defined as

$$\overline{dens}(M)=\underset{n\to \infty }{\lim \sup }\frac{card(M\cap [0,n-1])}{t}\mathrm{and}\underset{\_}{dens}(M)=\underset{n\to \infty }{\lim \inf }\frac{card(M\cap [0,n-1])}{t}$$

respectively. The conditions in the definition of the distributional chaos for operator$T$are equivalent to

$$\overline{dens}(\{n\in \mathbb{N}:{\Vert T_n(x)-T_n(y)\Vert }_k<\epsilon \})=1,\underset{\_}{dens}(\{n\in \mathbb{N}:{\Vert T_n(x)-T_n(y)\Vert }_k<\delta \})=0.$$

Theorem 3.

Let$\mathcal{T}$be a$C_0$-semigroup of operators on a Frechet space$X$. $x\in X$,$t_0>0$,$\forall k\in \mathbb{N}$, let$C_{t_0}^k=\underset{0\le t\le t_0}{\sup }{\Vert T{(x)}_t\Vert }_k$. Then for every$\epsilon ,\delta >0$and all$N>0$:

(i) 

$\mu (\{t\in [0,N]:{\Vert T_t(x)\Vert }_k>\delta \})\le t_0\left|\{s\in \mathbb{N}:s\le \frac{N}{t_0}+1,\hspace{0.17em}{\Vert T_{t_0}^{s-1}(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\}\right|$;

(ii) 

$t_0\left|\{s\in \mathbb{N}:s\le N,\hspace{0.17em}{\Vert T_{t_0}^s(x)\Vert }_k>\delta \}\right|\le \mu (\{t\in [0,N{t}_0]:{\Vert T_t(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\})$;

(iii) 

$\mu (\{t\in [0,N]:{\Vert T_t(x)\Vert }_k<\epsilon \})\le t_0\left|\{s\in \mathbb{N}:s\le \frac{N}{t_0}+1,\hspace{0.17em}{\Vert T_{t_0}^s(x)\Vert }_k<\epsilon C_{t_0}^k\}\right|$;

(iv) 

$t_0\left|\{s\in \mathbb{N}:s\le N,\hspace{0.17em}{\Vert T_{t_0}^s(x)\Vert }_k<\epsilon \}\right|\le \mu (\{t\in [0,(N+1)t_0]:{\Vert T_t(x)\Vert }_k<\epsilon C_{t_0}^k\})$.

Proof. 

(i) Let $A=\{t\le N:{\Vert T_t(x)\Vert }_k>\delta \}$, $B=\{s\in \mathbb{N}:\exists t^{\ast }\in A\cap [(s-1)t_0,s{t}_0]\}$, then,

$$B\subseteq \{s\in \mathbb{N}:1\le s\le \frac{N}{t_0}+1,{\Vert T_{t_0}^{s-1}(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\}$$

Indeed, if there exists $t^{\ast }\in [(s-1)t_0,s{t}_0]$ such that $t^{\ast }\le N$ and ${\Vert T_{t^{\ast }}(x)\Vert }_k>\delta$, then

$$1\le \frac{t^{\ast }}{t_0}\le s\le \frac{t^{\ast }}{t_0}+1\le \frac{N}{t_0}+1.$$

and because $t^{\ast }-(s-1)t_0\le t_0$, then

$$\begin{gathered} \delta <{\Vert T_{t^{\ast }}(x)\Vert }_k={\Vert T_{t^{\ast }-(s-1)t_0}{T}_{(s-1)t_0}(x)\Vert }_k\le (\underset{0\le t\le t_0}{\sup }{\Vert T_t(x)\Vert }_k){\Vert T_{(s-1)t_0}(x)\Vert }_k \\ =C_{t_0}^k{\Vert T_{(s-1)t_0}(x)\Vert }_k=C_{t_0}^k{\Vert T_{t_0}^{s-1}(x)\Vert }_k. \end{gathered}$$

That is,

$${\Vert T_{t_0}^{s-1}(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}.$$

Therefore,

$$\mu (A)\le {\displaystyle \sum _{s\in B}\mu ([(s-1)t_0,s{t}_0])}.$$

(ii) Let $M=\{s\in \mathbb{N}:s\le N,{\Vert T_{t_0}^s(x)\Vert }_k>\delta \}$. Then, for every $t\in [(s-1)t_0,s{t}_0]$, we have that

$$\delta <{\Vert T_{t_0}^s(x)\Vert }_k={\Vert T_{s{t}_0}(x)\Vert }_k={\Vert T_{s{t}_0-t}{T}_t(x)\Vert }_k\le C_{t_0}^k{\Vert T_t(x)\Vert }_k.$$

(The last inequality is right for the reason that $s{t}_0-t\le t_0$).

Hence,

$$\underset{s\in M}{\cup }[(s-1)t_0,s{t}_0]\subseteq \{t\in [0,N{t}_0]:{\Vert T_t(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\}.$$

Thus,

$$t_0\left|M\right|\le \mu (\{t\in [0,N{t}_0]:{\Vert T_t(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\}).$$

(iii) and (iv) can be obtained with analogous considerations.

This completes the proof.  □

Theorem 4.

Let$\mathcal{T}$be a$C_0$-semigroup of operators on a Frechet space$X$. $x\in X$,$t_0>0$,$\forall k\in \mathbb{N}$, let$C_{t_0}^k=\underset{0\le t\le t_0}{\sup }{\Vert T{(x)}_t\Vert }_k$. Then for$\forall \epsilon ,\delta >0$and all$N>0$:

(i) 

$\overline{Dens}(\{t\ge 0:{\Vert T_t(x)\Vert }_k>\delta \})\le \overline{dens}(\{s\in \mathbb{N}:{\Vert T_{t_0}^s(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\})$;

(ii) 

$\overline{dens}\left|\{s\in \mathbb{N}:\hspace{0.17em}{\Vert T_{t_0}^s(x)\Vert }_k>\delta \}\right|\le \overline{Dens}(\{t\ge 0:{\Vert T_t(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\})$;

(iii) 

$\overline{Dens}(\{t\ge 0:{\Vert T_t(x)\Vert }_k<\epsilon \})\le \overline{dens}\left|\{s\in \mathbb{N}:\hspace{0.17em}{\Vert T_{t_0}^s(x)\Vert }_k<\epsilon C_{t_0}^k\}\right|$;

(iv) 

$\overline{dens}(\{s\in \mathbb{N}:\hspace{0.17em}{\Vert T_{t_0}^s(x)\Vert }_k<\epsilon \})\le \overline{Dens}(\{t\ge 0:{\Vert T_t(x)\Vert }_k<\epsilon C_{t_0}^k\})$.

Proof. 

(i)’ By (i) of Theorem 3,

$$\begin{array}{cc}\overline{Dens}(\{t\ge 0:{\Vert T_t(x)\Vert }_k>\delta \})\hfill & =\underset{t\to \infty }{\lim \sup }\frac{1}{t}\mu (\{[0,t]\cap \{t\ge 0:{\Vert T_t(x)\Vert }_k>\delta \}\})\hfill \\ & =\underset{N\to \infty }{\lim \sup }\frac{1}{N}\mu (\{t\in [0,N]:{\Vert T_t(x)\Vert }_k>\delta \})\hfill \\ & \le \underset{N\to \infty }{\lim \sup }\frac{t_0}{N}\mu (\{s\in \mathbb{N}:s\le \frac{N}{t_0}+1,{\Vert T_{t_0}^{s-1}(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\})\hfill \\ & =\underset{N\to \infty }{\lim \sup }\frac{1}{N}\left|\{s\in \mathbb{N}:{\Vert T_{t_0}^s(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\}\cap [0,N]\right|\hfill \\ & =\overline{dens}(\{s\in \mathbb{N}:{\Vert T_{t_0}^s(x)\Vert }_k>\frac{\delta }{C_{t_0}^k}\}).\hfill \end{array}$$

(ii)’, (iii)’ and (iv)’ can be obtained with analogous considerations.

This completes the proof.  □

Theorem 5.

Let$\mathcal{T}={\{T_t\}}_{t\ge 0}$be a$C_0$-semigroup of operators on a Frechet space$X$. Then the following properties are equivalent.

(i) 

$\mathcal{T}$is distributionally chaotic;

(ii) 

$\forall t>0$, $T_t$is distributionally chaotic;

(iii) 

There exists$t_0>0$such that$T_{t_0}$is distributionally chaotic.

Proof. 

Let $S\subset X$ be a distributionally $\delta$-scrambled set for $\mathcal{T}$. Then, for $\forall x,y\in S:x\ne y$, there exists a $0<\delta <1$ such that

$$\underset{\_}{Dens}(\{s\ge 0:\rho (T_s(x),T_s(y))<\delta \})=0.$$

It means that

$$\underset{t\to \infty }{\lim \inf }\frac{\mu (\{s\ge 0:\rho (T_s(x),T_s(y))<\delta \}\cap [0,t])}{t}=0.$$

i.e.,

$$\underset{t\to \infty }{\lim \sup }\frac{\mu (\{s\ge 0:\rho (T_s(x),T_s(y))>\delta \}\cap [0,t])}{t}=1.$$

If ${\Vert T_s(x)-T_s(y)\Vert }_k>\frac{2^k\delta }{1-\delta }(\forall k\in \mathbb{N})$, then

$${\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}}·\frac{1}{1+{\Vert T_s(x)-T_s(y)\Vert }_k}<1-\delta .$$

So,

$$\rho (T_s(x),T_s(y))={\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}}·\frac{{\Vert T_s(x)-T_s(y)\Vert }_k}{1+{\Vert T_s(x)-T_s(y)\Vert }_k}=1-{\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}}·\frac{1}{1+{\Vert T_s(x)-T_s(y)\Vert }_k}>\delta .$$

Thus,

$$\overline{Dens}(\{s\ge 0:{\Vert T_s(x)-T_s(y)\Vert }_k>\frac{2^k\delta }{1-\delta }\})=1.$$

By (i)’ of Theorem 4, for every $t_0>0$, one has

$$\overline{dens}(\{s\in \mathbb{N}:{\Vert T_{t_0}^k(T_s(x)-T_s(y))\Vert }_k>\frac{2^k\delta }{C_{t_0}^k(1-\delta )}\})=1.$$

That is,

$$\underset{\_}{dens}(\{s\in \mathbb{N}:{\Vert T_{k{t}_0+s}(x)-T_{k{t}_0+s}(y)\Vert }_k<\frac{2^k\delta }{C_{t_0}^k(1-\delta )}\})=0.$$

On the other hand, for $\forall x,y\in S:x\ne y$ and every $0<\epsilon <1$, since $\overline{Dens}(\{s\ge 0:\rho (T_s(x),T_s(y))<\epsilon \})=1$, i.e.,

$$\underset{t\to \infty }{\lim \sup }\frac{\mu (\{s\ge 0:\rho (T_s(x),T_s(y))<\epsilon \}\cap [0,t])}{t}=1,$$

and

$$\rho (T_s(x),T_s(y))={\displaystyle \sum _{k=1}^{\infty }\frac{1}{2^k}}·\frac{{\Vert T_s(x)-T_s(y)\Vert }_k}{1+{\Vert T_s(x)-T_s(y)\Vert }_k}<\epsilon ,$$

then

$$\overline{Dens}(\{s\ge 0:{\Vert T_s(x)-T_s(y)\Vert }_k<\frac{\epsilon }{2^k(1-\epsilon )}\})=1$$

when ${\Vert T_s(x)-T_s(y)\Vert }_k<\frac{\epsilon }{2^k(1-\epsilon )}\hspace{0.17em}\hspace{0.17em}$$(\forall k\in \mathbb{N})$.

By (iii)’ of Theorem 4, for every $t_0>0$, one has

$$\overline{dens}(\{s\in \mathbb{N}:{\Vert T_{t_0}^k(T_s(x)-T_s(y))\Vert }_k<\frac{\epsilon C_{t_0}^k}{2^k(1-\epsilon )}\})=1.$$

For the arbitrariness of $\epsilon >0$, we have

$$\overline{dens}(\{s\in \mathbb{N}:{\Vert T_{k{t}_0+s}(x)-T_{k{t}_0+s}(y)\Vert }_k<\epsilon \})=1.$$

Thus, $S$ is a ${\delta }^{\prime }$-scrambled set for $T_t$, where ${\delta }^{\prime }=\frac{\delta }{C_{t_0}^k},\hspace{0.17em}t=k{t}_0+s\hspace{0.17em}(\forall t_0>0)$, i.e., for all $t>0$, $T_t$ is distributionally chaotic.

(ii) implies (iii). It is trivial.

(iii) implies (i). The proof is analogous to the first implication.

This completes the proof.  □

5. 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 ${\mu }_p(\mathcal{T})=1$. 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 ${\mu }_p(\mathcal{T})={\{T_t\}}_{t\ge 0}$ is equivalent to the distributional chaoticity of some $T_{t_0}\hspace{0.17em}(t_0>0)$. 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.