Kato Chaos in Linear Dynamics

Abstract

This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators $\overline{T}$. The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators $\overline{T}$ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator $\overline{T}$ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics.

1. Introduction

Chaos theory has been regarded as the ’science of surprises’, and chaos usually describes something wild or a stare of disorder. For a long time, there was no specific definition of chaos in mathematics, until 1975, when scholars Li and Yorke [1] first gave a rigorous mathematical definition of chaos, i.e., Li–Yorke chaos. A subset D of X is called a Li–Yorke scrambled set of f if any different points x and y from D satisfy

$$\underset{n\to \infty }{lim\; sup}d(f^n\left(x\right),f^n\left(y\right))>0,\underset{n\to \infty }{lim\; inf}d(f^n\left(x\right),f^n\left(y\right))=0.$$

The function f is chaotic in the sense of Li–Yorke if there exists an uncountable Li–Yorke scrambled set. Since then, the study of chaos has had a great influence on dynamical systems. In the development of chaos theory, scholars have put forward many different kinds of definitions of chaos according to the properties of iterates of mappings on a metric space and relevant properties. In 1989, Devaney gave another definition of chaos, which was known as Devaney chaos [2]. Schweitzer and Smítal introduced a new notion of chaos, which is called distributional chaos [3]. In 1996, a type of chaos known as Kato’s chaos or everywhere chaos was first introduced by H. Kato [4]. In a manner reminiscent of Li–Yorke’s chaos, an equivalent characterization of Kato’s chaos for a continuous map on a compact metric space was provided. For other types of chaos, refer to [5,6,7,8,9,10].

It is generally believed that chaos is intimately linked to nonlinearity. However, as early as 1929, Birkhoff obtained an example of a linear operator that possessed an important ingredient of chaos. Later, Godefroy and Shapiro [11] first brought Devaney chaos to linear dynamics. Scholars were then drawn to the study of several definitions of chaos in linear dynamical systems. Beauzamy [12] introduced irregular vectors for T in Banach space. Bernardes et al. [13] generalized irregular vectors to Fréchet space and obtained new characterizations of Li–Yorke chaos for linear operators on Banach space and Fréchet space. They obtained the conclusion that operator T is densely Li–Yorke chaotic if and only if T is generically w-Li–Yorke chaotic.

The connection between Li–Yorke chaos and the property of irregularity was identified by Bermúdez et al. [14]. Martínez-Giménez et al. [15] first studied distributional chaos in linear dynamics. An in-depth study of distributional chaos was conducted by Bernardes et al. [16].

Another interesting direction of research includes the notion of mean Li–Yorke chaos, which has been investigated by Bernardes et al. [17]. The relation between Li–Yorke chaos and mean Li–Yorke chaos in a sequence is discussed in [18]. A uniform treatment of Li–Yorke chaos, mean Li–Yorke chaos and distributional chaos for continuous endomorphisms of completely metrizable groups is given by Jiang [19]. For more recent work in linear dynamics, refer to [20,21,22,23]. The above works show that chaos in linear dynamical systems can also produce very complex behaviors.

Comparing the complexities between individual dynamics and collective dynamics is a popular topic in both linear and nonlinear dynamical systems. For a continuous map $f:X\to X$ defined on a topological space X, $(\mathbb{K}\left(X\right),\overline{f})$ is the collective dynamical system, where $\mathbb{K}\left(X\right)$ is a set of nonempty compact subsets of X endowed with the Vietoris topology, and $\overline{f}$ is the inducing map of f. Fedeli [24] investigated the relationships between the chaoticity of some set-valued discrete dynamical systems associated with $\overline{f}$ (collective chaos) and the chaoticity of f (individual chaos). Liu et al. [25] proved that if $\overline{f}$ is Li–Yorke sensitive, then f is Li–Yorke sensitve, and gave an example showing that Li–Yorke sensitivity of f dose not imply Li–Yorke sensitivity of $\overline{f}$. Jiang et al.’s [26] work concerned the sensitivity of the product system of set-valued non-autonomous discrete dynamical systems. The concepts of collective accessibility, collective sensitivity, collective infinite sensitivity and collective Li–Yorke sensitivity are defined and discussed in non-autonomous discrete systems [27]. Shao et al. [28] established topological (equi-)semiconjugacy and (equi-) conjugacy between induced non-autonomous set-valued systems and subshifts of finite type.

When it comes to linear dynamics, the investigation on collective dynamics was first started by Herzog and Lemmert [29]. Wu et al. [30] studied the hyperspace linear dynamics for connected compact sets. A study of several chaos types, including Devaney chaos and Li–Yorke chaos, and complexities, like mixing properties and topological ergodicity in hyperspaces associated with linear dynamical systems, was conducted by Bernardes et al. [31]. Barragán et al. [32] investigated the dynamical properties of the dynamical system $({\mathcal{SF}}_m^n\left(X\right),{\mathcal{SF}}_m^n\left(f\right)$. The relationships among the dynamical systems $(X,f)$, the hyperspace systems $(\mathcal{K}\left(X\right),\overline{f})$ and fuzzy systems$(\mathcal{F}\left(X\right),\widehat{f})$ were studied by Martínez-Giménez et al. [33].

This paper introduces the concept of Kato chaos into linear dynamics and its induced dynamics. The article is mainly arranged as follows: In Section 2, some preliminaries are introduced. In Section 3, we have an investigation on sensitivity, accessibility and Kato chaos for continuous linear operators. In Section 4, some efforts are made to discuss Kato chaos in the hyperspace. A conclusion is given in Section 5.

2. Preliminary

As usual, $\mathbb{N}$ denotes the collection of natural numbers and $\mathbb{Z}$ denotes the collection of all integers. Unless otherwise specified, X refers to an infinite-dimensional separable Fréchet space. A Fréchet space is defined by a vector space X endowed with semi-norms, which are increasing sequences ${\left(\parallel \dots \parallel \right)}_{k\in \mathbb{N}}$. The metric of the space is given as

$$\tilde{d}(x_1,x_2)=\sum _{k=1}^{\infty }\frac{1}{2^k}min\{1,{\parallel x_1-x_2\parallel }_k\},\mathrm{for} \mathrm{any}{x}_1,x_2\in X.$$

and such that, under the above metric, X is complete. The set of all continuous linear invertible operators on X is denoted by T(X). For $x\in X$, $\{x,Tx,T^{2}x,\cdots \}$ is called the orbit of x, short for $orb(x,T)$. If there is an $x\in X$ such that $orb(x,T)$ is dense in X, then T is hypercyclic.

Definition 1. 

A map is sensitive if there exists a $\delta >0$ and for any nonempty open subset $V\subset X$, we can find $x_1,x_2\in V$ and a natural number $n\ge 0$ such that $d(T^n\left(x_1\right),T^n\left(x_2\right))>\delta$.

Definition 2. 

A map is accessible if for any $ϵ>0$ and any nonempty open sets $U_1,U_2\subset X$, there are points $x_1\in U_1,x_2\in U_2$ and a positive integer n such that $d(T^n\left(x_1\right),T^n\left(x_2\right))<ϵ$.

If T is both sensitive and accessible, then T is called Kato chaotic.

For the basic definitions of hyperspace, denote that

$$\mathbb{K}\left(X\right)=\{D\subset X:D\mathrm{is} a \mathrm{nonempty} \mathrm{and} \mathrm{compact} \mathrm{subset} \mathrm{of}X\}.$$

Let $K_1,K_2\in \mathbb{K}\left(X\right)$; it defines $d_H$, the Hausdorff metric, on $\mathbb{K}\left(X\right)$ as follows:

$$d_H(K_1,K_2)=max\{\underset{a\in K_1}{max}\underset{b\in K_2}{min}d(a,b),\underset{b\in K_2}{max}\underset{a\in K_1}{min}d(a,b)\}.$$

For a finite collection $V_1,V_2,\dots ,V_r$ of nonempty open subsets of X, denote

$$\mathcal{V}(V_1,\cdots ,V_r)=\{D\in \mathbb{K}\left(X\right):D\subset {\cup }_{i=1}^r{V}_i\mathrm{and}D\cap V_i\ne \varnothing ,i=1,2,\cdots ,r\}.$$

The family $\left\{\mathcal{V}(V_1,\cdots ,V_r)\right|r\in \mathbb{N},V_1,V_2,\dots ,V_r\mathrm{are} \mathrm{nonempty} \mathrm{open} \mathrm{subsets} \mathrm{of}X\}$, which form a base for the topology of $\mathcal{K}\left(X\right)$ called vietoris topology.

With the vietoris topology, $(\mathbb{K}\left(X\right),d_H)$ is called hyperspace. The map on it denotes $\overline{T}$, which is induced by T, i.e., $\overline{T}\left(D\right)=T\left(D\right)$ for $D\in \mathbb{K}\left(X\right)$, where $T\left(D\right)=\{Td:d\in D\}$. It is well-known that if T is a continuous map, then $\overline{T}$ is also continuous.

3. Kato Chaos in Linear Dynamics

As can be seen in the definition of Kato chaos, sensitivity and accessibility constitute two important parts of Kato chaos. Therefore, in terms of the sensitivity and accessibility of Fréchet space, the following characterizations are obtained.

Lemma 1. 

Assume that $T\in T\left(X\right)$. T is sensitive if and only if there exists $\delta >0$; for each $ϵ>0$, there exists $x\in X$ with $d(x,0)<ϵ$ and $n\in \mathbb{N}$ such that $d(T^n\left(x\right),0)>\delta$.

Proof. 

For the necessity, it is obvious. Since T is sensitive, there is a $\delta >0$. For any $ϵ>0$, give an open set $U=\{x:d(0,x)<\frac{ϵ}{2}\}$; there are $y,z\in U$ such that $d(T^n\left(y\right),T^n\left(z\right))>\delta$ for some $n\in \mathbb{N}$. Put $x=y-z$, as we have done. For the sufficiency, there exists a $\delta >0$, for each real number $ϵ>0$ and each $y\in X$. From the hypothesis, there is $x\in X$ with $d(x,0)<ϵ$ and $d(T^n\left(x\right),0)>\delta$ for some $n\in \mathbb{N}$. Let $z=y+x$; we have $d(y,z)=d(x,0)<ϵ$ and $d(T^n\left(y\right),T^n\left(z\right))>\delta$. □

Corollary 1. 

Assume that $T\in T\left(X\right)$,

$$D=\{y\in X|{\left\{T^{n}y\right\}}_{n=0}^{\infty }\mathit{is} \mathit{unbounded}\}$$

If T is sensitive, then D constitutes a residual set of X.

Proof. 

According to Lemma 1, if T is sensitive, there is a $\delta >0$; for any $ϵ>0$, there exist $x\in X$ and $n\in \mathbb{N}$ with $d(x,0)<ϵ$ such that $d(T^n\left(x\right),0)>\delta$. Hence, ${\left(T^n\right)}_{n\in \mathbb{N}}$ is not equicontinuous. An application of the Banach–Steinhaus theorem gives the existence of a vector z such that orb$(z,T)$ is unbounded. From Proposition 5 of [13], vectors with unbounded orbits of T constitute a residual set of X. □

Theorem 1. 

Let $T:X\to X$ be a continuous linear operator defined on Fréchet space. If T is accessible, then

$$X_0=\{x\in X:\underset{n\to \infty }{lim\; inf}d(T^n\left(x\right),0)=0\}$$

is residual in X.

Proof. 

Put $\Delta \left(ϵ\right)=\{x\in X,d(x,0)<ϵ\}$. For any $x\in X$, assume that $U,W$ are neighborhoods of 0 which ensure that $U+U\subset W$. By the hypothesis, T is accessible. That is, for any $k\in \mathbb{N}$, there exist $y_1\in x+U$, $y_2\in U$ and $n\in \mathbb{N}$ such that $d(T^n\left(y_1\right),T^n\left(y_2\right))<1/k$. Let $y=y_1-y_2$. Note that $y\in {\bigcup }_{n=1}^{\infty }{T}^{-n}(\Delta (1/k))$. Then, ${\bigcup }_{n=1}^{\infty }{T}^{-n}(\Delta (1/k))$ is dense in X.

Let $D={\bigcap }_{k=1}^{\infty }{\bigcup }_{n=1}^{\infty }{T}^{-n}(\Delta (1/k))$; it can be verified that $D=X_0$. □

Let us recall the following definition from [13].

Give a Fréchet space X and an operator $T\in T\left(X\right)$; a Li–Yorke pair of T refers to a pair $(x,y)\in X\times X$ if

$$\underset{n\to \infty }{lim\; inf}d(T^n\left(x\right),T^n\left(y\right))=0$$

and

$$\underset{n\to \infty }{lim\; sup}d(T^n\left(x\right),T^n\left(y\right))>0.$$

A subset S of X is called a scrambled set if for any $x,y\in S(x\ne y)$, $(x,y)$ is a Li–Yorke pair for T.

The operator T is called to be Li–Yorke chaotic if there is an uncountable scrambled set for T. The operator T is densely Li–Yorke chaotic if there is an uncountable dense scrambled set for T. T is generically Li–Yorke chaotic if the scrambled set is residual in $X\times X$. T is called densely weak Li–Yorke chaotic if the set of all Li–Yorke pairs for T is dense in $X\times X$. T is called generically weak Li–Yorke chaotic if the set of Li–Yorke pairs for T is a residual set.

Definition 3. 

Suppose that $T:X\to X$ is a continuous linear operator defined on Fréchet space. A vector $x\in X$ is called an irregular vector for T if its orbit ${\left(T^{n}x\right)}_{n\in \mathbb{N}}$ is unbounded, while it has a subsequence converging to zero.

Definition 4. 

Suppose that $T:X\to X$ is a continuous linear operator defined on Fréchet space. The vector $x\in X$ is called a semi-irregular vector of T if the sequence ${\left(T^{n}x\right)}_{n\in \mathbb{N}}$ does not converge to zero, but has a subsequence that does.

Theorem 2 

([13]). Let X be a Fréchet space and T be a linear operator on it. Then, the following assertions are equivalent:

(i) 

T is densely Li–Yorke chaotic;

(ii) 

T is densely w-Li–Yorke chaotic;

(iii) 

T is generically w-Li–Yorke chaotic;

(iv) 

T admits a dense set of semi-irregular vectors;

(v) 

T admits a dense set of irregular vectors;

(vi) 

T admits a residual set of irregular vectors.

Theorem 3 

([4]). Let $T:X\to X$ be a map of a complete metric space without an isolated point. Then, T is Kato chaotic if and only if T satisfies the following condition: there is $\tau >0$ such that for each $ϵ>0$ and each compact subset $K\subset X$ there is a Cantor set C of X such that $d_H(K,C)<ϵ$ and if $x,y\in C$ with $x\ne y$, then

$$\underset{n\to \infty }{lim\; sup}d(T^n\left(x\right),T^n\left(y\right))\ge \tau$$

and

$$\underset{n\to \infty }{lim\; inf}d(T^n\left(x\right),T^n\left(y\right))=0.$$

Theorem 4. 

Suppose that X is a Fréchet space and $T\in T\left(X\right)$; then, the claims below are equivalent to each other:

1. 

operator T is densely Li–Yorke chaotic;

2. 

operator T is Kato chaotic.

Proof. 

(1) ⇒ (2). From Theorem 2, if T is densely Li–Yorke chaotic, then T admits a dense set of irregular vectors. Assume that D is a collection of irregular vectors and D is dense in X. Suppose $B=\left\{\right(a,b)\in X\times X|a-b\in D\}$; it is not hard to draw a conclusion that B is dense in $X\times X$.

First, we want to obtain the sensitivity of T. For any $x\in X$ and any neighborhood U of x in X, $U\times U$ is also a non-empty open set of $X\times X$. Therefore, there exists $(a_1,b_1)\in B$ such that $(a_1,b_1)\in U\times U$ ($a_1$ must not be equal to $b_1$, if so $a_1-b_1=0$ is not the irregular vector). Then, the sequence ${\left(T^n(a_1-b_1)\right)}_{n\in \mathbb{N}}$ is unbounded. That is to say that there is $3\tau >0$ and $n\in {\mathbb{N}}^{+}$ such that $d(T^n\left(a_1\right),T^n\left(b_1\right))=d(T^n(a_1-b_1),0)>3\tau$. This implies that either $d(T^n\left(a_1\right),T^n\left(x\right))>\tau$ or $d(T^n\left(x\right),T^n\left(b_1\right))>\tau$. Hence, T has sensitivity.

Next, we will prove that T is accessible. For any non-empty open sets $U,V\subset X$, there exists $(a_2,b_2)\in B$ such that $(a_2,b_2)\in U\times V$. Then, the sequence ${\left(T^n(a_2-b_2)\right)}_{n\in \mathbb{N}}$ has a subsequence converging to zero. That is to say, for any $ϵ>0$, there exists $n^{{}^{\prime }}\in {\mathbb{N}}^{+}$ such that $d(T^{n^{{}^{\prime }}}\left(a_2\right),T^{n^{{}^{\prime }}}\left(b_2\right))=d(T^{n^{{}^{\prime }}}(a_2-b_2),0)<ϵ$. Hence, T is accessible.

(2) ⇒ (1). For any $x\in X,x\ne 0$ and any $ϵ>0$, let $K=\{0,x\}$. According to Theorem 3, there is $\tau >0$ such that for $\frac{ϵ}{2}$, there exists a Cantor set C of X such that $d_H(K,C)<\frac{ϵ}{2}$ and for any $x,y\in C$ with $x\ne y$, we have

$$\underset{n\to \infty }{lim\; sup}d(T^n\left(x\right),T^n\left(y\right))\ge \tau$$

and

$$\underset{n\to \infty }{lim\; inf}d(T^n\left(x\right),T^n\left(y\right))=0.$$

This implies that there exist $y_1,y_2\in C,y_1\ne y_2$ such that $d(0,y_1)<\frac{ϵ}{2}$ and $d(x,y_2)<\frac{ϵ}{2}$. Let $y=y_1-y_2$; it can be easily verified that y is a semi-irregular vector and $d(y,x)\le d(y_2,x)+d(y,0)<ϵ$. Therefore, the set of semi-irregular vectors is dense in X. By Theorem 2, T is densely Li–Yorke chaotic. □

Examples of operators with Kato chaos are provided in the following:

Example 1. 

The operator T on $l_2$ is a bilateral weighted shift with respect to the canonical basis $\{e_n:n\in \mathbb{Z}\}$ and $T^{*}$ has an eigenvalue λ. If

$$T{e}_n=a_n{e}_{n+1}$$

where the weight sequence $\{a_n:n\in \mathbb{Z}\}$ is a bounded subset of $\mathbb{C}\setminus \left\{0\right\}$, take the weights

$$\left\{a_n\right\}=\{2,0.5,0.5,2,2,2,0.5,0.5,0.5,0.5\dots \}.$$

Then, T has a dense set of irregular vectors in $l_2$ [[34], Proposition 3.9]; by Theorem 2, it is densly Li–Yorke chaotic. Furthermore, it is Kato chaotic, and $T^{*}$ has an eigenvalue $\left|\lambda \right|<1$.

Example 2. 

Let $m_1=1,a_1=1+\frac{1}{2}$, and ${\mathcal{H}}_1={\mathbb{C}}^2$. Assume that we have selected $m_{n-1},a_{n-1}$ and ${\mathcal{H}}_{m-1}$. Let $p_n$ be an integer such that

$${(1+\frac{1}{2^{p_n}})}^{m_{n-1}+1}\le 2.$$

Let $a_n=1+\frac{1}{2^{p_n}}$ and $m_n>m_{n-1}$ be an integer such that $a_n^{m_n}\le n$.

Finally, take ${\mathcal{H}}_n={\mathbb{C}}^{m_n+1}$. For each n, we define the operator $A_n$ on ${\mathcal{H}}_n$ given by

$$\left(\begin{array}{cccccc}0& 0& 0& \cdots & 0& 0\\ a_n& 0& 0& \cdots & 0& 0\\ 0& a_n& 0& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \cdots & a_n& 0\end{array}\right)$$

(1)

By the (Proposition 3.10, [34]), the operator A has a dense set of irregular vectors. According to Theorem 2 and Theorem 4, it is Kato chaotic.

Corollary 2. 

Let X be a Fréchet space (or Banach space) and T be a linear operator on it. If T is hypercyclic, then T is chaotic in the sense of Kato.

Based on Theorem 4 and Remark 38 in reference [16], it is easy to give the following corollary:

Corollary 3. 

Suppose that $T:X\to X$ is a continuous linear operator defined on Fréchet space. Then, the claims below are equivalent to each other:

(1) 

T is Kato chaotic.

(2) 

$T^n$ is Kato chaotic for any $n\in \mathbb{N}$.

(3) 

$T^m$ is Kato chaotic for some $m\in \mathbb{N}$.

4. Kato Chaos in Hyperspace Induced by T

The main work of this section is to discuss Kato chaos in hyperspace induced by T.

Since sensitivity is a very important part of Kato chaos, we would like to gain insight into the connection between T and the induced maps $\overline{T}$. Let us first review some definitions of sensitivity.

For $U\subset X$ and $ϵ>0$, let

$$N_T(U,ϵ)=\{n\in \mathbb{N}:\mathrm{there} \mathrm{are}x,y\in U\mathrm{such} \mathrm{that}d(T^n\left(x\right),T^n\left(y\right))>ϵ\}.$$

Definition 5. 

An operator T is

(1) 

sensitive if there is an $ϵ>0$ for any nonempty open set $U\subset X$, $N_T(U,ϵ)\ne \varnothing$.

(2) 

multi-sensitive if there is $ϵ>0$ such that for every $k\ge 1$ and any nonempty open subset $U_1,U_2,\cdots ,U_k\subset X$, the set ${\displaystyle \bigcap _{1\le i\le k}}{N}_T(U_i,ϵ)$ is nonempty.

Theorem 5. 

Assume that X is a Fréchet space, $T\in T\left(X\right)$; then, the claims below are equivalent to each other:

(i) 

operator T is sensitive;

(ii) 

operator $\overline{T}$ is sensitive.

Proof. 

(ii) ⇒ (i) is obvious. We only need to prove (i) ⇒ (ii). Assume that $ϵ>0$ is the sensitive constant of T. For every nonempty open set

$$\mathbb{U}=\mathcal{V}(U_1,U_2,\cdots ,U_r)\subset \mathbb{K}\left(X\right),$$

where $r\ge 1$ and $U_1,U_2,\cdots ,U_r$ are nonempty open sets of X. Let $K=\{x_1,x_2,\cdots ,x_r\}\in \mathbb{U}$ such that $K\in \mathbb{K}\left(X\right),K\subset {\bigcup }_{i=1}^r{U}_i$ and $x_i=K\bigcap U_i\ne \varnothing ,i=1,\cdots ,r$. We can choose a sufficient small 0-neighborhood $W\ne \varnothing$ such that $K+W\subset {\bigcup }_{i=1}^r{U}_i$ and $x_i+W\subset U_i,i=1,\cdots ,r$. As T is sensitive, we have $x,y\in W$ and $n\in \mathbb{N}$ such that $d(T^n\left(x\right),T^n\left(y\right))>ϵ$. Notice that for any $1\le i\le r$, $x_i+x,x_i+y\in U_i$ and $d(T^n(x_i+x),T^n(x_i+y))>ϵ$. Let $K_1=\{x_1+x,x_2+x,\cdots ,x_r+x\}\in \mathbb{U}$; we want to construct $K_2\in \mathbb{U}$ such that $d_H({\overline{T}}_n\left(K_1\right),{\overline{T}}^n\left(K_2\right))>{ϵ}_0$, for some ${ϵ}_0>0$. Take $K_2={\left\{z_i\right\}}_{i=1}^r$ such that the following hold:

(1)

If $d(T^n(x_1+x),T^n(x_i+x))>\frac{ϵ}{2}$, then $z_i=x_i+x$;

(2)

If $d(T^n(x_1+x),T^n(x_i+x))\le \frac{ϵ}{2}$, then $z_i=x_i+y$.

Therefore, $K_2\in \mathbb{U}$ and $d_H({\overline{T}}^n\left(K_1\right),{\overline{T}}^n\left(K_2\right))>\frac{ϵ}{2}={ϵ}_0$. □

Remark 1. 

It should be emphasized that, in general, the above equivalence relation does not hold for nonlinear dynamics [35].

Theorem 6. 

Suppose that X is a Fréchet space and T is a continuous linear operator defined on X. Thus, T is sensitive if and only if T is multi-sensitive.

Proof. 

From the definitions, the multi-sensitivity of T clearly implies the sensitivity of T.

It is suffices to show the inverse, i.e., the sensitivity of T implies the multi-sensitivity of T.

For any given $r>0$ and any nonempty open sets $U_1,U_2,\cdots ,U_r$, there exist $x_1\in U_1,x_2\in U_2,\cdots ,x_r\in U_r$. We can choose a small enough open set W with $0\in W$ and $x_i+W\in U_i,1\le i\le r$. As T is sensitive, for a fix $\delta >0$, there exist two points $x,y\in W$ and a number $n\in \mathbb{N}$ such that $d(T^n\left(x\right),T^n\left(y\right))>\delta$. Hence, $d(T^n(x_i+x),T^n(x_i+y))>\delta$. That is to say that ${\bigcap }_{i=1}^r{N}_T(U_i,\delta )\ne \varnothing$. □

Lemma 2. 

Denote that $Pro{x}_n\left(X\right)=\{(x_1,x_2,\cdots ,x_n):\mathit{for} \mathit{every}ϵ>0\mathit{there} \mathit{exists}k\in \mathbb{N}\mathit{such} \mathit{that}d(T^k\left(x_i\right),T^k\left(x_j\right))<ϵ,i,j\in \{1,2,\cdots ,n\},i\ne j\}.$ For a linear dynamic $(X,T)$ where X is a Fréchet space and T is a linear operator, if $Pro{x}_n\left(X\right)$ is dense in $X^n$ for any $n\in \mathbb{N}$, then

1. 

$(X,T)$ is accessible.

2. 

$(\mathbb{K}\left(X\right),\overline{T})$ is accessible.

Proof. 

• Let $n=2$. $Pro{x}_2\left(X\right)$ is dense in $X^2$; according to the definition of accessibility, $(X,T)$ is accessible.
• Let ${\mathbb{U}}_H$ and ${\mathbb{V}}_H$ be a nonempty open set of $\mathbb{K}\left(X\right)$; note that ${\mathbb{U}}_H=\mathcal{V}(U_1,U_2,\cdots ,U_s)$, ${\mathbb{V}}_H=\mathcal{V}(V_1,V_2,\cdots ,V_t)$, $s,t\in {\mathbb{Z}}^{+}$. For any $n\in \mathbb{N}$, $Pro{x}_n\left(X\right)$ is dense in $X^n$. Put $n=s+t$; $Pro{x}_{s+t}\left(X\right)$ is dense in $X^{s+t}$. Hence, there is $(x_1,x_2,\cdots ,x_s,y_1,y_2,\cdots ,y_y)\in U_1\times U_2\times \cdots \times U_s\times V_1\times U_2\times \cdots \times V_t$ with the property that for any $ϵ>0$ there is a $k\in \mathbb{N}$ such that $d(T^k\left(x_i\right),T^k\left(y_j\right))<ϵ(1\le i\le s,1\le j\le t)$, where $x_i\in U_i$, $y_i\in V_j$ for $1\le i\le s,1\le j\le t$. Denote that $K_1=\{x_1,x_2,\cdots ,x_s\}$, $K_2=\{y_1,y_2,\cdots ,y_t\}$. It is clear that $K_1\in {\mathbb{U}}_H,K_2\in {\mathbb{V}}_H$ and we have $d_H({\overline{T}}^k\left(K_1\right),{\overline{T}}^k\left(K_2\right))<ϵ$

□

According to the Lemma 2, it is easy to have the following theorem:

Theorem 7. 

Assume that X is a Fréchet space and T is a linear operator defined on $X\to X$. If $Pro{x}_n\left(X\right)$ is dense in $X^n$ for any $n\in \mathbb{N}$, then the claims below are equivalent to each other:

(i) 

operator T is Kato chaotic.

(ii) 

operator $\overline{T}$ is Kato chaotic.

5. Conclusions

This paper has researched Kato chaocity for continuous linear operators in linear dynamics and its induced dynamics. It is obtained that the Kato chaocity is equivalent to dense Li–Yorke chaocity for a continuous linear operator defined on Fréchet space. In addition, a sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators $\overline{T}$ are equivalent. The research results of this paper enrich the study of chaos in linear dynamical systems. Another popular topic in linear dynamics is to compare individual dynamics and collective dynamics induced by convex space. What will happen to Kato chaos in a convex space? It is worth studying in the future.