Relationships among Various Chaos for Linear Semiflows Indexed with Complex Sectors

Abstract

In this paper, we investigate the relationships among point transitivity, topological transitivity, Li–Yorke chaos, and the existence of irregular vectors for a linear semiflow ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ indexed with a complex sector. We reveal the equivalence between topological transitivity and point transitivity for a linear semiflow ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, especially in case the range of some operator $T_t,t\in \mathsf{\Delta }$ is not dense. We also prove that Li–Yorke chaos is equivalent to the existence of a semi-irregular vector and that point transitivity is stronger than the existence of an irregular vector for any linear semiflow ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$. At last, unlike the conclusion for traditional linear dynamical systems, we show that there exists a Li–Yorke chaotic $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ without irregular vectors. The results and proof methods presented in this paper demonstrate the differences in the dynamical behavior between linear semiflows ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ and traditional linear systems with the acting semigroup $S={\mathbb{Z}}^{+}$ and $S={\mathbb{R}}^{+}$.

1. Introduction

During the last three decades, different phenomena associated with chaos induced by a single linear operator T, or by a strongly continuous semigroup of operators (i.e., $C_0$-semigroup) ${\left\{T_t\right\}}_{t\ge 0}$, on an infinite-dimensional vector space have been extensively investigated; see monographs [1,2] and the references therein. Among others, point transitivity (also called hypercyclicity) and topological transitivity are the fundamental and key notions to characterize linear chaos.

• A continuous operator T (A $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$) on a topological vector space X is called topologically transitive if for any pair of non-empty open sets $U,V\subset X$, the meeting time set $N(U,V):=\{n\in {\mathbb{Z}}^{+}|T^{n}U\cap V\ne \varnothing \}$ ($N(U,V):=\{t\in {\mathbb{R}}^{+}|T_{t}U\cap V\ne \varnothing \}$, respectively) is non-empty.
• A continuous operator T (A $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$) on a topological vector space X is called point-transitive or hypercyclic if there exists a point $x_0\in X$ such that the orbit $Orb(T,x_0):=\left\{T^n{x}_0\right|n\in {\mathbb{Z}}^{+}\}$ ($Orb({\left\{T_t\right\}}_{t\in {\mathbb{R}}^{+}},x_0):=\left\{T_t{x}_0\right|t\in {\mathbb{R}}^{+}\}$, respectively) is dense in X. Such a point $x_0$ is called a transitive point or a hypercyclic vector.

Early research on linear systems focused on individual linear operators (see [3,4,5,6]). Later, with the growing connection to the solution semigroups of differential equations, the dynamics of a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ on a Banach space was also systematically studied (see [7,8,9]). Recently, due to its applications in second-order elliptic and parabolic equations and connections with topological semiflows, another important case, a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space indexed with a complex sector $\mathsf{\Delta }:=\mathsf{\Delta }\left(\alpha \right)=\left\{r{e}^{i\theta }|r\ge 0,|\theta |\le \alpha \right\}$, where $0<\alpha \le {\displaystyle \frac{\pi }{2}}$, received considerable attention (see [10,11,12,13,14,15,16,17]).

In the studies on ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, topological transitivity and point transitivity remain core topics of investigation. The notions and definitions of topological transitivity and point transitivity can be easily generalized to the case ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, by replacing ${\mathbb{Z}}^{+}$ or ${\mathbb{R}}^{+}$ with $\mathsf{\Delta }$. In [10,11], Conejero and Peris explored whether there are mutual implications on topological transitivity between a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ and its subsemigroups or discretizations. In [12], Chaouchi et al. analyzed f-frequently hypercyclic, q-frequently hypercyclic and frequently hypercyclic $C_0$-semigroups indexed with complex sectors. In [14], He et al. studied the underlying relationships between transitivity and chaos for linear semiflows with abelian acting semigroups. In [13,15], He et al. characterized the $\mathcal{F}$-transitivity and the d$\mathcal{F}$-transitivity for translation semigroups and general composition semiflows ${\left\{T_{\phi (t,·)}\right\}}_{t\in S}$ on $L^p$-spaces. In [17], Liang et al. established the recurrent hypercyclicity criterion for the $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$.

Nevertheless, none of the aforementioned papers addressed the discussion of the equivalence between topological transitivity and point transitivity for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a separable space, which is important and fundamental in the study of traditional topological and linear semflows. Recall that $\pi :S\times X\to X,(t,x)\mapsto \pi (t,x)$ is a (topological) semiflow with the phase space X and with the phase semigroup S, denoted by $(\pi ,S,X)$, if S is an arbitrary additive topological semigroup with a neutral element e, X is a topological space, and the phase map $\pi :(t,x)\mapsto \pi (t,x)$ satisfies conditions:

(1)

$\pi :(t,x)\mapsto \pi (t,x)$ is jointly continuous;

(2)

$\pi (e,x)=x,\forall x\in X$;

(3)

$\pi (t,sx)=\pi (t+s,x),\forall s,t\in S,x\in X$.

For simplicity, we will denote the transition map ${\pi }_t:x\mapsto \pi (t,x)$ by $T_t$ and denote the semiflow $(\pi ,S,X)$ by ${\left\{T_t\right\}}_{t\in S}$. Moreover, ${\left\{T_t\right\}}_{t\in S}$ is called a linear semiflow if $T_t,\forall t\in S$ is a continuous linear operator on a topological vector space X. Our research subject in this paper, that is, a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, is a type of linear semiflows.

The equivalence between topological transitivity and point transitivity holds for traditional discrete topological systems $(\pi ,{\mathbb{Z}}^{+},X)$ and traditional continuous linear systems ${\left\{T_t\right\}}_{t\ge 0}$ on separable complete metric spaces (vector spaces in the linear case) without isolated points [18,19], but does not hold for a traditional continuous topological system $(\pi ,{\mathbb{R}}^{+},X)$. One simple counterexample is the additive action $\pi :(t,x)\mapsto t+x,\forall t\in {\mathbb{R}}^{+},x\in X$ on the space $X={\mathbb{R}}^{+}$. Furthermore, the equivalence still does not hold for general topological semiflows and linear semiflows (cf. [20] [Example 1.3] and [14] [Example 1.3]).

The reason for this difference is that, for traditional discrete topological systems and traditional continuous linear systems ${\left\{T_t\right\}}_{t\ge 0}$ on separable complete metric spaces (vector spaces in the linear case) without isolated points, each operator or mapping $T_t$ has a dense range, while this is not the case for a traditional continuous topological system $(\pi ,{\mathbb{R}}^{+},X)$ and a general topological or linear semiflow. In other words, the density of the operator’s range plays a crucial role in determining whether the equivalence between topological transitivity and point transitivity holds. In articles [10,11,12,13,14,15,17], the point-transitive or topologically transitive $C_0$-semigroups ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ are assumed to have each operator with a dense range. However, the condition that each operator in the $C_0$-semigroups ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ has a dense range does not always hold. We present a point-transitive (hypercyclic) $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ having some operators without dense range in the following.

Example 1.

Suppose Y is the Banach Space $C_0({\mathbb{R}}^{+},\mathbb{R})$, that is,

$$Y=C_0({\mathbb{R}}^{+},\mathbb{R}):=\left\{f:{\mathbb{R}}^{+}\to \mathbb{R}\mathrm{continuous}|\underset{t\to +\infty }{\lim }f\left(t\right)=0\right\},$$

with norm $\Vert f\Vert ={\sup }_{t\ge 0}\left|f\left(t\right)\right|$. Let $Q:=\left\{{\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n\cdots \right\}$ be the positive rational numbers sequence and $X:=\overline{\mathrm{span}\left\{e^{-\lambda t},\lambda \in Q\right\}}$ be a closed space in Y. Define a $C_0$-semigroup ${\left\{T_{\rho }^{\prime }\right\}}_{\rho \ge 0}$ on the space X as follows:

$$T_{\rho }^{\prime }\left(f\left(t\right)\right)=e^{-\rho t}f\left(t\right),\forall \rho ,t\ge 0,$$

where $f\in X.$ Let ${\left\{T_{\rho }^{″}\right\}}_{\rho \ge 0}$ be a hypercyclic (point-transitive) $C_0$-semigroup on the space X. Suppose $\mathsf{\Delta }:=\mathsf{\Delta }\left(\alpha \right)=\left\{r{e}^{i\theta }|r\ge 0,|\theta |\le \alpha \right\}$, where $0<\alpha \le {\displaystyle \frac{\pi }{2}}$. Define a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on X as follows:

$$T_{t}f=T_{{\rho }_1}^{\prime }{T}_{{\rho }_2}^{″}f,\mathrm{where}t={\rho }_1{e}^{i\alpha }+{\rho }_2{e}^{-i\alpha }\in \mathsf{\Delta }.$$

Then ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is point-transitive and $Rng\left(T_{\rho e^{i\alpha }}\right)$ is not dense for any $\rho >0$.

Proof. 

It is not hard to check that X is an infinite-dimensional separable Banach space.

At first, we will verify that the range of $T_{r{e}^{i\alpha }}\left(T_r^{\prime }\right)$ is not dense for any $r>0$. Fix arbitrary $r_1>0$. Then, $Rng\left(T_{r_1}^{\prime }\right)$ cannot approximate the function $e^{-\lambda t}\in X$, if $\lambda >0$ is sufficiently close to 0.

The existence of a hypercyclic $C_0$-semigroup ${\left\{T_{\rho }^{″}\right\}}_{\rho \ge 0}$ on the space X can be guaranteed by the theorem (see [21,22]) which states that every infinite-dimensional separable Banach space supports a hypercyclic operator (also supports a hypercyclic $C_0$-semigroup).

Lastly, the point transitivity of ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is clear since ${\left\{T_{\rho }^{″}\right\}}_{\rho \ge 0}$ is point-transitive. □

Now it is natural to ask if the equivalence between topological transitivity and point transitivity still holds for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ having some operators without dense range. Therefore, we propose the following question.

Question 1.

Is topological transitivity equivalent or inequivalent to point transitivity for any $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a separable Banach space?

In this article, we will answer this question affirmatively in Section 2. Moreover, Theorem 1 provides us with a more subtle insight into the transitivity of $C_0$-semigroups ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ and their subsemigroups. From this theorem, point-transitive (i.e., hypercyclic) $C_0$-semigroups ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on separable Banach spaces can be divided into two classes. The first class admits a hypercyclic $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ in which any subsemigroup restricted to a ray from the origin are not hypercyclic, while for any $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ in the second class, there exists one and only one subsemigroup restricted to a ray from the origin is point-transitive.

Li–Yorke chaos and sensitive dependence on initial conditions [23,24,25,26,27,28,29,30,31] are also basic notions in the study of topological and linear dynamics.

• A continuous map $f:X\to X$ on a metric space $(X,d)$ is called Li–Yorke chaotic if there exists an uncountable subset $\mathsf{\Gamma }\subset X$ such that for any distinct pair $(x,y)\in \mathsf{\Gamma }\times \mathsf{\Gamma }$,

  $$\underset{n\to \infty }{\lim \inf }d(f^{n}x,f^{n}y)=0,\underset{n\to \infty }{\lim \sup }d(f^{n}x,f^{n}y)>0.$$
  Such a set $\mathsf{\Gamma }$ is called a scramble set and $(x,y)\in \mathsf{\Gamma }\times \mathsf{\Gamma }$ is called a Li–Yorke chaotic pair.
• A continuous map $f:X\to X$ on a metric space $(X,d)$ is said to be sensitive to initial conditions if there exists $c>0$, such that for every $x\in X$ and any neighborhood U of x, there exists $y\in U$ and $n\in \mathbb{N}$, such that $d(f^{n}x,f^{n}y)>c$.

Generally, Li–Yorke chaos does not imply sensitivity on initial conditions, for instance, a Li–Yorke chaotic continuous map $f\in C\left(I\right)$ where I is an interval of the real line could be a constant on a certain subinterval in I. However, when the map is a continuous linear operator T on a Banach space $(X,||·|\left|\right)$, it is not hard to see that Li–Yorke chaos implies sensitivity, noting that sensitivity equals that ${\sup }_{n\in \mathbb{N}}\left|\right|T^n\left|\right|=\infty$. Indeed, in this case, Li–Yorke chaos is equivalent to the existence of semi-irregular vectors and irregular vectors [28,29]. Irregular vectors refer to vectors in a Banach space or Hilbert space that exhibit complex or non-convergent behavior under the repeated action of a linear operator or semigroup. We provide the strict definitions of irregular vectors and semi-irregular vectors in the following. Henceforth, X will denote a Banach space with norm $\left|\right|·\left|\right|$ and $L\left(X\right)$ will denote the space of continuous linear operators on X (or just referred to as operators).

Definition 1.

Given an operator $T\in L\left(X\right)$ on a Banach space $(X,||·|\left|\right)$, and a vector $x\in X$, we say that x is an irregular vector for T if

$$\underset{n\to \infty }{\lim \sup }\left|\right|T^{n}x\left|\right|=\infty ,\mathit{and}\underset{n\to \infty }{\lim \inf }\left|\right|T^{n}x\left|\right|=0.$$

Definition 2.

Given an operator $T\in L\left(X\right)$ on a Banach space $(X,||·|\left|\right)$, and a vector $x\in X$, we say that x is a semi-irregular vector for T if

$$\underset{n\to \infty }{\lim \sup }\left|\right|T^{n}x\left|\right|>0,\mathit{and}\underset{n\to \infty }{\lim \inf }\left|\right|T^{n}x\left|\right|=0.$$

On one hand, Li–Yorke chaos implies the existence of a semi-irregular vector, noting that the vector $z=x-y$ is semi-irregular if $(x,y)$ is a Li–Yorke pair for T. On the other hand, the existence of a semi-irregular vector implies Li–Yorke chaos. Suppose that x is a semi-irregular vector, then the segment $\mathsf{\Gamma }:=\left\{\lambda x;\left|\lambda \right|\le 1\right\}$ is a scrambled set for T, since any distinct pair $({\lambda }_{1}x,{\lambda }_{2}x)\in \mathsf{\Gamma }\times \mathsf{\Gamma }$ is a Li–Yorke pair. Therefore, Li–Yorke chaos is equivalent to the existence of a semi-irregular vector for a continuous linear operator.

Surprisingly enough, for continuous linear operators, Li–Yorke chaos further equals the existence of an irregular vector (see references [28,29]) which is apparently stronger than a semi-irregular vector. Clearly, definitions of Li–Yorke chaos, Li–Yorke pairs and (semi-)irregular vectors can be similarly given for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in {\mathbb{R}}^{+}}$ or ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a separable Banach space, with $t\in {\mathbb{R}}^{+}$ or $t\in \mathsf{\Delta }$ in replace of $n\in \mathbb{N}$. By adopting a similar method to that in [29] [Theorem 9], the conclusion that Li–Yorke chaos is equivalent to the existence of irregular vectors can be likewise established for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in {\mathbb{R}}^{+}}$. As for ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, it is natural to ask the following question.

Question 2.

Is Li–Yorke chaos equivalent to the existence of an irregular vector for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space X?

However, unlike the conclusion for a single operator T or a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in {\mathbb{R}}^{+}},$ we will show that the existence of irregular vectors implies Li–Yorke chaos, but not vice versa for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ in Section 3, which gives Question 2 a negative answer.

2. The Equivalence between Topological Transitivity and Point Transitivity

In this section, we will show that topological transitivity is equivalent to point transitivity for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a separable Banach space in Theorem 1.

At first, for the convenience of our readers, we recall the definition of a $C_0$-semigroup.

Definition 3.

A one-parameter family ${\left\{T_t\right\}}_{t\in S}$, where $S={\mathbb{R}}^{+}$ or $S=\mathsf{\Delta }$, of operators on X is called a strongly continuous semigroup of operators or a $C_0$-semigroup, if it satisfies the conditions:

(i) 

$T_0=I$;

(ii) 

$T_{t+s}=T_t{T}_s$ for all $s,t\in S$;

(iii) 

${\lim }_{s\to t}{T}_{s}x=T_{t}x$ for all $x\in X$, $t\in S$.

Readers can understand this definition in conjunction with the definition of a linear semiflow. We remind that a $C_0$-semigroup (i.e., a linear semiflow) ${\left\{T_t\right\}}_{t\in S}$ ($S={\mathbb{R}}^{+}$ or $\mathsf{\Delta }$) is locally equicontinuous, that is, for any $s>0$,

$$\underset{t\in S,\left|t\right|\le s}{\sup }\parallel T_t\parallel <\infty .$$

In the following, we give clear definitions of point transitivity and topological transitivity for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ as follows.

• A $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space X is called topologically transitive if for any pair of non-empty open sets $U,V\subset X$, the meeting time set $N(U,V):=\{t\in \mathsf{\Delta }|T_{t}U\cap V\ne \varnothing \}$ is non-empty.
• A $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space X is called point-transitive or hypercyclic if there exists a point $x_0\in X$ such that the orbit $Orb(T,x_0):=\left\{T_t{x}_0\right|t\in \mathsf{\Delta }\}$ is dense in X. Such a point $x_0$ is called a transitive point or a hypercyclic vector.

Before the main theorem (Theorem 1) in this section, let us give two useful lemmas for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ on a Banach space. We define ${\mathsf{\Delta }}_r:=\left\{t\in \mathsf{\Delta }:\left|t\right|\le r\right\}$ for any given $\mathsf{\Delta }$ and $r>0$.

Lemma 1.

Given a $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$ on a Banach space X, the following assertions are equivalent:

(i) 

Each operator $T_t\left(t>0\right)$ has a dense range;

(ii) 

There exists at least a real number $t_0>0$ such that the operator $T_{t_0}$ has a dense range;

(iii) 

There exists at least a real number $t_0>0$ such that the range of $T_{t_0}$ is somewhere dense, i.e., $intcl\left(Rng\right(T\left)\right)\ne \varnothing$.

Proof. 

The implications $\left(\mathrm{i}\right)\Rightarrow \left(\mathrm{ii}\right)\Rightarrow \left(\mathrm{iii}\right)$ are trivial. We only need to show that $\left(\mathrm{iii}\right)\Rightarrow \left(\mathrm{ii}\right)$ and $\left(\mathrm{ii}\right)\Rightarrow \left(\mathrm{i}\right)$.

$\left(\mathrm{iii}\right)\Rightarrow \left(\mathrm{ii}\right)$. We only need to note that $T_{t_0}X$ is somewhere dense in X indeed implies that it is dense in X since $T_{t_0}X$ is a linear subspace of X.

$\left(\mathrm{ii}\right)\Rightarrow \left(\mathrm{i}\right)$. For every $0<t\le t_0$, it is not hard to see that $T_{t_0}X=T_{t_0-t}{T}_{t}X=T_t{T}_{t_0-t}X$$\subset T_{t}X$, which implies that $T_t$ has a dense range.

For every $t>t_0$, we only need to note that there exists some $n\in \mathbb{N}$, such that $n{t}_0\ge t$ and then $T_{t}X$ contains $T_{n{t}_0}X$ which is dense in X. □

Lemma 2

(Lemma 2 [12]). Suppose that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is a hypercyclic $C_0$-semigroup on a Banach space X and $x\in X$ is a hypercyclic element of ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$. Then, the set $\left\{T_{t}x:t\in \mathsf{\Delta }\setminus {\mathsf{\Delta }}_s\right\}$ is dense in X for each $s>0$.

Now we give the following main theorem to show the equivalence between topological transitivity and point transitivity for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a separable Banach space.

Theorem 1.

Let ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ ($\mathsf{\Delta }=\mathsf{\Delta }\left(\alpha \right)$) be a $C_0$-semigroup on a separable Banach space X. Then, ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is point-transitive if and only if it is topologically transitive. Moreover, if ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is point-transitive, one and only one of the following conclusions holds true.

(a) 

The ranges of $T_{t{e}^{-i\alpha }}$ and $T_{t{e}^{i\alpha }}$ are both dense for any $t>0$. In this case, ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is topologically transitive.

(b) 

The range of $T_{r{e}^{i\alpha }}$ is nowhere dense, and the range of $T_{r{e}^{-i\alpha }}$ is dense for any $r>0$. In this case, ${\left\{T_{r{e}^{-i\alpha }}\right\}}_{r\ge 0}$ is topologically transitive (equals point-transitive).

(c) 

The range of $T_{r{e}^{-i\alpha }}$ is nowhere dense, and the range of $T_{r{e}^{i\alpha }}$ is dense for any $r>0$. In this case, ${\left\{T_{r{e}^{i\alpha }}\right\}}_{r\ge 0}$ is topologically transitive (equals point-transitive).

Proof. 

Firstly, it is trivial to prove that topological transitivity implies point transitivity. Indeed, a topological semiflow with a Polish phase space X is topologically transitive if and only if the set of transitive points is a dense $G_{\delta }-$set of X (cf. [32] and [Basic Fact 1] in [30]).

Secondly, we will prove that point transitivity implies topological transitivity. Note that for each nonzero complex number $t\in \mathsf{\Delta }$, either $T_{t}X\subset T_{r{e}^{i\alpha }}X$ or $T_{t}X\subset T_{r{e}^{-i\alpha }}X$ for some $r>0$. Since ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is point-transitive, the set $\left\{T_{t}x:t\in \mathsf{\Delta }\setminus {\mathsf{\Delta }}_s\right\}$ is dense for any $s>0$ from Lemma 2 in [12]. Then, we have that ${\bigcup }_{t\in \mathsf{\Delta }\setminus {\mathsf{\Delta }}_s}{T}_{t}X=\left({\bigcup }_{r>s}{T}_{r{e}^{i\alpha }}X\right)\bigcup \left({\bigcup }_{r>s}{T}_{r{e}^{-i\alpha }}X\right)$ is dense in X, which implies either ${\bigcup }_{r>s}{T}_{r{e}^{-i\alpha }}X$ or ${\bigcup }_{r>s}{T}_{r{e}^{i\alpha }}X$ is somewhere dense in X, which further implies that either $T_{s{e}^{-i\alpha }}X$ or $T_{s{e}^{i\alpha }}X$ is somewhere dense in X. From Lemma 1, it follows that one of the three statements below must hold true:

(i)

the ranges of $T_{r{e}^{-i\alpha }}$ and $T_{r{e}^{i\alpha }}$ are both dense for any $r>0$;

(ii)

the range of $T_{r{e}^{i\alpha }}$ is nowhere dense, and the range of $T_{r{e}^{-i\alpha }}$ is dense for any $r>0$;

(iii)

the range of $T_{r{e}^{-i\alpha }}$ is nowhere dense, and the range of $T_{r{e}^{i\alpha }}$ is dense for any $r>0$.

We only need to prove that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is topological transitive in case $\left(\mathrm{i}\right)$, and that ${\left\{T_{r{e}^{-i\alpha }}\right\}}_{r\ge 0}$ (${\left\{T_{r{e}^{i\alpha }}\right\}}_{r\ge 0}$) is topological transitive in case $\left(\mathrm{ii}\right)$ ($\left(\mathrm{iii}\right)$, respectively).

In case $\left(\mathrm{i}\right)$, each operator $T_t\left(t\in \mathsf{\Delta }\right)$ has a dense range. In this case, the proof is relatively straightforward. In fact, a point-transitive surjective topological semiflow with a Polish phase space and an abelian acting semigroup must be topologically transitive from [20] [Theorem 4.1]. We will give a simple proof here for completeness. Let $x_0\in X$ be a transitive point. We will show that $T_{t_0}{x}_0,\forall t_0\in \mathsf{\Delta }$, is a transitive point. Fix arbitrary $t_0\in \mathsf{\Delta }$. Note that $T_{t_0}$ has a dense range and $x_0$ is a transitive point; hence, the orbit of $T_{t_0}{x}_0$,

$$\mathrm{Orb}({\left\{T_t\right\}}_{t\in \mathsf{\Delta }},T_{t_0}{x}_0):=\left\{T_s{T}_{t_0}{x}_0|s\in \mathsf{\Delta }\right\}=\left\{T_{t_0}{T}_s{x}_0|s\in \mathsf{\Delta }\right\}$$

is dense in X. Therefore, each point $T_{t_0}{x}_0,\forall t_0\in \mathsf{\Delta }$, is transitive, which means that the set of all transitive points of ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is dense and further implies that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is topological transitive.

Now, let us proceed to the proof when the condition $\left(\mathrm{ii}\right)$ or $\left(\mathrm{iii}\right)$ holds true. In these two cases, we assume that X is infinite-dimensional, observing that if ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is point-transitive and X is finite-dimensional, then each operator $T_t\left(t\in \mathsf{\Delta }\right)$ must be subjective, which comes to the case $\left(\mathrm{i}\right)$.

In case $\left(\mathrm{ii}\right)$, we will show that for any nonempty open subsets $U,V\subset X$, there exists some $t=r{e}^{-i\alpha }$ such that $T_{t}U\cap V\ne \varnothing$. Let $x\in U,y\in V$. From the local equicontinuity of ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, one can deduce that there exist some $\delta >0$, and neighborhoods $U_x^{\prime }\subset U,V_y^{\prime }\subset V$ of $x,y$, respectively, such that

$$T_{r{e}^{i\theta }}{U}_x^{\prime }\subset U\mathrm{and}{T}_{r{e}^{i\theta }}{V}_y^{\prime }\subset V,\forall 0\le r\le 2\delta ,-\alpha \le \theta \le \alpha .$$

(1)

Denote

$$L_{\delta }:=\left\{r{e}^{i\alpha }+\rho e^{-i\alpha }|0\le r\le \delta ,\rho \ge 0\right\}.$$

Then

$$L_{\delta }^c=\left\{r{e}^{i\alpha }+\rho e^{-i\alpha }|r>\delta ,\rho \ge 0\right\},$$

which means that $T_{t}X\subset T_{\delta e^{i\alpha }}X$, for any $t\in L_{\delta }^c$. Since the range of $T_{\delta e^{i\alpha }}$ is nowhere dense, the union set ${\bigcup }_{t\in L_{\delta }^c}{T}_{t}X$ ($\subset T_{\delta e^{i\alpha }}X$) is also nowhere dense. Then, ${\bigcup }_{t\in L_{\delta }}{T}_t{x}_0$ is dense, and there exists $t_1=r_1{e}^{i\alpha }+{\rho }_1{e}^{-i\alpha }\in L_{\delta }$, and $t_2=r_2{e}^{i\alpha }+{\rho }_2{e}^{-i\alpha }\in L_{\delta }$, such that $T_{t_1}{x}_0\in U_x^{\prime }$ and $T_{t_2}{x}_0\in V_y^{\prime }.$

We note that $|t_1|$ and $|t_2|$ can be taken sufficiently large, because neither $U_x^{\prime }$ nor $V_y^{\prime }$ can be contained in the set ${\bigcup }_{t\in \mathsf{\Delta },\left|t\right|\le s_0}{T}_t{x}_0$, for any $s_0>0$ which is indeed a compact set in X. Hence, ${\rho }_1$ and ${\rho }_2$ can be taken sufficiently large. Without loss of generality, we assume that ${\rho }_2>{\rho }_1$.

If $r_1\ge r_2$, let $t_2^{\prime }=r_1{e}^{i\alpha }+{\rho }_2{e}^{-i\alpha }$. From (1), we have that

$$T_{t_2^{\prime }}{x}_0=T_{t_2+(r_1-r_2)e^{i\alpha }}{x}_0=T_{(r_1-r_2)e^{i\alpha }}{T}_{t_2}{x}_0\in V.$$

Note, that $T_{t_2^{\prime }}{x}_0=T_{t_2^{\prime }-t_1}{T}_{t_1}{x}_0\in V,\mathrm{and}\mathrm{that}{T}_{t_1}{x}_0\in U$, which means that

$$t_2^{\prime }-t_1=({\rho }_2-{\rho }_1)e^{-i\alpha }\in N(U,V).$$

If $r_1<r_2$, let $t_1^{\prime }=r_2{e}^{i\alpha }+{\rho }_1{e}^{-i\alpha }$. Through a similar approach, one can prove that $t_2-t_1^{\prime }=({\rho }_2-{\rho }_1)e^{-i\alpha }\in N(U,V)$. Therefore, we can conclude that ${\left\{T_{r{e}^{-i\alpha }}\right\}}_{r\ge 0}$ is topological transitive, and hence point-transitive.

In case $\left(\mathrm{iii}\right)$, the proof is similar to that in case $\left(\mathrm{ii}\right)$, so we omit the details here and the proof of the theorem is finished. □

Remark 1.

Theorem 1 provides us with a more subtle insight into the transitivity of $C_0$-semigroups ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ and their autonomous discretizations. From this theorem, point-transitive or topological transitive $C_0$-semigroups ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ can be divided into two classes. The first class is that each operator $T_t,\forall t\in \mathsf{\Delta }$, has a dense range. In this case, Conejero and Peris [11] showed that there is a hypercyclic (i.e., point-transitive) semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, whose autonomous discretizations are not hypercyclic. In the second case, one and only one of the autonomous discretizations ${\left\{T_{r{e}^{-i\alpha }}\right\}}_{r\ge 0}$ and ${\left\{T_{r{e}^{i\alpha }}\right\}}_{r\ge 0}$ is point-transitive.

3. The Relationships among Hypercyclicity, Li–Yorke Chaos and Existence of Irregular Vectors

In this section, we study the relationships among point transitivity (hypercyclicity), Li–Yorke chaos and the existence of irregular vectors for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$. Specifically, we will show that just like the conclusion for a single operator T or $C_0$-semigroup ${\left\{T_t\right\}}_{t\ge 0}$, Li–Yorke chaos is equivalent to the existence of a semi-irregular vector and that hypercyclicity is stronger than the existence of irregular vector for any linear semiflow ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ as some basic facts. Next, in Theorem 2 we will show that there exists a Li–Yorke chaotic $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ without irregular vectors, which gives Question 2 a negative answer.

At first, for the convenience of our readers, we offer clear definitions of Li–Yorke chaos, irregular vectors and semi-irregular vectors for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ as follows.

• A $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space $(X,||·|\left|\right)$ is called Li–Yorke chaotic if there exists an uncountable subset $\mathsf{\Gamma }\subset X$, such that for any distinct pair $(x,y)\in \mathsf{\Gamma }\times \mathsf{\Gamma }$,

  $$\underset{t\in \mathsf{\Delta },\left|t\right|\to \infty }{\lim \inf }\left|\right|T_{t}x-T_{t}y\left|\right|=0,\underset{t\in \mathsf{\Delta },\left|t\right|\to \infty }{\lim \sup }\left|\right|T_{t}x-T_{t}y\left|\right|>0.$$
  Such a set $\mathsf{\Gamma }$ is called a scramble set and $(x,y)\in \mathsf{\Gamma }\times \mathsf{\Gamma }$ is called a a Li–Yorke chaotic pair.
• A $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space $(X,||·|\left|\right)$ is said to be sensitive to initial conditions if there exists $c>0$, such that for every $x\in X$ and any neighborhood U of x, there exists $y\in U$ and $t\in \mathsf{\Delta }$, such that $\left|\right|T_{t}x-T_{t}y\left|\right|>c$.
• Given a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space $(X,||·|\left|\right)$ and a vector $x\in X$, we say that x is an irregular vector for ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, if

  $$\underset{t\in \mathsf{\Delta },\left|t\right|\to \infty }{\lim \sup }\left|\right|T_{t}x\left|\right|=\infty ,\mathrm{and}\underset{t\in \mathsf{\Delta },\left|t\right|\to \infty }{\lim \inf }\left|\right|T_{t}x\left|\right|=0.$$
• Given a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a Banach space $(X,||·|\left|\right)$ and a vector $x\in X$, we say that x is a semi-irregular vector for ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, if

  $$\underset{t\in \mathsf{\Delta },\left|t\right|\to \infty }{\lim \sup }\left|\right|T_{t}x\left|\right|>0,\mathrm{and}\underset{t\in \mathsf{\Delta },\left|t\right|\to \infty }{\lim \inf }\left|\right|T_{t}x\left|\right|=0.$$

Next, we provide some basic facts in the following.

Basic Fact 1.

Let ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ be a $C_0$-semigroup on a Banach space X. Then, the following assertions are equivalent:

(i) 

${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is Li–Yorke chaotic;

(ii) 

${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ admits a Li–Yorke pair;

(iii) 

${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ admits a semi-irregular vector.

Proof. 

The proof of this fact is very simple, and we provide it here merely for the sake of completeness.

$\left(\mathrm{i}\right)\Rightarrow \left(\mathrm{ii}\right)$ is trivial.

$\left(\mathrm{ii}\right)\Rightarrow \left(\mathrm{iii}\right)$: Suppose that $(x,y)$ is a Li–Yorke pair for ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, then $z=x-y$ is semi-irregular.

$\left(\mathrm{iii}\right)\Rightarrow \left(\mathrm{ii}\right)$: Suppose that x is a semi-irregular vector, then the segment $\mathsf{\Gamma }:=\left\{\lambda x;\left|\lambda \right|\le 1\right\}$ is a scrambled set for ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$. □

Basic Fact 2.

Let ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ be a $C_0$-semigroup on a Banach space X. If ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is hypercyclic with a hypercyclic vector $x\in X$, then x is irregular.

Proof. 

From Lemma 2 in [12], any hypercyclic vector of ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ must be irregular. □

Basic Fact 3.

Let ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ be a $C_0$-semigroup on a Banach space X. Then, ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is sensitive to initial conditions if and only if ${\sup }_{t\in \mathsf{\Delta }}\left|\right|T_t\left|\right|=\infty$.

Basic Fact 4.

Let ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ be a $C_0$-semigroup on a Banach space X. If ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ admits an irregular vector, then it is sensitive.

Next, we will show that there exists a Li–Yorke chaotic $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ which is not sensitive and admits no irregular vector.

Theorem 2.

There exists a Li–Yorke chaotic $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ which is not sensitive to initial conditions and admits no irregular vector.

Proof. 

Let

$$\rho \left(\tau \right):=\left\{\begin{array}{ccc}{\displaystyle \frac{1}{r_1}}\hfill & & \mathrm{if}\tau \in \left\{r_1{e}^{i\alpha }+r{e}^{-i\alpha }:r\ge 0\right\}\mathrm{for}\mathrm{some}{r}_1>1,\hfill \\ 1\hfill & & \mathrm{if}\tau \in \left\{r_1{e}^{i\alpha }+r{e}^{-i\alpha }:0\le r_1\le 1,r\ge 0\right\},\hfill \end{array}\right.$$

$p>0$, and $X=L_{\rho }^p(\mathsf{\Delta },\mathbb{K}):=\left\{f:\mathsf{\Delta }\to \mathbb{K}:f\mathrm{measurable}\mathrm{and}{\int }_{\mathsf{\Delta }}{\left|f\left(\tau \right)\right|}^p\rho \left(\tau \right)d\tau <\infty \right\}$ with the norm $\parallel f\parallel :={\left({\int }_{\mathsf{\Delta }}{\left|f\left(\tau \right)\right|}^p\rho \left(\tau \right)d\tau \right)}^{1/p}.$ Given $f\in X$ and $t\in \mathsf{\Delta }$, define $T_{t}f$ as follows

$$T_{t}f\left(x\right):=\left\{\begin{array}{ccc}f(x-t)\hfill & & \mathrm{if}x\in t+\mathsf{\Delta },\hfill \\ 0\hfill & & \mathrm{otherwise}.\hfill \end{array}\right.$$

Now, we show that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is Li–Yorke chaotic but not sensitive. Choose a function $f\in X$ defined by

$$f\left(x\right):=\left\{\begin{array}{ccc}1\hfill & & \mathrm{if}x\in {\mathsf{\Delta }}_1,\hfill \\ 0\hfill & & \mathrm{otherwise}.\hfill \end{array}\right.$$

Then, one can easily obtain that

$$\parallel T_{r{e}^{-i\alpha }}f\parallel =\parallel f\parallel \mathrm{and}\parallel T_{r{e}^{i\alpha }}f\parallel \le {\left({\displaystyle \frac{1}{r}}\right)}^{1/p}\parallel f\parallel ,\forall r>0,$$

which means that f is a semi-irregular vector for ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, and that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is Li–Yorke chaotic.

On the other hand, for any $t\in \mathsf{\Delta }$, we have that

$$\parallel T_{t}f\parallel ={\left({\int }_{\mathsf{\Delta }+t}{\left|f(\tau -t)\right|}^p\rho \left(\tau \right)d\tau \right)}^{1/p}={\left({\int }_{\mathsf{\Delta }}{\left|f\left(\tau \right)\right|}^p\rho (\tau +t)d\tau \right)}^{1/p}\le \parallel f\parallel ,$$

which implies that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is bounded and not sensitive. Now, we have shown that ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ is Li–Yorke chaotic but not sensitive and hence admits no irregular vector from Basic Fact 4.

□

4. Concluding Remarks

Research on topological transitivity and point transitivity including their equivalence are key topics in the investigation of topological semiflows and linear semiflows. The density of the operator’s range plays a crucial role in determining whether the equivalence between topological transitivity and point transitivity holds. In the previous studies, point-transitive or topologically transitive linear semiflows ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ are assumed to have each operator with a dense range. In this paper, we give a point-transitive linear semiflow (i.e., $C_0$-semigroup) ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ having some operators without a dense range. Furthermore, we show that topological transitivity is equivalent to point transitivity for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ on a separable Banach space even if some operator $T_t(t\in \mathsf{\Delta })$ does not have a dense range, which is little known in the current literature. Li–Yorke chaos and sensitive dependence on initial conditions are also fundamental notions in the study of topological and linear dynamics. For a single operator T or a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in {\mathbb{R}}^{+}}$ on a Banach space, Li–Yorke chaos implies sensitivity and is equivalent to the existence of an irregular vector. However, for a $C_0$-semigroup ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$, we show that the existence of irregular vectors implies Li–Yorke chaos, but not vice versa. All these results and their proofs reveal the differences in the dynamical behavior between linear semiflows ${\left\{T_t\right\}}_{t\in \mathsf{\Delta }}$ and traditional linear systems with the acting semigroup $S={\mathbb{Z}}^{+}$ and $S={\mathbb{R}}^{+}$.