1 Introduction
Dynamical systems is a branch of mathematics concerned with the asymptotic behavior of systems. The study of orbits generated by the system plays an important role for exploring the complexity of the asymptotic behavior. Plenty of results related to properties of orbits have been obtained for various systems under different settings. For the dynamical system generated by a linear operator on some linear space with certain structure, it seems that there is no complexity due to the linearity. However, even for such a system, there can be abundant complex phenomena, meaning that it is significant and interesting to consider linear dynamics.
One of the central concepts in the study of linear dynamics is the so-called hypercyclicity, which is closely related to numerous important properties of linear dynamics. Let X be a Banach space and
$T\colon X\to X$
be a bounded linear operator. A vector
$x\in X$
is said to be hypercyclic if the orbit of x,
$\{T^nx\colon n\geq 0\}$
, is dense in X, and the operator T is said to be hypercyclic if there exists some hypercyclic vector in X. The set of all hypercyclic vectors of T is denoted by
$HC(T)$
. Note that the concept of hypercyclicity is in fact the so-called transitivity for general topological dynamical systems. It is important to point out that linear structure can be made full use of for linear dynamics, compared to general dynamical systems, which means that we can consider the properties of linear dynamics from some new points of view. For a complete discussion on the dynamics of operators, the reader can refer to [Reference Bayart and Matheron3] and [Reference Grosse-Erdmann and Peris12] and references therein.
As mentioned above, hypercyclic operators play a significant role in linear dynamics, making it useful and interesting to study their properties from various perspectives. In [Reference Li14] the first named author of this studied the behaviors of orbits in the mean sense of a hypercyclic operator on a Banach space. In this article, another aspect of the hypercyclic operator is considered, and we will study the density properties of orbits for such operators.
First of all, let us recall some concepts related to the topic of this article. For a subset A of
$\mathbb {N}$
, the upper density and lower density of A are defined by
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(A)=\limsup_{n\to\infty} \frac{ \#(A\cap [1,n])}{n}, \end{align*}$$
and
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(A)=\liminf_{n\to\infty} \frac{ \#(A\cap [1,n])}{n}, \end{align*}$$
respectively, where
$\#(\cdot )$
denotes the number of elements of a finite set. If
$\operatorname {\mathrm {\overline {dens}}}(A)=\operatorname {\mathrm {\underline {dens}}}(A)$
, then we say that the density of A exists, and denote it by
$\operatorname {\mathrm {dens}}(A)$
.
Let X be a Banach space with the norm
$\Vert \cdot \Vert $
and
$T\colon X\to X$
be a bounded linear operator from X to itself. This article focuses on the following properties of hypercyclic operators in terms of density. Following [Reference Bermúdez, Bonilla, Martínez-Giménez and Peris5], a vector
$x\in X$
is said to be distributionally irregular of type
$1$
for T if there exist two subsets A and B of
$\mathbb {N}$
such that
$\operatorname {\mathrm {\overline {dens}}}(A)=\operatorname {\mathrm {\overline {dens}}}(B)=1$
,
$\lim \limits _{A\ni n\to \infty } \Vert T^n x\Vert =0$
, and
$\lim \limits _{B\ni n\to \infty } \Vert T^n x\Vert =\infty $
. Inspired by the concept of distributional chaos of type
$2\frac {1}{2}$
given in [Reference da Costa Bernardes, Bonilla, Peris and Wu7], we introduce the concept of distributionally irregular of type
$2\frac {1}{2}$
. A vector
$x\in X$
is said to be distributionally irregular of type
$2\frac {1}{2}$
for T if there exist two subsets A and B of
$\mathbb {N}$
such that
$\operatorname {\mathrm {\overline {dens}}}(A)\operatorname {\mathrm {\overline {dens}}}(B)>0$
,
$\lim \limits _{A\ni n\to \infty } \Vert T^n x\Vert =0$
and
$\lim \limits _{B\ni n\to \infty } \Vert T^n x\Vert =\infty $
. A vector
$x\in X$
is said to be asymptotic to zero with density one if there exists a subset A of
$\mathbb {N}$
with
$\operatorname {\mathrm {dens}}(A)=1$
such that
$\lim \limits _{A\ni n\to \infty }\Vert T^n x\Vert =0$
, and divergent to infinity with density one if there exists a subset B of
$\mathbb {N}$
with
$\operatorname {\mathrm {dens}}(B)=1$
such that
$\lim \limits _{B\ni n\to \infty }\Vert T^n x\Vert =\infty $
.
In addition, we say that a property P on vectors in X is generic if the collection of vectors which satisfy the property P contains a dense
$G_\delta $
subset of X. The following theorem is our main result in this article.
Theorem 1.1 If T is a hypercyclic operator on a separable Banach space X, then exactly one of the following four assertions holds:
-
(1) every vector in X is asymptotic to zero with density one;
-
(2) generic vectors in X are distributionally irregular of type
$1$
; -
(3) generic vectors in X are distributionally irregular of type
$2\frac {1}{2}$
, and no hypercyclic vector is distributionally irregular of type
$1$
; -
(4) every hypercyclic vector in X is divergent to infinity with density one.
The article is organized as follows. Theorem 1.1 will be proved in Section 2. In Section 3, we present some examples concerned with weighted backward shifts on
$\ell ^p$
to show that all four cases in Theorem 1.1 can occur. In Section 4, similar results for
$C_0$
-semigroups will be discussed.
2 Proof of theorem 1.1
The aim of this section is to prove Theorem 1.1. We begin with the following standard result on density; therefore, its proof is omitted.
Lemma 2.1 Let
$(a_n)_n$
be a sequence in
$\mathbb {R}$
. Then,
-
(1) there exists a subset A of
$\mathbb {N}$
with
$\operatorname {\mathrm {\overline {dens}}}(A)=1$
such that
$\lim _{A\ni n\to \infty }|a_n|=0$
if and only if for every
$\varepsilon>0$
,
$\operatorname {\mathrm {\overline {dens}}}( \{n\in \mathbb {N}: |a_n|<\varepsilon \} )=1$
; -
(2) there exists a subset B of
$\mathbb {N}$
with
$\operatorname {\mathrm {dens}}(B)=1$
such that
$\lim _{B\ni n\to \infty }|a_n|=0$
if and only if for every
$\varepsilon>0$
,
$\operatorname {\mathrm {dens}}( \{n\in \mathbb {N}: |a_n|<\varepsilon \} )=1$
.
By a topological dynamical system, we mean a pair
$(X,T)$
, where X is a metric space with the metric d and
$T\colon X\to X$
is a continuous map. Following [Reference Fomin10] and [Reference Li, Tu and Ye15], we say that a topological dynamical system
$(X,T)$
is mean-L-stable if for every
$\varepsilon>0$
there exists a
$\delta>0$
such that for any
$x,y\in X$
with
$d(x,y)<\delta $
, one has
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon d(T^nx,T^ny)\geq \varepsilon\})<\varepsilon, \end{align*}$$
and mean-L-unstable if there exists a
$\delta>0$
such that for any non-empty open subset U of X there exist
$x,y\in U$
such that
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon d(T^nx,T^ny)> \delta\})\geq \delta. \end{align*}$$
We now state the following dichotomy theorem.
Theorem 2.2 [Reference Jiang and Li13, Theorem 5.13]
Let T be a bounded linear operator on a Banach space X. Then, T is either mean-L-stable or mean-L-unstable.
We have an equivalent characterization concerned with mean-L-stability in terms of density under some conditions, the following proposition plays a key role in such a characterization.
Proposition 2.3 Let T be a bounded linear operator on a Banach space X. Assume that T is mean-L-stable. Then we have the following:
-
(1) For a vector
$x\in X$
, if
$\liminf _{n\to \infty }\Vert T^nx\Vert = 0$
, then x is asymptotic to zero with density one. -
(2) The collection of vectors in X which are asymptotic to zero with density one is a closed subspace of X.
Proof
-
(1) Let
$x\in X$
be a vector satisfying
$\liminf _{n\to \infty }\Vert T^nx\Vert = 0$
. Fix any
$\varepsilon>0$
. Note that T is mean-L-stable, which means for any
$0<\varepsilon '<\varepsilon $
there exists
$\delta>0$
such that for any x,
$y\in X$
with
$\Vert x-y\Vert <\delta $
, we have By the assumption on x, there exists
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^nx- T^ny\Vert \geq \varepsilon'\})<\varepsilon'. \end{align*}$$
$k\in \mathbb {N}$
such that
$\Vert T^kx\Vert <\delta $
. Then for
$T^kx$
and
$\mathbf {0}$
, we know that implying that
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon\Vert T^{n+k}x\Vert \geq \varepsilon'\})<\varepsilon', \end{align*}$$
Noticing that
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^{n+k}x\Vert < \varepsilon'\})\geq1-\varepsilon'. \end{align*}$$
$\varepsilon '<\varepsilon $
we have Letting
$$ \begin{align*} \operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^{n}x\Vert <\varepsilon\}) &\geq \operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^{n}x\Vert <\varepsilon'\})\\ &\geq\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^{n+k}x\Vert < \varepsilon'\})>1-\varepsilon'. \end{align*} $$
$\varepsilon '\to 0$
we obtain meaning that
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^{n}x\Vert <\varepsilon\})\geq 1, \end{align*}$$
Noticing the arbitrariness of
$$\begin{align*}\operatorname{\mathrm{dens}}(\{n\in\mathbb{N}\colon \Vert T^{n}x\Vert <\varepsilon\})=1. \end{align*}$$
$\varepsilon $
and using Lemma 2.1 (2) we know that there exists a subset A of
$\mathbb {N}$
with
$\operatorname {\mathrm {dens}}(A)=1$
such that
$\lim _{A\ni n\to \infty } \Vert T^n x\Vert =0$
. Then x is asymptotic to zero with density one.
-
(2) Let
$X_0$
be the collection of vectors in X which are asymptotic to zero with density one. For any
$x, y\in X_0$
, there exist two subsets
$A_x, A_y\subset \mathbb {N}$
with
$\operatorname {\mathrm {dens}}(A_x)= 1=\operatorname {\mathrm {dens}}(A_y)$
such that Let
$$\begin{align*}\lim_{A_x\ni n\to\infty} \Vert T^n x\Vert =0 \ \text{and} \ \lim_{A_y\ni n\to\infty} \Vert T^n y\Vert =0. \end{align*}$$
$A=A_x\cap A_y$
. Since
$\operatorname {\mathrm {dens}}(A_x)=\operatorname {\mathrm {dens}}(A_y)=1$
, it is easy to see that
$\operatorname {\mathrm {dens}}(A)= 1$
. On the other hand, clearly we have which means that
$$ \begin{align*} 0\leq&\lim_{A\ni n\to\infty} \Vert T^n (x+y)\Vert \leq\lim_{A\ni n\to\infty}\left(\Vert T^n x\Vert+\Vert T^n y)\Vert\right)\\ =&\lim_{A\ni n\to\infty}\Vert T^n x\Vert+\lim_{A\ni n\to\infty}\Vert T^n y\Vert=0, \end{align*} $$
$x+y\in X_0$
. It is easy to check that
$rx\in X_0$
for any
$r\in \mathbb {K}$
, where
$\mathbb {K}=\mathbb {R}\text { or }\mathbb {C}$
. Now we have proved that
$X_0$
is a subspace. In the following we will prove that
$X_0$
is closed. Let
$\{x_k\}_{k\in \mathbb {N}}$
be a sequence in
$X_0$
with
$\lim _{k\to \infty }x_k= x\in X$
. Fix any
$\varepsilon>0$
. For any
$0<\varepsilon '<\frac {\varepsilon }{2}$
, noticing that T is mean-L-stable, there exists
$\delta>0$
such that for any y,
$z\in X$
with
$\Vert y-z\Vert <\delta $
, that is
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon\Vert T^n(y-z)\Vert\geq\varepsilon'\})<\varepsilon', \end{align*}$$
Choose
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon\Vert T^n(y-z)\Vert<\varepsilon'\})\geq 1-\varepsilon'. \end{align*}$$
$m>0$
such that
$\Vert x_m-x\Vert <\delta $
. Then we have Since
$$ \begin{align*} &\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon\Vert T^n(x_m-x)\Vert<\varepsilon'\})\ge 1-\varepsilon'. \end{align*} $$
$x_m\in X_0$
, we know that
$\operatorname {\mathrm {dens}}(\{n\in \mathbb {N}\colon \Vert T^nx_m \Vert <\varepsilon '\})=1$
. Then we have Letting
$$ \begin{align*} \operatorname{\mathrm{\underline{dens}}}(\{n\in \mathbb{N}\colon \Vert T^nx \Vert <\varepsilon)\} &\ge \operatorname{\mathrm{\underline{dens}}}\big(\{n\in \mathbb{N}: \Vert T^nx_m \Vert <\varepsilon'\} \\ &\qquad\qquad \cap \{ n\in \mathbb{N}: \Vert T^nx_m-T^nx \Vert <\varepsilon'\} \big)\\ &\ge 1-\varepsilon'. \end{align*} $$
$\varepsilon '\to 0$
, we obtain that This shows that
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in \mathbb{N}: \Vert T^nx \Vert <\varepsilon)\}=1. \end{align*}$$
$x\in X_0$
and then
$X_0$
is closed.
Now we can give the following characterization of mean-L-stability for hypercyclic operators.
Proposition 2.4 Let
$T\colon X\to X$
be a hypercyclic operator on a separable Banach space X. Then T is mean-L-stable if and only if every vector in X is asymptotic to zero with density one.
Proof “
$\Rightarrow $
” For every hypercyclic vector
$x\in X$
, as
$\liminf _{n\to \infty } \Vert T^nx\Vert =0$
, by Proposition 2.3 (1), x is asymptotic to zero with density one. By the fact that
$HC(T)$
is dense in X and Proposition 2.3 (2), every vector in X is asymptotic to zero with density one.
“
$\Leftarrow $
” By the assumption and Lemma 2.1 (2), for any
$z\in X$
and any
$\varepsilon>0$
we have
$$\begin{align*}\operatorname{\mathrm{dens}}(\{n\in\mathbb{N}\colon\Vert T^nz \Vert<\varepsilon\})=1. \end{align*}$$
Then for any x,
$y\in X$
and any
$\varepsilon>0$
we have
$$ \begin{align*} \operatorname{\mathrm{dens}}(\{n\in\mathbb{N}\colon\Vert T^n(x-y)\Vert<\varepsilon\}) =1. \end{align*} $$
The arbitrariness of
$\varepsilon $
shows that T is mean-L-stable.
A characterization of mean-L-unstability can also be formulated, involving the so-called distributional sensitivity. We say that a topological dynamical system
$(X,T)$
is distributionally sensitive if there exists
$\delta>0$
such that for any
$x\in X$
and
$\varepsilon>0$
, there exists
$y\in X$
with
$d(x,y)<\varepsilon $
such that
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon d(T^nx,T^ny)> \delta\})=1. \end{align*}$$
In addition, for a bounded linear operator T on a Banach space X, we say that a vector
$x\in X$
is distributionally unbounded if there exists a subset B of
$\mathbb {N}$
with
$\operatorname {\mathrm {\overline {dens}}}(B)=1$
and
$\lim _{B\ni n\to \infty } \Vert T^nx \Vert =\infty $
. The following result follows from [Reference Jiang and Li13, Proposition 5.33].
Proposition 2.5 Let T be a bounded linear operator on a Banach space X. If T is distributionally sensitive, then generic vectors in X are distributionally unbounded.
The following result is essentially contained in [Reference da Costa Bernardes, Bonilla, Müller and Peris6, Proposition 8], see [Reference Jiang and Li13, Theorem 5.41] for the version here, which establishes a characterization of mean-L-unstability.
Theorem 2.6 Let T be a bounded linear operator on a Banach space X. Then T is mean-L-unstable if and only if it is distributionally sensitive.
Now we can give the following dichotomy for hypercyclic operators.
Proposition 2.7 Let T be a hypercyclic operator on a separable Banach space X. Then either every vector in X is asymptotic to zero with density one, or generic vectors in X are distributionally unbounded.
Proof By Theorem 2.2 we know that T is either mean-L-stable or mean-L-unstable. On the one hand, if T is mean-L-stable, note that T is hypercyclic, by Proposition 2.4 we know that every vector in X is asymptotic to zero with density one. On the other hand, if T is mean-L-unstable, by Theorem 2.6 we know that T is distributionally sensitive. Then by Proposition 2.5 we obtain that generic vectors in X are distributionally unbounded.
Let T be a hypercyclic operator on a Banach space X. For the proof of our main result (Theorem 1.1), we also need a parameter
$c(T) \in [0, 1]$
associated with T, which was firstly introduced by Grivaux and Matheron in [Reference Grivaux and Matheron11]. For each
$R>0$
, define
$$\begin{align*}c_R(T) =\sup_{x\in HC(T)} \operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^nx\Vert<R\}), \end{align*}$$
and
$$\begin{align*}c(T)=\sup_{R>0} c_R(T). \end{align*}$$
Remark 2.8 It is shown in [Reference Grivaux and Matheron11, Remark 4.6] that there exists a dense
$G_\delta $
subset
$X_0$
of
$HC(T)$
such that for any
$x\in X_0$
and
$R>0$
,
$\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^nx\Vert <R\})=c(T)$
. Since
$HC(T)$
is a dense
$G_\delta $
subset of X,
$X_0$
is also a dense
$G_\delta $
subset X.
Concerned with
$c(T)$
, we have the following proposition, which is an important tool in the proof of Theorem 1.1.
Proposition 2.9 Let T be a hypercyclic operator on a separable Banach space X. Then
$$\begin{align*}c(T)=\sup\{ \lambda\in[0,1]\colon \exists x\in HC(T)\, \&\, R>0 \text{ s.t. } \operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^nx\Vert<R\})\geq \lambda\} \end{align*}$$
and for every hypercyclic vector
$y\in HC(T)$
and
$R>0$
,
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^ny\Vert\geq R\})\geq 1-c(T). \end{align*}$$
Proof Let
$$\begin{align*}A= \{ \lambda\in[0,1]\colon \exists x\in HC(T) \, \&\, R>0 \text{ s.t. } \operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^nx\Vert<R\})\geq \lambda\}. \end{align*}$$
For any
$\lambda \in A$
, there exist
$x\in HC(T)$
and
$R>0$
such that
$\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^nx\Vert <R\})\geq \lambda $
. Let
$\lambda _0=\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^nx\Vert <R\})$
. It is clear that
$\lambda \leq \lambda _0$
and
$\lambda _0\in A$
. By the definition of
$c_R(T)$
and
$c(T)$
, one has
$\lambda _0\leq c_R(T)\leq c(T)$
. Then
$c(T)$
is a upper bound for A. For every
$\varepsilon>0$
, there exists
$R_0>0$
such that
$c_{R_0}(T)>c(T)-\varepsilon $
and then there exists
$x\in HC(T)$
such that
$\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^nx\Vert <R\})\geq c(T)-\varepsilon $
. So
$\max \{ c(T)-\varepsilon , 0\}\in A$
. This implies that
$c(T)=\sup A$
.
Fix
$y\in HC(T)$
and
$R>0$
. One has
$\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^ny\Vert <R\})\in A$
and then
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^ny\Vert<R\})\leq c(T). \end{align*}$$
Therefore,
$$ \begin{align*} \operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^ny\Vert\geq R\})&\geq 1- \operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^ny\Vert<R\})\\ &\geq 1-c(T).\\[-34pt] \end{align*} $$
Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1
If T is mean-L-stable, then by Proposition 2.4 every vector in X is asymptotic to zero with density one. So, in this case, assertion (1) holds.
If T is mean-L-unstable and
$c(T)=1$
, then combining Proposition 2.5 and Theorem 2.6 there exists a dense
$G_\delta $
subset
$X_1$
of X such that every vector in
$X_1$
is distributionally unbounded, and by Remark 2.8, there exists a dense
$G_\delta $
subset
$X_2$
of X such that for every
$x\in X_2$
and
$R>0$
,
$\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^nx\Vert <R\})=1$
. Let
$X_0=X_1\cap X_2$
. Then
$X_0$
is a dense
$G_\delta $
subset of X and every vector in
$X_0$
is distributionally irregular of type
$1$
. So, in this case, assertion (2) holds.
If T is mean-L-unstable and
$0<c(T)<1$
, then there exists a dense
$G_\delta $
subset
$X_1$
of X such that every vector in
$X_1$
is distributionally unbounded, and by Remark 2.8, there exists a dense
$G_\delta $
subset
$X_2$
of X such that for every
$x\in X_2$
and
$R>0$
,
$\operatorname {\mathrm {\overline {dens}}}(\{n\in \mathbb {N}\colon \Vert T^nx\Vert <R\})=c(T)>0$
. Let
$X_0=X_1\cap X_2$
. Then
$X_0$
is a dense
$G_\delta $
subset of X and every vector in
$X_0$
is distributionally irregular of type
$2\frac {1}{2}$
. By Proposition 2.9, for every hypercyclic vector
$y\in HC(T)$
and
$R>0$
,
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^ny\Vert\geq R\})\geq 1-c(T)>0. \end{align*}$$
So every hypercyclic vector in X is not distributionally irregular of type
$1$
. So, in this case, assertion (3) holds.
If T is mean-L-unstable and
$c(T)=0$
, then by Proposition 2.9, for every hypercyclic vector
$y\in HC(T)$
and
$R>0$
,
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert T^ny\Vert\geq R\})\geq 1-c(T)=1. \end{align*}$$
So every hypercyclic vector in X is divergent to infinity with density one. So, in this case, assertion (4) holds.
Remark 2.10 By the proof of Theorem 1.1, we can replace the case (3) in Theorem 1.1 by
-
(3’) there exists a dense
$G_\delta $
subset
$X_0$
of
$ X$
and a constant
$c\in (0,1)$
such that for every
$x\in X_0$
, there exist two subsets A and B of
$\mathbb {N}$
with
$\operatorname {\mathrm {\overline {dens}}}(A)=c$
and
$\operatorname {\mathrm {\overline {dens}}}(B)=1$
such that
$\lim \limits _{A\ni n\to \infty } \Vert T^n x\Vert =0$
and
$\lim \limits _{B\ni n\to \infty } \Vert T^n x\Vert =\infty $
, and for any hypercyclic vector
$y\in HC(T)$
there exists a subset C of
$\mathbb {N}$
with
$\operatorname {\mathrm {\underline {dens}}} (C)=1-c$
such that
$\lim \limits _{C\ni n\to \infty } \Vert T^n y\Vert =\infty $
.
3 Some examples
In this section, we will present some examples concerned with weighted backward shifts on
$\ell ^p$
to show that all four cases in Theorem 1.1 can occur.
For a positive weight sequence
$v=(v_j)_{j\geq 1}$
and
$1\leq p<\infty $
, let
$$\begin{align*}\ell^p(v,\mathbb{N})=\bigg\{(x_j)_{j\geq 1} \colon \sum_{j=1}^\infty |x_j|^p v_j<\infty\bigg\} \end{align*}$$
be the weighted
$\ell ^p$
-space. The norm on
$\ell ^p(v,\mathbb {N})$
is defined as follows:
$$\begin{align*}\Vert x\Vert = \bigg(\sum_{j=1}^\infty |x_j|^p v_j\bigg)^{\frac{1}{p}},\quad \forall x=(x_j)_{j\geq 1}\in \ell^p(v,\mathbb{N}). \end{align*}$$
The unilateral backward shift B on
$\ell ^p(v,\mathbb {N})$
is defined as
$$\begin{align*}(Bx)_j=x_{j+1},\quad \forall x=(x_j)_{j\geq 1}\in \ell^p(v,\mathbb{N}),\ j\geq 1. \end{align*}$$
It is not hard to show that the map
$B\colon \ell ^p(v,\mathbb {N})\to \ell ^p(v,\mathbb {N})$
is well-defined (and, equivalently, continuous) if and only if
$\sup _{j\geq 1}\frac {v_j}{v_{j+1}}<\infty $
.
Example 3.1 By [Reference Martínez-Giménez, Oprocha and Peris16, Theorem 2.1], there exists a positive weight sequence v such that the operator
$B\colon \ell ^p(v,\mathbb {N})\to \ell ^p(v,\mathbb {N})$
is mixing,
$1\leq p<\infty $
, and satisfies the case (1) of Theorem 1.1.
Remark 3.2 By [Reference Bermúdez, Bonilla, Martínez-Giménez and Peris5, Theorem 35], in every infinite-dimensional separable Banach space there exists a hypercyclic and distributionally chaotic operator which admits a dense distributionally irregular manifold, which satisfies the case (2) of Theorem 1.1.
Now we consider the bilateral case. For a positive weight sequence
$v=(v_j)_{j\in \mathbb {Z}}$
and
$1\leq p<\infty $
, let
$$\begin{align*}\ell^p(v,\mathbb{Z})=\bigg\{(x_j)_{j\in\mathbb{Z}} \colon \sum_{j\in \mathbb{Z}} |x_j|^p v_j<\infty\bigg\}, \end{align*}$$
be the weighted
$\ell ^p$
-space. The norm on
$\ell ^p(v,\mathbb {Z})$
is defined as follows:
$$\begin{align*}\Vert x\Vert = \bigg(\sum_{j\in\mathbb{Z}} |x_j|^p v_j\bigg)^{\frac{1}{p}},\quad \forall x=(x_j)_{j\in\mathbb{Z}}\in \ell^p(v,\mathbb{Z}). \end{align*}$$
The bilateral backward shift B on
$\ell ^p(v,\mathbb {Z})$
is defined as
$$\begin{align*}(Bx)_j=x_{j+1},\quad \forall x=(x_j)_{j\in\mathbb{Z}}\in \ell^p(v,\mathbb{Z}),\ j\in\mathbb{Z}. \end{align*}$$
Similarly, the map
$B\colon \ell ^p(v,\mathbb {Z})\to \ell ^p(v,\mathbb {Z})$
is well-defined (and, equivalently, continuous) if and only if
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}<\infty $
.
Remark 3.3 By [Reference Martínez-Giménez, Oprocha and Peris16, Example 3.5], there exists a positive weight sequence v such that the operator
$B\colon \ell ^p(v,\mathbb {Z})\to \ell ^p(v,\mathbb {Z})$
is hypercyclic,
$1\leq p<\infty $
, and every non-zero vector in X is distributionally irregular of type
$1$
. Note that this is a special situation satisfying case (2) in Theorem 1.1.
Remark 3.4 Bayart and Ruzsa in [Reference Bayart and Ruzsa4, Section 6] constructed a frequently hypercyclic weighted shift on
$c_0(\mathbb {Z})$
with
$0<c(T)<1$
, see also the proof of Theorem 1.7 in [Reference Grivaux and Matheron11]. This meets the case (3) in Theorem 1.1.
In the following example, we will construct a hypercyclic operator T on the weighted
$\ell ^p$
-space such that
$c(T)\in (0,1)$
and for every non-zero vector
$x\in X$
there exists
$\tau _x>0$
such that
$\operatorname {\mathrm {\underline {dens}}}(\{j\geq 0\colon \Vert T^j x\Vert \geq \tau _x\})\geq 3/7$
.
Example 3.5 Let
$a_k=8^k$
for each
$k\in \mathbb {N}$
. Define a positive weight sequence
$v=(v_j)_{j\in \mathbb {Z}}$
as follows:
$$\begin{align*}v_j= \begin{cases} \frac{1}{j}, & j\in\mathbb{N},\\ 1, & 0\leq -j\leq a_1,\\ \frac{1}{2^{a_k+j}} & a_k< -j \leq a_k+k,\\ \frac{1}{2^k}, & a_k+k < -j \leq 2 a_k-k,\\ \frac{1}{2^{2a_k+j}} & 2a_k-k< -j \leq 2a_k,\\ 1 & 2a_k<-j\leq a_{k+1}. \end{cases} \end{align*}$$
Consider the weighted
$\ell ^p$
-space
$\ell ^p(v,\mathbb {Z})$
and the bilateral backward shift B on
$\ell ^p(v,\mathbb {Z})$
. As
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}\leq 2$
, B is well-defined. By [Reference Grosse-Erdmann and Peris12, Theorem 4.12], the bilateral backward shift B is hypercyclic.
Consider the vector
$e_0$
in
$\ell ^p(v,\mathbb {Z})$
, i.e.,
$e_0(0)=1$
and
$e_0(i)=0$
for all
$i\in \mathbb {Z}\setminus \{0\}$
. For any
$n\in \mathbb {N}$
,
$\Vert B^n e_0\Vert =v_{-n}$
. Note that
$$\begin{align*}\{n\in\mathbb{N}\colon v_{-n}=1\} \supset \bigcup_{k=1}^\infty [2a_k,a_{k+1}] \end{align*}$$
and
$$\begin{align*}\{n\in\mathbb{N}\colon v_{-n}\leq \tfrac{1}{2^m}\} \supset \bigcup_{k=m}^\infty [a_k+k,2a_k-k]. \end{align*}$$
So
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n e_0\Vert\geq 1\})\geq \tfrac{3}{7}, \end{align*}$$
and for every
$\varepsilon>0$
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n e_0\Vert\leq \varepsilon\})\geq \tfrac{4}{7}. \end{align*}$$
For every non-zero
$x\in \ell ^p(v,\mathbb {Z})$
, there exists some
$k\in \mathbb {Z}$
such that
$(B^k x)(0)>0$
, then there exists
$\tau _x>0$
such that
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n x\Vert\geq \tau_x\})\geq \tfrac{3}{7}. \end{align*}$$
Note that the collection
$$\begin{align*}\{x\in X\colon \operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n x\Vert\leq \varepsilon\})\geq \tfrac{4}{7}, \forall \varepsilon>0\} \end{align*}$$
is a dense
$G_\delta $
subset of X, as it contains all the vectors with only finite non-zero coordinates.
Remark 3.6 Menet in [Reference Menet17] constructed a Devaney chaotic operator on
$\ell ^p$
such that for any non-zero vector x, there exists
$\tau>0$
such that
$\operatorname {\mathrm {\underline {dens}}}(\{j\geq 0\colon \Vert T^j x\Vert \geq \tau \})=1$
. Then
$c(T)=0$
. This satisfies case (4) in Theorem 1.1.
The hypercyclic operator constructed in the following example is adapted from that in [Reference Li14], which has the property that every non-zero vector goes to infinity with density one.
Example 3.7 Let
$a_k=8^k$
for each
$k\in \mathbb {N}$
. Define a weight sequence
$v=(v_j)_{j\in \mathbb {Z}}$
as follows:
$$\begin{align*}v_j= \begin{cases} \frac{1}{j}, & j\in\mathbb{N},\\ 1, & 0\leq -j\leq a_1,\\ 2^{k -j-a_k} & a_k< -j \leq a_k+2k,\\ 2^{-2k -j-a_k} & a_k+2k< -j \leq a_k+3k+1,\\ 2^{k+1} & a_k+3k+1 <-j\leq a_{k+1}. \end{cases} \end{align*}$$
Consider the weighted
$\ell ^p$
-space
$\ell ^p(v,\mathbb {Z})$
and the bilateral backward shift B on
$\ell ^p(v,\mathbb {Z})$
. As
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}\leq 2$
, B is well-defined. By [Reference Grosse-Erdmann and Peris12, Theorem 4.12], the bilateral backward shift B is hypercyclic.
Consider the vector
$e_0$
in
$\ell ^p(v,\mathbb {Z})$
, i.e.,
$e_0(0)=1$
and
$e_0(i)=0$
for all
$i\in \mathbb {Z}\setminus \{0\}$
. For any
$n\in \mathbb {N}$
,
$\Vert B^n e_0\Vert =v_{-n}$
. Note that for every
$m\in \mathbb {N}$
$$\begin{align*}\{n\in\mathbb{N}\colon v_{-n}\geq 2^{m+1}\} \supset \bigcup_{k=m}^\infty [a_k+3k+1,a_{k+1}]. \end{align*}$$
So for every
$m\in \mathbb {N}$
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n e_0\Vert\geq 2^{m+1} \})=1. \end{align*}$$
For every non-zero
$x\in \ell ^p(v,\mathbb {Z})$
, there exists some
$k\in \mathbb {Z}$
such that
$(B^k x)(0)>0$
, then for every
$R>0$
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n x\Vert\geq R\})=1. \end{align*}$$
4 Results for
$C_0$
-semigroups
The dynamics of
$C_0$
-semigroups were systematically studied for the first time in [Reference Desch, Schappacher and Webb9]. For further details, see [Reference Grosse-Erdmann and Peris12, Chapter 7]. In this section, we discuss results for
$C_0$
-semigroups that are analogous to those presented in the previous sections.
We say that a one-parameter family
$(T_t)_{t\geq 0}$
of bounded linear operators on a Banach space X is a
$C_0$
-semigroup if
$T_0=I$
,
$T_{t+s}=T_tT_s$
for all
$t,s\geq 0$
and
$\lim _{t\to s}T_t x=T_sx$
for all
$x\in X$
and
$s\geq 0$
. It is easy to see that such a
$C_0$
-semigroup is locally equicontinuous, that is, for any
$s>0$
,
$$\begin{align*}\sup_{t\in [0,s]} \Vert T_t\Vert<\infty. \end{align*}$$
Let
$(T_t)_{t\ge 0}$
be a
$C_0$
-semigroup on a Banach space X and
$x\in X$
. The orbit of x under
$(T_t)_{t\ge 0}$
is defined by
$orb(x, (T_t)_{t\geq 0})=\{T_tx: t\ge 0\}$
. We say that
$(T_t)_{t\ge 0}$
is hypercyclic if there is some
$x\in X$
whose orbit under
$(T_t)_{t\ge 0}$
is dense in X. In such case, x is called a hypercyclic vector for
$(T_t)_{t\ge 0}$
.
The following result concerned with the equivalent characterizations of hypercyclicity for
$C_0$
-semigroups was proved in [Reference Conejero, Müller and Peris8], see [Reference Grosse-Erdmann and Peris12, Theorem 7.26] for the version stated here.
Theorem 4.1 Let
$(T_t)_{t\geq 0}$
be a
$C_0$
-semigroup on a Banach space X and
$x\in X$
. Then the following assertions are equivalent:
-
(1) x is hypercyclic for
$(T_t)_{t\geq 0}$
; -
(2) x is hypercyclic for
$T_s$
for some
$s>0$
; -
(3) x is hypercyclic for
$T_s$
for every
$s>0$
.
Let
$A\subset [0,\infty )$
be a Lebesgue measurable set. The lower and upper densities of A are defined as
$$\begin{align*}\operatorname{\mathrm{\underline{Dens}}}(A)=\liminf_{t\to\infty}\frac{\mu(A\cap[0,t])}{t}\text{ and } \operatorname{\mathrm{\overline{Dens}}}(A)=\limsup_{t\to\infty}\frac{\mu(A\cap[0,t])}{t}, \end{align*}$$
where
$\mu $
is the Lebesgue measure on
$[0,\infty )$
. If
$\operatorname {\mathrm {\underline {Dens}}}(A)=\operatorname {\mathrm {\overline {Dens}}}(A)$
, then we say that the density of A exists, and denote it by
$\operatorname {\mathrm {Dens}}(A)$
.
The following result on the relationship between “
$\mathrm {Dens}$
” and “
$\mathrm {dens}$
” is essential contained in [Reference Albanese, Barrachina, Mangino and Peris1, Lemma 2.4], see also [Reference da Costa Bernardes, Bonilla, Peris and Wu7, Lemma 3].
Lemma 4.2 Let
$(T_t)_{t\geq 0}$
be a
$C_0$
-semigroup on a Banach space X. For each
$s>0$
, let
$C_s= \sup _{t\in [0,s]} \Vert T_t\Vert $
. Then for every
$x\in X$
and
$\varepsilon>0$
,
-
(1)
$\operatorname {\mathrm {\underline {Dens}}}(\{t\geq 0\colon \Vert T_tx\Vert <\varepsilon \})\leq \operatorname {\mathrm {\underline {dens}}}(\{j\in \mathbb {N}\colon \Vert (T_s)^j x\Vert <C_s \varepsilon \})$
; -
(2)
$\operatorname {\mathrm {\underline {dens}}}(\{j\in \mathbb {N}\colon \Vert (T_s)^jx\Vert < \varepsilon \})\leq \operatorname {\mathrm {\underline {Dens}}}(\{t\geq 0\colon \Vert T_tx\Vert <C_s\varepsilon \})$
; -
(3)
$\operatorname {\mathrm {\overline {Dens}}}(\{t\geq 0\colon \Vert T_tx\Vert <\varepsilon \})\leq \operatorname {\mathrm {\overline {dens}}}(\{j\in \mathbb {N}\colon \Vert (T_s)^jx\Vert <C_s \varepsilon \})$
; -
(4)
$\operatorname {\mathrm {\overline {dens}}}(\{j\in \mathbb {N}\colon \Vert (T_s)^jx\Vert < \varepsilon \})\leq \operatorname {\mathrm {\overline {Dens}}}(\{t\geq 0\colon \Vert T_tx\Vert <C_s\varepsilon \})$
.
Similar to that for a single operator, we can define mean-L-stability and mean-L-unstability for a
$C_0$
-semigroup as follows. Let
$(T_t)_{t\ge 0}$
be a
$C_0$
-semigroup on a Banach space X. We say that X is mean-L-stable for
$(T_t)_{t\ge 0}$
if for every
$\epsilon>0$
there exists a
$\delta>0$
such that for any
$x, y\in X$
with
$d(x,y)<\delta $
, one has
$$\begin{align*}\operatorname{\mathrm{\overline{Dens}}}(\{t\ge 0: \Vert T_tx-T_ty\Vert \ge \epsilon \} )<\epsilon, \end{align*}$$
and mean-L-unstable for
$(T_t)_{t\ge 0}$
if there exists a
$\delta>0$
such that for any non-empty open subset U of X there exist
$x, y\in U$
such that
$$\begin{align*}\operatorname{\mathrm{\overline{Dens}}}(\{t\ge 0: \Vert T_tx - T_ty\Vert>\delta \}) \ge \delta. \end{align*}$$
The following proposition gives equivalent characterizations for mean-L-stability (resp., mean-L-unstability).
Proposition 4.3 Let
$(T_t)_{t\geq 0}$
be a
$C_0$
-semigroup on a Banach space X. Then the following assertions are equivalent:
-
(1) the
$C_0$
-semigroup
$(T_t)_{t\geq 0}$
is mean-L-stable (reps. mean-L-unstable); -
(2) the operator
$T_s$
is mean-L-stable (reps. mean-L-unstable) for some
$s>0$
; -
(3) the operator
$T_s$
is mean-L-stable (reps. mean-L-unstable) for every
$s>0$
.
Proof
$(3) \Rightarrow (2)$
: It is clear.
$(1) \Rightarrow (3)$
: For every
$s>0$
, and let
$C_s=\sup _{t\in [0,s]}\Vert T_t \Vert $
. Since
$(T_t)_{t>0}$
is mean-L-stable, then for any
$\epsilon>0$
, there exists a
$\delta>0$
, such that for any
$x,y\in X$
with
$d(x,y)<\delta $
,
$$\begin{align*}\operatorname{\mathrm{\overline{Dens}}}(\{t\ge 0: \Vert T_tx- T_ty\Vert \ge \tfrac{\epsilon}{C_s} \} )<\epsilon, \end{align*}$$
then by Lemma 4.2, one has
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{j\in\mathbb{N}\colon \Vert (T_s)^jx-(T_s)^jy\Vert \ge \varepsilon\})\leq \operatorname{\mathrm{\overline{Dens}}}(\{t\geq 0\colon \Vert T_tx-T_ty\Vert\ge \tfrac{\epsilon}{C_s}\})<\epsilon. \end{align*}$$
This shows that
$T_s$
is mean-L-stable.
$(2) \Rightarrow (1) $
: We may assume that
$T_s$
is mean-L-stable for some
$s>0$
. Put
$C_s=\sup _{t\in [0,s]}\Vert T_s \Vert $
. Then for any
$\epsilon>0$
, there exists a
$\delta>0$
such that for any
$x,y\in X$
with
$d(x,y)<\delta $
,
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{j\in \mathbb{N}: \Vert (T_s)^jx-(T_s)^jy \Vert \ge \tfrac{\epsilon}{C_s} \} )<\epsilon, \end{align*}$$
thus by Lemma 4.2, one has
$$\begin{align*}\operatorname{\mathrm{\overline{Dens}}}(\{t\ge 0: \Vert T_tx-T_ty \Vert \ge \epsilon \}) \le \operatorname{\mathrm{\overline{dens}}}(\{j\in \mathbb{N}: \Vert (T_s)^jx-(T_s)^jy \Vert \ge \tfrac{\epsilon}{C_s} \} )<\epsilon. \end{align*}$$
This shows that
$(T_t)_{t\ge 0}$
is mean-L-stable.
By a similar proof, Proposition 4.3 for mean-L-unstable case holds.
In order to obtain a parallel result to Theorem 1.1, a similar parameter need to be introduced as follows. If
$(T_t)_{t\geq 0}$
is a hypercyclic
$C_0$
-semigroup on a Banach space X, for each
$R>0$
, define
$$\begin{align*}c_R((T_t)_{t\geq 0}) =\sup_{x\in HC((T_t)_{t\geq 0})} \operatorname{\mathrm{\overline{Dens}}}(\{t\in\mathbb{R}_{\geq0 }\colon \Vert T_tx\Vert<R\}), \end{align*}$$
and
$$\begin{align*}c((T_t)_{t\geq 0} )=\sup_{R>0} c_R((T_t)_{t\geq 0}). \end{align*}$$
The following proposition plays a similar role in the proof of Theorem 4.5 as that Proposition 2.9 in the proof of Theorem 1.1, and can be proved in a similar way as Proposition 4.3, hence we omit the proof.
Proposition 4.4 Let
$(T_t)_{t\geq 0}$
be a hypercyclic
$C_0$
-semigroup on a Banach space X. Then
$c((T_t)_{t\geq 0} )=c(T_s)$
for every
$s>0$
.
We also need to introduce some concepts in terms of density for
$C_0$
-semigroups, as we did in Section 1 for a single operator. Let
$(T_t)_{t\geq 0}$
be a
$C_0$
-semigroup on a Banach space X. A vector
$x\in X$
is said to be distributionally irregular of type
$1$
for
$(T_t)_{t\geq 0}$
if there exist two measurable subsets A and B of
$\mathbb {R}_{\geq 0}$
such that
$\operatorname {\mathrm {\overline {Dens}}}(A)=\operatorname {\mathrm {\overline {Dens}}}(B)=1$
,
$\lim \limits _{A\ni t\to \infty } \Vert T_t x\Vert =0$
and
$\lim \limits _{B\ni t\to \infty } \Vert T_t x\Vert =\infty $
. A vector
$x\in X$
is said to be distributionally irregular of type
$2\frac {1}{2}$
for
$(T_t)_{t\geq 0}$
if there exist two measurable subsets A and B of
$\mathbb {R}_{\geq 0}$
such that
$\operatorname {\mathrm {\overline {Dens}}}(A)\operatorname {\mathrm {\overline {Dens}}}(B)>0$
,
$\lim \limits _{A\ni t\to \infty } \Vert T_t x\Vert =0$
and
$\lim \limits _{B\ni t\to \infty } \Vert T_t x\Vert =\infty $
. A vector
$x\in X$
is said to be asymptotic to zero with density one if there exists a measurable subset A of
$\mathbb {R}_{\geq 0}$
with
$\operatorname {\mathrm {Dens}}(A)=1$
such that
$\lim \limits _{A\ni t\to \infty }\Vert T_t x\Vert =0$
, and divergent to infinity with density one if there exists a measurable subset A of
$\mathbb {R}_{\geq 0}$
with
$\operatorname {\mathrm {Dens}}(A)=1$
such that
$\lim \limits _{A\ni t\to \infty }\Vert T_t x\Vert =\infty $
.
Now we are ready to state our main result for the case of
$C_0$
-semigroups, whose proof is similar to that of Theorem 1.1, and we omit it.
Theorem 4.5 If
$(T_t)_{t\geq 0}$
is a hypercyclic
$C_0$
-semigroup on a Banach space X, then exactly one of the following four assertions holds:
-
(1) every vector in X is asymptotic to zero with density one;
-
(2) generic vectors in X are distributionally irregular of type
$1$
; -
(3) generic vectors in X are distributionally irregular of type
$2\frac {1}{2}$
, and no hypercyclic vector is distributionally irregular of type
$1$
; -
(4) every hypercyclic vector in X is divergent to infinity with density one.
There are some examples in [Reference Albanese, Barrachina, Mangino and Peris1] to show that the case (2) in Theorem 4.5 can occur. In the following, we will give more examples to show that the other three cases in Theorem 4.5 can occur.
4.1 Unilateral translation semigroup
By an admissible weight function on
$\mathbb {R}_{\geq 0}$
, we mean a measurable function
$\rho \colon \mathbb {R}_{\geq 0}\to \mathbb {R}$
satisfying the following two conditions:
-
(1)
$\rho (t)>0$
for all
$t\in \mathbb {R}_{\geq 0}$
; -
(2) there exist constants
$M\geq 1$
and
$\omega>0$
such that
$\rho (s)\leq Me^{\omega t} \rho (t+s)$
for all
$s\in \mathbb {R}_{\geq t}$
and all
$t>0$
.
Let
$1\leq p<\infty $
and
$\rho $
be an admissible weight function on
$\mathbb {R}_{\geq 0}$
. We consider the separable Banach space of p-integrable functions in the Lebesgue sense as
$$\begin{align*}L_\rho^p(\mathbb{R}_{\geq 0}):=\{ f \colon \mathbb{R}_{\geq 0}\to \mathbb{R}\text{ measurable}, \Vert f\Vert_p<\infty\}, \end{align*}$$
where
$$\begin{align*}\Vert f\Vert_p = \bigg(\int_{\mathbb{R}_{\geq 0}} |f(s)|^p \rho(s){\mathop{}\!\mathrm{d}} s\bigg)^{1/p}. \end{align*}$$
The translation semigroup on
$L_\rho ^p(\mathbb {R}_{\geq 0})$
defined by
$(T_t f)(x)=f(x+t)$
,
$t,x\in \mathbb {R}_{\geq 0}$
, is a well-defined
$C_0$
-semigroup by the definition of admissible weight.
Let
$v=(v_j)_{j\in \mathbb {N}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {N}}\frac {v_j}{v_{j+1}}<\infty $
. Following [Reference Barrachina and Peris2], we construct an admissible weight function induced by v. For each
$t\in (n-1,n]$
for some
$n\in \mathbb {N}$
, let
$\rho _v(t)=v_n$
, and
$\rho _v(0)=v_1$
. Then it is easy to see that
$\rho _v$
is an admissible weight function.
For each
$f\in L_\rho ^p (\mathbb {R}_{\geq 0})$
and
$n\in \mathbb {N}$
, define
$x_n^f=\Big (\!\int _{[n-1,n]} |f(t)|^p {\mathop {}\!\mathrm {d}} t\Big )^{1/p}$
. Let
$x^f=(x_n^f)_{n\in \mathbb {N}}$
. Then,
$$\begin{align*}\Vert x^f\Vert_p^p = \sum_{n=1}^\infty |x^f_n|^p v_n = \sum_{n=1}^\infty \int_{[n-1,n]} |f(t)|^p \rho(t) {\mathop{}\!\mathrm{d}} t =\int_{\mathbb{R}_{\geq 0}} |f(t)|^p \rho(t) {\mathop{}\!\mathrm{d}} t =\Vert f\Vert_p^p. \end{align*}$$
Combining Examples 4.4, 4.7 and 7.10 in [Reference Grosse-Erdmann and Peris12], we have the following result.
Proposition 4.6 Let
$v=(v_j)_{j\in \mathbb {N}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {N}}\frac {v_j}{v_{j+1}}<\infty $
. If
$(\ell ^p(v),B)$
is hypercyclic (resp., mixing), then
$(L_{\rho _v}^{p} (\mathbb {R}_{\geq 0}), (T_t)_{t\geq 0})$
is hypercyclic (resp., mixing).
We now have the following result, which, when combined with Example 3.1 from the previous section, shows that case (1) in Theorem 4.5 can occur.
Proposition 4.7 Let
$v=(v_j)_{j\in \mathbb {N}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {N}}\frac {v_j}{v_{j+1}}<\infty $
. If
$(\ell ^p(v,\mathbb {N}),B)$
is mixing and every vector is asymptotic to zero with density one, then
$(L_{\rho _v}^p (\mathbb {R}_{\geq 0}), (T_t)_{t\geq 0})$
is mixing and every vector is asymptotic to zero with density one.
4.2 Bilateral translation semigroup
Similar to that in previous subsection, by an admissible weight function on
$\mathbb {R}$
, we mean a measurable function
$\rho \colon \mathbb {R}\to \mathbb {R}$
satisfying the following two conditions:
-
(1)
$\rho (t)>0$
for all
$t\in \mathbb {R}$
; -
(2) there exist constants
$M\geq 1$
and
$\omega>0$
such that
$\rho (s)\leq Me^{\omega t} \rho (t+s)$
for all
$s\in \mathbb {R}$
and all
$t>0$
.
Let
$1\leq p<\infty $
and
$\rho $
be an admissible weight function on
$\mathbb {R}$
. Just as that in Subsection 4.1, we consider the separable Banach space of p-integrable functions in the Lebesgue sense as
$$\begin{align*}L_\rho^p(\mathbb{R}):=\{ f \colon \mathbb{R}\to \mathbb{R}\text{ measurable}, \Vert f\Vert_p<\infty\}, \end{align*}$$
where
$$\begin{align*}\Vert f\Vert_p = \bigg(\int_{\mathbb{R}} |f(s)|^p \rho(s){\mathop{}\!\mathrm{d}} s\bigg)^{1/p}. \end{align*}$$
Similarly, the translation semigroup on
$L_\rho ^p(\mathbb {R})$
defined by
$(T_t f)(x)=f(x+t)$
,
$t,x\in \mathbb {R}$
, is a well-defined
$C_0$
-semigroup by the definition of admissible weight.
Let
$v=(v_j)_{j\in \mathbb {Z}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}<\infty $
. For each
$t\in (n-1,n]$
for some
$n\in \mathbb {Z}$
, let
$\rho _v(t)=v_n$
. Clearly
$\rho _v$
is an admissible weight function.
For each
$f\in L_\rho ^p (\mathbb {R})$
and
$n\in \mathbb {Z}$
, define
$x_n^f=\Big (\int _{[n-1,n]} |f(t)|^p {\mathop {}\!\mathrm {d}} t\Big )^{1/p}$
. Let
$x^f=(x_n^f)_{n\in \mathbb {Z}}$
. Similar to the discussion in the previous subsection, we know that
$\Vert x^f\Vert _p^p =\Vert f\Vert _p^p$
. Then similar to the above construction, we have the following:
Proposition 4.8 Let
$v=(v_j)_{j\in \mathbb {Z}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}<\infty $
. If
$(\ell ^p(v,\mathbb {Z}),B)$
is hypercyclic and every non-zero vector is distributionally irregular of type 1, then
$(L_{\rho _v}^p (\mathbb {R}), (T_t)_{t\geq 0})$
is hypercyclic and every non-zero vector is distributionally irregular of type 1.
Proposition 4.9 Let
$v=(v_j)_{j\in \mathbb {Z}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}<\infty $
. If
$(\ell ^p(v,\mathbb {Z}),B)$
is hypercyclic and for every non-zero vector
$x\in \ell ^p(v,\mathbb {Z})$
there exists
$\tau _x>0$
such that
$$\begin{align*}\operatorname{\mathrm{\underline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n x\Vert\geq \tau_x\})\geq \tfrac{3}{7}, \end{align*}$$
and for any vector with only finite non-zero coordinates
$x\in \ell ^p(v,\mathbb {Z})$
,
$$\begin{align*}\operatorname{\mathrm{\overline{dens}}}(\{n\in\mathbb{N}\colon \Vert B^n x\Vert\leq \varepsilon\})\geq \tfrac{4}{7}, \end{align*}$$
then
$(L_{\rho _v}^p (\mathbb {R}), (T_t)_{t\geq 0})$
is hypercyclic and for every non-zero function
$f\in L_{\rho _v}^p (\mathbb {R})$
there exists
$\tau _x>0$
such that
$$\begin{align*}\operatorname{\mathrm{\underline{Dens}}}(\{t\geq 0\colon \Vert T_t f\Vert\geq \tau_x\})\geq \tfrac{3}{7}, \end{align*}$$
for any function f with only bounded support
$$\begin{align*}\operatorname{\mathrm{\overline{Dens}}}(\{t\geq 0 \colon \Vert T_t f\Vert\leq \varepsilon\})\geq \tfrac{4}{7}. \end{align*}$$
By combining the above result with Example 3.5 from the previous section, we see that case (3) in Theorem 4.5 can occur.
Proposition 4.10 Let
$v=(v_j)_{j\in \mathbb {Z}}$
be a positive weight sequence with
$\sup _{j\in \mathbb {Z}}\frac {v_j}{v_{j+1}}<\infty $
. If
$(\ell ^p(v,\mathbb {Z}),B)$
is hypercyclic and every non-zero vector is divergent to infinity with density one, then
$(L_{\rho _v}^p (\mathbb {R}), (T_t)_{t\geq 0})$
is hypercyclic and every non-zero vector is divergent to infinity with density one.
By combining the above result with Example 3.7 from the previous section, we see that case (4) in Theorem 4.5 can occur.
Acknowledgement
The authors would like to thank the referee for the careful reading and helpful suggestions.
