Supercyclic Weighted Translations on Quotient Spaces

Abstract

In this note, we give the sufficient and necessary condition for weighted translations on the Orlicz spaces of quotient spaces to be supercyclic. By applying this characterization of supercyclicity, the descriptions of hypercyclicity, topological mixing and Cesàro hypercyclicity on such spaces are obtained as well.

1. Introduction

A bounded linear operator T on a Banach space X is called hypercyclic if there exists a vector $x\in X$ such that the orbit $\{x,Tx,\cdots ,T^{n}x,\cdots \}$ is dense in X. If the orbit $\{\alpha T^{n}x:\alpha \in \mathbb{C},n\in \mathbb{N}\cup \left\{0\right\}\}$ is dense in X, then T is said to be supercyclic, where x is called a supercyclic vector of T. Hypercyclicity and supercyclicity have been studied by many authors (for instance [1,2,3,4,5,6]). Indeed, in [6], Salas characterized supercyclic bilateral weighted shifts in terms of weights. Later, the necessary and sufficient condition for weighted translations on locally compact groups to be supercyclic were obtained in [7]. Recently, the spectrum of supercyclic and hypercyclic operators was investigated in [1]. In [2], supercyclic properties of certain differentiation operators were studied. We refer to these two classical books [8,9] on this topic of hypercyclicity and linear dynamics.

Hypercyclicity arises from the invariant closed subset problem in analysis. Indeed, every nonzero vector in a Banach space X is hypercyclic if, and only if, X does not admit nontrivial invariant closed subsets. Additionally, hypercyclicity and topological transitivity are equivalent for an operator T on a Banach space X when X is separable. Topological transitivity is one of the ingredients to form chaos.

In fact, one can also consider hypercyclicity and supercyclicity for sequences of operators. A sequence of operators ${\left(T_n\right)}_{n\in \mathbb{N}}$ on X is hypercyclic if the set $\left\{x,T_{1}x,\cdots ,T_{n}x,\cdots \right\}$ is dense in X, and in this case, x is a hypercyclic vector of ${\left(T_n\right)}_{n\in \mathbb{N}}$. It is known that ${\left(T_n\right)}_{n\in \mathbb{N}}$ is hypercyclic with a dense set of hypercyclic vectors if, and only if, this sequence is topologically transitive. A sequence of operators ${\left(T_n\right)}_{n\in \mathbb{N}}$ is called topologically transitive if for any nonempty open sets U and V, we have $T_n\left(U\right)\cap V\ne ⌀$ for some $n\in \mathbb{N}$. Moreover, if $T_n\left(U\right)\cap V\ne ⌀$ is satisfied from some $n\in \mathbb{N}$ onwards, then ${\left(T_n\right)}_{n\in \mathbb{N}}$ is topologically mixing. If ${\left(T_n\right)}_{n\in \mathbb{N}}$ is generated by a single operator T, that is, $T_n:=T^n$ for each n, then T is topologically transitive or mixing instead of the sequence ${\left(T_n\right)}_{n\in \mathbb{N}}$.

Let $\Phi$ be a Young function, and let $G/H$ be a quotient space. In Theorem 1 of our paper, we will give the sufficient and necessary condition for the weighted translation operators on the Orlicz space $L^{\Phi }(G/H)$ to be supercyclic. In the proof of Theorem 1, we will show this sequence ${\left({\alpha }_n{T}^n\right)}_{n\in \mathbb{N}}$ is topologically transitive, which says ${\left({\alpha }_n{T}^n\right)}_{n\in \mathbb{N}}$ is hypercyclic, and therefore T is supercyclic. The obtained results subsume [7] (Theorem 2.1) in two directions. First, they extend the result from locally compact groups to quotient spaces. Secondly, we consider the Orlicz space, which is a generalization of the Lebesgue space.

In the following, we recall some preliminaries for further study. Let G be a locally compact group, and let H be a closed subgroup of G. Let $G/H:=\{xH:x\in G\}$ be the set of all left cosets of H in G. The quotient topology on $G/H$ is defined in this way: $E\subseteq G/H$ is open if, and only if, ${\pi }^{-1}\left(E\right):=\{x\in G:xH\in E\}$ is open in G, where $\pi$ is the quotient map. The set $G/H$ equipped with the quotient topology is called the quotient space. By ([10] Proposition 2.2), the quotient space $G/H$ is locally compact and Hausdorff (see also [11]).

Assume that ${\Delta }_G\left(x\right)={\Delta }_H\left(x\right)$ for all $x\in H$, where ${\Delta }_G$ and ${\Delta }_H$ are the modular functions on G and H, respectively. Then, by ([10], Theorem 2.42), there exists a G-invariant Radon measure $\mu$ on $G/H$. In other words, for each $a\in G$ and each Borel set $E\subseteq G/H$, we have $\mu \left(aE\right)=\mu \left(E\right)$, where

$$aE:=\{\left(ax\right)H:x\in {\pi }^{-1}\left(E\right)\}.$$

Next, we recall that a continuous convex function $\Phi :[0,\infty )\to [0,\infty )$ is called a Young function if $\Phi \left(0\right)=\underset{x\to 0}{lim}\Phi \left(x\right)=0$ and $\underset{x\to \infty }{lim}\Phi \left(x\right)=\infty$. Then the complimentary $\Psi$ of a Young function $\Phi$ is defined by

$$\Psi \left(s\right):=sup\{ts-\Phi (t):t\ge 0\}\in [0,\infty ],\mathrm{for}s\ge 0.$$

For a Young function $\Phi$ and $v\in [0,\infty )$, ${\Phi }^{-1}\left(v\right)$ is defined by

$${\Phi }^{-1}\left(v\right)=sup\{u\ge 0:\Phi \left(u\right)\le v\}.$$

In addition, the Orlicz space $L^{\Phi }(G/H)$ is the set of all measurable functions

$f:G/H\to \mathbb{R}$ with

$${\int }_{G/H}\Phi \left(\alpha \right|f\left|\right)d\mu \le \infty$$

for some $\alpha >0$. For each $f\in L^{\Phi }(G/H)$, we define

$${\parallel f\parallel }_{\Phi }:=sup\left\{{\int }_{G/H}\left|fv\right|d\mu :v\in B_{\Psi }\right\},$$

where $B_{\Psi }$ is the set of all measurable functions v with ${\int }_{G/H}\Psi \left(\right|v\left|\right)d\mu \le 1$. Then $(L^{\Phi }(G/H),\parallel ·{\parallel }_{\Phi })$ is a Banach space [12]. Moreover, if $\Phi$ is ${\Delta }_2$-regular, then $L^{\Phi }(G/H)$ is separable and infinite-dimensional, and the set $C_c(G/H)$ of all continuous compact supported functions on $G/H$ is dense in $L^{\Phi }(G/H)$ [12]. A Young function $\Phi$ is ${\Delta }_2$-regular if there exist a constant $M>0$ and $t_0\ge 0$ such that $\Phi \left(2t\right)\le M\Phi \left(t\right)$ for $t\ge t_0$ [12].

Remark 1.

By the left-invariance of the measure μ regarding the action of G on $G/H$, for every $b\in G$ and $v\in B_{\Psi }$, we have

$$\begin{array}{cc}\hfill {\int }_{G/H}\Psi \left(\right|v\left(bxH\right)\left|\right)d\mu \left(xH\right)& ={\int }_{G/H}\Psi \left(\right|v\left(xH\right)\left|\right)d\mu \left(b^{-1}xH\right)\hfill \\ \hfill & ={\int }_{G/H}\Psi \left(\right|v\left(xH\right)\left|\right)d\mu \left(xH\right).\hfill \end{array}$$

Hence, setting $v_b\left(xH\right):=v\left(bxH\right)$ for all $xH\in G/H$, we have

$$B_{\Psi }=\{v_b:v\in B_{\Psi }\}$$

for all $b\in G$. Also, according to [12], Proposition 4, p. 61 and [12], Corollary 7, p. 78, for each non-null measurable set $A\subseteq G/H$ with finite measure, we have

$${\displaystyle \frac{1}{\parallel {\chi }_A{\parallel }_{\Phi }}}\le {\Phi }^{-1}\left({\displaystyle \frac{1}{\parallel {\chi }_A{\parallel }_1}}\right).$$

Now we are ready to define weighted translation operators on $L^{\Phi }(G/H)$. A weight on $G/H$ is a continuous function $w:G/H\to (0,\infty )$. Then the weighted translation operator $T_{a,w}$ on $L^{\Phi }(G/H)$ is given by

$$T_{a,w}\left(f\right)\left(xH\right):=w\left(xH\right)f\left(\left(a^{-1}x\right)H\right)$$

where $f\in L^{\Phi }(G/H)$ and $a,x\in G$. Additionally, since $\mu$ is a G-invariant measure on $G/H$, $T_{a,w}$ is a bounded linear operator.

In Section 2, we will give the characterization of supercyclicity for the weighted translation operator $T_{a,w}$ in Theorem 1, which entails three consequences. The first one is the description of a hypercyclic weighted translation operator $T_{a,w}$ in Corollary 1. The second one is to give a sufficient and necessary condition for $T_{a,w}$ to be topologically mixing in Corollary 2. In the last one, Cesàro hypercyclicity of weighted translation operators is characterized in Corollary 3 together with a concrete and significant example.

2. Main Results

In this section, we will give the sufficient and necessary conditions for a weighted translation $T_{a,w}$ on the Orlicz space of a quotient space $G/H$ to be supercyclic. The necessary condition requires the H-aperiodic property. The description of aperiodicity was first characterized in [13], Lemma 2.1.

Definition 1.

Let G be a locally compact group, and let H be a closed subgroup of G. We say that an element $a\in G$ is H-aperiodic if for each compact subset $K\subseteq G/H$, there is a constant $N>0$ such that $K\cap a^{n}K=⌀$ (equivalently, $K\cap a^{-n}K=⌀$) for all $n\ge N$.

Example 1.

Assume that $G:=\mathrm{SL}(2,\mathbb{R})$ and

$$H:=\left\{\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]:\theta \in [0,2\pi ]\right\}.$$

Then H is a compact subgroup of G, and ${\Delta }_G\left(x\right)={\Delta }_H\left(x\right)=1$ for all $x\in H$. Also, $G/H\cong \{z\in \mathbb{C}:\mathrm{Im}(z)>0\}$ by the mapping

$$gH\mapsto {\displaystyle \frac{ai+b}{ci+d}},G/H\to \{z\in \mathbb{C}:\mathrm{Im}\left(z\right)>0\}$$

where $g:=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in G$. In this setting, an element $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in G$ is H-aperiodic if for every compact $K\subseteq \{z\in \mathbb{C}:\mathrm{Im}(z)>0\}$, there is some $N>0$ such that

$${\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]}^n\left(z\right)\notin K(n\ge N\mathrm{and}z\in K),$$

where

$$\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \eta \end{array}\right]\left(z\right):={\displaystyle \frac{\alpha z+\beta }{\gamma z+\eta }}$$

for all $\left[\begin{array}{cc}\alpha & \beta \\ \gamma & \eta \end{array}\right]\in G$. In particular, ${\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]}^n=\left[\begin{array}{cc}1& 2n\\ 0& 1\end{array}\right]$ and

$${\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]}^{n}K=\left\{z+2n:z\in K\right\}.$$

Hence $\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ is H-aperiodic by the fact that the compact subsets of $\{z\in \mathbb{C}:\mathrm{Im}(z)>0\}$ are bounded.

Theorem 1.

Let G be a locally compact group, and let H be a closed subgroup of G. Let $a\in G$ be H-aperiodic, and let Φ be a ${\Delta }_2$-regular Young function. Let w be a weight on $G/H$ such that ${\displaystyle \parallel \frac{1}{w}{\parallel }_{sup}<\infty }$. Let $T_{a,w}$ be a weighted translation on $L^{\Phi }(G/H)$. Then the following conditions are equivalent.

(i)

$T_{a,w}$ is supercyclic on $L^{\Phi }(G/H)$.

(ii)

For each compact subset $K\subseteq G/H$ with $\mu \left(K\right)>0$, there exist a sequence $\left(E_k\right)$ of Borel subsets of K, a sequence $\left({\alpha }_n\right)\subseteq \mathbb{R}\setminus \left\{0\right\}$, and a strictly increasing sequence $\left(n_k\right)\subseteq \mathbb{N}$ such that $\mu \left(K\right)={lim}_{k\to \infty }\mu \left(E_k\right)$ and

$$\underset{k\to \infty }{lim}{∥{\left.{\phi }_{n_k}\right|}_{E_k}∥}_{\Phi }=\underset{k\to \infty }{lim}{∥{\left.{\tilde{\phi }}_{n_k}\right|}_{E_k}∥}_{\Phi }=0,$$

where

$${\phi }_{n_k}\left(xH\right):={\alpha }_{n_k}w\left(a^{n_k}xH\right)w\left(a^{n_k-1}xH\right)\cdots w\left(axH\right)$$

and

$${\tilde{\phi }}_{n_k}\left(xH\right):={\displaystyle \frac{1}{{\alpha }_{n_k}w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(n_k-1)}xH\right)}}$$

for all $xH\in G/H$.

Proof. 

$\left(\mathrm{ii}\right)\Rightarrow \left(\mathrm{i}\right)$. Assume the condition (ii) holds. Let $U,V\subseteq L^{\Phi }(G/H)$ be nonempty open sets. Since $\mu$ is a Radon measure on $G/H$ and $\Phi$ is ${\Delta }_2$-regular, the set $C_c(G/H)$ of all continuous compact supported functions on $G/H$ is dense in $L^{\Phi }(G/H)$. This implies that there exist $f\in U\bigcap C_c(G/H)$ and $g\in V\bigcap C_c(G/H)$. Put

$$K:=\mathrm{supp}\left(f\right)\bigcup \mathrm{supp}\left(g\right).$$

Then K is a compact subset of $G/H$. Hence there exists a sequence $\left(E_k\right)$ of Borel subsets of K, $\left({\alpha }_n\right)\subseteq \mathbb{R}\setminus \left\{0\right\}$ and $\left(n_k\right)\subseteq \mathbb{N}$ satisfying the condition (ii). Hence, given $\epsilon >0$, there exists an $N\in \mathbb{N}$ such that

$$\parallel {\phi }_{n_k}{{|}_{E_k}\parallel }_{\infty }\le {\displaystyle \frac{\epsilon }{{\parallel f\parallel }_{\Phi }}},and{∥{\left.{\tilde{\phi }}_{n_k}\right|}_{E_k}∥}_{\infty }\le {\displaystyle \frac{\epsilon }{{\parallel g\parallel }_{\Phi }}}$$

for each $k\ge N$. Therefore, by Remark 1, for each $k\ge N$, we have the next equalities’ chain:

$$\begin{array}{c}{∥{\alpha }_{n_k}{T}_{a,w}^{n_k}\left(f{\chi }_{E_k}\right)∥}_{\Phi }\hfill \\ =\underset{v\in B_{\Psi }}{sup}{\int }_{a^{n_k}{E}_k}\left|{\alpha }_{n_k}w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-\left(n_k-1\right)}xH\right)v\left(xH\right)\right|\left|f\left(a^{-n_k}xH\right)\right|d\mu \left(xH\right)\hfill \\ =\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}\left|{\alpha }_{n_k}w\left(a^{n_k}xH\right)w\left(a^{n_k-1}xH\right)\cdots w\left(axH\right)v\left(a^{n_k}xH\right)\right|\left|f\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ =\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}\left|{\alpha }_{n_k}w\left(a^{n_k}xH\right)w\left(a^{n_k-1}xH\right)\cdots w\left(axH\right)v\left(xH\right)\right|\left|f\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ =\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}{\phi }_{n_k}\left(xH\right)\left|v\left(xH\right)f\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ \le \parallel {\phi }_{n_k}{|}_{E_k}{\parallel }_{\infty }\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}\left|v\left(xH\right)f\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ \le \parallel {\phi }_{n_k}{|}_{E_k}{\parallel }_{\infty }\underset{v\in B_{\Psi }}{sup}{\int }_{G/H}\left|v\left(xH\right)f\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ = \parallel {\phi }_{n_k}{|}_{E_k}{\parallel }_{\infty }{\parallel f\parallel }_{\Phi }\le \epsilon ,\hfill \end{array}$$

which enables

$$\underset{k\to \infty }{lim}{∥{\alpha }_{n_k}{T}_{a,w}^{n_k}\left(f{\chi }_{E_k}\right)∥}_{\Phi }=0.$$

On the other hand, due to ${\displaystyle {\parallel \frac{1}{w}\parallel }_{sup}<\infty }$, one can define the inverse $S_{a,w}$ of $T_{a,w}$ by

$$S_{a,w}\left(f\right)\left(xH\right)={\displaystyle \frac{f\left(axH\right)}{w\left(axH\right)}}$$

for all $f\in L^{\Phi }(G/H)$ and $x\in G$. Then we obtain

$$\underset{k\to \infty }{lim}{∥{\displaystyle \frac{1}{{\alpha }_{n_k}}}{S}_{a,w}^{n_k}\left(g{\chi }_{E_k}\right)∥}_{\Phi }=0$$

by applying the similar calculation for $S_{a,w}$ below:

$$\begin{array}{cc}& {∥{\displaystyle \frac{1}{{\alpha }_{n_k}}}{S}_{a,w}^{n_k}\left(g{\chi }_{E_k}\right)∥}_{\Phi }\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{a^{-n_k}{E}_k}{\displaystyle \frac{1}{\left|{\alpha }_{n_k}w\left(axH\right)w\left(a^{2}xH\right)\cdots w\left(a^{n_k}xH\right)\right|}}\left|v\left(xH\right)g\left(a^{n_k}xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}{\displaystyle \frac{1}{\left|{\alpha }_{n_k}w\left(a^{1-n_k}xH\right)w\left(a^{2-n_k}xH\right)\cdots w\left(xH\right)\right|}}\left|v\left(a^{-n_k}xH\right)g\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}{\displaystyle \frac{\left|v\right(xH\left)\right|}{\left|{\alpha }_{n_k}w\left(a^{1-n_k}xH\right)w\left(a^{2-n_k}xH\right)\cdots w\left(xH\right)\right|}}\left|g\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}\left|v\left(xH\right){\tilde{\phi }}_{n_k}\left(xH\right)\right|\left|g\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & \le {\parallel {\tilde{\phi }}_{n_k}{|}_{E_k}\parallel }_{\infty }\underset{v\in B_{\Psi }}{sup}{\int }_{E_k}\left|v\left(xH\right)\right|\left|g\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & \le {\parallel {\tilde{\phi }}_{n_k}{|}_{E_k}\parallel }_{\infty }\underset{v\in B_{\Psi }}{sup}{\int }_{G/H}\left|v\left(xH\right)\right|\left|g\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & = {\parallel {\tilde{\phi }}_{n_k}{|}_{E_k}\parallel }_{\infty }{\parallel g\parallel }_{\Phi }\hfill \\ & <\epsilon .\hfill \end{array}$$

Now, let

$$v_k:=f{\chi }_{E_k}+{\displaystyle \frac{1}{{\alpha }_{n_k}}}{S}_{a,w}^{n_k}\left(g{\chi }_{E_k}\right)$$

for each $k\in \mathbb{N}$. Then

$${∥v_k-f∥}_{\Phi }\le {\parallel f\parallel }_{\infty }{\displaystyle \frac{1}{{\Phi }^{-1}\left(\frac{1}{\mu (K\setminus E_k)}\right)}}+{∥{\displaystyle \frac{1}{{\alpha }_{n_k}}}{S}_{a,w}^{n_k}\left(g{\chi }_{E_k}\right)∥}_{\Phi },$$

and

$${∥{\alpha }_{n_k}{T}_{a,w}^{n_k}{v}_k-g∥}_{\Phi }\le {∥{\alpha }_{n_k}{T}_{a,w}^{n_k}\left(f{\chi }_{E_k}\right)∥}_{\Phi }+{\parallel g\parallel }_{\infty }{\displaystyle \frac{1}{{\Phi }^{-1}\left(\frac{1}{\mu (K\setminus E_k)}\right)}}.$$

Hence there exists $k\in \mathbb{N}$ such that

$$\left({\alpha }_{n_k}{T}_{a,w}^{n_k}\right)\left(U\right)\cap V\ne ⌀,$$

which implies the sequence ${\left({\alpha }_n{T}_{a,w}^n\right)}_{n\in N}$ is topologically transitive and hypercyclic. Therefore $T_{a,w}$ is supercyclic.

$\left(\mathrm{i}\right)\Rightarrow \left(\mathrm{ii}\right)$. Assume that K is a compact subset of $G/H$ with $\mu \left(K\right)>0$. Since a is H-aperiodic, there exists a constant $M>0$ such that $K\cap a^{n}K=⌀$ (equivalently, $K\cap a^{-n}K=⌀$) for $n\ge M$. For the compact set K, one has ${\chi }_K\in L^{\Phi }(G/H)$. In addition, by the assumption of supercyclicity of $T_{a,w}$, for each $0<\epsilon <1$, there exist a function $f\in L^{\Phi }(G/H)$, an $m\ge M$ and an $\alpha \in \mathbb{R}\setminus \left\{0\right\}$ such that

$${∥f-{\chi }_K∥}_{\Phi }\le {\epsilon }^2\mathrm{and}{∥\alpha T_{a,w}^{m}f+{\chi }_K∥}_{\Phi }\le {\epsilon }^2.$$

Put $A:=\{xH\in K:|f\left(xH\right)-1|\ge \epsilon \}$. Then by Remark 1,

$${\epsilon }^2\ge {∥f-{\chi }_K∥}_{\Phi }\ge {∥{\chi }_A(f-{\chi }_K)∥}_{\Phi }\ge \epsilon {\parallel {\chi }_A\parallel }_{\Phi }\ge {\displaystyle \frac{\epsilon }{{\Phi }^{-1}\left(\frac{1}{\parallel {\chi }_A{\parallel }_1}\right)}}.$$

Hence ${\displaystyle \frac{1}{{\Phi }^{-1}\left(\frac{1}{\mu \left(A\right)}\right)}\le \epsilon }$ and ${\displaystyle \mu \left(A\right)\le \frac{1}{\Phi \left(\frac{1}{\epsilon }\right)}}$. Similarly, if

$$B:=\left\{x\in K:\left|\alpha T_{a,w}^{m}f\left(xH\right)+1\right|\ge \epsilon \right\},$$

then

$${\epsilon }^2\ge {∥\alpha T_{a,w}^{m}f+{\chi }_K∥}_{\Phi }\ge \epsilon {\parallel {\chi }_B\parallel }_{\Phi }$$

and $\mu \left(B\right)\le {\displaystyle \frac{1}{\Phi \left(\frac{1}{\epsilon }\right)}}$. Now, define

$$E:=\left\{xH\in K:\left|f\right(xH)-1|\le \epsilon \right\}\bigcap \left\{xH\in K:\left|\alpha T_{a,w}^{m}f\left(xH\right)+1\right|\le \epsilon \right\}.$$

Then ${\displaystyle \mu (K\setminus E)\le \frac{2}{\Phi \left(\frac{1}{\epsilon }\right)}}$. On the other hand, by Remark 1 and H-aperiodicity of $a\in G$,

$$\begin{array}{cc}\hfill {\epsilon }^2& \ge {∥\alpha T_{a,w}^{m}f+{\chi }_K∥}_{\Phi }\hfill \\ & \ge {∥{\chi }_{a^{m}E}(\alpha T_{a,w}^{m}f+{\chi }_K)∥}_{\Phi }\hfill \\ & ={∥\alpha {\chi }_{a^{m}E}{T}_{a,w}^{m}f∥}_{\Phi }\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{a^{m}E}\left|\alpha T_{a,w}^{m}f\left(xH\right)\right|\left|v\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{a^{m}E}\left|\alpha w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(m-1)}xH\right)\right|\left|f\left(a^{-m}xH\right)\right|\left|v\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_E\left|\alpha w\left(a^{m}xH\right)w\left(a^{m-1}xH\right)\cdots w\left(axH\right)\right|\left|f\left(xH\right)\right|\left|v\left(a^{m}xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_E\left|\alpha w\left(a^{m}xH\right)w\left(a^{m-1}xH\right)\cdots w\left(axH\right)\right|\left|f\left(xH\right)\right|\left|v\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & \ge (1-\epsilon )\underset{v\in B_{\Psi }}{sup}{\int }_E{\phi }_m\left(xH\right)\left|v\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & =(1-\epsilon )\parallel {\phi }_m{{|}_E\parallel }_{\Phi },\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\epsilon }^2& \ge {∥f-{\chi }_K∥}_{\Phi }\hfill \\ & \ge {∥{\chi }_{a^{-m}E}(f-{\chi }_K)∥}_{\Phi }\hfill \\ & ={∥{\chi }_{a^{-m}E}f∥}_{\Phi }\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_{a^{-m}E}{\displaystyle \frac{\left|v\right(xH\left)\right|}{\left|\alpha w\left(axH\right)w\left(a^{2}xH\right)\cdots w\left(a^{m}xH\right)\right|}}\left|\alpha T_{a,w}^{m}f\left(a^{m}xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_E{\displaystyle \frac{\left|v\right(a^{-m}xH\left)\right|}{\left|\alpha w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(m-1)}xH\right)\right|}}\left|\left(\alpha T_{a,w}^{m}f\right)\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & =\underset{v\in B_{\Psi }}{sup}{\int }_E{\displaystyle \frac{\left|v\right(xH\left)\right|}{\left|\alpha w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(m-1)}xH\right)\right|}}\left|\left(\alpha T_{a,w}^{m}f\right)\left(xH\right)\right|d\mu \left(xH\right)\hfill \\ & \ge (1-\epsilon )\underset{v\in B_{\Psi }}{sup}{\int }_E{\tilde{\phi }}_m\left(xH\right)\left|v\left(xH\right)\right|d\mu \left(xH\right).\hfill \end{array}$$

Then the condition (ii) follows. □

In the following, we will apply the above result to describe hypercyclicity, topological mixing and Cesàro hypercyclicity for $T_{a,w}$. First, letting ${\alpha }_n=1$ for each n in Theorem 1, we have the characterization of hypercyclicity on $L^{\Phi }(G/H)$.

Corollary 1.

Let G be a locally compact group, and let H be a closed subgroup of G. Let $a\in G$ be H-aperiodic, and let Φ be a ${\Delta }_2$-regular Young function. Let w be a weight on $G/H$ such that ${\displaystyle \parallel \frac{1}{w}{\parallel }_{sup}<\infty }$. Let $T_{a,w}$ be a weighted translation on $L^{\Phi }(G/H)$. Then the following conditions are equivalent.

(i)

$T_{a,w}$ is hypercyclic on $L^{\Phi }(G/H)$.

(ii)

For each compact subset $K\subseteq G/H$ with $\mu \left(K\right)>0$, there exist a sequence $\left(E_k\right)$ of Borel subsets of K and a strictly increasing sequence $\left(n_k\right)\subseteq \mathbb{N}$ such that $\mu \left(K\right)={lim}_{k\to \infty }\mu \left(E_k\right)$ and

$$\underset{k\to \infty }{lim}{∥{\left.{\phi }_{n_k}\right|}_{E_k}∥}_{\Phi }=\underset{k\to \infty }{lim}{∥{\left.{\tilde{\phi }}_{n_k}\right|}_{E_k}∥}_{\Phi }=0,$$

where

$${\phi }_{n_k}\left(xH\right):=w\left(a^{n_k}xH\right)w\left(a^{n_k-1}xH\right)\cdots w\left(axH\right)$$

and

$${\tilde{\phi }}_{n_k}\left(xH\right):={\displaystyle \frac{1}{w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(n_k-1)}xH\right)}}$$

for all $xH\in G/H$.

By replacing $\left(n_k\right)$ with the full sequence $\left(n\right)$ in Corollary 1, one can obtain the result of topological mixing below.

Corollary 2.

Let G be a locally compact group, and let H be a closed subgroup of G. Let $a\in G$ be H-aperiodic, and let Φ be a ${\Delta }_2$-regular Young function. Let w be a weight on $G/H$ such that ${\displaystyle \parallel \frac{1}{w}{\parallel }_{sup}<\infty }$. Let $T_{a,w}$ be a weighted translation on $L^{\Phi }(G/H)$. Then the following conditions are equivalent.

(i)

$T_{a,w}$ is topologically mixing on $L^{\Phi }(G/H)$.

(ii)

For each compact subset $K\subseteq G/H$ with $\mu \left(K\right)>0$, there exists a sequence $\left(E_n\right)$ of Borel subsets of K such that $\mu \left(K\right)={lim}_{n\to \infty }\mu \left(E_n\right)$ and

$$\underset{n\to \infty }{lim}{∥{\left.{\phi }_n\right|}_{E_n}∥}_{\Phi }=\underset{n\to \infty }{lim}{∥{\left.{\tilde{\phi }}_n\right|}_{E_n}∥}_{\Phi }=0,$$

where

$${\phi }_n\left(xH\right):=w\left(a^{n}xH\right)w\left(a^{n-1}xH\right)\cdots w\left(axH\right)$$

and

$${\tilde{\phi }}_n\left(xH\right):={\displaystyle \frac{1}{w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(n-1)}xH\right)}}$$

for all $xH\in G/H$.

Another application is to characterize Cesàro hypercyclicity of $T_{a,w}$ on $L^{\Phi }(G/H)$. An operator T on a Banach space X is called Cesàro hypercyclic if there exists $x\in X$ such that the set $\left\{({\displaystyle \frac{1}{n}}{\sum }_{j=1}^{n-1}{T}^j)x:n\ge 1\right\}$ is dense in X, which was first introduced by León-Saavedra in [14]. It was shown in [14] that T is Cesàro hypercyclic if, and only if, this sequence $\left\{({\displaystyle \frac{1}{n}}{T}^n)x:n\ge 1\right\}$ is hypercyclic. Using this equivalence and putting ${\alpha }_n={\displaystyle \frac{1}{n}}$ for each n in Theorem 1, the characterization for $T_{a,w}$ on $L^{\Phi }(G/H)$ to be Cesàro hypercyclic can be obtained immediately.

Corollary 3.

Let G be a locally compact group, and let H be a closed subgroup of G. Let $a\in G$ be H-aperiodic, and let Φ be a ${\Delta }_2$-regular Young function. Let w be a weight on $G/H$ such that ${\displaystyle {\displaystyle \parallel }\frac{1}{w}{\parallel }_{sup}<\infty }$. Let $T_{a,w}$ be a weighted translation on $L^{\Phi }(G/H)$. Then the following conditions are equivalent.

(i)

$T_{a,w}$ is Cesàro hypercyclic on $L^{\Phi }(G/H)$.

(ii)

For each compact subset $K\subseteq G/H$ with $\mu \left(K\right)>0$, there exists a sequence $\left(E_k\right)$ of Borel subsets of K and a strictly increasing sequence $\left(n_k\right)\subseteq \mathbb{N}$ such that $\mu \left(K\right)={lim}_{k\to \infty }\mu \left(E_k\right)$ and

$$\underset{k\to \infty }{lim}{∥{\left.{\phi }_{n_k}\right|}_{E_k}∥}_{\Phi }=\underset{k\to \infty }{lim}{∥{\left.{\tilde{\phi }}_{n_k}\right|}_{E_k}∥}_{\Phi }=0,$$

where

$${\phi }_{n_k}\left(xH\right):={\displaystyle \frac{1}{n_k}}w\left(a^{n_k}xH\right)w\left(a^{n_k-1}xH\right)\cdots w\left(axH\right)$$

and

$${\tilde{\phi }}_{n_k}\left(xH\right):={\displaystyle \frac{1}{\frac{1}{n_k}w\left(xH\right)w\left(a^{-1}xH\right)\cdots w\left(a^{-(n_k-1)}xH\right)}}$$

for all $xH\in G/H$.

Example 2.

Let $\mathcal{U}$ be the upper half plane in $\mathbb{C}$. Then the measure $d\mu :={\displaystyle \frac{dxdy}{y^2}}$ on $\mathcal{U}$ is invariant regarding the action of $\mathrm{SL}(2,\mathbb{R})$ on $\mathcal{U}$. By Example 1, we know $g=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right]$ is H-aperiodic where

$$H:=\left\{\left[\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right]:\theta \in [0,2\pi ]\right\}.$$

Assume that Φ is a ${\Delta }_2$-regular Young function, and w is a weight on $\mathcal{U}$ with $\parallel {\displaystyle \frac{1}{w}}{\parallel }_{sup}<\infty$. Then by Corollary 3, the operator $T_{g,w}$ is Cesàro hypercyclic if for each compact subset $K\subseteq \mathcal{U}$ with $\mu \left(K\right)>0$, there exists a sequence $\left(E_k\right)$ of Borel subsets of K and a strictly increasing sequence $\left(n_k\right)\subseteq \mathbb{N}$ such that $\mu \left(K\right)={lim}_{k\to \infty }\mu \left(E_k\right)$ and

$$\underset{k\to \infty }{lim}{∥{\left.{\phi }_{n_k}\right|}_{E_k}∥}_{\Phi }=\underset{k\to \infty }{lim}{∥{\left.{\tilde{\phi }}_{n_k}\right|}_{E_k}∥}_{\Phi }=0,$$

where

$${\phi }_{n_k}\left(z\right):={\displaystyle \frac{1}{n_k}}w\left(z+2n_k\right)w\left(z+2(n_k-1)\right)\cdots w(z+2)$$

and

$${\tilde{\phi }}_{n_k}\left(z\right):={\displaystyle \frac{n_k}{w\left(z\right)w\left(z-2\right)w\left(z-4\right)\cdots w\left(z-2(n_k-1)\right)}}$$

for all $z\in \mathcal{U}$.

Next, we will give the concrete Young function and weight to satisfy the conditions above. Let $\Phi \left(x\right)=x^{2}lnx$ for all $x\ge e$, and let Φ be a convex function with $\Phi \left(0\right)=0$ on $[0,e)$. Then Φ is a ${\Delta }_2$-regular Young function. Define the function f by

$$f\left(x\right):=\left\{\begin{array}{cc}2\hfill & \mathrm{if}x\le 1;\hfill \\ 1+{\displaystyle \frac{1}{x^2}}\hfill & \mathrm{if}x>1,\hfill \end{array}\right.$$

and put $w\left(z\right):=f\left(\mathrm{Re}\right(z\left)\right)$ for all $z\in \mathcal{U}$. Then, w is bounded and also away from zero. Let $K\subset \mathcal{U}$ be compact. Then K is closed and bounded. Hence, there is some $M>0$ such that for every $z\in K$, $\left|\mathrm{Re}\right(z\left)\right|\le M$. Let $E_k:=K$ for all $k\in N$. Then clearly ${lim}_{k\to \infty }\mu \left(E_k\right)=\mu \left(K\right)$.

To complete this example, we next claim that ${lim}_{k\to \infty }{\parallel {\phi }_{n_k}\parallel }_{\Phi }=0$. Because of the boundedness of K, there is some $J_0\in \mathbb{N}$ such that for all $j\ge J_0$ and $z\in K$, we have $\mathrm{Re}\left(z\right)+2j>1$. Note that there exists a constant $C_K>0$ such that for every $z\in K$,

$$\begin{array}{cc}\hfill Q_n\left(z\right):& =\prod _{j=1}^{n}w(z+2j)\hfill \\ \hfill & =\prod _{j=1}^{n}f(\mathrm{Re}\left(z\right)+2j)\hfill \\ \hfill & =\left(\prod _{j=1}^{J_0-1}f(\mathrm{Re}\left(z\right)+2j)\right)·\left(\prod _{j=J_0}^{n}f(\mathrm{Re}\left(z\right)+2j)\right)\hfill \\ \hfill & \le C_K·\left(\prod _{j=J_0}^{n}f(\mathrm{Re}\left(z\right)+2j)\right)\hfill \\ \hfill & =C_K·\prod _{j=J_0}^n\left(1+{\displaystyle \frac{1}{{(\mathrm{Re}\left(z\right)+2j)}^2}}\right).\hfill \end{array}$$

This implies that $Q_n\left(z\right)$ is uniformly bounded on K by some constant $M_K$. Hence, for each $z\in K$ and $k\in \mathbb{N}$,

$$|{\phi }_{n_k}\left(z\right)|={\displaystyle \frac{1}{n_k}}\left|Q_{n_k}\left(z\right)\right|\le {\displaystyle \frac{M_K}{n_k}}.$$

Therefore, ${\phi }_{n_k}\left(z\right)$ converges to 0 uniformly on K, and so

$$\underset{k\to \infty }{lim}{\parallel {\phi }_{n_k}\parallel }_{\Phi }=0.$$

Similarly, there is $J_1\in \mathbb{N}$ such that for all $j\ge J_1$ and $z\in K$, $\mathrm{Re}\left(z\right)-2j\le 1$. Hence, there is some constant $C_K^{\prime }>0$ such that for every $z\in K$ and $n\in \mathbb{N}$,

$$\begin{array}{cc}\hfill {\tilde{Q}}_n\left(z\right):& =\prod _{j=0}^{n-1}w(z-2j)\hfill \\ \hfill & =\prod _{j=0}^{n-1}f(\mathrm{Re}\left(z\right)-2j)\hfill \\ \hfill & =\left(\prod _{j=0}^{J_1-1}f(\mathrm{Re}\left(z\right)-2j)\right)·\left(\prod _{j=J_1}^{n-1}f(\mathrm{Re}\left(z\right)-2j)\right)\hfill \\ \hfill & \le C_K^{\prime }·\left(\prod _{j=J_1}^{n-1}f(\mathrm{Re}\left(z\right)-2j)\right)\hfill \\ \hfill & =C_K^{\prime }{2}^{n-J_1},\hfill \end{array}$$

and for every $z\in K$ and $k\in \mathbb{N}$,

$$|{\tilde{Q}}_{n_k}\left(z\right)|\ge C_K^{\prime }·2^{n_k-J_1}={\displaystyle \frac{C_K^{\prime }}{2^{J_1}}}·2^{n_k}.$$

Hence

$$|{\tilde{\phi }}_{n_k}\left(z\right)|={\displaystyle \frac{n_k}{|{\tilde{P}}_{n_k}\left(z\right)|}}\le {\displaystyle \frac{n_k·2^{J_1}}{C_K^{\prime }·2^{n_k}}},$$

which implies that ${\tilde{\phi }}_{n_k}\left(z\right)$ converges to 0 uniformly on K. Therefore,

$$\underset{k\to \infty }{lim}{\parallel {\tilde{\phi }}_{n_k}\parallel }_{\Phi }=0.$$

By Corollary 3, one can conclude that $T_{g,w}$ is Cesàro hypercyclic on $L^{\Phi }(\mathrm{SL}(2,\mathbb{R})/H)$.

3. Conclusions

In this paper, we mainly tackle the notion of supercyclicity, which is a generalization of hypercyclicity. Hypercyclicity is related to the invariant closed subset problem in analysis, and equivalent to topological transitivity if the Banach space is separable. Topological mixing is a stronger notion than topological transitivity, which is also one of the conditions to form chaos. The obtained results in this paper give sufficient and necessary conditions for weighted translation operators on the Orlicz space of quotient spaces to be supercyclic. By applying the main result, the characterizations of hypercyclicity, topological mixing and Cesàro hypercyclicity for weighted translations on such spaces are obtained as well, together with a concrete example. All these results generalize some work on the Lebesgue space of locally compact groups to the new setting of the Orlicz spaces on quotient spaces.