Vishvesh Kumar, Seyyed Mohammad Tabatabaie
In this paper, we characterize hypercyclic sequences of weighted translation operators on an Orlicz space in the context of locally compact hypergroups.
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Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is investigated. The results are achieved by establishing an analogy between frequent hypercyclicity for the translation semigroup and for weighted pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts in spaces of vanishing sequences. Frequent hypercylic translation semigroups in weighted $L^p$-spaces are also characterized.
We prove the existence of common hypercyclic, entire functions for certain uncountable families of traslation type operators with relative large gaps.
We show that families of translation operators, where the translates grow exponentially fast, do not admit common hypercyclic functions. The result is close to be optimal.
We prove the existence of common hypercyclic entire functions for uncountable families of translation type operators. Contrary to our previous work [34], here the parameter which reflects the uncountable family lies on the unit circle. On the other hand the sequence of translations is in a sense less restrictive if it is to be compared with the corresponding sequence in [34]. Our result largely extends a result of Costakis in [19].
We use the generization of Weyl's equidistribution theorem to characterize several necessary conditions of hypercyclic weighted translation operators with periodic element.
Given a locally compact group $G$ and $1\leq p<\infty$, a sufficient condition ensuring the \emph{disjoint hypercyclicity} of finitely many weighted translations on $L^p(G)$ was investigated in this paper.
We show that the non-zero multiples of the derivative operator and the non-zero multiples of non-trivial translation operators on the space of entire functions share a common hypercyclic subspace, i.e. a closed infinite-dimensional subspace in which each non-zero vector has a dense orbit for each of these operators.
We analyze $f$-frequently hypercyclic, $q$-frequently hypercyclic ($q> 1$) and frequently hypercyclic $C_{0}$-semigroups ($q=1$) defined on complex sectors, working in the setting of separable infinite-dimensional Fréchet spaces. Some structural results of ours are given for a general class of ${\mathcal F}$-frequently hypercyclic $C_{0}$-semigroups, as well. We investigate generalized frequently hypercyclic translation semigroups and generalized frequently hypercyclic semigroups induced by semiflows on weighted function spaces. Several illustrative examples are presented.
Frequently hypercyclic $C_0$-semigroups indexed with complex sectors
本文研究了以复扇区为指标集的强连续算子半群的频繁超循环性，提出了修正定义，建立了充要条件，并导出了 $L^p_ρ(Δ, \mathbb{K})$ 空间上平移半群的实用判据与显式示例。
It is known that homogeneous polynomials on Banach spaces cannot be hypercyclic, but there are examples of hypercyclic homogeneous polynomials on some non-normable Fréchet spaces. We show the existence of hypercyclic polynomials on $H(\mathbb C)$, by exhibiting a concrete polynomial which is also the first example of a frequently hypercyclic homogeneous polynomial on any $F$-space. We prove that the homogeneous polynomial on $ H(\mathbb C)$ defined as the product of a translation operator and the evaluation at 0 is mixing, frequently hypercyclic and chaotic. We prove, in contrast, that some na…
We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of $p$-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.
We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\mbox{cos}(D)$, $De^D$, or $e^D-aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. Non-trivial translations on the space of complex-valued, smooth functions on the real line do su…
We generalize a result for the translation $C_0$-semigroup on $L^p(\R_+,μ)$ about the equivalence of being chaotic and satisfying the Frequent Hypercyclicity criterion due to Mangino and Peris to certain weighted composition $C_0$-semigroups. Such $C_0$-semigroups appear in a natural way when dealing with initial value problems for linear first order partial differential operators. We apply our result to the linear von Foerster-Lasota equation arising in mathematical biology. Weighted composition $C_0$-semigroups on Sobolev spaces are also considered.
Luh hypercyclic vector for composition operator
本文构造单连通域上全纯函数，使其在复合算子下所有导数与反导数轨道稠密，证明Luh超循环向量构成稠密线性流形，并研究加权复合算子生成的余弦算子函数的超循环性条件。
Hereditarily frequently hypercyclic operators and disjoint frequent hypercyclicity
该研究引入遗传频繁超循环性概念，证明其相关算子性质与构造反例，并解决不交频繁超循环性的多个问题，部分结果在Furstenberg族框架下得到。
Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a (bounded or unbounded) scalar type spectral operator $A$ in a complex Banach space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of the exponentials of such an operator, which, under a certain condition on the spectrum of the operator $A$, coincides with the $C_0$-semigroup generated by $A$. The spectrum of $A$ lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group $\left\{e^{tA}\right\}_{t\in {\mathbb R}}…
We show that if $E$ is an arbitrary $(DFN)$-space, then every nontrivial convolution operator on the Fréchet nuclear space $\mathcal{H}(E)$ is mixing, in particular hypercyclic. More generally we obtain the same conclusion when $E=F^{\prime}_c,$ where $F$ is a separable Fréchet space with the approximation property. On the opposite direction we show that a translation operator on the space $\mathcal{H}(\mathbb{C}^{\mathbb{N}}) $ is never hypercyclic.
Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,ω)=\{f:\int|fω|^p<\infty\}$, where $ω:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on $ω$, $G$ acts on $L^p(G,ω)$ by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in $L^p(G,ω)$? H. Salas (1995) gave a criterion of hypercyclicity in the case $G=\mathbb Z$ . Under mild assumptions, we present a corresponding characterization for a general locally compact group $G$. …
On the dynamics of composition operators: supercyclicity, odometers and translations
本文研究可测函数Banach空间上复合算子的动力学性质，特别通过里程计诱导的复合算子给出新例子和反例，并得到L_p空间上复合算子的超循环性和频繁超循环性的一般结论。
In this paper we characterize hypercyclic translation operators on the space of all compact linear operators on a Hilbert space H. Also, we give some sufficient condition for a related cosine operator function to be chaotic or topologically transitive
Universal holomorphic maps with slow growth II. functional analysis methods
借助超循环算子理论，本文证明了复欧几里得空间间具有慢增长率的超循环全纯映射的抽象丰富性，并证明了从$\mathbb{C}^n$到连通Oka流形$Y$的全纯映射空间中存在稠密的公共频繁超循环元素。
$J$-class weighted translations on locally compact groups
本文利用扩张极限集研究局部紧群上加权平移算子的J-类性质，给出其成为J-类算子的充要条件，并区分了J-类与超循环行为的边界。
Dynamics of the translation semigroup on directed metric trees
研究有向度量树上的加权L^p空间中左翻译半群的动力学，推导出强连续性的必要充分条件，并通过树结构中权重的渐近衰减特性刻画超循环性和弱混属性。
Dynamical properties and some classes of non-porous subsets
本文在一般勒贝格空间中引入多类非σ-多孔子集，并证明Lp空间上加权平移算子列的非超循环向量集不是σ-多孔的。
A classical result of Godefroy and Shapiro states that every nontrivial convolution operator on the space $\mathcal{H}(\mathbb{C}^n)$ of entire functions of several complex variables is hypercyclic. In sharp contrast with this result Fávaro and Mujica show that no translation operator on the space $\mathcal{H}(\mathbb{C}^\mathbb{N})$ of entire functions of infinitely many complex variables is hypercyclic. In this work we study the linear dynamics of convolution operators on $\mathcal{H}(\mathbb{C}^\mathbb{N})$. First we show that no convolution operator on $\mathcal{H}(\mathbb{C}^\mathbb{N})$ …
Multiresolution analysis on spectra of hermitian matrices
本文在复 Hermite 矩阵空间上构造适应酉群共轭不变性的多分辨率分析，利用超群平移刻画并证明径向标准正交小波基的存在性。
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