Frédéric Bayart, Fernando Costa, Quentin Menet
We investigate the existence of a common hypercyclic vector for a family $(T_λ)_{λ\in Λ}$ of hypercyclicoperators acting on the same Banach space $X$. We give positive and negative results involving the dimension of $Λ$ and the regularity of each map $λ\in Λ\mapsto T_λ^n x$, $x\in X$, $n\in\mathbb N$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the existence of a common hypercyclic vector for different families of composition operators.
We show that every hypercyclic operator on a real locally convex space admits a dense, invariant linear manifold of hypercyclic vectors.
We prove that the space $l^2$ contains a dense set of vectors which are hypercyclic simultaneously for all multiples of the backward shift operator by constants of absolute value greater than 1.
We provide necessary and sufficient conditions on the existence of common hypercyclic vectors for multiples of the backward shift operator along sparse powers. Our main result strongly generalizes corresponding results which concern the full orbit of the backward shift. Some of our results are valid in a more general context, in the sense that they apply for a wide class of hypercyclic operators.
We show that, under suitable conditions, an operator acting like a shift on some sequence space has a frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. We also apply it to every operator satisfying the Frequent Hypercyclicity Criterion, recovering a result of Murillo and Peris.
Common frequently hypercyclic random vectors
研究可数家族加权逆向移位在ℓ_p空间中的共同频繁超循环向量，利用概率技术提出存在性准则并补充不存在性结果，应用于多项式加权逆向移位家族，提供共同频繁超循环向量的条件。
We show that several convolution operators on the space of entire functions, such as the MacLane operator, support a dense hypercyclic algebra that is not finitely generated. Birkhoff's operator also has this property on the space of complex-valued smooth functions on the real line.
Bonet, Frerick, Peris and Wengenroth constructed a hypercyclic operator on the locally convex direct sum of countably many copies of the Banach space $\ell_1$. We extend this result. In particular, we show that there is a hypercyclic operator on the locally convex direct sum of a sequence $\{X_n\}_{n\in\N}$ of Fréchet spaces if and only if each $X_n$ is separable and there are infinitely many $n\in\N$ for which $X_n$ is infinite dimensional. Moreover, we characterize inductive limits of sequences of separable Banach spaces which support a hypercyclic operator.
We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators.
We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset $Λ$ of the complex plane for the family {$λ$T : $λ$ $\in$ $Λ$} to have a common frequently hypercyclic vector. In passing, this permits us to eas…
We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator $T$ on a complex Fréchet space $X$ and a set $Λ\subseteq \R_+\times\C$ which is not of zero three-dimensional Lebesgue measure, the family $\{aT+bI:(a,b)\inΛ\}$ has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. I…
Let $D$ be the differentiation operator $Df=f'$ acting on the Fréchet space $\H$ of all entire functions in one variable with the standard (compact-open) topology. It is known since 1950's that the set $H(D)$ of hypercyclic vectors for the operator $D$ is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepúlveda whether the set $H(D)$ contains (up to the zero function) a non-trivial subalgebra of $\H$ or an infinite dimensional closed linear subspace of $\H$. In the present article both questions are answered affirmatively.
On hypercyclic and (common) upper frequently hypercyclic subspaces
本文研究了 unilateral weighted backward shift 在 $\ell_p$ 上的超循环空间性质，证明了存在不包含频繁超循环向量的超循环子空间，并解决了 Bès 和 Menet 2015 年提出的问题。
Motivated by a question posed by Sophie Grivaux concerning the regularity of the orbits of frequently hypercylic operators, we show the following: for any operator $T$ on a separable metrizable and complete topological vector space $X$ which is both frequently hypercyclic and piecewise syndetic hypercyclic, the lower density and upper Banach density of the recurrence set $\{n\geq 1: T^n x\in U\}$ are different, for any hypercyclic vector $x\in X$ for $T$, and a certain collection of non-empty open sets $U\subseteq X$. As an immediate consequence we got a sufficient condition for a chaotic oper…
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。
Density properties of orbits for a hypercyclic operator on a Banach space
研究Banach空间上超循环算子轨道的密度性质，证明四种情况中仅有一种成立，并通过例子展示所有情况均可能出现。
We study the existence of hypercyclic algebras for convolution operators $Φ(D)$ on the space of entire functions whose symbol $Φ$ has unimodular constant term. In particular, we provide new eigenvalue criteria for the existence of densely strongly algebrable sets of hypercyclic vectors.
Considering a family of upper frequently hypercyclic operators we care about the existence of vectors which are upper frequently hypercyclic for any operator of this family. We establish sufficient conditions for a family of operators to have these vectors called common upper frequently hypercyclic vectors. Using this result, we then give a construction of such vectors. Finally we derive some applications to families of weighted shifts.
A main objective of the present paper is to develop the theory of hypercyclicity and supercyclicity of linear operators on topological vector space over non-Archimedean valued fields. We show that there does not exist any hypercyclic operator on finite dimensional spaces. Moreover, we give sufficient and necessary conditions of hypercyclicity (resp. supercyclicity) of linear operators on separable $F$-spaces. It is proven that a linear operator $T$ on topological vector space $X$ is hypercyclic (supercyclic) if it satisfies Hypercyclic (resp. Supercyclic) Criterion. We consider backward shifts…
We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group ${T_t}_{t\in\C^n}$ with holomorphic dependence on the parameter $t$. This result covers those existing in the literature. We also describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups ${T_t}_{t\geq 0}$.
The question of whether a hypercyclic operator $T$ acting on a Fr{é}chet algebra $X$ admits or not an algebra of hypercyclic vectors (but 0) has been addressed in the recent literature. In this paper we give new criteria and characterizations in the context of convolution operators acting on $H(\mathbb C)$ and backward shifts acting on a general Fr{é}chet sequence algebra.Analogous questions arise for stronger properties like frequent hypercyclicity. In this trend we give a sufficient condition for a weighted backward shift to admit an upper frequently hypercyclic algebra and we find a weighte…
In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is rel…
A hypercyclicity criterion for non-metrizable topological vector spaces
该研究提出非可度量化序列可分拓扑向量空间上算子序列超循环性的充分条件，并用两类无法通过F-空间比较原理推导超循环性的算子实例验证，解决了相关公开问题。
Let (k(n)) n=1,2,... be a strictly increasing sequence of positive integers . We consider a specific sequence of differential operators Tk(n),λ , n=1,2,... on the space of entire functions , that depend on the sequence (k(n)) n=1,2,... and the non-zero complex number λ . We establish the existence of an entire function f , such that for every positive number λ the set {Tk(n),λ ,n=1,2,... } is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane . This provides the best possible strenghthened version of a corresponding r…
Trichotomy for the orbits of a hypercyclic operator on a Banach space
本文在可分 Banach 空间上建立了超循环算子轨道的三分法，区分均值渐近于零、绝对均值不规则和均值发散到无穷三种情形，并用加权后移算子说明三种情况均可实现。
We give a short proof of the existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Frechet space. A similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.
Let $(T\_λ)\_{λ\inΛ}$ be a family of operators acting on a $F$-space $X$, where the parameter space $Λ$ is a subset of $\mathbb R^d$. We give sufficient conditions on the family to yield the existence of a vector $x\in X$ such that, for any $λ\inΛ$, the set $\big\{T\_λ^n x;\ n\geq 1\big\}$ is dense in $X$. We obtain results valid for any value of $d\geq 1$ whereas the previously known results where restricted to $d=1$. Our methods also shed new light on the one-dimensional case.
We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T; that there exist frequently hypercyclic operators on the sequence space c_0 admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vect…
Hypercyclic algebras for weighted shifts on trees
本文研究有向树序列空间上加权后向移位算子的超循环向量代数存在性，给出 $c_0(V)$、$\ell^1(V)$ 上的代数结构结论及 $\ell^p(V)$ 上的充要条件，并探讨了混合性及无根树情形。
We prove that a bounded operator $T$ on a separable Banach space $X$ satisfying a strong form of the Frequent Hypercyclicity Criterion (which implies in particular that the operator is universal in the sense of Glasner and Weiss) admits frequently hypercyclic vectors with irregularly visiting orbits, i.e. vectors $x\in X$ such that the set $\mathcal{N}_T(x,U)=\{n\ge 1\,;\,T^{n}x\in U\}$ of return times of $x$ into $U$ under the action of $T$ has positive lower density for every non-empty open set $U\subseteq X$, but there exists a non-empty open set $U_0\subseteq X$ such that $\nt{x}{U_0}$ has…
Extension of hypercyclic and frequently hypercyclic subspaces
研究满足Kitai准则或频繁超循环准则的算子T，证明超循环向量和频繁超循环向量集合中的线性结构可扩展至连续统维数，并给出有限维子空间一般可扩展的条件。
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.
In this paper, we generalize to the context of algebras some recent results on the existence of common hypercyclic vectors for families of products of backward shift operators. We also give, in a multi-dimensional setting, a positive answer to a question raised by F. Bayart, D. Papathanasiou and the author about the existence of a common hypercyclic algebra on $\ell_1(\mathbb{N})$ with the convolution product for the family of backward shifts $(B_{w(λ)})_{λ>0}$ induced by the weights $w_n(λ)=1+λ/n$.
Existence of hypercyclic algebras in Fréchet algebras
本文证明每个可分无穷维Fréchet代数存在支撑稠密不变超循环代数的连续线性算子，肯定回答了Bayart等人的问题，并在有连续范数时构造了核扰动形式。
On frequently supercyclic operators and an F_Γ-hypercyclicity criterior with applications
本文引入F_Γ-超循环算子与F-超循环标量集概念，给出识别判据，证明伪移位算子在无界Γ下满足性质，并刻画超循环标量集的充要条件，部分解答了2016年提出的开放问题。
Existence and non-existence in common hypercyclicity and its frequent variants
研究公共超循环性及其频繁变体中的公共向量存在性，通过Lipschitz常数函数的渐近行为确定存在与不存在的尖锐阈值，并给出加权移位下的最优结果。
On the complexity of upper frequently hypercyclic vectors
本文研究上频繁超循环向量的复杂性，通过纯拓扑证明将 $\mathrm{UFHC}(T)$ 改进为 $G_{δσ}$-集，并构造反例否定其始终为 $G_δ$-集的猜想，同时探讨了不同拓扑下的[非]等价性。
We study the dynamics induced by an $m$-linear operator. We answer a question of Bès and Conejero showing an example of an $m$-linear hypercyclic operator acting on a Banach space. Moreover we prove the existence of $m$-linear hypercyclic operators on arbitrary infinite dimensional separable Banach spaces. We also prove an existence result about symmetric bihypercyclic bilinear operators, answering a question by Grosse-Erdman and Kim.
We study hypercyclicity properties of a family of non-convolution operators defined on spaces of holomorphic functions on $\mathbb{C}^N$. These operators are a composition of a differentiation operator and an affine composition operator, and are analogues of operators studied by Aron and Markose on $H(\mathbb{C})$. The hypercyclic behavior is more involved than in the one dimensional case, and depends on several parameters involved.
Chaotic linear dynamics deals primarily with various topological ergodic properties of semigroups of continuous linear operators acting on a topological vector space. We treat questions of characterizing which of the spaces from a given class support a semigroup of prescribed shape satisfying a given topological ergodic property. In particular, we characterize countable inductive limits of separable Banach spaces that admit a hypercyclic operator, show that there is a non-mixing hypercyclic operator on a separable infinite dimensional complex Fréchet space $X$ if and only if $X$ is non-isomorp…
Strong universality, recurrence, and analytic P-ideals in dynamical systems
本文研究动力系统中由解析P-理想刻画的强泛性与强回复性，建立其与通常泛性、回复性及轨道聚点性质之间的结构关系，并给出Fréchet空间和Banach空间线性动力系统中关于零轨道稠密性、上频繁超循环的等价刻画。
In this paper, We study dimension-free $L^p$ estimates for UMD lattice-valued $q$-variations of Hardy-Littlewood averaging operators associated with the Euclidean balls.
This paper systematically studies finite rank dimension groups, as well as finite dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite dimensional ordered real vector …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。