AI 中文总结
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。
AI 中文摘要
给定一个作用在同一个$F$-空间$X$上、由参数集$\Lambda \subset \mathbb{R}^d$（$d\geq1$）索引的有界线性算子族$\mathcal T = (T_\lambda)_{\lambda\in\Lambda}$，我们探讨了同时满足以下三个动力学性质之一（对$\mathcal T$的所有元素而言）的向量的存在性与不存在性：超循环性、上频繁超循环性和频繁超循环性。我们的方法依赖于将$\mathcal T$关联到一个“Lipschitz常数函数”$F$，其增长编码了算子与参数集$\Lambda$之间的相互作用。我们证明了$F$的渐近行为决定了公共向量存在与不存在之间的尖锐阈值。在超循环性情形下，我们确定了$\sum 1/F(n)^s$（其中$s$是$\Lambda$的Hausdorff维数）的发散作为存在的自然条件，并证明了补充的不存在性结果，这些结果对加权移位而言是最优的。对于上频繁超循环性，我们获得了在$d$维参数集情形下以形如$\sum 1/F(\gamma^n)$的级数表述的不存在性判据。在一维情形下，这些判据是最优的，并使我们能够将Mestiri关于对数型增长的已知存在性结果与超出该尺度的不存在性进行对比。对于频繁超循环性，我们建立了一个二分法，表明公共向量本质上仅在$F$有界时才存在，而任何无界增长都会阻止其存在。我们的结果适用于广泛的参数集类别，包括自相似分形，为增长条件和$\Lambda$的几何特征如何决定公共动力学行为提供了统一视角。最后，在加权移位的具体背景下，我们证明了如果两个算子的权重乘积比率序列具有两个不同的非零聚点，则它们不一定共享相同的频繁超循环向量，从而确立了Grivaux、Matheron和Menet一个结果的最优性。
英文摘要
Given a family $\mathcal T = (T_λ)_{λ\inΛ}$ of bounded linear operators acting on the same $F$-space $X$ and indexed by a set of parameters $Λ\subset \mathbb{R}^d$, $d\geq1$, we explore the existence and non-existence of vectors that simultaneously satisfy, for all elements of $\mathcal T$, one of the following three dynamical properties: hypercyclicity, upper frequent hypercyclicity, and frequent hypercyclicity. Our approach relies on associating to $\mathcal T$ a "Lipschitz constant function'' $F$, whose growth encodes the interaction between the operators and the parameter set $Λ$. We show that the asymptotic behavior of $F$ determines sharp thresholds between existence and non-existence of common vectors. In the hypercyclicity setting, we identify the divergence of $\sum 1/F(n)^s$, where $s$ is the Hausdorff dimension of $Λ$, as a natural condition for existence, and prove complementary non-existence results, which are optimal for weighted shifts. For upper frequent hypercyclicity, we obtain criteria of non-existence in terms of series of the form $\sum 1/F(γ^n)$ for parameter sets in $d$-dimensions. Considered in the one-dimensional case, these criteria are optimal and allow us to contrast known existence results of Mestiri for logarithmic-type growth with non-existence beyond this scale. For frequent hypercyclicity, we establish a dichotomy showing that common vectors exist essentially only when $F$ is bounded, while any unbounded growth prevents their existence. Our results apply to a broad class of parameter sets, including self-similar fractals, providing a unified perspective on how growth conditions and geometric features of $Λ$ determine common dynamical behavior. Finally, in the specific context of weighted shifts, we prove that two operators do not necessarily share the same frequently hypercyclic vectors if the sequence of their weight products ratios admits two distinct non-zero cluster points, establishing the optimality of a result of Grivaux, Matheron and Menet.
