线性动力学中的观测方案与不规则性
Observation Schemes and Irregularity in Linear Dynamics
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中文总结 AI 辅助
该研究构建以观测方案空间为核心的线性动力学不规则性结构框架，建立相关准则得到不规则向量的线性结构结果，并在密度假设下得到不规则性全局行为三分法与观测方案类刚性现象。
中文摘要 AI 辅助
我们构建了一个以观测方案空间$\Upsilon$为核心的线性动力学不规则性结构框架。该方法将底层动力学行为与观测机制分离开来，提供了一个统一框架，使得Li–Yorke混沌、平均Li–Yorke混沌、分布混沌等经典概念作为对应狄拉克测度和Cesàro平均的特例出现。我们建立了两个确保大型线性结构存在的抽象准则，得到了绝对$(μ_m)$-不规则向量与分布$(μ_m)$-不规则向量的稠密可线性化与空间化结果。此外，在自然的密度假设下，我们得到了描述整个$\Upsilon$上不规则性全局行为的三分法，以及生成各类混沌行为的观测方案类的刚性现象。
英文摘要
We develop a structural framework for irregularity in linear dynamics centered on the space $Υ$ of observation schemes. This approach separates the underlying dynamical behavior from the observation mechanism and provides a unified setting in which classical notions such as Li--Yorke chaos, mean Li--Yorke chaos, and distributional chaos arise as particular cases corresponding to Dirac measures and Cesàro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(μ_m)$-irregular and distributionally $(μ_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $Υ$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.
