AI 中文总结
本文研究非自治动力系统的各类邻近关系，引入两种弱于经典交换性的渐近交换性概念，建立邻近关系的不变性等性质，推导相关系统邻近对的联系，通过示例验证结果的必要性与尖锐性。
AI 中文摘要
本文研究非自治动力系统的各类邻近关系，引入逐点渐近交换性与渐近交换性概念，证明二者严格弱于经典交换性。建立条件使邻近性、 syndetic邻近性、区域邻近性等不同邻近关系呈现不变性与结构性质，关联邻近性与等度连续性、弱混合性，刻画邻近关系变为平凡或极大的条件。对由一致强收敛序列生成的系统，推导该系统、其乘积系统与对应极限系统的邻近对间的联系，提供若干示例说明假设的必要性与结果的尖锐性。
英文摘要
In this paper, we investigate various forms of proximal relations for non-autonomous dynamical systems. In particular, we introduce the notion of pointwise asymptotic commutativity and asymptotic commutativity, and show that these are strictly weaker than classical commutativity. We establish conditions under which different proximal relations, including proximality, syndetic proximality, and regional proximality, exhibit invariance and structural properties. In particular, we relate proximality with equicontinuity and weak mixing, and characterize conditions where proximal relations become trivial or maximal. Further, for systems generated by uniformly and strongly convergent sequences, we derive connections between proximal pairs of the system, its product systems, and the corresponding limiting system. Several examples are provided to demonstrate the necessity of the assumptions and the sharpness of the results.
