AI 中文总结
本文引入F-空间概念与对应的格罗莫夫-豪斯多夫距离，定义动力系统间的新距离，证明其保持两类稳定性，研究豪斯多夫映射性质，还改进距离并计算环面平移的精确距离。
AI 中文摘要
我们引入F-空间的概念——即配备广义伪度量族与标记点的集合，并构建F-空间的范畴性格罗莫夫-豪斯多夫距离。基于此，我们提出了动力系统之间距离的新定义，这里的动力系统广义上指参数化映射族。研究重点为局域动力学：我们证明，关于该新距离的极限转换下，李雅普诺夫稳定性与渐近稳定性均得以保持（后者需满足共同吸引半径的条件）。研究确定，稳定系统类是紧动力系统空间中的闭且无处稠密子集。接下来，我们研究豪斯多夫映射的性质，该映射在子集族上诱导出F-空间结构。在最后部分，我们引入动力系统的另一种改进版格罗莫夫-豪斯多夫距离，并利用它计算非共振向量的环面平移之间的精确距离。
英文摘要
We introduce the notion of an F-space - a set equipped with a family of generalized pseudometrics and marked points, and construct the categorical Gromov-Hausdorff distance for F-spaces. Based on this, we propose a new definition of the distance between dynamical systems, understood in a broad sense as parameterized families of maps. The main focus is on local dynamics: we prove the preservation of Lyapunov stability and asymptotic stability under limit transitions with respect to the new distance (in the latter case, assuming a common radius of attraction). It is established that the class of stable systems is a closed and nowhere dense subset in the space of compact dynamical systems. Next, we investigate the properties of the Hausdorff map, which induces an F-space structure on the family of subsets. In the final part, we introduce another modified version of the Gromov-Hausdorff distance for dynamical systems and use it to calculate the exact distance between torus translations for a non-resonant vector.
Comments30 pages
