Limit shadowing of topological chains in dynamical systems on uniform spaces
一致空间上动力系统拓扑链的极限跟踪性质

摘要

本文聚焦紧致 Hausdorff 一致空间上拓扑动力系统的核心性质关联问题，针对经典度量空间框架下动力系统跟踪性与混合性结论的局限性，以一致空间为推广载体，系统引入拓扑链、拓扑链混合、拓扑混合及极限跟踪等基础概念，依托一致结构的对称性、复合性等核心性质，结合周期伪轨构造方法完成严谨理论推导，最终证明紧致 Hausdorff 一致空间中，动力系统具备极限跟踪性质且拓扑链混合的充要条件为系统拓扑混合。该成果突破度量空间的约束，将经典结论推广至更具一般性的一致空间范畴，丰富了拓扑动力系统链式性质与跟踪性质的理论体系，为非度量一致空间上动力系统的深入研究提供了关键理论支撑。

Abstract

This paper focuses on the relationship among core properties of topological dynamical systems on compact Hausdorff uniform spaces. To overcome the limitations of classical results on shadowing and mixing in metric spaces, we adopt uniform spaces as a generalized framework. We systematically introduce fundamental concepts such as topological chains, topological chain mixing, topological mixing, and limit shadowing. Leveraging the symmetry, composition, and other essential properties of uniform structures, combined with periodic pseudo-orbit construction methods, we carry out rigorous theoretical derivations. Ultimately, we prove that in a compact Hausdorff uniform space, a dynamical system possesses the limit shadowing property and is topologically chain mixing if and only if it is topologically mixing. This result breaks the constraints of metric spaces, extends classical conclusions to the more general setting of uniform spaces, enriches the theoretical system of chain properties and shadowing properties in topological dynamics, and provides key theoretical support for further research on dynamical systems on non-metric uniform spaces.