拓扑动力系统中几类回复性的研究

回复性的研究是拓扑动力系统中最重要的课题之一，探讨回复时间集与组合性质的关联已成为该领域的国际前沿热点话题。本项目将围绕这一话题，运用Ellis半群理论来研究拓扑半流的几类回复性质，及其对应的组合性质。.具体来说，本项目的主要讨论对象是离散半群中几类具有组合性质的子集（如syndetic集，thick集，IP-集，central集等），以及其对应的回复性和乘积回复性，并在已有工作的基础上，引入新思路和新方法，证明这几类子集在离散半群中的动力系统实现定理。基于此，本项目将给出拓扑半流的distal点，（IP-，C-）回复点与（IP-，C-）乘积回复点之间关系的刻画，并进一步研究它们的相对情形。另外本项目还将研究极小半流的若干扩充问题，以及几类万有极小半流的相关性质。

Recurrence is one of the most important topics in topological dynamical systems, the research on the relationship between the returning-time sets and combinatorial properties has become a hot issue in the international frontiers of this fields. This project will focus on this topic, using Ellis semigroup theory to study some recurrent properties of topological semiflows and their corresponding combinatorial properties. .Specifically, the main discussion objects of this project are the subsets with combinatorial properties of discrete semigroups (for example syndetic sets，thick sets，IP-sets，central sets and so on), and the related recurrence and product recurrence. On the basis of the previous work, new ideas and methods are introduced to prove the dynamical realizations of the above subsets of discrete semigroups. Based on this, the project will give the relationship between distal points，(IP-，C-) recurrent points and (IP-，C-) product recurrent points of topological semiflows, and further study their relative cases. In addition, this project will study extensions of minimal semiflows, and the related properties of some universal minimal semiflows.