论文标题
半流程的代数：两个拓扑的故事
The algebra of semi-flows: a tale of two topologies
论文作者
论文摘要
为了捕获动力系统的全球结构，我们根据适当构造的拓扑来重新重新制定动态，我们称之为流动拓扑；我们称此过程拓扑化。这在双期拓扑空间方面产生了半流量的描述，其第一个拓扑与（相）空间相对应，第二个拓扑对应于流动拓扑。通过离散化，即定义和检查适当的有限子结构的研究促进了拓扑的研究。通过离散相关的流量拓扑来使其离散化提供了一种优雅的解决方案。我们介绍了Morse的预订，这是一个更一般的双人离散化的实例，该实例综合了空间和流动拓扑，并编码了动力学的方向性。我们描述了如何使用适当的（CO）同源信息来增强Morse预购，以描述动态的不变性；该合奏提供了半流量的代数化。该论文的主要成分的说明是通过应用于离散抛物线流的应用程序提供的。代数化在包含寄生抛物线流的摩尔斯理论信息方面，代数为正编织带来了一个新的不变。
To capture the global structure of a dynamical system we reformulate dynamics in terms of appropriately constructed topologies, which we call flow topologies; we call this process topologization. This yields a description of a semi-flow in terms of a bi-topological space, with the first topology corresponding to the (phase) space and the second to the flow topology. A study of topology is facilitated through discretization, i.e. defining and examining appropriate finite sub-structures. Topologizing the dynamics provides an elegant solution to their discretization by discretizing the associated flow topologies. We introduce Morse pre-orders, an instance of a more general bi-topological discretization, which synthesize the space and flow topologies, and encode the directionality of dynamics. We describe how Morse pre-orders can be augmented with appropriate (co)homological information in order to describe invariance of the dynamics; this ensemble provides an algebraization of the semi-flow. An illustration of the main ingredients of the paper is provided by an application to the theory of discrete parabolic flows. Algebraization yields a new invariant for positive braids in terms of a bi-graded differential module which contains Morse theoretic information of parabolic flows.
