论文标题
在组合矢量字段上对简单复合物创建半流量
Creating Semiflows on Simplicial Complexes from Combinatorial Vector Fields
论文作者
论文摘要
Robin Forman引入的简单复合物上的组合矢量场发现了近年来的许多应用。但是，它们与经典动力系统的关系还不太清楚。在最近的工作中，表明，对于有限的简单复合物上的每个组合矢量场，都可以在基础多层X上构建多价离散时间动力系统，该动力学与Conley Index理论意义上的组合流相同。但是，福尔曼（Forman）对组合流的原始描述似乎是由流动的概念，即连续时间动力学系统的动机。在本文中，结果表明，可以在X上构造半流，该半节具有与基础组合矢量场相同的动力学。动力学行为的等效性是在Conley-Morse图的意义上建立的，并使用拓扑空间X的平铺，这使得可以直接构建所有涉及的孤立不变集纯粹基于组合信息的隔离不变集。
Combinatorial vector fields on simplicial complexes as introduced by Robin Forman have found numerous and varied applications in recent years. Yet, their relationship to classical dynamical systems has been less clear. In recent work it was shown that for every combinatorial vector field on a finite simplicial complex one can construct a multivalued discrete-time dynamical system on the underlying polytope X which exhibits the same dynamics as the combinatorial flow in the sense of Conley index theory. However, Forman's original description of combinatorial flows appears to have been motivated more directly by the concept of flows, i.e., continuous-time dynamical systems. In this paper, it is shown that one can construct a semiflow on X which exhibits the same dynamics as the underlying combinatorial vector field. The equivalence of the dynamical behavior is established in the sense of Conley-Morse graphs and uses a tiling of the topological space X which makes it possible to directly construct isolating blocks for all involved isolated invariant sets based purely on the combinatorial information.
