由组合向量场构造单纯复形上的半流
Creating Semiflows on Simplicial Complexes from Combinatorial Vector Fields
摘要：
由Robin Forman引入的单纯复形上的组合向量场近年来得到了广泛的应用。然而，它们与经典动力系统的关系还不太清楚。最近的工作表明，对于有限单纯复形上的任意组合向量场，可以在其基础多面体X上构造一个多值离散时间动力系统，该系统表现出与Conley指标理论意义下的组合流相同的动力学性质.然而，福尔曼对组合流的原始描述似乎更直接地受到流的概念的激励，即，连续时间动力系统在本文中，它表明，人们可以构建一个半流X表现出相同的动力学作为潜在的组合向量场。动力学行为的等价性建立在Conley-Morse图的意义上，并使用拓扑空间X的平铺，这使得有可能直接构造隔离块的所有涉及的孤立不变集的基础上纯粹的组合信息。
英文摘要：
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.
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