复旗流形上的吸引子网络
Attractor Networks on Complex Flag Manifolds
摘要：
Robbin和Salamon证明了吸引-排斥网络和李雅普诺夫映射是等价的概念，并以射影空间上的线性流为例说明了这一点。在这些例子中，不动点相对于Smale序是线性排序的，这使得吸引-排斥网络过于简单。在本文中，我们提供了一类例子，其中的吸引-排斥网络和它的晶格结构可以明确地确定，即使Smale顺序是不全的。它们与复旗流形上的特殊流动有关。在这个过程中，我们表明，Smale秩序的不动点的集合上，可以确定与著名的Bruhat秩序。这也可以从Kazhdan和Lusztig的结果导出，但我们使用Palis的Lambda引理给出了一个新的证明。为了读者的方便，我们还介绍了旗流形通过初等动力系统只使用最低限度的李理论。
英文摘要：
Robbin and Salamon showed that attractor-repellor networks and Lyapunov maps are equivalent concepts and illustrate this with the example of linear flows on projective spaces. In these examples the fixed points are linearly ordered with respect to the Smale order which makes the attractor-repellor network overly simple. In this paper we provide a class of examples in which the attractor-repellor network and its lattice structure can be explicitly determined even though the Smale order is not total. They are associated with special flows on complex flag manifolds. In the process we show that the Smale order on the set of fixed points can be identified with the well-known Bruhat order. This could also be derived from results of Kazhdan and Lusztig, but we give a new proof using the Lambda-Lemma of Palis. For the convenience of the reader we also introduce the flag manifolds via elementary dynamical systems using only a minimum of Lie theory.
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