Poincaré映射与悬流：一个范畴注记
Poincaré maps and suspension flows: a categorical remark
摘要：
庞加莱映射和悬浮流是研究动力系统的基本构造的例子。本研究的目的是表明，如果动力系统的范畴是适当的设置，这些结构定义了一个伴随函子对。首先，我们考虑在不一定光滑的拓扑流形上的流范畴中构造Poincaré映射。我们表明，众所周知的结果可以推广和庞加莱映射的建设是函的，如果一个类别的流与全球庞加莱部分是适当的定义。接下来，我们考虑悬浮流的构造及其功能性。最后，我们考虑了Poincaré映射和悬浮流的构造的伴随性。通过考虑流的自然性，我们可以得出结论：流的拓扑等价或拓扑共轭的概念不足以描述映射动力系统与具有整体Poincaré截面的流之间的对应关系.我们定义了另一类具有全局Poincaré截面的流，并证明了如果考虑这两类流，则悬挂函子和Poincaré映射函子形成伴随等价。从而得到了映射动力系统与具有整体Poincaré截面的流之间的范畴对应.这将使我们能够更好地理解地图动力系统和流动之间的联系。
英文摘要：
Poincaré maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincaré maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincaré maps is functorial, if a category of flows with global Poincaré sections is adequately defined. Next, we consider the construction of suspension flows and its functoriality. Finally, we consider the adjointness of the constructions of Poincaré maps and suspension flows. By considering the naturality, we can conclude that the concepts of topological equivalence or topological conjugacy of flows are not sufficient to describe the correspondence between map dynamical systems and flows with global Poincaré sections. We define another category of flows with global Poincaré sections and show that the suspension functor and the Poincaré map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence between map dynamical systems and flows with global Poincaré sections is obtained. This will enable us to better understand the connection between map dynamical systems and flows.
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