A categorical view of Poincare maps and suspension flows
A categorical view of Poincare maps and suspension flows
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庞加莱图和悬浮流的分类视图
DOI：
10.1080/14689367.2022.2027346
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
Suda Tomoharu
中科院分区：
文献类型：
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作者：
山本 浩太郎;Rodriguez Lopez Carlos Eduardo;Grzech Dagny;鵜崎 真妃;Caputi Lorenzo;三村 徹郎;山崎 真巳;O'Connor Sarah E.;Suda Tomoharu
庞加莱图和悬浮流是动力系统研究中基本结构的例子。本研究旨在证明，如果动力系统的范畴被适当地设定，这些结构定义了伴随函子对。首先，我们考虑了不一定是光滑的拓扑流形上流动范畴中的庞加莱映射的构造。我们证明，如果具有全局庞加莱截面的一类流被充分定义，那么众所周知的结果可以推广，并且庞加莱映射的构造是泛函的。接下来，我们考虑悬浮流的构造及其功能。最后，我们考虑了庞加莱映射和悬浮流构造的伴随性。通过考虑其自然性，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全局庞加莱截面的流之间的对应关系。我们定义了另一类具有全局庞加莱截面的流，并证明了如果考虑这类流，悬浮函子和庞加莱映射函子形成伴随等价。因此，得到了映射动力系统与具有全局庞加莱截面的流之间的绝对对应关系。这将使我们能够更好地理解地图动态系统和流之间的联系。
Poincar\'e 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 Poincar\'e 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 Poincar\'e maps is functorial, if a category of flows with global Poincar\'e sections is adequately defined. Next, we consider the construction of suspension flows and its functoriality. Finally, we consider the adjointness of the constructions of Poincar\'e 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 Poincar\'e sections. We define another category of flows with global Poincar\'e sections and show that the suspension functor and the Poincar\'e map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence between map dynamical systems and flows with global Poincar\'e sections is obtained. This will enable us to better understand the connection between map dynamical systems and flows.