References & Citations
Mathematics > Dynamical Systems
Title: Poincaré maps and suspension flows: a categorical remark
(Submitted on 14 Jul 2021 (v1), last revised 19 Feb 2022 (this version, v2))
Abstract: 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.
Submission history
From: Tomoharu Suda [view email][v1] Wed, 14 Jul 2021 09:11:40 GMT (14kb)
[v2] Sat, 19 Feb 2022 02:56:38 GMT (14kb)
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