Poincaré maps and suspension flows: a categorical remark

Abstract

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.