Poincar\'e maps and suspension flows: a categorical remark

TL;DR

This paper establishes a categorical framework linking Poincaré maps and suspension flows, showing they form an adjoint pair of functors, thus deepening the understanding of the relationship between discrete and continuous dynamical systems.

Contribution

It generalizes the functorial construction of Poincaré maps and suspension flows to topological flows and demonstrates their adjoint equivalence in a suitable categorical setting.

Findings

Poincaré maps are functorial in the category of flows with global sections.

Suspension flows form a functor that is adjoint to the Poincaré map functor.

A categorical equivalence between map systems and flows with global sections is established.

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.