Specific properties of the ODE's flow in dimension two versus dimension three

TL;DR

This paper explores the asymptotic behavior of flows generated by regular vector fields on tori, highlighting differences between two and three dimensions, with a focus on the Herman rotation set and specific flow examples.

Contribution

It revisits the Franks-Misiurewicz theorem in 2D, provides detailed examples like the Stepanoff flow, and discusses extensions to higher dimensions.

Findings

Herman rotation set in 2D is a closed line segment.

Complete analysis of Stepanoff flow with specific vector fields.

Extensions of flow properties to higher dimensions.

Abstract

This paper deals with the asymptotics of the ODE's flow induced by a regular vector field b on the d-dimensional torus R d /Z d. First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of R 2. Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field b = a $\zeta$, where $\zeta$ is a constant vector in R 2. Furthermore, several extensions of the