Smale flows on $\mathbb{S}^2\times\mathbb{S}^1$

TL;DR

This paper provides a complete classification of Smale flows on the manifold ^2^1 using abstract Lyapunov graphs, establishing necessary and sufficient conditions for their correspondence.

Contribution

It introduces a combinatorial classification framework for Smale flows on ^2^1 based on abstract Lyapunov graphs, filling a gap in dynamical systems theory.

Findings

Complete classification of Smale flows on ^2^1

Necessary and sufficient conditions for Lyapunov graphs

Framework for analyzing dynamical systems on product manifolds

Abstract

In this paper, we use abstract Lyapunov graphs as a combinatorial tool to obtain a complete classification of Smale flows on $\mathbb{S}^2\times\mathbb{S}^1$. This classification gives necessary and sufficient conditions that must be satisfied by an abstract Lyapunov graph in order for it to be associated to a Smale flow on $\mathbb{S}^2\times\mathbb{S}^1$.