Lyapunov graphs of nonsingular Smale flows on $S^{1}\times S^{2}$

TL;DR

This paper characterizes the Lyapunov graphs associated with nonsingular Smale flows on the 3-manifold S^1 x S^2, extending previous work on S^3, and analyzes the role of singular vertices in these graphs.

Contribution

It provides necessary and sufficient conditions for Lyapunov graphs to correspond to nonsingular Smale flows on S^1 x S^2, advancing understanding of flow dynamics on this manifold.

Findings

Characterization of Lyapunov graphs for S^1 x S^2

Conditions for association with nonsingular Smale flows

Analysis of singular vertices in Lyapunov graphs

Abstract

In this paper, following J. Franks' work on Lyapunov graphs of nonsingular Smale flows on $S^3$, we study Lyapunov graphs of nonsingular Smale flows on $S^1 \times S^2$. More precisely, we determine necessary and sufficient conditions on an abstract Lyapunov graph to be associated with a nonsingular Smale flow on $S^1 \times S^2$. We also study the singular type vertices in Lyapunov graphs of nonsingular Smale flows on 3-manifolds.