Depth $0$ Nonsingular Morse Smale flows on $S^3$

TL;DR

This paper introduces weighted Lyapunov graphs to analyze nonsingular Morse-Smale flows on the 3-sphere, classifies depth 0 flows without heteroclinic trajectories, and provides tools for topological classification and comparison.

Contribution

It develops the concept of weighted Lyapunov graphs for NMS flows on S^3, enabling classification and comparison of depth 0 flows without heteroclinic trajectories.

Findings

WLG detects indexed links of NMS flows.

Classification of all depth 0 NMS flows with up to 4 periodic orbits.

A simplified Umanskii Theorem for topological equivalence.

Abstract

In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on $S^3$. WLG is quite sensitive to NMS flows on $S^3$. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tools to describe nonsingular Morse-Smale flows without heteroclinic trajectories connecting saddle orbits (abbreviated as depth $0$ NMS flows). It mainly contains the following several directions: \begin{enumerate} \item we use WLG to list depth $0$ NMS flows on $S^3$; \item with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of depth $0$ NMS flows; \item to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two depth $0$ NMS flows on $S^3$ are topological equivalence; \item under these theories, we classify (up to topological equivalence) all depth 0 NMS flows on $S^3$ with periodic orbits number no more than 4. \end{enumerate}