Behavior $0$ nonsingular Morse Smale flows on $S^3$

Bin Yu

doi:10.3934/dcds.2016.36.509

Article Contents

2016, Volume 36, Issue 1: 509-540. Doi: 10.3934/dcds.2016.36.509

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Behavior $0$ nonsingular Morse Smale flows on $S^3$

Bin Yu^1,

1.
Department of Mathematics, Tongji University, Shanghai 200092

Received: May 31, 2014

Revised: November 30, 2014

Published: May 2017

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Abstract

In this paper, we first develop 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 behavior $0$ NMS flows). It mainly contains the following several directions:
1. we use WLG to list behavior $0$ NMS flows on $S^3$;
2. with the help of WLG, comparing with Wada's algorithm, we provide a direct description about the (indexed) link of behavior $0$ NMS flows;
3. to overcome the weakness that WLG can't decide topologically equivalent class, we give a simplified Umanskii Theorem to decide when two behavior $0$ NMS flows on $S^3$ are topological equivalence;
4. under these theories, we classify (up to topological equivalence) all behavior 0 NMS flows on $S^3$ with periodic orbits number no more than 4.

Keywords:

Behavior $0$ nonsingular Morse-Smale flows,
weighted Lyapunov graphs,
links,
topologically equivalent classes.

Mathematics Subject Classification: Primary: 37D15, 37C15; Secondary: 37C15, 57M25.

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