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Links of closed orbits of nonsingular Morse-Smale flows
| Content Provider | Semantic Scholar |
|---|---|
| Author | Sasano, Kazuhiro |
| Copyright Year | 1983 |
| Abstract | The links which consist of closed orbits of certain nonsingular MorseSmale flows on S3 are determined. Recently various properties of the links of closed orbits of flows have been investigated by several persons (for instance, [F2], [BW], [M]). In this paper we determine the link which consists of closed orbits of certain nonsingular Morse-Smale flows on S3. (For definitions see below.) Precisely speaking we prove THEOREM. Suppose that a nonsingular Morse-Smale flow on S3 has a single closed orbit h0 of index 2, a single closed orbit h, +I of index 0, and n closed orbits h1, h2 ., h, ofindex!. (A) If all of the closed orbits of index 1 are untwisted, then the link consisting of all closed orbits is trivial. (B) If all of the closed orbits of index 1 are twisted, then by reordering hI, h2,. . , hn appropriately, we find k such that (a) h h and h h + I make the Hopf link, (b) for any i k, hJ+ is a (2, qj)-cable of hj, where p, and qJ are arbitrary odd integers. (See Figure 1.) COROLLARY. Suppose that a nonsingular Morse-Smale flow on S3 has three closed orbits. Then there exists just one closed orbit of index i for each i (O < i < 2), and the closed orbits make either the trivial link or the link as in Figure 2, where h is the orbit of index 1 and L1 is a (2, p)-cable of h ( p = odd), and one of LI and L2 is of index 0 and the other is of index 2. The author is very grateful to Professors Itiro Tamura, Shigenori Matsumoto and Koich Yano for their advice and encouragement, Professor Nagayoshi Iwahori for his advice on Algebra, and Noriko Maruyama for her encouragement. Finally the author thanks the referee for his useful suggestions. Received by the editors June 28, 1982. 1980 Mathemnatics Subject Classification. Primary 58F09; Secondary 57M25. |
| Starting Page | 727 |
| Ending Page | 734 |
| Page Count | 8 |
| File Format | PDF HTM / HTML |
| DOI | 10.1090/S0002-9939-1983-0702309-0 |
| Volume Number | 88 |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1983-088-04/S0002-9939-1983-0702309-0/S0002-9939-1983-0702309-0.pdf |
| Alternate Webpage(s) | https://doi.org/10.1090/S0002-9939-1983-0702309-0 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
