Nonsingular Morse--Smale Flows with Three Periodic Orbits on Orientable $3$-Manifolds
Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 426-443.

Voir la notice de l'article provenant de la source Math-Net.Ru

The topological equivalence of nonsingular Morse–Smale flows under assumptions of various generality has been considered in many works (see, e.g., [1]–[4]). However, in the case of a small number of periodic orbits, it is possible to significantly simplify the known invariants and, most importantly, bring the classification problem to implementation by describing the admissibility of the obtained invariants. In the recent paper [5], an exhaustive classification of flows with two orbits on any closed $n$-manifolds was obtained. The present paper gives a complete topological classification for flows with three periodic orbits on orientable $3$-manifolds.
Keywords: nonsingular flow, Morse–Smale flow, topological classification.
@article{MZM_2022_112_3_a10,
     author = {O. V. Pochinka and D. D. Shubin},
     title = {Nonsingular {Morse--Smale} {Flows} with {Three} {Periodic} {Orbits} on {Orientable} $3${-Manifolds}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {426--443},
     publisher = {mathdoc},
     volume = {112},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a10/}
}
TY  - JOUR
AU  - O. V. Pochinka
AU  - D. D. Shubin
TI  - Nonsingular Morse--Smale Flows with Three Periodic Orbits on Orientable $3$-Manifolds
JO  - Matematičeskie zametki
PY  - 2022
SP  - 426
EP  - 443
VL  - 112
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a10/
LA  - ru
ID  - MZM_2022_112_3_a10
ER  - 
%0 Journal Article
%A O. V. Pochinka
%A D. D. Shubin
%T Nonsingular Morse--Smale Flows with Three Periodic Orbits on Orientable $3$-Manifolds
%J Matematičeskie zametki
%D 2022
%P 426-443
%V 112
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a10/
%G ru
%F MZM_2022_112_3_a10
O. V. Pochinka; D. D. Shubin. Nonsingular Morse--Smale Flows with Three Periodic Orbits on Orientable $3$-Manifolds. Matematičeskie zametki, Tome 112 (2022) no. 3, pp. 426-443. http://geodesic.mathdoc.fr/item/MZM_2022_112_3_a10/

[1] J. Franks, “Nonsingular smale flows on $s^3$”, Topology, 24:3 (1985), 265–282 | DOI | MR

[2] Ya. L. Umanskii, “Neobkhodimye i dostatochnye usloviya topologicheskoi ekvivalentnosti trekhmernykh dinamicheskikh sistem Morsa–Smeila s konechnym chislom osobykh traektorii”, Matem. sb., 181:2 (1990), 212–239 | MR | Zbl

[3] A. O. Prishlyak, “Polnyi topologicheskii invariant potokov Morsa–Smeila i razlozhenie na ruchki trekhmernykh mnogoobrazii”, Fundament. i prikl. matem., 11:4 (2005), 185–196 | MR | Zbl

[4] Yu. Bin, “Behavior 0 nonsingular morse-smale flows on $s^3$”, Discrete Contin. Dyn. Syst., 36:1 (2016), 509 | MR

[5] O. V. Pochinka, D. D. Shubin, “Non-singular morse-smale flows on $n$-manifolds with attractor-repeller dynamics”, Nonlinearity, 35:3 (2022), 1485 | DOI | MR

[6] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 | DOI | MR

[7] B. Campos, A. Cordero, J. Martínez Alfaro, P. Vindel, “Nms flows on three-dimensional manifolds with one saddle periodic orbit”, Acta Math. Sin. (Engl. Ser.), 20:1 (2004), 47–56 | DOI | MR

[8] D. D. Shubin, “Topologiya nesuschikh mnogoobrazii nesingulyarnykh potokov s tremya neskruchennymi orbitami”, Izvestiya vuzov. PND, 29:6 (2021), 863–868 | DOI

[9] D. Rolfsen, Knots and Links, Publish or Perish, Houston, TX, 1990 | MR

[10] M. C. Irwin, “A classification of elementary cycles”, Topology, 9:1 (1970), 35–47 | DOI | MR