Geodesic
Minimizing the number of heteroclinic curves of a 3-diffeomorphism with fixed points with pairwise different Morse
Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 311-317 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider Morse–Smale $3$-diffeomorphisms whose nonwandering set consists of exactly four fixed points with pairwise distinct Morse indices. The question of which closed $3$-manifolds admit such diffeomorphisms remains open. The set of these manifolds is known to contain all lens spaces. Moreover, on all manifolds except $\mathbb{S}^2\times\mathbb{S}^1$, such diffeomorphisms have heteroclinic curves. We prove that the number of heteroclinic diffeomorphism curves on a given manifold can be minimized by reducing to finitely many noncompact heteroclinic curves that are orientable intersections of invariant saddle manifolds. This result paves the way to an exhaustive description of closed $3$-manifolds that the diffeomorphisms in question.
Keywords: heteroclinic curves, orientable intersection, Morse–Smale diffeomorphisms. 
@article{TMF_2023_215_2_a10,
     author = {O. V. Pochinka and E. A. Talanova},
     title = {Minimizing the~number of heteroclinic curves of a~3-diffeomorphism with fixed points with pairwise different {Morse}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {311--317},
     year = {2023},
     volume = {215},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a10/}
}
TY  - JOUR
AU  - O. V. Pochinka
AU  - E. A. Talanova
TI  - Minimizing the number of heteroclinic curves of a 3-diffeomorphism with fixed points with pairwise different Morse
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2023
SP  - 311
EP  - 317
VL  - 215
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a10/
LA  - ru
ID  - TMF_2023_215_2_a10
ER  - 
%0 Journal Article
%A O. V. Pochinka
%A E. A. Talanova
%T Minimizing the number of heteroclinic curves of a 3-diffeomorphism with fixed points with pairwise different Morse
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2023
%P 311-317
%V 215
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a10/
%G ru
%F TMF_2023_215_2_a10
O. V. Pochinka; E. A. Talanova. Minimizing the number of heteroclinic curves of a 3-diffeomorphism with fixed points with pairwise different Morse. Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 2, pp. 311-317. http://geodesic.mathdoc.fr/item/TMF_2023_215_2_a10/

[1] O. Pochinka, E. Talanova, D. Shubin, Knot is a complete invariant of a Morse–Smale 3-diffeomorphism with four fixed points, arXiv: 2209.04815

[2] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “Novye sootnosheniya dlya sistem Morsa–Smeila s trivialno vlozhennymi odnomernymi separatrisami”, Matem. sb., 194:7 (2003), 25–56 | DOI | DOI | MR | Zbl

[3] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, “O diffeomorfizmakh Morsa–Smeila s chetyrmya periodicheskimi tochkami na zamknutykh orientiruemykh mnogoobraziyakh”, Matem. zametki, 74:3 (2003), 369–386 | DOI | DOI | MR | Zbl

[4] V. Z. Grines, T. V. Medvedev, O. V. Pochinka, Dynamical Systems on 2-and 3-Manifolds, Developments in Mathematics, 46, Springer, Cham, 2016 | DOI | MR

[5] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, O. V. Pochinka, “Globalnye attraktor i repeller diffeomorfizmov Morsa–Smeila”, Differentsialnye uravneniya i topologiya. II, Trudy MIAN, 271, MAIK “Nauka/Interperiodika”, M., 2010, 111–133 | DOI | MR

[6] V. I. Shmukler, O. V. Pochinka, “Bifurkatsii, menyayuschie tip geteroklinicheskikh krivykh $3$-diffeomorfizma Morsa–Smeila”, TVIM, 2021, no. 1, 101–114

[7] D. Rolfsen, “Knots and links”, Mathematics Lecture Series, 7, Publish or Perish Press, Berkeley, CA, 1976 | MR | Zbl

[8] A. Hatcher, Notes on Basic 3-Manifold Topology, 2007 https://pi.math.cornell.edu/<nobr>$\sim$</nobr> hatcher/3M/3M.pdf