Geodesic
On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set
Matematičeskie trudy, Tome 21 (2018) no. 2, pp. 163-180 
V. Z. Grines; E. Ya. Gurevich; E. V. Zhuzhoma; V. S. Medvedev. On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set. Matematičeskie trudy, Tome 21 (2018) no. 2, pp. 163-180. http://geodesic.mathdoc.fr/item/MT_2018_21_2_a7/
@article{MT_2018_21_2_a7,
     author = {V. Z. Grines and E. Ya. Gurevich and E. V. Zhuzhoma and V. S. Medvedev},
     title = {On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set},
     journal = {Matemati\v{c}eskie trudy},
     pages = {163--180},
     year = {2018},
     volume = {21},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2018_21_2_a7/}
}
TY  - JOUR
AU  - V. Z. Grines
AU  - E. Ya. Gurevich
AU  - E. V. Zhuzhoma
AU  - V. S. Medvedev
TI  - On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set
JO  - Matematičeskie trudy
PY  - 2018
SP  - 163
EP  - 180
VL  - 21
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/MT_2018_21_2_a7/
LA  - ru
ID  - MT_2018_21_2_a7
ER  - 
%0 Journal Article
%A V. Z. Grines
%A E. Ya. Gurevich
%A E. V. Zhuzhoma
%A V. S. Medvedev
%T On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set
%J Matematičeskie trudy
%D 2018
%P 163-180
%V 21
%N 2
%U http://geodesic.mathdoc.fr/item/MT_2018_21_2_a7/
%G ru
%F MT_2018_21_2_a7

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

We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension $4$ and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly $\mu$ node and $\nu$ saddle equilibrium points of indices equal to either $1$ or $n-1$. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius $g=\frac{\nu-\mu+2}2$. We also suggest an algorithm for constructing gradient-like flows on closed manifolds of dimension $3$ and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices.

[1] Andronov A. A., Pontryagin L. S., “Grubye sistemy”, DAN SSSR, 14:5 (1937), 247–251 | MR

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

[3] Grines V. Z., Zhuzhoma E. V., Medvedev V. S., “O strukture nesuschego mnogoobraziya dlya sistem Morsa — Smeila bez geteroklinicheskikh peresechenii”, Tr. MIRAN, 297, 2017, 201–210 | Zbl

[4] Keldysh L. V., “Topologicheskie vlozheniya v evklidovo prostranstvo”, Tr. MIAN SSSR, 81, 1966, 3–184

[5] Milnor Dzh., Uolles A., Differentsialnaya topologiya. Nachalnyi kurs, Mir, M., 1972 | MR

[6] Khirsh M., Differentsialnaya topologiya, 1979, M.

[7] Shilnikov L. P., Shilnikov A. L., Turaev D. V., Chua L., Metody kachestvennoi teorii v nelineinoi dinamike, v. 1, In-t kompyut. issled., M.–Izhevsk, 2004

[8] Bonatti C., Grines V., Medvedev V., Pecou E., “Three-Dimensional manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves”, Topology Appl., 117:3 (2002), 335–344 | DOI | MR | Zbl

[9] Daverman R. J., Venema G. A., Embeddings in Manifolds, Graduate Studies in Mathematics, 106, Amer. Math. Soc., Providence, RI, 2009 | DOI | MR | Zbl

[10] Grines V. Z., Gurevich E. A., Pochinka O. V., “Topological classification of Morse–Smale diffeomorphisms without heteroclinic intersection”, J. Math. Sci., 208:1 (2015), 81–90 | DOI | MR | Zbl

[11] Kervaire M. A., Milnor J. W., “Groups of homotopy spheres. I”, Ann. Math., 2:77 (1963), 504–537 | DOI | MR | Zbl

[12] Lee J., Introduction to Smooth Manifolds, Graduate Texts in Mathematics, 218, Springer-Verlag, New York, 2012 | DOI | MR

[13] Matsumoto Y., An Introduction to Morse Theory, Translations of Mathematical Monographs, 208, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[14] Smale S., “Morse inequalities for a dynamical system”, Bull. Amer. Math. Soc., 66 (1960), 43–49 | DOI | MR | Zbl

[15] Smale S., “On gradient dynamical systems”, Ann. of Math. (2), 74:1 (1961), 199–206 | DOI | MR | Zbl

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

[17] Zhuzhoma E. V., Medvedev V. S., “Morse–Smale systems with few non-wandering points”, Topology Appl., 160:3 (2013), 498–507 | DOI | MR | Zbl