Geodesic
The symmetry groups of bifurcations of integrable Hamiltonian systems
Sbornik. Mathematics, Tome 205 (2014) no. 11, pp. 1668-1682 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two-dimensional atoms are investigated; these are used to code bifurcations of the Liouville foliations of nondegenerate integrable Hamiltonian systems. To be precise, the symmetry groups of atoms with complexity at most 3 are under study. Atoms with symmetry group $\mathbb Z_p\oplus\mathbb Z_q$ are considered. It is proved that $\mathbb Z_p\oplus\mathbb Z_q$ is the symmetry group of a toric atom. The symmetry groups of all nonorientable atoms with complexity at most 3 are calculated. The concept of a geodesic atom is introduced. Bibliography: 9 titles.
Keywords: integrable systems, atoms, finite groups. 
@article{SM_2014_205_11_a5,
     author = {E. I. Orlova},
     title = {The symmetry groups of bifurcations of integrable {Hamiltonian} systems},
     journal = {Sbornik. Mathematics},
     pages = {1668--1682},
     year = {2014},
     volume = {205},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2014_205_11_a5/}
}
TY  - JOUR
AU  - E. I. Orlova
TI  - The symmetry groups of bifurcations of integrable Hamiltonian systems
JO  - Sbornik. Mathematics
PY  - 2014
SP  - 1668
EP  - 1682
VL  - 205
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2014_205_11_a5/
LA  - en
ID  - SM_2014_205_11_a5
ER  - 
%0 Journal Article
%A E. I. Orlova
%T The symmetry groups of bifurcations of integrable Hamiltonian systems
%J Sbornik. Mathematics
%D 2014
%P 1668-1682
%V 205
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2014_205_11_a5/
%G en
%F SM_2014_205_11_a5
E. I. Orlova. The symmetry groups of bifurcations of integrable Hamiltonian systems. Sbornik. Mathematics, Tome 205 (2014) no. 11, pp. 1668-1682. http://geodesic.mathdoc.fr/item/SM_2014_205_11_a5/

[1] A. T. Fomenko, “Teoriya Morsa integriruemykh gamiltonovykh sistem”, Dokl. AN SSSR, 287:5 (1986), 1071–1075 | MR | Zbl

[2] A. T. Fomenko, “Simplekticheskaya topologiya vpolne integriruemykh gamiltonovykh sistem”, UMN, 44:1(265) (1989), 145–173 | MR | Zbl

[3] Nguyen Tien Zung, “Decomposition of nondegenerate singularities of integrable Hamiltonian systems”, Lett. Math. Phys., 33:3 (1995), 187–193 | DOI | MR | Zbl

[4] A. T. Fomenko, A. Konyaev, “Algebra and geometry through Hamiltonian systems”, Continuous and distributed systems. Theory and applications, Solid Mech. Appl., 211, eds. M. Z. Zgurovsky, V. A. Sadovnichiy, Springer, Cham, 2014, 3–21 | DOI

[5] A. T. Fomenko, A. Yu. Konyaev, “New approach to symmetries and singularities in integrable Hamiltonian systems”, Topology Appl., 159:7 (2012), 1964–1975 | DOI | MR | Zbl

[6] A. A. Oshemkov, “Funktsii Morsa na dvumernykh poverkhnostyakh. Kodirovanie Osobennostei”, Novye rezultaty v teorii topologicheskoi klassifikatsii integriruemykh sistem, Sbornik statei, Tr. MIAN, 205, Nauka, M., 1994, 131–140 ; A. A. Oshemkov, “Morse functions on two-dimensional surfaces. Encoding of singularities”, Proc. Steklov Inst. Math., 205 (1995), 119–127 | MR | Zbl

[7] E. A. Kudryavtseva, A. T. Fomenko, “Lyubaya konechnaya gruppa yavlyaetsya gruppoi simmetrii nekotoroi karty (“atoma”-bifurkatsii)”, Vestn. Mosk. un-ta. Ser. 1 Matem. Mekh., 2013, no. 3, 21–29 ; E. A. Kudryavtseva, A. T. Fomenko, “Any finite group is the group of symmetries of some map (“atom”-bifurcations)”, Moscow Univ. Math. Bull., 68, no. 3, 148–155 | MR

[8] A. V. Bolsinov, A. T. Fomenko, Integriruemye gamiltonovy sistemy. Geometriya, topologiya, klassifikatsiya, v. 1, 2, Izd. dom “Udmurtskii universitet”, Izhevsk, 1999, 444 s., 447 pp. ; A. V. Bolsinov, A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Chapman Hall/CRC, Boca Raton, FL, 2004, xvi+730 pp. | MR | Zbl | DOI | MR | Zbl

[9] E. A. Kudryavtseva, A. T. Fomenko, “Gruppy simmetrii pravilnykh funktsii Morsa na poverkhnostyakh”, Dokl. RAN, 446:6 (2012), 615–617 | MR | Zbl