Geodesic
Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on~$\mathbb C^2$
Informatics and Automation, Nonlinear analytic differential equations, Tome 254 (2006), pp. 111-129.

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

This paper is a survey of the present state of the problems related to the generic properties of foliations defined on $\mathbb C^2$ by algebraic differential equations. We prove that the properties of density, absolute rigidity, and existence of a countable set of complex limit cycles are inherent in all equations except possibly for the union of some real algebraic set and real analytic set of codimension at least two in the space of coefficients. 
@article{TRSPY_2006_254_a3,
     author = {A. A. Shcherbakov},
     title = {Dynamics of {Local} {Groups} of {Conformal} {Mappings} and {Generic} {Properties} of {Differential} {Equations} on~$\mathbb C^2$},
     journal = {Informatics and Automation},
     pages = {111--129},
     publisher = {mathdoc},
     volume = {254},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a3/}
}
TY  - JOUR
AU  - A. A. Shcherbakov
TI  - Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on~$\mathbb C^2$
JO  - Informatics and Automation
PY  - 2006
SP  - 111
EP  - 129
VL  - 254
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a3/
LA  - ru
ID  - TRSPY_2006_254_a3
ER  - 
%0 Journal Article
%A A. A. Shcherbakov
%T Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on~$\mathbb C^2$
%J Informatics and Automation
%D 2006
%P 111-129
%V 254
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a3/
%G ru
%F TRSPY_2006_254_a3
A. A. Shcherbakov. Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on~$\mathbb C^2$. Informatics and Automation, Nonlinear analytic differential equations, Tome 254 (2006), pp. 111-129. http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a3/

[1] Ilyashenko Yu.S., “Primer uravnenii $dw/dz=P_n(z,w)/Q_n(z,w)$, imeyuschikh schetnoe chislo predelnykh tsiklov i skol ugodno bolshoi zhanr po Petrovskomu–Landisu”, Mat. sb., 80:3 (1969), 388–404 | Zbl

[2] Ilyashenko Yu.S., “Topologiya fazovykh portretov analiticheskikh differentsialnykh uravnenii na kompleksnoi proektivnoi ploskosti”, Tr. sem. im. I.G. Petrovskogo, 4 (1978), 83–136 | MR | Zbl

[3] Khudai-Verenov M.G., “Ob odnom svoistve reshenii odnogo differentsialnogo uravneniya”, Mat. sb., 56:3 (1962), 301–308 | MR | Zbl

[4] Myuller B., “O plotnosti reshenii odnogo uravneniya v $\mathbb C\mathrm P^2$”, Mat. sb., 98:3 (1975), 363–377 | MR | Zbl

[5] Pyartli A.S., “Ratsionalnye differentsialnye uravneniya s kommutativnoi gruppoi monodromii na beskonechnosti”, Tr. Mosk. mat. o-va, 61 (2000), 75–106 | MR | Zbl

[6] Scherbakov A.A., “O plotnosti orbity psevdogruppy konformnykh otobrazhenii i obobschenii teoremy Khudai-Verenova”, Vestn. Mosk. un-ta. Matematika. Mekhanika, 1982, no. 4, 10–15 | Zbl

[7] Scherbakov A.A., “Topologicheskaya klassifikatsiya rostkov konformnykh otobrazhenii s tozhdestvennoi lineinoi chastyu”, Vestn. Mosk. un-ta. Matematika. Mekhanika, 1982, no. 3, 52–57 | Zbl

[8] Scherbakov A.A., “Topologicheskaya i analiticheskaya sopryazhennost dlya nekommutativnykh grupp rostkov konformnykh otobrazhenii”, Tr. sem. im. I.G. Petrovskogo, 10 (1984), 170–196 | Zbl

[9] Scherbakov A.A., “O kompleksnykh predelnykh tsiklakh uravneniya $dw/dz=P_n/Q_n$”, UMN, 41:1 (1986), 211–212 | MR

[10] Belliart M., Liousse I., Loray F., “Sur l'existence de points fixes attractifs pour les sous-groupes de $\mathrm{Aut}(\mathbb C,0)$”, C. r. Acad. sci. Paris Sér. 1, 324:4 (1997), 443–446 | MR | Zbl

[11] Cerveau D., Moussu R., “Groupes d'automorphismes de $(\mathbb C,0)$ et équations différentielles $ydy +\dots =0$”, Bull. Soc. math. France, 116:4 (1988), 459–488 | MR | Zbl

[12] Camacho C., Sad P., “Topological classification and bifurcations of holomorphic flows with resonances in $\mathbb C^2$”, Invent. math., 67 (1982), 447–472 | DOI | MR | Zbl

[13] Gomez-Mont X., Wirtz B., “On fixed points of conformal pseudogroups”, Bol. Soc. Brasil. Mat., 26:2 (1995), 201–209 | DOI | MR | Zbl

[14] Gomez-Mont X., Ortiz-Bobadilla L., Sistemas dinamicos holomorfos en superficies, Aport. Mat. Notas Invest., 3, Soc. Mat. Mexicana, México, 1989 | MR | Zbl

[15] Elizarov P.M., Ilyashenko Yu.S., Shcherbakov A.A., Voronin S.M., “Finitely generated groups of germs of one-dimensional conformal mappings, and invariants for complex singular points of analytic foliations of the complex plane”, Nonlinear Stokes phenomena, Adv. Sov. Math., 14, Amer. Math. Soc., Providence (RI), 1993, 57–105 | MR | Zbl

[16] Ilyashenko Yu., “Centennial history of Hilbert's 16th problem”, Bull. Amer. Math. Soc., 39:3 (2002), 301–354 | DOI | MR | Zbl

[17] Ilyashenko Yu.S., Pyartli A.S., “The monodromy group at infinity of a generic polynomial vector field on the complex projective plane”, Russ. J. Math. Phys., 2:3 (1994), 275–315 | MR | Zbl

[18] Loray F., Rebelo J., “Minimal, rigid foliations by curves on $\mathbb C\mathrm P^n$”, J. Eur. Math. Soc., 5 (2003), 147–201 | DOI | MR | Zbl

[19] Lins Neto A., Sad P., Azevedo Scárdua B., “On topological rigidity of projective foliations”, Bull. Soc. math. France, 126:3 (1998), 381–406 | MR | Zbl

[20] Nakai I., “Separatrices for non-solvable dynamics on $\mathbb C,0$”, Ann. Inst. Fourier, 44:2 (1994), 569–599 | MR | Zbl

[21] Ortiz-Bobadilla L., “Quadratic vector fields in $\mathbb C\mathrm P^2$ with two suddle–node type singularities at infinity”, J. Dyn. and Control Syst., 1:3 (1995), 295–317 | DOI | MR | Zbl

[22] Shcherbakov A.A., Rosales-Gonzáles E., Ortiz-Bobadilla L., “Countable set of limit cycles for the equation $dw/dz=P_n (z,w)/Q_n (z,w)$”, J. Dyn. and Control Syst., 4:4 (1998), 539–581 | DOI | MR | Zbl