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 Mat. Sb. (N.S.), 1983, Volume 121(163), Number 1(5), Pages 111–126 (Mi msb2156)

Remarks on the orbital analytic classification of germs of vector fields

P. M. Elizarov, Yu. S. Ilyashenko

Abstract: Associated to the germ of a holomorphic vector field on $\mathbf C^2$ whose linear part belongs to a Siegel domain, is the germ of a conformal map $(\mathbf C,0)\to(\mathbf C,0)$; the latter is the monodromy transformation induced by a circuit around the singular point on a separatrix.
It is proved that the monodromy transformations are moduli for the orbital analytic classification of germs of vector fields at a singular point: two vector field germs with the same linear part of Siegel type are orbitally analytically equivalent if and only if for each of the germs one can choose a local separatrix such that these separatrices are tangent at zero and such that the monodromy maps corresponding to them are analytically equivalent.
Moduli for the orbital analytic classification of vector field germs in higher-dimensional spaces are also constructed, and a new proof of the theorem about the topological classification of vector fields with saddle resonant singular points is given.
Bibliography: 24 titles.

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English version:
Mathematics of the USSR-Sbornik, 1984, 49:1, 111–124

Bibliographic databases:

UDC: 517.9+517.5
MSC: Primary 34A20, 34A25; Secondary 34C05

Citation: P. M. Elizarov, Yu. S. Ilyashenko, “Remarks on the orbital analytic classification of germs of vector fields”, Mat. Sb. (N.S.), 121(163):1(5) (1983), 111–126; Math. USSR-Sb., 49:1 (1984), 111–124

Citation in format AMSBIB
\Bibitem{EliIly83} \by P.~M.~Elizarov, Yu.~S.~Ilyashenko \paper Remarks on the orbital analytic classification of germs of vector fields \jour Mat. Sb. (N.S.) \yr 1983 \vol 121(163) \issue 1(5) \pages 111--126 \mathnet{http://mi.mathnet.ru/msb2156} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=699741} \zmath{https://zbmath.org/?q=an:0541.32003} \transl \jour Math. USSR-Sb. \yr 1984 \vol 49 \issue 1 \pages 111--124 \crossref{https://doi.org/10.1070/SM1984v049n01ABEH002700} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Martinet J., Ramis J., “Analytical Classification of 1st-Order Resonant Non-Linear Differential-Equations”, Ann. Sci. Ec. Norm. Super., 16:4 (1983), 571–621
2. P. M. Elizarov, “Orbital analytic nonequivalence of saddle resonance vector fields in $(\mathbf C^2,0)$”, Math. USSR-Sb., 51:2 (1985), 533–547
3. Yu. S. Ilyashenko, “Dulac's memoir “On limit cycles” and related problems of the local theory of differential equations”, Russian Math. Surveys, 40:6 (1985), 1–49
4. Dixon P., Esterle J., “Michael Problem and the Poincaré-Fatou-Bieberbach Phenomenon”, Bull. Amer. Math. Soc., 15:2 (1986), 127–187
5. S. I. Trifonov, “Divergence of Dulac's rows”, Math. USSR-Sb., 69:1 (1991), 37–56
6. M. Ya. Zhitomirskii, “Degeneracies of differential 1-forms and Pfaffian structures”, Russian Math. Surveys, 46:5 (1991), 53–90
7. Gong X., “Conformal Maps, Monodromy Transformations, and Non-Reversible Hamiltonian Systems”, Math. Res. Lett., 7:4 (2000), 471–476
8. Helena Reis, “Equivalence and semi-completude of foliations”, Nonlinear Analysis: Theory, Methods & Applications, 64:8 (2006), 1654
9. Rosales-Gonzalez E., “On Rigidity of Germs of Holomorphic Dicritic Foliations and Formal Normal Forms.”, Singularities in Geometry and Topology, 2005, eds. Brasselet J., Damon J., Trang L., Oka M., World Scientific Publ Co Pte Ltd, 2007, 705–722
10. Camara L., Scardua B., “On the Integrability of Holomorphic Vector Fields”, Discret. Contin. Dyn. Syst., 25:2 (2009), 481–493
11. Ortiz-Bobadilla L., Rosales-Gonzalez E., Voronin S.M., “Analytic Classification of Foliations Induced By Germs of Holomorphic Vector Fields in (C-N,0) With Non-Isolated Singularities”, J. Dyn. Control Syst., 25:3 (2019), 491–516
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