Remarks on the orbital analytic classification of germs of vector fields

P. M. Elizarov ; Yu. S. Ilyashenko

Sbornik. Mathematics, Tome 49 (1984) no. 1, pp. 111-124

Résumé

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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