Geodesic
Normal Forms of Families of Maps in the Poincar\'e Domain
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear analytic differential equations, Tome 254 (2006), pp. 101-110.

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An analog of Brushlinskaya's theorem about normal forms of deformations of vector fields in the Poincaré domain is proved; namely, it is proved that for each analytic map whose linear part at a fixed point belongs to the Poincaré domain and has different eigenvalues, the analytic normal form of a deformation of this map is polynomial and contains (in addition to the linear part) only monomials that are resonant for the unperturbed map. A global (with respect to the parameter) version of this theorem is also proved. 
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I. S. Gorbovitskii. Normal Forms of Families of Maps in the Poincar\'e Domain. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear analytic differential equations, Tome 254 (2006), pp. 101-110. http://geodesic.mathdoc.fr/item/TM_2006_254_a2/

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[2] Ilyashenko Yu.S., Pyartli A.S., “Materializatsiya rezonansov Puankare i raskhodimost normalizuyuschikh ryadov”, Tr. sem. im. I.G. Petrovskogo, 7 (1981), 3–49 | MR | Zbl

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