Geodesic
Normal Forms of Families of Maps in the Poincar\'e Domain
Informatics and Automation, 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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     author = {I. S. Gorbovitskii},
     title = {Normal {Forms} of {Families} of {Maps} in the {Poincar\'e} {Domain}},
     journal = {Informatics and Automation},
     pages = {101--110},
     publisher = {mathdoc},
     volume = {254},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a2/}
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I. S. Gorbovitskii. Normal Forms of Families of Maps in the Poincar\'e Domain. Informatics and Automation, Nonlinear analytic differential equations, Tome 254 (2006), pp. 101-110. http://geodesic.mathdoc.fr/item/TRSPY_2006_254_a2/