Normal Forms of Families of Maps in the Poincaré Domain

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.