# Arbeiten zur deskriptiven Mengenlehre (Mathematische by A.A. Ljapunow, E.A. Stschegolkow and W.J. Arsenin

By A.A. Ljapunow, E.A. Stschegolkow and W.J. Arsenin

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Repeating this process inductively, we can construct an infinite sequence of polynomial maps H1 , H2 , . . , Hm , . . and the formal fields F1 = F , F2 , . . , Fm , . . such that Fm = Ax + (terms of order m and more), while Hm conjugates Fm with Fm+1 . Thus the composition H (m) = Hm ◦ · · · ◦ H1 conjugates the initial field F1 with the field Fm+1 without nonlinear terms up to order m. It remains to notice that by construction of Hm+1 the composition H (m+1) = Hm+1 ◦ H (m) has the same terms of order m as H (m) itself.

The Poincar´e–Dulac paradigm does the rest of the proof. This general statement immediately implies a number of corollaries. For example, if A is diagonal matrix with the spectrum {λ1 , . . , λn }, then A∗ is ¯1, . . , λ ¯ n }. As was already also diagonal with the conjugate eigenvalues {λ ¯ α − noted, restriction of adA∗ on Dm is diagonal with the eigenvalues λ, ¯ k = λ, α − λk . 12 yields the usual Poincar´e– Dulac form. 13. If A = ( −1 0 2 plane R with the coordinates (x, y), then ker adA∗ = ker adA and the entire formal normal form, including the linear part, commutes with the rotation ∂ ∂ vector field A = x ∂y − y ∂x .

Can be removed from the expansion by applying a suitable formal conjugacy. 14) with G (x) = G(x) + Rm (x) + · · · , where Rm is a homogeneous vector field of order m, if and only if G(x) + Pm (G(x)) = G(x + Pm (x)) + Rm (x + Pm (x)), which after collection of terms of order m yields the identity P (M x) = M P (x) + R(x), P = Pm , R = R m . 2). 16) can be studied by the methods similar to the operator adA . If M is a diagonal matrix with the diagonal entries µ1 , . . , µn , then all monomials Fkα of the standard basis in Dm are eigenvectors for SM with the eigenvalues µj − µα = µj − µα1 1 · · · µαnn .

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