Download Lectures on analytic diff. equations (web draft, March 2006) by Ilyashenko Yu., Yakovenko S. PDF

By Ilyashenko Yu., Yakovenko S.

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8, all finite truncations j k N are nilpotent. Define the operator H −1 as the series H−1 = E − N + N2 − N3 ± · · · . 3) 26 1. Normal forms and desingularization This series converges (in fact, stabilizes) after truncation to any finite order because of the above nilpotency, hence by definition converges to an operator on C[[x]] satisfying the identities H ◦ H−1 = H−1 ◦ H = E. It is an homomorphism of algebra(s), since for any a, b ∈ C[[x]] and their images a = Ha, b = Hb which also can be chosen arbitrarily, we have H(ab) = a b and therefore H−1 (a b ) = H−1 H(ab) = ab = (H−1 a )(H−1 b ).

5) 3. Formal flows and embedding 27 is a formal vector field: F(f g) = lim t−1 (Ht (f g) − f g) t→0 = lim t−1 Ht f Ht g + f Ht g − f Ht g − f g t→0 = lim Ht g · t−1 (Ht f − f ) + lim f · t−1 (Ht g − g) t→0 t→0 = g Ff + f Fg. 9. A holomorphic one-parametric subgroup of formal maps {H t } ⊆ Diff[[Cn , 0]] is a formal flow of the formal vector field F , if F= dHt dt ∈ D[[Cn , 0]]. 6), called the generator of the one-parametric subgroup. 19, showing that, conversely, any formal vector field generates an holomorphic one-parametric subgroup of formal maps.

Then the linear space Dm of homogeneous vector fields of degree m can be naturally identified with the tensor product Dm = Hm ⊗C Cn and inherits the standard Hermitian structure for which the monomials ϕα ⊗ek = √1α! Fαk form an orthonormal basis. Given a matrix A ∈ Matn (C), denote by A∗ the adjoint matrix obtained from A by transposition and complex conjugacy: a∗ij = a ¯ji . If A(x) = Ax is n the corresponding linear vector field on C and, respectively, A∗ (x) = A∗ x, then both act as linear (differential) operators A = aij xi ∂x∂ j and A∗ = a ¯ji xi ∂x∂ j on Hm .

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