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4. Resonant normal forms: Poincar´ e–Dulac paradigm. , to eliminate some of the Taylor terms, when resonances between eigenvalues occur. -b In this resonant case the operator adA = [A, ·] of commutation with the linear part may be no longer surjective and in general the condition Vm = 0 meaning absence of terms of order m after the transformation, cannot be achieved. 5) Dr af tv er (the sum not necessary should be direct). 7 (Poincar´e–Dulac paradigm). 5), then any formal vector field F (x) = Ax + · · · with the linearization matrix A is formally conjugated to some formal vector field whose all nonlinear terms of degree m belong to the subspace Nm .
Zn ]], defined as = sup |Mf (z)| = |Mf (ρ, . . , ρ)| |z|<ρ +∞. 1) F ρ po s For a formal vector function F = (F1 , . . , Fn ) the majorant norm is = F1 ρ + · · · + Fn ρ. 2) ty The majorant space Bρ is the subspace of formal (vector) functions from C[[x]] having finite majorant ρ-norm. ρ is complete. 8. The space Bρ with the majorant norm · ew ar e Proof. If ρ = 1, this is obvious: B1 is the space of infinite absolutely converging sequences {cα }, and hence is isomorphic to the standard Lebesgue space 1 which is complete.
This allows to define the sum of the series exp tF as a linear operator Ht : C[[x]] → C[[x]] via its finite truncations of all orders. The group property Ht+s = Ht ◦ Hs follows from the formal identity exp(t + s) = exp t · exp s, since tF and sF obviously commute. It remains to 1. , Ht (f g) = Ht f Ht g for any two series f, g ∈ C[[x]]. Fk (f g) = k! q! Fp f Fq g. p+q=k ge nu in e Denote the multiplication operation in C[[x]] by the asterisk (not to be confused with composition of homomorphisms), we have for any f, g ∈ C[[x]] the iterated Leibnitz rule Substituting this identity into the exponential series, we have = k p+q=k tp p p!
