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By Ilyashenko Yu.,Yakovenko S.

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24), formal maps act on formal vector fields. In this section we investigate the formal normal forms to which a formal vector field can be brought by a suitable formal isomorphism. 1. 24) holds on the level of formal series. sio n The fields are formally equivalent if and only if the corresponding derivations F, F of the algebra C[[x]] are conjugated by a suitable isomorphism H ∈ Diff[[Cn , 0]]: H ◦ F = F ◦ H. Dr af tv er Obviously, two holomorphically equivalent (holomorphic) germs of vector fields are formally equivalent.

14, for two Lie algebras gl(n, R) ⊂ gl(n, C) and the respective Lie groups GL(n, R) ⊂ GL(n, C), the exponent is surjective on the ambient (bigger) group but not on the subgroup. 17. 10) is a derivation, log H ∈ Der(F). This follows from the fact that log and exp are mutually inverse on sufficiently small neighborhoods of E and 0 respectively. However, the size of this neighborhood depends on F. 6. Embedding in the formal flow. 15, one can prove the following general result obtained by F. Takens in 1974, see [Tak01].

Then H is invertible in Diff[[C , 0]]. ty If A = E is the identity matrix and H = (h1 , . . , hn ), hi (x) = xi + vi (x) mod mk+1 , where vi are homogeneous polynomials of degree k 2, then the formal inverse map H −1 = (h1 , . . , hn ) has the components hi (x) = xi − vi (x) mod mk+1 . of Clearly, it the first assertion of the Theorem follows from the second assertion applied to the formal map A−1 H. e The following definition will play important role throughout this section. 7. A finite-dimensional linear operator A : Cn → Cn is unipotent, if A − E is nilpotent, (A − E)n = 0.

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