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In this paper, we will consider (germs of) holomorphic mappings of the form (f(z), λ 1 w 1 (1 + g 1 (z)),...,λ n w n (1 + g n (z))), defined in a neighborhood of the origin in C n+1 . Most of our interest is in those mappings where f(z) = z + a m z m +... is a germ tangent to the identity and g i (0) = 0 for i = 1,..., n, and i i 6 C possess no resonances, for these are the so-called Poincare-Dulac normal forms of the mappings (z + 0(2), λ 1 ω + O(2) λ n ω + 0(2)). We construct formal normal forms for these mappings and discuss a condition which tests for the convergence or divergence of the conjugating maps, giving specific examples.

Further reductions of Poincaré-Dulac normal forms in ${\mathbf{C}}^{n+1}$