On multi-dimensional Fatou bifurcation

If f:(Cn,0)→(Cn,0) is a germ of a holomorphic map, such that zero is an isolated fixed point for all of its iterates, then by Nm(f) we denote the maximal number of periodic orbits of period m that can be born from the fixed point zero under a small perturbation of f. In [8] G.Y. Zhang (2007) proves that the linear part of a germ determines some natural restrictions on the possible sequences of numbers N1(f),N2(f),… (cf. Theorem 1.4). In this paper we show that when the linear part of f is diagonalizable, and all its eigenvalues are roots of unity of pairwise relatively prime degrees greater than 1, then in the case when n⩽2, there are no other restrictions on these sequences, except the ones given by the theorem of Zhang. We also show that this is not the case for n⩾3. In order to complete the proofs of these results, we establish an efficient way of computing the numbers Nm(f) for a large class of germs. An important tool that we use is the factor maps corresponding to the resonant normal forms.