  Let $f:(\mathbb{C}^n,0)\mapsto(\mathbb{C}^n,0)$ be a germ of an
$n$-dimensional holomorphic map. Assume that the origin is an isolated fixed
point of each iterate of $f$. Then $\{\mathcal{N}_q(f)\}_{q=1}^{\infty}$, the
sequence of the maximal number of periodic orbits of period $q$ that can be
born from the fixed point zero under a small perturbation of $f$, is well
defined. According to Shub-Sullivan, Chow-Mallet-Paret-Yorke and G. Y. Zhang,
the linear part of the holomorphic germ $f$ determines some natural
restrictions on the sequence(cf. Theorem 1.1). Later, I. Gorbovickis proves
that when the linear part of $f$ is contained in a certain large class of
diagonal matrices, it has no other restrictions on the sequence only when the
dimension $n\leq2$ (cf. Theorem 1.3). In this paper for the general case we
obtain a sufficient and necessary condition that the linear part of $f$ has no
other restrictions on the sequence $\{\mathcal{N}_q(f)\}_{q=1}^{\infty}$,
except the ones given by Theorem 1.1.

Dynamical Systems