Obstacles to periodic orbits hidden at fixed point of holomorphic maps

Abstract

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 $n\le 2$ (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.

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