论文标题
隐藏在全体形态图的固定轨道的障碍物
Obstacles to periodic orbits hidden at fixed point of holomorphic maps
论文作者
论文摘要
令$ f：（\ mathbb {c}^n，0）\ mapsto（\ mathbb {c}^n，0）$是$ n $ dimensional-demensional Holomorphic Map的胚芽。假设原点是$ f $的每个迭代的孤立固定点。然后，$ \ {\ Mathcal {n} _q（f）\} _ {q = 1}^{\ infty} $，在$ f $的小范围内，可以从固定点零的零点零诞生的周期$ q $的最大周期性轨道数量的顺序。根据Shub-Sullivan，Chow-Mallet-Paret-Yorke和G. Y. Zhang的说法，Holomorthic Germ $ F $的线性部分决定了对序列的一些自然限制（参见定理1.1）。后来，I。Gorbovickis证明，当$ f $的线性部分包含在某个大型的对角线矩阵中时，仅当尺寸$ n \ leq2 $（参见定理1.3）时，它没有其他限制。在本文中，对于一般情况，我们获得了足够且必要的条件，即$ f $的线性部分对序列$ \ {\ MATHCAL {n} _Q（f）\} _ {q = 1}^{\ infty} $没有其他限制，除了由Theorem 1.1给出。
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.
