Fatou sets in complex dynamics on projective spaces
Fatou sets in complex dynamics on projective spaces
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Fatou 在射影空间上设置了复杂的动力学
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发表时间：
1994
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通讯作者：
T. Ueda
中科院分区：
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作者：
T. Ueda
复射影空间上由全纯映射定义的复动力系统的理论，推广了一元有理函数的迭代理论，已有几位作者[FS 1]，[FS 2]，[HP]，[U3]进行了研究。关于Julia集和Fatou集，在一定程度上进行了与单变量情况的类比。也有许多问题，我们遇到的第一次在高维的情况下。本文证明了复射影空间上次数大于1的复动力系统的Fatou集的两个基本结果：Fatou集是伪凸的，因此是Stein（定理2.3）; Fatou集是Carath'eodory双曲的，因此是小林双曲的（定理2.5和2.6）。利用后一个定理，我们可以得到一些类似于一维情形的结果。证明了吸引周期点的直接盆中含有临界点。在附加条件下，对二维抛物周期点也证明了同样的结果。为了证明上述基本定理，我们采用Hubbard和Papadopol [HP]提出的方法。也就是说，我们考虑，对于一个全纯映射，
The theory of complex dynamical systems defined by holomorphic maps on complex projective spaces, which generalizes the iteration theory of one variable rational functions, has been studied by several authors [FS1], [FS2], [HP], [U3]. Concerning Julia sets and Fatou sets, analogies to the one variable case are pursued to some extent. There are also many problems which we encounter first in higher dimensional case. In this paper, we prove two fundamental results on Fatou sets for complex dynamical systems of degree greater than 1 on complex projective spaces: Fatou sets are pseudoconvex, hence Stein (Theorem 2.3); Fatou sets are Carath’eodory hyperbolic, hence Kobayashi hyperbolic (Theorems 2.5 and 2.6). With the latter theorem, we can derive some results analogous to the one dimensional case. It is proved that the immediate basin of an attractive periodic point contains critical points. The same result is proved for a parabolic periodic point in two dimensional case under an additional condition. TO prove the above fundamental theorems we employ the method originated by Hubbard and Papadopol [HP]. Namely we consider, for a holomorphic map