Escaping points of entire functions of small growth
Escaping points of entire functions of small growth
复制标题
小增长全功能的逃生点
DOI：
10.1007/s00209-008-0339-0
复制
发表时间：
2008
影响因子：
0.8
通讯作者：
G. Stallard
中科院分区：
文献类型：
--
作者：
P. Rippon;G. Stallard
设 f 为超越整函数，并设 I(f) 表示迭代下逃逸至无穷大的点集。我们给出的条件确保对于某些函数，I(f) 是连通的。特别是，我们证明，如果 f 的阶数为零且增长足够小，或者阶数小于 1/2 且有规律的增长，则 I(f) 是连通的。这表明，对于这些函数，Eremenko 的 I(f) 没有有界分量的猜想是正确的。我们还给出了与 I(f) 相关的新准则，该准则足以确保 f 不具有无界 Fatou 分量。
Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.