与排斥周期点相关的正规定则

Normality Concerning Repelling Periodic Points

本文研究了与排斥周期点相关的正规族问题,主要得到以下结论:(1)设F是区域D内的全纯函数族,k≥2是正整数.如果对任意的f∈F,f(z)-z的零点重级≥3,且fk在区域D内至多有3k-1个排斥不动点,则F在区域D内正规.且定理中的条件都是必要的;(2)设F是区域D内的亚纯函数族,k≥3是正整数.如果对任意的f∈F,f(z)-z的零点与极点重级均≥3,且f^(k)在区域D内至多有2k-1个排斥不动点,则F在区域D内正规.

We studied the normality concerning repelling periodic poits,and obtained two results as follow:(1)Let F be a family of holomorphic functions in a domain D,and let k≥2 be a positive integer.If,for each f E F,all zeros of f(z)-z have multiplicity at least 3,and its iteration fk has at most 3k-1 distinct repelling fixed points in D,then F is normal in D.There are examples show that all conditions are necessary in this result;(2)Let F be a family of meromorphic functions in a domain D,and let k≥3 be a positive integer.If,for each f ∈ F,all zeros and poles of f(z)-z have multiplicity at least 3,and its iteration f^(k) has at most 2k-1 distinct repelling fixed points in D,then F is normal in D.