In this article, two results concerning the periodic points and normality of meromorphic functions are obtained: (i) the exact lower bound for the numbers of periodic points of rational functions with multiple fixed points and zeros is proven by letting R(z) be a non-polynomial rational function, and if all zeros and poles of R(z) − z are multiple, then R^k(z) has at least k + 1 fixed points in the complex plane for each integer k ≥ 2; (ii) a complete solution to the problem of normality of meromorphic functions with periodic points is given by letting ℱ be a family of meromorphic functions in a domain D, and letting k ≥ 2 be a positive integer. If, for each f ∈ ℱ, all zeros and poles of f(z) − z are multiple, and its iteration f^k has at most k distinct fixed points in D, then ℱ is normal in D. Examples show that all of the conditions are the best possible.

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关于多重亚纯函数的周期点和正态性

在本文中，获得了关于亚纯函数的周期点和正态性的两个结果：（i）通过使R（z）为a，证明具有多个固定点和零的有理函数的周期点数的确切下界。非多项式有理函数，并且如果R（z）-z的所有零点和极点都为多个，则对于每个整数k，R ^k（z）在复平面上至少具有k + 1个固定点≥2; （ii）向与周期点亚纯函数的常态的问题的完整解决方案通过让ℱ是亚纯函数域中的给定家族d，并让ķ ≥2是一个正整数。如果，对于每个˚F＆Element; ℱ，全零和的极˚F（ż） - Ž是多重的，和它的迭代˚F ^ķ具有至多ķ在不同的固定点d，再ℱ处于正常d。例子表明，所有条件都是最好的。