Normal families: New perspectives
Normal families: New perspectives
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DOI：
10.1090/s0273-0979-98-00755-1
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发表时间：
1998-07
影响因子：
1.3
通讯作者：
L. Zalcman
中科院分区：
文献类型：
--
作者：
L. Zalcman
本文调查了表征平面域上亚纯函数正常族的引理的一些令人惊讶的应用。其中包括 (i) 皮卡德定理、(ii) 哥德堡定理（C 上的亚纯函数，它是一阶代数微分方程的有限阶解）和 (iii) Fatou-Julia 定理（d ≥ 2 有理函数的 Julia 集是排斥周期点的闭包）的简短而有效的证明。我们还讨论了布洛赫原理，并为与该原理相关的海曼的一些问题提供了简单的解决方案。二十多年前，在部分解释布洛赫原理的过程中，我证明了一个描述平面域上全纯和亚纯函数正常族特征的小引理[68]。多年来，引理不断发展，并且在灵巧的手中，被证明具有惊人的多功能性，可应用于函数论和相关领域的各种主题。随着人们对正常家庭重新产生兴趣（很大程度上源于它们在复杂动态中发挥的重要作用），调查单变量理论中一些最引人注目的应用似乎是明智的，目的是让尽可能广泛的受众能够使用这项技术。这就是本报告的目的。该理论的一个令人愉快的方面是，明智地应用引理常常会得出简洁得几乎神奇的证明。在这种情况下，我们并没有努力抵制写出完整证明的诱惑。除了函数论的基本知识之外，几乎不需要任何其他知识就可以理解接下来的内容，因此鼓励读者鼓起勇气，努力读完。现在我们开始讲我们的故事。 1. 设 D 为复平面 C 中的域。我们将关注解析映射（即亚纯函数） f ： (D, | |R2) → (Ĉ, χ) 编辑于 1997 年 10 月 15 日收到，修订后的形式于 1998 年 5 月 26 日。1991 年数学学科分类。初级30D45；次级 30D35、34A20、58F23。
This paper surveys some surprising applications of a lemma characterizing normal families of meromorphic functions on plane domains. These include short and efficient proofs of generalizations of (i) the Picard Theorems, (ii) Gol’dberg’s Theorem (a meromorphic function on C which is the solution of a first-order algebraic differential equation has finite order), and (iii) the Fatou-Julia Theorem (the Julia set of a rational function of degree d ≥ 2 is the closure of the repelling periodic points). We also discuss Bloch’s Principle and provide simple solutions to some problems of Hayman connected with this principle. Over twenty years ago, on the way to a partial explication of the phenomenon known as Bloch’s Principle, I proved a little lemma characterizing normal families of holomorphic and meromorphic functions on plane domains [68]. Over the years, the lemma has grown and, in dextrous hands, proved amazingly versatile, with applications to a wide variety of topics in function theory and related areas. With the renewed interest in normal families (arising largely from the important role they play in complex dynamics), it seems sensible to survey some of the most striking of these applications to the one-variable theory, with the aim of making this technique available to as broad an audience as possible. That is the purpose of this report. One pleasant aspect of the theory is that judicious application of the lemma often leads to proofs which seem almost magical in their brevity. In such cases, we have made no effort to resist the temptation to write out complete proofs. Hardly anything beyond a basic knowledge of function theory is required to understand what follows, so the reader is urged to take courage and plough on through. And now we turn to our tale. 1. Let D be a domain in the complex plane C. We shall be concerned with analytic maps (i.e., meromorphic functions) f : (D, | |R2) → (Ĉ, χ) Received by the editors October 15, 1997, and, in revised form, May 26, 1998. 1991 Mathematics Subject Classification. Primary 30D45; Secondary 30D35, 34A20, 58F23.