Normal families: New perspectives

Abstract

Abstract. 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 1 (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.

Dynamic rays of bounded-type entire functions

“…(Compare Figure 1.) More generally, given any point z ∈ J(p), we can find a sequence of rescalings of iterates of p that converges to an entire function f : C → C of finite order by the "Zalcman lemma" (see e.g. [Zal98]). Again, since J(p) is connected, we have f ∈ B. Theorem 1.2 implies that, for all such functions and their finite compositions, the escaping set consists of rays.…”

Normal criteria for families of meromorphic functions

“…Lemma 6 (Zalcman's Lemma, see [8]). Let F be a family of meromorphic functions defined in the unit disc △.…”

A Sectorial Theorem on Completely Ramified Rational Functions

“…An application of Zalcman's rescaling lemma [17] to Theorem 2.2 shows the following. If q = 4 and if all but finitely many f ∈ F have no simple island over any of the D i then, with at most finitely many exceptions, all f ∈ F have simply connected islands over every D i .…”