An Ahlfors Islands Theorem for Non-archimedean Meromorphic Functions

Ahlfors’ contribution to the theory of meromorphic functions

This is an expanded version of one of the Lectures in memory of Lars Ahlfors in Haifa in 1996. Some mistakes are corrected and references added. This article is an exposition for non-specialists of Ahlfors’ work in the theory of meromorphic functions. When the domain is not specified we mean meromorphic functions in the complex plane C. The theory of meromorphic functions probably begins with t...

Generation of the Ahlfors Five Islands Theorem

NEW PROOF OF THE AHLFORS FIVE ISLANDS THEOREM 339 from

Let D j, j = 1 , . . . , 5, be simply-connected domains on the Riemann sphere with piecewise analytic boundary and pairwise disjoint closures. Let D C C be a domain and denote by 3rA(D) = f a (D, {Dj}~=I) the family of all meromorphic functions f : D ~ C with the property that no subdomain of D is mapped conformally onto one of the domains Dj by f . (If there is such a subdomain, then it is cal...

Non-archimedean Nevanlinna Theory in Several Variables and the Non-archimedean Nevanlinna Inverse Problem

Cartan’s method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects...

The Role of the Ahlfors Five Islands Theorem in Complex Dynamics

The Ahlfors five islands theorem has become an important tool in complex dynamics. We discuss its role there, describing how it can be used to deal with a variety of problems. This includes questions concerning the Hausdorff dimension of Julia sets, the existence of singleton components of Julia sets, and the existence of repelling periodic points. We point out that for many applications a simp...