Five islands set

The five-island theorem is a theorem of function theory , which goes back to Lars Ahlfors , more precisely the value distribution theory of meromorphic functions . It improves on Picard's Little Theorem and is considered to be one of the main applications of Ahlfors' geometric approach to value distribution theory.

statement

Let be a transcendent meromorphic function and let five simply connected domains of the complex plane with disjoint closed hulls . Then there is one and for each a simply connected region such that there is a conformal mapping of on . ${\displaystyle f}$ ${\displaystyle D_{1},\ldots ,D_{5}}$ ${\displaystyle j\in \{1,2,3,4,5\}}$ ${\displaystyle R>0}$ ${\displaystyle G\subset \left\{z\in \mathbb {C} \colon \parallel z\parallel \geq R\right\}}$ ${\displaystyle f\mid _{G}}$ ${\displaystyle G}$ ${\displaystyle D_{j}}$

literature

• L. Ahlfors: Sur les fonctions inverses des fonctions méromorphes. Compt. Rend. Acad. Sci. Paris 194, 1145-1147, 1932.
• L. Ahlfors: About the circles that are simply covered by a Riemann surface. Comm. Math. Helv. 5, 28-38, 1933. Online
• W. Bergweiler: The role of the Ahlfors Five Islands Theorem in complex dynamics. Conf. Geom. Dyn. 4, 22-34, 2000. Online .

Web links

• Ahlfors Five Islands Theorem (MathWorld)