The role of the Ahlfors five islands theorem in complex dynamics

Bergweiler, Walter

doi:10.1090/S1088-4173-00-00057-6

The role of the Ahlfors five islands theorem in complex dynamics
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by Walter Bergweiler

Conform. Geom. Dyn. 4 (2000), 22-34

DOI: https://doi.org/10.1090/S1088-4173-00-00057-6

Published electronically: March 14, 2000

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Abstract:

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 simplified version of the Ahlfors five islands theorem suffices, and we give an elementary proof of this version.

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