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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

The role of the Ahlfors five islands theorem in complex dynamics


Author: Walter Bergweiler
Journal: Conform. Geom. Dyn. 4 (2000), 22-34
MSC (2000): Primary 30C25; Secondary 30D05, 30D45, 37F10
Published electronically: March 14, 2000
MathSciNet review: 1741773
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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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Additional Information

Walter Bergweiler
Affiliation: Mathematisches Seminar, Christian–Albrechts–Universität zu Kiel, Ludewig–Meyn–Str. 4, D–24098 Kiel, Germany
Email: bergweiler@math.uni-kiel.de

DOI: http://dx.doi.org/10.1090/S1088-4173-00-00057-6
PII: S 1088-4173(00)00057-6
Keywords: Ahlfors theory, covering surface, island, Bloch principle, rescaling lemma, normal family, complex dynamics, Julia set, periodic point, Hausdorff dimension.
Received by editor(s): October 29, 1999
Published electronically: March 14, 2000
Article copyright: © Copyright 2000 American Mathematical Society