The role of the Ahlfors five islands theorem in complex dynamics. (English) Zbl 0954.30012
Summary: 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.
MSC:
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 30C25 | Covering theorems in conformal mapping theory |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
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