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Existence of herman rings for meromorphic functions
Abstract:We apply the Shishikura surgery construction to transcendental maps in order to obtain examples of meromorphic functions with Herman rings, in a variety of possible arrangements. We give a sharp bound on the maximum possible number of such rings that a meromorphic function may have, in terms of the number of poles. Finally we discuss the possibility of having “unbounded” Herman rings (i.e., with an essential singularity in the boundary), and give some examples of maps with this property.
Keywords:Holomorphic dynamics  Transcendental maps  Julia set  Fatou set  Herman rings  Quasi-conformal surgery  2000 Mathematics Subject Classification: Primary 37F10  Secondary 37F45
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