On the connectivity of the Julia sets of meromorphic functions

We prove that every transcendental meromorphic map f with disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton’s method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we sh...

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Vydáno v:Inventiones mathematicae Ročník 198; číslo 3; s. 591 - 636
Hlavní autoři: Barański, Krzysztof, Fagella, Núria, Jarque, Xavier, Karpińska, Bogusława
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014
Springer Nature B.V
Springer Verlag
Témata:
Absorption
Boundaries
Discs
Disks
Entire functions
Funcions de variables complexes
Funcions enteres
Functions of complex variables
Mathematical analysis
Mathematics
Mathematics and Statistics
Meromorphic functions
Newton methods
Texts
30D20
Primary 30D05
37F10
ISSN:0020-9910, 1432-1297
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Shrnutí:We prove that every transcendental meromorphic map f with disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton’s method for finding zeroes of an entire map is connected. Moreover, extending a result of Cowen for holomorphic self-maps of the disc, we show the existence of absorbing domains for holomorphic self-maps of hyperbolic regions, whose iterates tend to a boundary point. In particular, the results imply that periodic Baker domains of Newton’s method for entire maps are simply connected, which solves a well-known open question.
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ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-014-0504-5