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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Bibliographic Details
Published in:Inventiones mathematicae Vol. 198; no. 3; pp. 591 - 636
Main Authors: Barański, Krzysztof, Fagella, Núria, Jarque, Xavier, Karpińska, Bogusława
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014
Springer Nature B.V
Springer Verlag
Subjects:
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
Online Access:Get full text
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Summary: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