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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Veröffentlicht in:	Inventiones mathematicae Jg. 198; H. 3; S. 591 - 636
Hauptverfasser:	Barański, Krzysztof, Fagella, Núria, Jarque, Xavier, Karpińska, Bogusława
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014 Springer Nature B.V Springer Verlag
Schlagworte:	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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Abstract	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.
AbstractList	(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We prove that every transcendental meromorphic map ... 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. 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. 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. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We prove that every transcendental meromorphic map ... 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.[PUBLICATION ABSTRACT]
Author	Karpińska, Bogusława Fagella, Núria Barański, Krzysztof Jarque, Xavier
Author_xml	– sequence: 1 givenname: Krzysztof surname: Barański fullname: Barański, Krzysztof organization: Institute of Mathematics, University of Warsaw – sequence: 2 givenname: Núria surname: Fagella fullname: Fagella, Núria email: fagella@maia.ub.es organization: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona – sequence: 3 givenname: Xavier surname: Jarque fullname: Jarque, Xavier organization: Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona – sequence: 4 givenname: Bogusława surname: Karpińska fullname: Karpińska, Bogusława organization: Faculty of Mathematics and Information Science, Warsaw University of Technology
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Cites_doi	10.3934/dcds.2006.15.379 10.4064/fm215-2-5 10.1090/S0002-9947-99-01927-3 10.5802/aif.2184 10.1017/S014338570000612X 10.4064/-23-1-229-235 10.24033/bsmf.998 10.2307/1971408 10.1088/0951-7715/14/3/301 10.1007/BF02797685 10.1007/978-3-642-13171-4_5 10.2307/1971308 10.1017/S0143385706000162 10.1201/b10617-11 10.5802/aif.2277 10.1155/IMRN/2006/65498 10.1007/978-1-4612-4364-9 10.1090/S0002-9947-1981-0607108-9 10.1007/s002220100149 10.1017/S0305004106009315 10.1201/b10617-9 10.4064/fm-154-3-207-260 10.1080/10236190903203846 10.1090/S0002-9947-96-01511-5 10.1112/plms/pdn012 10.5565/PUBLMAT_47103_09 10.1090/chel/371 10.1007/BF01389368 10.1080/17476939808815123
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Snippet	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... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) We prove that every transcendental meromorphic map ... with disconnected Julia set... (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).We prove that every transcendental meromorphic map ... with disconnected Julia set... 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...
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SubjectTerms	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
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