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On the connectivity of the Julia sets of meromorphic functions

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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.

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Acknowledgments

We wish to thank the Institut de Matemàtiques de la Universitat de Barcelona for its hospitality.

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Authors and Affiliations

  1. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 , Warszawa, Poland

    Krzysztof Barański

  2. Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, 08007 , Barcelona, Catalonia, Spain

    Núria Fagella & Xavier Jarque

  3. Faculty of Mathematics and Information Science, Warsaw University of Technology, Pl. Politechniki 1, 00-661 , Warszawa, Poland

    Bogusława Karpińska

Authors
  1. Krzysztof Barański
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  2. Núria Fagella
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  3. Xavier Jarque
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  4. Bogusława Karpińska
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Corresponding author

Correspondence to Núria Fagella.

Additional information

K. Barański is partially supported by Polish NCN Grant N N201 607940. N. Fagella and X. Jarque were partially supported by the Catalan grant 2009SGR-792, and by the Spanish Grants MTM-2006-05849 and MTM-2008-01486 Consolider (including a FEDER contribution) and MTM2011-26995-C02-02. B. Karpińska is partially supported by Polish NCN Grant N N201 607940 and Polish PW Grant 504G 1120 0011 000.

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Barański, K., Fagella, N., Jarque, X. et al. On the connectivity of the Julia sets of meromorphic functions. Invent. math. 198, 591–636 (2014). https://doi.org/10.1007/s00222-014-0504-5

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