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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We wish to thank the Institut de Matemàtiques de la Universitat de Barcelona for its hospitality.
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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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DOI: https://doi.org/10.1007/s00222-014-0504-5