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 2014-12, Vol.198 (3), p.591-636 |
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| container_end_page | 636 |
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| container_issue | 3 |
| container_start_page | 591 |
| container_title | Inventiones mathematicae |
| container_volume | 198 |
| creator | Barański, Krzysztof Fagella, Núria Jarque, Xavier Karpińska, Bogusława |
| description | 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. |
| doi_str_mv | 10.1007/s00222-014-0504-5 |
| format | Article |
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f
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| ispartof | Inventiones mathematicae, 2014-12, Vol.198 (3), p.591-636 |
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| language | eng |
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| source | SpringerNature Complete Journals; Recercat |
| 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 |
| title | On the connectivity of the Julia sets of meromorphic functions |
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