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On the Connectivity of the Julia sets of meromorphic functions
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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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BARANSKI, Krzysztof, et al. On the Connectivity of the Julia sets of meromorphic functions. Inventiones Mathematicae. 2014. Vol. 198, num. 3, pags. 591-636. ISSN 0020-9910. [consulted: 24 of October of 2026]. Available at: https://hdl.handle.net/2445/63074