In this paper we present a unified proof of the fact that the Julia set of Newton’s method applied to a holomorphic function on the complex plane (a polynomial of degree larger than 1 or a transcendental entire function) is connected. The result was recently completed by the authors’ previous work, as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this note we present a unified, direct and reasonably self-contained proof which works in all situations alike.
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Krzysztof Barański, Núria Fagella, Xavier Jarque, Bogusława Karpińska, Connectivity of Julia sets of Newton maps: a unified approach. Rev. Mat. Iberoam. 34 (2018), no. 3, pp. 1211–1228DOI 10.4171/RMI/1022