Revista Matemática Iberoamericana

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Volume 34, Issue 3, 2018, pp. 1211–1228
DOI: 10.4171/RMI/1022

Published online: 2018-08-27

Connectivity of Julia sets of Newton maps: a unified approach

Krzysztof Barański[1], Núria Fagella[2], Xavier Jarque[3] and Bogusława Karpińska

(1) University of Warsaw, Poland
(2) Universitat de Barcelona, Spain
(3) Universitat de Barcelona, Spain

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.

Keywords: Newton’s method, root-finding algorithm, iteration, Julia set

Barański Krzysztof, Fagella Núria, Jarque Xavier, Karpińska Bogusława: Connectivity of Julia sets of Newton maps: a unified approach. Rev. Mat. Iberoam. 34 (2018), 1211-1228. doi: 10.4171/RMI/1022