Walter Bergweiler; David Drasin; James K. Langley
Baker domains for Newton’s method
(Domaines de Baker et la méthode de Newton)
Annales de l'institut Fourier, 57 no. 3 (2007), p. 803-814, doi: 10.5802/aif.2277
Article PDF | Reviews MR 2336830 | Zbl 1122.30019
Class. Math.: 30D05, 37F10, 65H05
Keywords: Baker domain, Newton’s method, iteration, Julia set, Fatou set, asymptotic value
Résumé - Abstract
For an entire function $f$ let $N(z) = z - f(z)/f^{\prime}(z)$ be the Newton function associated to $f$. Each zero $\xi $ of $f$ is an attractive fixed point of $N$ and is contained in an invariant component of the Fatou set of the meromorphic function $N$ in which the iterates of $N$ converge to $\xi $. If $f$ has an asymptotic representation $f(z) \sim \exp ( - z^n ) , \, n \in \mathbb{N}$, in a sector $| \arg z | < \varepsilon $, then there exists an invariant component of the Fatou set where the iterates of $N$ tend to infinity. Such a component is called an invariant Baker domain.
A question in the opposite direction was asked by A. Douady: if $N$ has an invariant Baker domain, must $0$ be an asymptotic value of $f$? X. Buff and J. Rückert have shown that the answer is positive in many cases.
Using results of Balašov and Hayman, it is shown that the answer is negative in general: there exists an entire function $f$, of any order between $\frac{1}{2}$ and $1$, and without finite asymptotic values, for which the Newton function $N$ has an invariant Baker domain.
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