Baker domains for Newton’s method

Bergweiler, Walter, Drasin, David, and Langley, James K.. "Baker domains for Newton’s method." Annales de l’institut Fourier 57.3 (2007): 803-814. <http://eudml.org/doc/10242>.

@article{Bergweiler2007,
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 | &lt; \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.},
affiliation = {Christian–Albrechts–Universität zu Kiel Mathematisches Seminar Ludewig–Meyn–Str. 4 24098 Kiel (Germany); Purdue University Department of Mathematics West Lafayette, IN 47907 (USA); University of Nottingham School of Mathematical Sciences NG7 2RD (United Kingdom)},
author = {Bergweiler, Walter, Drasin, David, Langley, James K.},
journal = {Annales de l’institut Fourier},
keywords = {Baker domain; Newton’s method; iteration; Julia set; Fatou set; asymptotic value; Newton's method},
language = {eng},
number = {3},
pages = {803-814},
publisher = {Association des Annales de l’institut Fourier},
title = {Baker domains for Newton’s method},
url = {http://eudml.org/doc/10242},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Bergweiler, Walter
AU - Drasin, David
AU - Langley, James K.
TI - Baker domains for Newton’s method
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 803
EP - 814
AB - 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 | &lt; \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.
LA - eng
KW - Baker domain; Newton’s method; iteration; Julia set; Fatou set; asymptotic value; Newton's method
UR - http://eudml.org/doc/10242
ER -