# Baker domains for Newton’s method

• [1] Christian–Albrechts–Universität zu Kiel Mathematisches Seminar Ludewig–Meyn–Str. 4 24098 Kiel (Germany)
• [2] Purdue University Department of Mathematics West Lafayette, IN 47907 (USA)
• [3] University of Nottingham School of Mathematical Sciences NG7 2RD (United Kingdom)
• Volume: 57, Issue: 3, page 803-814
• ISSN: 0373-0956

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## Abstract

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For an entire function $f$ let $N\left(z\right)=z-f\left(z\right)/{f}^{\prime }\left(z\right)$ 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 $f\left(z\right)\sim exp\left(-{z}^{n}\right),\phantom{\rule{0.166667em}{0ex}}n\in ℕ$, in a sector $|argz|<\epsilon$, 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.

## How to cite

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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 -

## References

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1. S. K. Balašov, On entire functions of finite order with zeros on curves of regular rotation, Math. USSR Izvestija (1973), 601-627 Zbl0283.30028
2. W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (1993), 151-188 Zbl0791.30018MR1216719
3. W. Bergweiler, F. V. Haeseler, H. Kriete, H. G. Meier, N. Terglane, Newton’s method for meromorphic functions, Complex Analysis and its Applications 305 (1994), 147-158, YangC. C.C. C. Zbl0810.30017
4. X. Buff, J. Rückert, Virtual immediate basins of Newton maps and asymptotic values, Int. Math. Res. Not. (2006), 1-18 Zbl1161.37327MR2211149
5. W. K. Hayman, On integral functions with distinct asymptotic values, Proc. Cambridge Philos. Soc. (1969), 301-315 Zbl0179.10902MR244487
6. W. K. Hayman, Subharmonic functions, 2 (1989), Academic Press, London Zbl0699.31001MR1049148
7. R. Nevanlinna, Eindeutige analytische Funktionen, (1953), Springer, Göttingen, Heidelberg Zbl0050.30302MR57330
8. M. Tsuji, Potential theory in modern function theory, (1959), Maruzen, reprint by Chelsea Zbl0087.28401MR114894

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