# Baker domains for Newton’s method

Walter Bergweiler^{[1]}; David Drasin^{[2]}; James K. Langley^{[3]}

- [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)

Annales de l’institut Fourier (2007)

- Volume: 57, Issue: 3, page 803-814
- ISSN: 0373-0956

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topBergweiler, 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 | < \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 | < \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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- X. Buff, J. Rückert, Virtual immediate basins of Newton maps and asymptotic values, Int. Math. Res. Not. (2006), 1-18 Zbl1161.37327MR2211149
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- W. K. Hayman, Subharmonic functions, 2 (1989), Academic Press, London Zbl0699.31001MR1049148
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