With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article | Next article Walter Bergweiler; David Drasin; James K. LangleyBaker 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, 65H05Keywords: Baker domain, Newton’s method, iteration, Julia set, Fatou set, asymptotic value Résumé - AbstractFor 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. Bibliography[1] S. K. Balašov, “On entire functions of finite order with zeros on curves of regular rotation”, Math. USSR Izvestija (1973) no. 7, p. 601-627, translation form Izv. Akad. Nauk. SSSR, Ser. Mat. 37 (1973) no 3 Article |  Zbl 0283.30028[2] W. Bergweiler, “Iteration of meromorphic functions”, Bull. Amer. Math. Soc. (1993) no. 29, p. 151-188 Article |  MR 1216719 |  Zbl 0791.30018[3] W. Bergweiler, F. V. Haeseler, H. Kriete, H. G. Meier & N. Terglane, Newton’s method for meromorphic functions, in C. C. Yang, G. C. Wen, K. Y. Li, Y. M. Chiang, ed., Complex Analysis and its Applications, Pitman Res., 1994, p. 147-158  Zbl 0810.30017[4] X. Buff & J. Rückert, “Virtual immediate basins of Newton maps and asymptotic values”, Int. Math. Res. Not. (2006) no. 65498, p. 1-18  MR 2211149 |  Zbl 05039691[5] W. K. Hayman, On integral functions with distinct asymptotic values, in Proc. Cambridge Philos. Soc., 1969, p. 301-315  MR 244487 |  Zbl 0179.10902[6] W. K. Hayman, Subharmonic functions, London Math. Soc. Monographs 2, Academic Press, 1989  MR 1049148 |  Zbl 0699.31001[7] R. Nevanlinna, Eindeutige analytische Funktionen, Springer, 1953  MR 57330 |  Zbl 0050.30302[8] M. Tsuji, Potential theory in modern function theory, Maruzen, 1959  MR 114894 |  Zbl 0087.28401 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310