no. 2
Immediate and Virtual Basins of Newton’s Method for Entire Functions
[Domaines d’attraction immédiats et virtuels de la méthode de Newton pour les applications entières]
Mayer, Sebastian1 ; Schleicher, Dierk2
1 Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen (Germany)
2 International University Bremen, School of Engineering and Science, Postfach 750 561, 28725 Bremen (Germany)
Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 325-336.

Nous étudions la méthode bien connue de Newton pour trouver les racines des applications holomorphes entières. Notre résultat principal est que le domaine d’attraction immédiat de chaque racine est simplement connexe et non borné. D’ailleurs, nous introduisons les “domaines immédiats virtuels” dans lesquels la dynamique converge vers l’infini ; nous démontrons aussi qu’ils sont simplement connexes.

We investigate the well known Newton method to find roots of entire holomorphic functions. Our main result is that the immediate basin of attraction for every root is simply connected and unbounded. We also introduce “virtual immediate basins” in which the dynamics converges to infinity; we prove that these are simply connected as well.

DOI : 10.5802/aif.2184
Classification : 30D05, 37F10, 37N30
Mots clés : Newton method, entire functions, immediate basin, virtual basins
Mayer, Sebastian 1 ; Schleicher, Dierk 2

1 Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen (Germany)
2 International University Bremen, School of Engineering and Science, Postfach 750 561, 28725 Bremen (Germany) 
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Mayer, Sebastian; Schleicher, Dierk. Immediate and Virtual Basins of Newton’s Method for Entire Functions. Annales de l'Institut Fourier, Tome 56 (2006) no. 2, pp. 325-336. doi : 10.5802/aif.2184. http://www.numdam.org/articles/10.5802/aif.2184/

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