Immediate and Virtual Basins of Newton’s Method for Entire Functions
Annales de l'Institut Fourier, Volume 56 (2006) no. 2, p. 325-336

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.

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.

DOI : https://doi.org/10.5802/aif.2184
Classification:  30D05,  37F10,  37N30
Keywords: Newton method, entire functions, immediate basin, virtual basins
@article{AIF_2006__56_2_325_0,
     author = {Mayer, Sebastian and Schleicher, Dierk},
     title = {Immediate and Virtual Basins of Newton's Method for Entire Functions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {2},
     year = {2006},
     pages = {325-336},
     doi = {10.5802/aif.2184},
     mrnumber = {2226018},
     zbl = {1103.30015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2006__56_2_325_0}
}
Mayer, Sebastian; Schleicher, Dierk. Immediate and Virtual Basins of Newton’s Method for Entire Functions. Annales de l'Institut Fourier, Volume 56 (2006) no. 2, pp. 325-336. doi : 10.5802/aif.2184. http://www.numdam.org/item/AIF_2006__56_2_325_0/

[1] Bergweiler, W. Iteration of meromorphic functions, Bulletin of the American Mathematical Society, Tome 29 (1993), pp. 151-188 | Article | MR 1216719 | Zbl 0791.30018

[2] Bergweiler, W.; Terglane, N. Weakly repelling fixpoints and the connectivity of wandering domains, Transactions of the American Mathematical Society, Tome 348 (1996) no. 1, pp. 1-12 | Article | MR 1327252 | Zbl 0842.30021

[3] Buff, X.; Rückert, J. Virtual immediate basins of Newton maps and asymptotic values (International Mathematics Research Notes, to appear) | Zbl 1161.37327

[4] Çilingir, F. On infinite area for complex exponential function, Chaos, Solitons and Fractals, Tome 22 (2004), pp. 1189-1198 | Article | MR 2078842 | Zbl 1063.30026

[5] Cowen, C. C. Iteration and the solution of functional equations for functions analytic in the unit disk, Transactions of the AMS, Tome 265 (1981) no. 1, pp. 69-95 | Article | MR 607108 | Zbl 0476.30017

[6] Haruta, Mako E. Newton’s method on the complex exponential function, Transactions of the AMS, Tome 351 (1999) no. 6, pp. 2499-2513 | Article | MR 1422898 | Zbl 0922.58068

[7] Hubbard, John; Schleicher, Dierk; Sutherland, Scott How to find all roots of complex polynomials by newton’s method, Inventiones Mathematicae, Tome 146 (2001), pp. 1-33 | Article | MR 1859017 | Zbl 1048.37046

[8] Mayer, Sebastian Newton’s method for entire functions, Technische Universität München (2002) (Diplomarbeit)

[9] Przytycki, Feliks Remarks on the simple connectedness of basins of sinks for iterations of rational maps, Dynamical Systems and Ergodic Theory, K. Krzyzewski. Polish Scientific Publishers, Warszawa (1989), pp. 229-235 | MR 1102717 | Zbl 0703.58033

[10] Rückert, Johannes; Schleicher, Dierk Combinatorial structure of immediate basins of Newton maps (Manuscript, submitted. ArXiv math.DS/0505652)

[11] Schleicher, Dierk On the number of iterations of Newton’s method for complex polynomials, Ergodic Theory Dyn. Syst., Tome 22 (2002) no. 3, pp. 935-945 | Article | MR 1908563 | Zbl 1011.37024

[12] Shishikura, Mitsuhiro The connectivity of the Julia set and fixed points (1990) (Preprint IHES, 37)

[13] Smale, Steven On the efficiency of algorithms of analysis, Bulletin of the American Mathematical Society, Tome 13 (1985) no. 2, pp. 87-121 | Article | MR 799791 | Zbl 0592.65032