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/

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

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

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

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

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

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

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

[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 | Zbl

[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., Volume 22 (2002) no. 3, pp. 935-945 | DOI | MR | Zbl

[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, Volume 13 (1985) no. 2, pp. 87-121 | DOI | MR | Zbl

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