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.
Keywords: Newton method, entire functions, immediate basin, virtual basins
Mayer, Sebastian 1; Schleicher, Dierk 2
@article{AIF_2006__56_2_325_0, author = {Mayer, Sebastian and Schleicher, Dierk}, title = {Immediate and {Virtual} {Basins} of {Newton{\textquoteright}s} {Method} for {Entire} {Functions}}, journal = {Annales de l'Institut Fourier}, pages = {325--336}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {2}, year = {2006}, doi = {10.5802/aif.2184}, mrnumber = {2226018}, zbl = {1103.30015}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2184/} }
TY - JOUR AU - Mayer, Sebastian AU - Schleicher, Dierk TI - Immediate and Virtual Basins of Newton’s Method for Entire Functions JO - Annales de l'Institut Fourier PY - 2006 SP - 325 EP - 336 VL - 56 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2184/ DO - 10.5802/aif.2184 LA - en ID - AIF_2006__56_2_325_0 ER -
%0 Journal Article %A Mayer, Sebastian %A Schleicher, Dierk %T Immediate and Virtual Basins of Newton’s Method for Entire Functions %J Annales de l'Institut Fourier %D 2006 %P 325-336 %V 56 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2184/ %R 10.5802/aif.2184 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2184/
[1] Iteration of meromorphic functions, Bulletin of the American Mathematical Society, Volume 29 (1993), pp. 151-188 | DOI | MR | Zbl
[2] 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] Virtual immediate basins of Newton maps and asymptotic values (International Mathematics Research Notes, to appear) | Zbl
[4] On infinite area for complex exponential function, Chaos, Solitons and Fractals, Volume 22 (2004), pp. 1189-1198 | DOI | MR | Zbl
[5] 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] Newton’s method on the complex exponential function, Transactions of the AMS, Volume 351 (1999) no. 6, pp. 2499-2513 | DOI | MR | Zbl
[7] How to find all roots of complex polynomials by newton’s method, Inventiones Mathematicae, Volume 146 (2001), pp. 1-33 | DOI | MR | Zbl
[8] Newton’s method for entire functions, Technische Universität München (2002) (Diplomarbeit)
[9] 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] Combinatorial structure of immediate basins of Newton maps (Manuscript, submitted. ArXiv math.DS/0505652)
[11] 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] The connectivity of the Julia set and fixed points (1990) (Preprint IHES, 37)
[13] On the efficiency of algorithms of analysis, Bulletin of the American Mathematical Society, Volume 13 (1985) no. 2, pp. 87-121 | DOI | MR | Zbl
Cited by Sources: