Immediate and Virtual Basins of Newton’s Method for Entire Functions

Sebastian Mayer[1]; Dierk Schleicher[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)

Annales de l’institut Fourier (2006)

  • Volume: 56, Issue: 2, page 325-336
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Mayer, Sebastian, and Schleicher, Dierk. "Immediate and Virtual Basins of Newton’s Method for Entire Functions." Annales de l’institut Fourier 56.2 (2006): 325-336. <http://eudml.org/doc/10148>.

@article{Mayer2006,
abstract = {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.},
affiliation = {Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen (Germany); International University Bremen, School of Engineering and Science, Postfach 750 561, 28725 Bremen (Germany)},
author = {Mayer, Sebastian, Schleicher, Dierk},
journal = {Annales de l’institut Fourier},
keywords = {Newton method; entire functions; immediate basin; virtual basins; entire function; virtual basin},
language = {eng},
number = {2},
pages = {325-336},
publisher = {Association des Annales de l’institut Fourier},
title = {Immediate and Virtual Basins of Newton’s Method for Entire Functions},
url = {http://eudml.org/doc/10148},
volume = {56},
year = {2006},
}

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
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 2
SP - 325
EP - 336
AB - 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.
LA - eng
KW - Newton method; entire functions; immediate basin; virtual basins; entire function; virtual basin
UR - http://eudml.org/doc/10148
ER -

References

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  2. W. Bergweiler, N. Terglane, Weakly repelling fixpoints and the connectivity of wandering domains, Transactions of the American Mathematical Society 348 (1996), 1-12 Zbl0842.30021MR1327252
  3. X. Buff, J. Rückert, Virtual immediate basins of Newton maps and asymptotic values Zbl1161.37327
  4. F. Çilingir, On infinite area for complex exponential function, Chaos, Solitons and Fractals 22 (2004), 1189-1198 Zbl1063.30026MR2078842
  5. C. C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Transactions of the AMS 265 (1981), 69-95 Zbl0476.30017MR607108
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  7. John Hubbard, Dierk Schleicher, Scott Sutherland, How to find all roots of complex polynomials by newton’s method, Inventiones Mathematicae 146 (2001), 1-33 Zbl1048.37046MR1859017
  8. Sebastian Mayer, Newton’s method for entire functions, (2002) 
  9. Feliks Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, Dynamical Systems and Ergodic Theory (1989), 229-235, K. Krzyzewski. Polish Scientific Publishers, Warszawa Zbl0703.58033MR1102717
  10. Johannes Rückert, Dierk Schleicher, Combinatorial structure of immediate basins of Newton maps Zbl1042.30012
  11. Dierk Schleicher, On the number of iterations of Newton’s method for complex polynomials, Ergodic Theory Dyn. Syst. 22 (2002), 935-945 Zbl1011.37024MR1908563
  12. Mitsuhiro Shishikura, The connectivity of the Julia set and fixed points, (1990) Zbl1180.37074
  13. Steven Smale, On the efficiency of algorithms of analysis, Bulletin of the American Mathematical Society 13 (1985), 87-121 Zbl0592.65032MR799791

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