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On the Julia set of König's root-finding algorithms

Author: Gerardo Honorato
Journal: Proc. Amer. Math. Soc. 141 (2013), 3601-3607
MSC (2010): Primary 37F10, 30D05, 37F50; Secondary 65H04
Published electronically: July 1, 2013
MathSciNet review: 3080182
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Abstract: As is well known, the Julia set of Newton's method applied to complex polynomials is connected. The family of König's root-finding algorithms is a natural generalization of Newton's method. We show that the Julia set of König's root-finding algorithms of order $ \sigma \geq 3$ applied to complex polynomials is not always connected.

References [Enhancements On Off] (What's this?)

  • 1. N. Argiropoulos, V. Drakopoulos and A. Böhm, Julia and Mandelbrot-like sets for higher order König rational iterative maps, Fractal Frontier, M. M. Novak and T. G. Dewey, eds., World Scientific, Singapore, 1997, 169-178. MR 1636268 (99g:30028)
  • 2. X. Buff and C. Henriksen, On K $ \ddot {\textrm {o}}$nig's root-finding algorithms, Nonlinearity, vol. 16, 2003, 989-1015. MR 1975793 (2004c:37086)
  • 3. J. Hubbard, D. Schleicher and S. Sutherland, How to find all roots of complex polynomials by Newton's method, Invent. Math., vol. 146, Number 1, 2001, 1-33. MR 1859017 (2002i:37059)
  • 4. S. Mayer and D. Schleicher, Immediate and virtual basins of Newton's method for entire functions, Ann. Inst. Fourier (Grenoble), vol. 56, no. 2, 2006, 325-336. MR 2226018 (2007b:30028)
  • 5. H. Meier, On the connectedness of the Julia-set for rational functions, Preprint no. 4, RWTH Aachen, 1989.
  • 6. F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, Dynamical Systems and Ergodic Theory, ed. by K. Krzyzewski, Polish Scientific Publishers (Warsawa), 1989, 229-235. MR 1102717 (92e:58180)
  • 7. W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 0210528 (35:1420)
  • 8. T. Scavo and J. Thoo, On the geometry of Halley's method, Amer. Math. Monthly, vol. 102, Number 5, 1995, 417-426. MR 1327786 (96f:01019)
  • 9. J. Schiff, Normal Families, Universitext, Springer Verlag, New York, 1993. MR 1211641 (94f:30046)
  • 10. M. Shishikura, Connectivity of the Julia set and fixed point, in Complex Dynamics: Families and Friends, D. Schleicher, ed., 257-276, A K Peters, Wellesley, MA, 2009. MR 2508260 (2010d:37093)
  • 11. L. Tan, Branched coverings and cubic Newton maps, Fund. Math., vol. 154, 1997, 207-260. MR 1475866 (2000e:37051)

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Additional Information

Gerardo Honorato
Affiliation: Instituto Nacional de Matemática Pura e Aplicada (IMPA), Estr. D. Castorina 110, 22460-320 Rio de Janeiro, Brazil

Keywords: Root--finding algorithms, complex dynamics
Received by editor(s): December 12, 2011
Received by editor(s) in revised form: January 5, 2012
Published electronically: July 1, 2013
Additional Notes: The author was supported in part by the Research Network on Low Dimensional Dynamics, PBCT ACT-17-CONICYT, FONDECYT 3120016, Chile and CNPq (The Brazilian National Research Council)
Dedicated: Dedicated to the memory of Sergio Plaza S.
Communicated by: Bryna Kra
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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