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Multiple attractors in Newton's method

Published online by Cambridge University Press:  19 September 2008

Mike Hurley
Mike Hurley
Affiliation:
Department of Mathematics and Statistics, Case Western Reserve University, Cleveland, Ohio 44106, USA
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Abstract

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For each d ≥ 2 there exists a polynomial p with real coefficients such that the associated Newton function z–[p(z)/p′(z)] has 2d–2 distinct attracting periodic orbits in the complex plane. According to a theorem of G. Julia, this is the maximal number of attracting orbits that any rational function of degree d can possess.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 4 , December 1986 , pp. 561 - 569
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[B1]Barna, B.. Uber das Newtonsche Verfahren zur Annaherung von wurzeln algebraischen Gleichungen. Publ. Math. Debrecen. 2 (1951), 50–63.CrossRefGoogle Scholar
[B2–4]Barna, B.. Uber die Divergenzpunkte des Newtonschen Verfahrens zur bestimmung von wurzeln algebraischen Gleichungen: I. Publ. Math. Debrecen, 3 (1953), 109–118.Google Scholar
Uber die Divergenzpunkte des Newtonschen Verfahrens zur bestimmung von wurzeln algebraischen Gleichungen: II. Publ. Math. Debrecen, 4 (1956), 384–397.Google Scholar
Uber die Divergenzpunkte des Newtonschen Verfahrens zur bestimmung von wurzeln algebraischen Gleichungen: III. Publ. Math. Debrecen, 8 (1961), 193, 207.Google Scholar
[Bla]Blanchard, P.. Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85–141.CrossRefGoogle Scholar
[CoM]Cosnard, M. & Masse, C.. Convergence presque partout de la methode de Newton. C. R. Acad. Sci. Paris 297 (1983), 549–552.Google Scholar
[CuGS]Curry, J., Garnett, L. & Sullivan, D.. On the iteration of a rational function: computer experiments with Newton's method. Comm. Math. Phys. 91 (1983), 267–277.CrossRefGoogle Scholar
[H]Hurley, M.. Attracting orbits in Newton's method. Trans. Amer. Math. Soc. To appear.Google Scholar
[HM]Hurley, M. & Martin, C.. Newton's algorithm and chaotic dynamical systems. SIAM J. Math. Anal. 15 (1984), 238–252.CrossRefGoogle Scholar
[J]Julia, G.. Memoire sur l'iteration des fonctions rationnelles. J. Math. 8 (1918), 47–245.Google Scholar
[SaU]Saari, D. & Urenko, J.. Newton's method, circle maps, and chaotic motion. Amer. Math. Monthly 91 (1985), 3–17.CrossRefGoogle Scholar
[Sm]Smale, S.. The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. 4 (1981), 1–36.CrossRefGoogle Scholar
[Sul]; Sullivan, D.. Conformal dynamical systems, pp. 725–752 in Geometric Dynamics, Palis, J. edit., (Springer Lect. Notes in Math. 1007). Springer–Verlag, New York, 1983.CrossRefGoogle Scholar
[W]Wong, S.. Newton's Method and symbolic dynamics. Proc. Amer. Math. Soc. 91 (1984), 245–253.Google Scholar
See also Wong, S.. The instability of the period two cycles of Newton's Method, preprint.Google Scholar