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Multiple attractors in Newton's method

  • Mike Hurley (a1)
Abstract
Abstract

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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[Bla] P. Blanchard . Complex analytic dynamics on the Riemann sphere. Bull. Amer. Math. Soc. 11 (1984), 85141.

[CuGS] J. Curry , L. Garnett & D. Sullivan . On the iteration of a rational function: computer experiments with Newton's method. Comm. Math. Phys. 91 (1983), 267277.

[HM] M. Hurley & C. Martin . Newton's algorithm and chaotic dynamical systems. SIAM J. Math. Anal. 15 (1984), 238252.

[SaU] D. Saari & J. Urenko . Newton's method, circle maps, and chaotic motion. Amer. Math. Monthly91 (1985), 317.

[Sm] S. Smale . The fundamental theorem of algebra and complexity theory. Bull. Amer. Math. Soc. 4 (1981), 136.

[Sul]; D. Sullivan . Conformal dynamical systems, pp. 725752 in Geometric Dynamics, J. Palis edit., (Springer Lect. Notes in Math. 1007). Springer–Verlag, New York, 1983.

[W] S. Wong . Newton's Method and symbolic dynamics. Proc. Amer. Math. Soc. 91 (1984), 245253.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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