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Newton’s Method and Complex Dynamical Systems

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Newton’s Method and Dynamical Systems

Abstract

This article is devoted to the discussion of Newton’s method. Beginning with the old results of A.Cayley and E.Schrôder we proceed to the theory of complex dynamical systems on the sphere, which was developed by G.Julia and R.Fatou at the beginning of this century, and continued by several mathematicians in recent years.

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References

  1. L.V.Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966

    MATH  Google Scholar 

  2. L.V.Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, 1966

    Google Scholar 

  3. L.deBranges, A proof of the Bieberbach conjecture, Acta Mathematika, 154:1–2, 1985

    Article  MathSciNet  Google Scholar 

  4. L.deBranges, A proof of the Bieberbach conjecture, Acta Mathematika, 154:1985, 137–152

    Article  MathSciNet  Google Scholar 

  5. H.Brolin, Invariant sets under iteration of rational functions, Arkiv för Matematik 6, 1967, 103–141

    Article  MathSciNet  Google Scholar 

  6. P.Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the AMS, Vol. 11, Number 1, July 1984, 85–141

    Article  MathSciNet  MATH  Google Scholar 

  7. I.N.Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc., 49, 1984, 563–576

    Article  MathSciNet  MATH  Google Scholar 

  8. I.N.Baker, Some entire functions with multiply connected wandering domains, Erg.Th.Dyn.Sys.5, 1985, 163–169

    MATH  Google Scholar 

  9. LN.Baker, An entire function which has wandering domains, J. Austral. Math. Soc., 22, 1976, 173–176

    Article  MATH  Google Scholar 

  10. B.Barna, Über das Newtonsche Verfahren zur Annäherung von Wurzeln algebraischer Gleichungen, Publ.Math. Debrecen, 2, 1951, 50–63

    MathSciNet  MATH  Google Scholar 

  11. Über die Divergenzpunkte des Newtonschen Verfahrens zur von Wurzeln algebraischer Gleichungen, I, II, in, Publ.Math. Debrecen, 3, 1953, 109–118, Publ.Math. Debrecen, 4, 1956, 384–397, Publ.Math. Debrecen, 8, 1961, 193–207

    Google Scholar 

  12. B.Bielefeld, Y.Fisher, F.v.Haeseler, Computing the Laurent Series of the map Ψ: C \D̄→Ç\D, Max-Planck-Institut für Mathematik, Bonn, MPI/88–46

    Google Scholar 

  13. D.Braess, Über die Einzugsgebiete der Nullstellen von Polynomen beim Newton-Verfahren, Numer. Math., 29, 1977, 123–132

    Article  MathSciNet  MATH  Google Scholar 

  14. B.Branner, J.H.Hubbard, Iteration of cubic polynomials I, Acta Mathematica 160, 1988, 143–206

    Article  MathSciNet  MATH  Google Scholar 

  15. Boettcher, Bulletin of the Kasan Math. Society, vol. 14, 1905, 176

    Google Scholar 

  16. L.Bers, On Sullivan’s proof of the finiteness theorem, Amer. J. Math., 109, No. 5, 1987, 833–852

    Article  MathSciNet  MATH  Google Scholar 

  17. A.D.Brjuno, Convergence of transformations of differential equations to normal forms, Dokl.Akad.Nauk.URSS 165, 1965, 987–989

    MathSciNet  Google Scholar 

  18. A.D.Brjuno Analytic form of differential equations, Trans. Moscow Math.Soc. 25, 1971, 131–288

    MathSciNet  Google Scholar 

  19. A.D.Brjuno Analytic form of differential equations, Trans. Moscow Math.Soc. 26, 1972, 199–239

    Google Scholar 

  20. C.Camacho, On the local structure of conformai mappings and holomorphic vector fields in C2, Société Mathématique de France, Astérisque 59–60 (1978), 83–93

    MathSciNet  Google Scholar 

  21. A.Cayley, The Newton-Fourier imaginary problem, Amer. J. Math. II, 1879, 97

    Article  MathSciNet  Google Scholar 

  22. A.Cayley, Application of the Newton-Fourier Mathod to an imaginary root of equation, Quart. J. of Pure and App. Math. XVI, 1879, 179–185

    Google Scholar 

  23. A.Cayley, Sur les racines d’une équation algébrique, CRAS 110, 1890, 215–218

    Google Scholar 

  24. C.Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73, 1913, 323–370

    Article  MathSciNet  MATH  Google Scholar 

  25. M.Cosnard, C.Masse, Convergence presque partout de la méthode de Newton, CRAS Paris, t. 297 (14 novembre 1983), Série 1–549

    Google Scholar 

  26. H.Cremer, Zum Zentrumsproblem, Math.Ann., 98, 1928, 151–163

    Article  MathSciNet  Google Scholar 

  27. H.Cremer, Über die Häufigkeit der Nichtzentren, Math.Ann., 115, 1938, 573–580

    Article  MathSciNet  Google Scholar 

  28. H.Cremer, Über die Iteration rationaler Funktionen, Jber. d. Dt. Math.-Verein, 33, 1925, 185–210

    MATH  Google Scholar 

  29. H.Cremer, Über Konvergenz und Zentrumproblem, Math.Ann. 110, 1935, 739–744

    Article  MathSciNet  Google Scholar 

  30. J.Curry, L.Garnett, D.Sullivan, On the iteration of rational functions: Computer experiments with Newton’s Method, Commun. Math. Phys., 91, 1983, 267–277

    Article  MathSciNet  MATH  Google Scholar 

  31. RX.Devaney, Introduction to Chaotic Dynamical Systems, Benjamin-Cummings, Menlo Park, 1986

    MATH  Google Scholar 

  32. A.Douady, Systèmes dynamiques holomorphes, exposé no. 599, Séminaire N.Bourbaki 1982/83, Astérisque 105–106, 1983, 39–63

    MathSciNet  Google Scholar 

  33. A.Douady, Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, 39éme anneé, 1986–87, n°677

    Google Scholar 

  34. A.Douady, Chirugie sur les applications holomorphes, ICM 86, Berkeley

    Google Scholar 

  35. A.Douady, J.H.Hubbard, Itération des polynômes quadratiques complexes, C.R.Acad.Sci.Paris, 294, 1982, 123–126

    MathSciNet  MATH  Google Scholar 

  36. A.Douady, J.H.Hubbard, On the dynamics of polynomial-like mappings, Ann.Sci.École Norm.Sup., 4e série, t.18, 1985, 287–343

    MathSciNet  MATH  Google Scholar 

  37. A.Douady, J.H.Hubbard, Etude dynamiques des polynômes complexes I, II, Publ.Math.d’Orsay, 84–02, 1984, 85–02, 1985

    Google Scholar 

  38. P. Fatou, Sur les équations fonctionelles, Bull.Soc.Math.France, 47, 1919, 161–271; 48, 1920, 33–94 and 208–304

    Google Scholar 

  39. P.Fatou, Sur les équations fonctionelles, Bull.Soc.Math.France, 47, 1919, 161–271

    MathSciNet  Google Scholar 

  40. P.Fatou, Sur les équations fonctionelles, Bull.Soc.Math.France, 48, 1920, 33–94

    MathSciNet  Google Scholar 

  41. P.Fatou, Sur les équations fonctionelles, Bull.Soc.Math.France, 48, 1920, 208–304

    MathSciNet  Google Scholar 

  42. P.Fatou, Sur l’itération des fonctions transcendantes entières, Acta Math. 47, 1926, 337–370

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations, J. Statistical Physics 19, 1978, 25–52

    Article  MathSciNet  MATH  Google Scholar 

  44. S. Großmann, S. Thomae, Invariant Distributions and Stationary Correlation Functions of One-Dimensional Discrete Processes, Zeitsch. f. Naturforschg. 32a, 1977, 1353–1363

    Google Scholar 

  45. J. Guckenheimer, Endomorphisms of the Riemann sphere, Proc. of the Symp. of Pure Math., vol. 14, ed. S.S. Chern and S. Smale, AMS, Providence, R.I., 1970, 95–123

    Google Scholar 

  46. F.v. Haeseler, Über sofortige Attraktionsgebiete superattraktiver Zykel, Thesis, Universität Bremen, 1985

    Google Scholar 

  47. M.R. Herman, Are the critical points on the boundaries of singular domains, Institut Mittag-Leffler, Report Nr. 14,1984, and Comm.Math.Phys. 99, 1985, 563–612

    Google Scholar 

  48. M.R. Herman, Exemples de fractions rationelles ayant une orbit dense sur la sphère de Riemann, Bull.Soc.Math.France, 112, 1984, 93–142

    MathSciNet  MATH  Google Scholar 

  49. M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ.Math. IHES 49, 1979, 5–233

    MathSciNet  MATH  Google Scholar 

  50. M.R. Herman, Recent results and some open questions on Siegel’s Linearization Theorem of germs of complex analytic diffeomorphisms of Cn near a fixed point, Proceedings VIII Internat. Conf. Math.-Phys., World Science Publ., Singapore, 1987, 138–184

    Google Scholar 

  51. M. Hurley, C. Martin, Newton’s Algorithm and Chaotic Dynamical Systems, SLAM J. Math. Anal., Vol. 15, No. 2, March 1984, 238–252

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Hurley, Multiple attractors in Newton’s Method, Ergod. Th. & Dynam. Sys., 6, 1986, 561–569

    Article  MathSciNet  MATH  Google Scholar 

  53. G. Julia, Mémoire sur l’itération des fonctions rationelles, J. de. Math, pures et appliquées, ser. 8.1, 1918, 47–245

    Google Scholar 

  54. M.S. Lattès, Sur l’itération des substitutions rationelles et les fonctions de Poincaré, C.R.A.S., 166, 1918, 26–28

    MATH  Google Scholar 

  55. O. Letho, K. Virtanen, Quasiconformal Mappings in the Plane, Springer Verlag, 1973

    Google Scholar 

  56. T.Y. Li, J.A. Yorke, Period three implies chaos, Amer.Math.Monthly, 82, 1975, 985–992

    Article  MathSciNet  MATH  Google Scholar 

  57. B.B. Mandelbrot, Fractal aspects of the iteration of z ↦ λz(1 − z) for complex λ and z, Ann.N.Y.Acad.Sci., 375, 1980, 249–259

    Article  Google Scholar 

  58. B.B. Mandelbrot, On the dynamics of iterated maps VIII. The map z ↦ λ(z + 1/z), from linear to planar chaos, and the measurement of chaos, in “Chaos and Statistical Methods”, ed. Y. Kuramoto, Springer, 1984

    Google Scholar 

  59. J. Martinet, Normalisation des champ de vecteurs, d’après Brjuno, Sém. Bourbaki, exp. 564, Lect. Notes in Math., Springer-Verlag 901, 1981, 55–70

    MathSciNet  Google Scholar 

  60. M. Misiurewicz, On iterates of e z, Ergod. Th. & Dynam. Sys. 1, 1981, 103–106

    Article  MathSciNet  MATH  Google Scholar 

  61. J.K. Moser, C.L. Siegel, Lectures on celestial mechanics, Springer-Verlag, Grundlehren Bd. 187, 1971

    Google Scholar 

  62. R. Mañé, P. Sad, D. Sullivan, On the dynamics of rational maps, Ann.Sci.Ecole Norm.Sup., 4e serie, t.16, 1983, 193–217

    Google Scholar 

  63. C. McMullen, Families of rational maps and iterative root-finding algorithms, Annals of Math. 125, 1987, 467–493

    Article  MathSciNet  MATH  Google Scholar 

  64. P.J. Myrberg, Iteration der reellen Polynome zweiten Grades, Ann. Acad. Sci. Fennicae, A.I. no. 256, 1958

    Google Scholar 

  65. P.J. Myrberg, Iteration der reellen Polynome zweiten Grades II, Ann. Acad. Sci. Fennicae, A.I. no. 268, 1959

    Google Scholar 

  66. P.J. Myrberg, Iteration der reellen Polynome zweiten Grades HI, Ann. Acad. Sci. Fennicae, A.I. no. 336/3, 1964

    Google Scholar 

  67. H.-O. Peitgen, D. Saupe, F.v. Haeseler, Newton’s Method and Julia sets, in Dynamische Eigenschaften nichtlinearer Differenzengleichungen und ihre Anwendungen in der Ökonomie, G. Gabisch and H.v. Trotha (eds.), 1985, GMD Studien 97 preprint Univ. Bremen, 1983

    Google Scholar 

  68. H.-O. Peitgen, D. Saupe, F.v. Haeseler, Cayley’s problem and Julia sets, Math. Intell., vol.6, Nr.2, 1984, 11–20

    Article  MATH  Google Scholar 

  69. H.-O. Peitgen, P.H. Richter, The Beauty of Fractals, Springer-Verlag, 1986

    MATH  Google Scholar 

  70. H.-O. Peitgen, D. Saupe, The Science of Fractal Images, Springer-Verlag, 1988

    MATH  Google Scholar 

  71. O.Perron, Die Lehre von den Kettenbrüchen, 2 Bde, B.G. Teubner, Stuttgart, 1977

    Google Scholar 

  72. C.H.Pommerenke, On conformal mappings and iteration of rational functions, Complex Variables, 1986, Vol. 5, 117–126

    Google Scholar 

  73. C.H.Pommerenke, The Bieberbach conjecture, Math. Intell., Vol. 7, No.2, 1985, 23–25

    Article  MathSciNet  Google Scholar 

  74. C.H.Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975

    MATH  Google Scholar 

  75. P.H.Richter, H.-J.Scholz, Der Goldene Schnitt in der Natur, in Ordnung aus dem Chaos, B.O.Küppers, Piper, München, Zürich, 1987

    Google Scholar 

  76. J.F.Ritt, On the iteration of rational functions, Trans, of the AMS, Vol.21, 1920, 348–356

    Article  MathSciNet  MATH  Google Scholar 

  77. H.Rüssmann, Kleine Nenner II. Bemerkungen zur Newtonschen Methode, Nachr.Akad.Wiss.Göttingen, MathPhys.Kl., 1972,1–20

    Google Scholar 

  78. D.G.Saari, J.B.Urenko, Newton’s Method, Circle Maps, and Chaotic Motion, Amer. Math. Monthly, January 1984, 3–17

    Google Scholar 

  79. D.Saupe, Discrete versus continuous Newton’s Method: A case study, Acta Applic. Math. 13, 1988

    Google Scholar 

  80. E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen, Math.Ann., 2, 1870, 317–365

    Article  MathSciNet  Google Scholar 

  81. E.Schröder,Über iterierte Funktionen, Math.Ann., 3, 1871, 296–322

    Article  Google Scholar 

  82. M.Shishikura, On the quasi-conformal surgery of the rational functions, Ann.Sci.Ecole Norm.Sup., 4e série, t.20, 1987, 1–29

    MathSciNet  MATH  Google Scholar 

  83. M.Shub, S.Smale, On the existence of generally convergent alhorithms, J. Complexity 1, 1986, 2–11

    Article  MathSciNet  Google Scholar 

  84. T.Schneider, Einführung in die transzendenten Zahlen, Springer Verlag, Berlin Göttingen Heidelberg, 1957

    MATH  Google Scholar 

  85. C.L.Siegel, Iteration of analytic functions, Ann.Math., 43,1942,607–612

    Article  MATH  Google Scholar 

  86. S. Smale, On the complexity of algorithms of analysis, BAMS 13, 1985, 87–121

    Article  MathSciNet  MATH  Google Scholar 

  87. D.Sullivan, Quasiconformal homeomorphisms and dynamics I, Ann.Math., 122, 1985,401–418

    Article  MATH  Google Scholar 

  88. D.Sullivan, Quasiconformal homeomorphism and dynamics II,III, preprint MES

    Google Scholar 

  89. D.Sullivan, Conformai dynamical systems, Lect.Notes in Math., 1007, 1983

    Google Scholar 

  90. H.Töpfer, Über die Iteration der ganzen transzendenten Funktionen insbesondere von sin und cos, Math. Ann. 117, 1940, 65–84

    Article  Google Scholar 

  91. S.Ushiki, H.-O.Peitgen, F.v.Haeseler, Hyperbolic components of rational fractions z → λz(l + 1/z), The Theory of Dynamical Systems and its Applications to Non-linear Problems, World Sci.Publ., 1984, 61–70

    Google Scholar 

  92. J.-C.Yoccoz, Théorème de Siegel pour les polynômes quadratiques, manuscript, 1985

    Google Scholar 

  93. J.-C.Yoccoz, Conjugaison différentiate des difféomorphismes du cercle dont le nombre de rotation v/’erifie une condition diophantienne, Ann.Sc.E.N.S,4ème série, 17, 1984, 333–359

    MathSciNet  MATH  Google Scholar 

  94. J.-C.Yoccoz, C 1-conjugaison des difféomorphismes du cercle, Lect. Notes in Math., Springer-Verlag, 1007, 1983, 814–827

    Google Scholar 

  95. J.-C.Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de (C,0), CRAS Paris, t.306, Série I, 1988, p. 55–58,

    Google Scholar 

  96. E.R.Vrscay, Julia sets and Mandelbrot-like sets associated with higher order Schröder rational iteration functions: A computer assisted study, Math, of Comp. 46, Nr. 173,1986,151–169

    MathSciNet  MATH  Google Scholar 

  97. E.R.Vrscay, W.J.Gilbert, Extraneous Fixed Points, Basin Boundaries and Chaotic Dynamics for Schröder and König Rational Iteration Functions, Numer.Math. 52, 1988, 1–16

    Article  MathSciNet  MATH  Google Scholar 

  98. S.Wong, Newton’s Method and Symbolic Dynamics, Proc. Amer. Math. Soc. 91, 1984, 245–253

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Dynamische Systeme, Universität Bremen, 2800, Bremen 33, Germany

    F. v. Haeseler & H.-O. Peitgen

  2. Department of Mathematics, UCSC, Santa Cruz, CA, 95064, USA

    H.-O. Peitgen

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  1. F. v. Haeseler
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  2. H.-O. Peitgen
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Editors and Affiliations

  1. Institute for Dynamical Systems, University of Bremen, Germany

    Heinz-Otto Peitgen

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v. Haeseler, F., Peitgen, HO. (1988). Newton’s Method and Complex Dynamical Systems. In: Peitgen, HO. (eds) Newton’s Method and Dynamical Systems. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2281-5_1

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  • DOI: https://doi.org/10.1007/978-94-009-2281-5_1

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