Acta Applicandae Mathematica

, Volume 13, Issue 1–2, pp 3–58 | Cite as

Newton's method and complex dynamical systems

  • F. v. Haeseler
  • H. O. Peitgen
Article

Abstract

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

AMS subject classifications (1980)

30D05 65H05 

Key words

Newton method iteration of rational functions Mandelbrot set polynomial-like mappings 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    L.V. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1966Google Scholar
  2. [2]
    L.V.Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, 1966Google Scholar
  3. [3]
    L. deBranges, A proof of the Bieberbach conjecture, Acta Mathematika, 154:1–2, 1985, 137–152Google Scholar
  4. [4]
    H. Brolin, Invariant sets under iteration of rational functions, Arkiv för Matematik 6, 1967, 103–141Google Scholar
  5. [5]
    P.Blanchard, Complex analytic dynamics on the Riemann sphere, Bulletin of the AMS, Vol. 11, Number 1, July 1984, 85–141Google Scholar
  6. [6]
    I.N. Baker, Wandering domains in the iteration of entire functions, Proc. London Math. Soc., 49, 1984, 563–576Google Scholar
  7. [7]
    I.N. Baker, Some entire functions with multiply connected wandering domains, Erg.Th.Dyn.Sys. 5, 1985, 163–169Google Scholar
  8. [8]
    I.N. Baker, An entire function which has wandering domains, J. Austral. Math. Soc., 22, 1976, 173–176Google Scholar
  9. [9]
    B. Barna, Über das Newtonsche Verfahren zur Annäherung von Wurzeln algebraischer Gleichungen, Publ.Math. Debrecen, 2, 1951, 50–63Google Scholar
  10. [10]
    Über die Divergenzpunkte des Newtonschen Verfahrens zur Bestimmung von Wurzeln algebraischer Gleichungen, I, II, III, Publ.Math. Debrecen, 3, 1953, 109–118, Publ.Math. Debrecen, 4, 1956, 384–397, Publ.Math. Debrecen, 8, 1961, 193–207Google Scholar
  11. [11]
    B. Bielefeld, Y.Fisher, F.V.Haeseler, Computing the Laurent Series of the map ψ: C\ D → C\D, Max-Planck-Institut für Mathematik, Bonn, MPI/88-46Google Scholar
  12. [12]
    D. Braess, Über die Einzugsgebiete der Nullstellen von Polynomen beim Newton-Verfahren, Numer. Math., 29, 1977, 123–132Google Scholar
  13. [13]
    B. Branner, J.H. Hubbard, Iteration of cubic polynomials I, Acta Mathematica 160, 1988, 143–206Google Scholar
  14. [14]
    Boettcher, Bulletin of the Kasan Math. Society, vol. 14, 1905, 176Google Scholar
  15. [15]
    L. Bers, On Sullivan's proof of the finiteness theorem, Amer. J. Math., 109, No. 5, 1987, 833–852Google Scholar
  16. [16]
    A.D. Brjuno, Convergence of transformations of differential equations to normal forms, Dokl.Akad.Nauk.URSS 165, 1965, 987–989Google Scholar
  17. [17]
    A.D. Brjuno Analytic form of differential equations, Trans. Moscow Math.Soc. 25, 1971, 131–288; 26, 1972, 199–239Google Scholar
  18. [18]
    C. Camacho, On the local structure of conformal mappings and holomorphic vector fields in C2, Société Mathématique de France, Astérisque 59–60 (1978), 83–93Google Scholar
  19. [19]
    A. Cayley, The Newton-Fourier imaginary problem, Amer. J. Math. II, 1879, 97Google Scholar
  20. [20]
    A. Cayley, Application of the Newton-Fourier Mathod to an imaginary root of equation, Quart. J. of Pure and App. Math. XVI, 1879, 179–185Google Scholar
  21. [21]
    A. Cayley, Sur les racines d'une équation algébrique, CRAS 110, 1890, 215–218Google Scholar
  22. [22]
    C. Carathéodory, Über die Begrenzung einfach zusammenhängender Gebiete, Math. Ann. 73, 1913, 323–370Google Scholar
  23. [23]
    M.Cosnard, C.Masse, Convergence presque partout de la méthode de Newton, CRAS Paris, t. 297 (14 novembre 1983), Série I-549Google Scholar
  24. [24]
    H. Cremer, Zum Zentrumsproblem, Math.Ann., 98, 1928, 151–163Google Scholar
  25. [25]
    H. Cremer, Über die Häufigkeit der Nichtzentren, Math.Ann., 115, 1938, 573–580Google Scholar
  26. [26]
    H. Cremer, Über die Iteration rationaler Funktionen, Jber. d. Dt. Math.-Verein, 33, 1925, 185–210Google Scholar
  27. [27]
    H. Cremer, Über Konvergenz und Zentrumproblem, Math.Ann. 110, 1935, 739–744Google Scholar
  28. [28]
    J. Curry, L. Garnett, D. Sullivan, On the iteration of rational functions: Computer experiments with Newton's Method, Commun. Math. Phys., 91, 1983,267–277Google Scholar
  29. [29]
    R.L. Devaney, Introduction to Chaotic Dynamical Systems, Benjamin-Cummings, Menlo Park, 1986Google Scholar
  30. [30]
    A.Douady, Systèmes dynamiques holomorphes, exposé no. 599, Séminaire N.Bourbaki 1982/83, Astérisque 105–106, 1983, 39–63Google Scholar
  31. [31]
    A.Douady, Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, 39éme anneé, 1986–87, no677Google Scholar
  32. [32]
    A.Douady, Chirugie sur les applications holomorphes, ICM 86, BerkeleyGoogle Scholar
  33. [33]
    A. Douady, J.H. Hubbard, Itération des polynômes quadratiques complexes, C.R.Acad.Sci.Paris, 294, 1982, 123–126Google Scholar
  34. [34]
    A.Douady, J.H.Hubbard, On the dynamics of polynomial-like mappings, Ann.Sci.École Norm.Sup., 4e série, t. 18,1985, 287–343Google Scholar
  35. [35]
    A.Douady, J.H. Hubbard, Etude dynamiques des polynômes complexes I,II, Publ.Math.d'Orsay, 84-02, 1984, 85-02, 1985Google Scholar
  36. [36]
    P. Fatou, Sur les équations fonctionelles, Bull.Soc.Math.France, 47, 1919, 161–271; 48, 1920, 33–94 and 208–304Google Scholar
  37. [37]
    P. Fatou, Sur l'itération des fonctions transcendantes entières, Acta Math. 47, 1926, 337–370Google Scholar
  38. [38]
    M. Feigenbaum, Quantitative Universality for a Class of Nonlinear Transformations, J. Statistical Physics 19, 1978, 25–52Google Scholar
  39. [39]
    S. Großmann, S. Thomae, Invariant Distributions and Stationary Correlation Functions of One-Dimensional Discrete Processes, Zeitsch. f. Naturforschg. 32a, 1977, 1353–1363Google Scholar
  40. [40]
    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, 95–123Google Scholar
  41. [41]
    F.v.Haeseler, Über sofortige Attraktionsgebiete superattraktiver Zykel, Thesis, Universität Bremen, 1985Google Scholar
  42. [42]
    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, 563–612Google Scholar
  43. [43]
    M.R. Herman, Exemples de fractions rationelles ayant une orbit dense sur la sphère de Riemann, Bull.Soc.Math.France, 112, 1984, 93–142Google Scholar
  44. [44]
    M.R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ.Math. IHES 49, 1979, 5–233Google Scholar
  45. [45]
    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, 138–184Google Scholar
  46. [46]
    M. Hurley, C. Martin, Newton's Algorthm and Chaotic Dynamical Systems, SIAM J. Math. Anal., Vol. 15, No. 2, March 1984, 238–252Google Scholar
  47. [47]
    M. Hurley, Multiple attractors in Newton's Method, Ergod. Th. & Dynam. Sys., 6, 1986, 561–569Google Scholar
  48. [48]
    G. Julia, Mémoire sur l'itération des fonctions rationelles, J. de. Math. pures et appliquées, ser. 8. 1, 1918, 47–245Google Scholar
  49. [49]
    M.S. Lattés, Sur l'itération des substitutions rationelles et les fonctions de Poincaré, C.R.A.S., 166, 1918, 26–28Google Scholar
  50. [50]
    O.Letho,K. Virtanen, Quasiconformal Mappings in the Plane, Springer Verlag, 1973Google Scholar
  51. [51]
    T.Y. Li, J.A. Yorke, Period three implies chaos, Amer.Math.Monthly, 82, 1975, 985–992Google Scholar
  52. [52]
    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, 249–259Google Scholar
  53. [53]
    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, 1984Google Scholar
  54. [54]
    J.Martinet, Normalisation des champ de vecteurs, d'après Brjuno, Sém. Bourbaki, exp. 564, Lect. Notes in Math., Springer-Verlag 901, 1981, 55–70Google Scholar
  55. [55]
    M. Misiurewicz, On iterates ofe z, Ergod. Th. & Dynam. Sys. 1, 1981, 103–106Google Scholar
  56. [56]
    J.K.Moser, C.L. Siegel, Lectures on celestial mechanics, Springer-Verlag, Grundlehren Bd. 187, 1971Google Scholar
  57. [57]
    R. Mañé, P. Sad, D. Sullivan, On the dynamics of rational maps, Ann.Sci.Ecole Norm.Sup., 4e serie, t. 16,1983, 193–217Google Scholar
  58. [58]
    C. McMullen, Families of rational maps and iterative root-finding algorithms, Annals of Math. 125, 1987, 467–493Google Scholar
  59. [59]
    P.J. Myrberg, Iteration der reellen Polynome zweiten Grades, Ann. Acad. Sci. Fennicae, A.I. no. 256, 1958Google Scholar
  60. [60]
    P.J. Myrberg, Iteration der reellen Polynome zweiten Grades II, Ann. Acad. Sci. Fennicae, A.I. no. 268, 1959Google Scholar
  61. [61]
    P.J. Myrberg, Iteration der reellen Polynome zweiten Grades III, Ann. Acad. Sci. Fennicae, A.I. no. 336/3, 1964Google Scholar
  62. [62]
    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, 1983Google Scholar
  63. [63]
    H.-O. Peitgen, D. Saupe, F.v. Haeseler, Cayley's problem and Julia sets, Math. Intell., vol. 6, Nr. 2, 1984, 11–20Google Scholar
  64. [64]
    H.-O.Peitgen, P.H. Richter, The Beauty of Fractals, Springer-Verlag, 1986Google Scholar
  65. [65]
    H.-O.Peitgen, D.Saupe, The Science of Fractal Images, Springer-Verlag, 1988Google Scholar
  66. [66]
    O. Perron, Die Lehre von den Kettenbrüchen, 2 Bde, B.G. Teubner, Stuttgart, 1977Google Scholar
  67. [67]
    C.H.Pommerenke, On conformal mappings and iteration of rational functions, Complex Variables, 1986, Vol. 5, 117–126Google Scholar
  68. [68]
    C.H. Pommerenke, The Bieberbach conjecture, Math. Intell., Vol. 7, No.2, 1985, 23–25Google Scholar
  69. [69]
    C.H. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975Google Scholar
  70. [70]
    P.H.Richter, H.-J.Scholz, Der Goldene Schnitt in der Natur, in Ordnung aus dem Chaos, B.O.Küppers, Piper, München, Zürich, 1987Google Scholar
  71. [71]
    J.F. Ritt, On the iteration of rational functions, Trans. of the AMS, Vol.21, 1920, 348–356Google Scholar
  72. [72]
    H.Rüssmann, Kleine Nenner II. Bemerkungen zur Newtonschen Methode, Nachr.Akad.Wiss.Göttingen, Math.Phys.Kl., 1972,1–20Google Scholar
  73. [73]
    D.G.Saari, J.B.Urenko, Newton's Method, Circle Maps, and Chaotic Motion, Amer. Math. Monthly, January 1984, 3–17Google Scholar
  74. [74]
    D. Saupe, Discrete versus continuous Newton's Method: A case study, Acta Applic. Math. 13, 1988Google Scholar
  75. [75]
    E. Schröder, Über unendlich viele Algorithmen zur Auflösung der Gleichungen, Math.Ann., 2, 1870, 317–365Google Scholar
  76. [76]
    E. Schröder, Über iterierte Funktionen, Math.Ann., 3, 1871, 296–322Google Scholar
  77. [77]
    M. Shishikura, On the quasi-conformal surgery of the rational functions, Ann.Sci.Ecole Norm.Sup., 4e série, t.20, 1987, 1–29Google Scholar
  78. [78]
    M. Shub, S. Smale, On the existence of generally convergent alhorithms, J. Complexity 1, 1986, 2–11Google Scholar
  79. [79]
    T. Schneider, Einführung in die transzendenten Zahlen, Springer Verlag, Berlin Göttingen Heidelberg, 1957Google Scholar
  80. [80]
    C.L. Siegel, Iteration of analytic functions, Ann.Math., 43, 1942, 607–612Google Scholar
  81. [81]
    S. Smale, On the complexity of algorithms of analysis, BAMS 13, 1985, 87–121Google Scholar
  82. [82]
    D. Sullivan, Quasiconformal homeomorphisms and dynamics I, Ann.Math., 122, 1985, 401–418Google Scholar
  83. [83]
    D.Sullivan, Quasiconformal homeomorphism and dynamics, II,III, preprint IHESGoogle Scholar
  84. [84]
    D.Sullivan, Conformal dynamical systems, Lect.Notes in Math., 1007, 1983Google Scholar
  85. [85]
    H. Töpfer, Über die Iteration der ganzen transzendenten Funktionen insbesondere von sin und cos, Math. Ann. 117, 1940, 65–84Google Scholar
  86. [86]
    S.Ushiki, H.-O.Peitgen, F.v.Haeseler, Hyperbolic omponents of rational fractionsz→λz(1+1/z), The Theory of Dynamical Systems and its Applications to Non-linear Problems, World Sci.Publ., 1984, 61–70Google Scholar
  87. [87]
    J.-C.Yoccoz, Théorème de Siegel pour les polynômes quadratiques, manuscript, 1985Google Scholar
  88. [88]
    J.-C. Yoccoz, Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation v/'erifie une condition diophantienne, Ann.Sc.E.N.S,4ème série, 17, 1984, 333–359Google Scholar
  89. [89]
    J.-C.Yoccoz,C 1-conjugaison des difféomorphismes du cercle, Lect. Notes in Math., Springer-Verlag, 1007, 1983, 814–827Google Scholar
  90. [90]
    J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de (C, 0), CRAS Paris, t.306, Série I, 1988, p. 55–58,Google Scholar
  91. [91]
    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, 151–169Google Scholar
  92. [92]
    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, 1–16Google Scholar
  93. [93]
    S. Wong, Newton's Method and Symbolic Dynamics, Proc. Amer. Math. Soc. 91, 1984, 245–253Google Scholar

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • F. v. Haeseler
    • 1
  • H. O. Peitgen
    • 1
    • 2
  1. 1.Institut für Dynamische Systeme Universität BremenBremen 33FRG
  2. 2.Department of Mathematics UCSCSanta CruzUSA

Personalised recommendations