Periodic Fatou Components and Singularities of the Inverse Function

  • Walter Bergweiler
Conference paper
Part of the NATO Science Series II: Mathematics, Physics and Chemistry book series (NAII, volume 147)

Abstract

The periodic components of the Fatou set of a rational or entire function are closely connected to the singularities of the inverse function. This article is a survey of classical as well as more recent results concerning this relation.

Keywords

complex dynamics rational or entire functions periodic Fatou components singularities of inverse functions 

Mathematics Subject Classification (2000)

30D05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Baker, I.N. (1970) Limit functions and sets of non-normality in iteration theory, Ann. Acad. Sci. Fenn., Ser. A, I. Math., Vol. 467, pp. 3–11.MATHGoogle Scholar
  2. [2]
    Baker, I.N. (1995) The domains of normality of an entire function, Ann. Acad. Sci. Fenn., Ser. A, I. Math., Vol. 1, pp. 277–283.Google Scholar
  3. [3]
    Baker, I.N. (1984) Wandering domains in the iteration of entire functions, Proc. London Math. Soc., (3), Vol. 49, pp. 563–576.MATHMathSciNetGoogle Scholar
  4. [4]
    Baker, I.N. and Domínguez, P. (1999) Boundaries of unbounded Fatou components of entire functions, Ann. Acad. Sci. Fenn., Ser. A, I. Math., Vol. 24, pp. 437–464.Google Scholar
  5. [5]
    Baker, I.N. and Weinreich, J. (1991) Boundaries which arise in the iteration of entire functions, Rev. Roumaine Math. Pures Appl., Vol. 36, pp. 413–420.MathSciNetGoogle Scholar
  6. [6]
    Barański, K. and Fagella, N. (2001) Univalent Baker domains, Nonlinearity, Vol. 14, pp. 411–429.MathSciNetGoogle Scholar
  7. [7]
    Bargmann, D. (2001) Normal families of covering maps, J. Anal. Math., Vol. 85, pp. 291–306.MATHMathSciNetGoogle Scholar
  8. [8]
    Bargmann, D. (1999) Iteration of inner functions and boundaries of components of the Fatou set, Berichtsreihe Math. Sem. Kiel, Preprint 99–4.Google Scholar
  9. [9]
    Beardon, A.F. (1991) Iteration of rational functions, Springer-Verlag, New York.Google Scholar
  10. [10]
    Bergweiler, W. (1993) Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.), Vol. 29, pp. 151–188.MATHMathSciNetGoogle Scholar
  11. [11]
    Bergweiler, W. (1993) Newton’s method and a class of meromorphic functions without wandering domains, Ergodic Theory Dynam. Systems, Vol. 13, pp. 231–247.MATHMathSciNetGoogle Scholar
  12. [12]
    Bergweiler, W. (1995) Invariant domains and singularities, Math. Proc. Cambridge Philos. Soc., Vol. 117, pp. 525–532.MATHMathSciNetGoogle Scholar
  13. [13]
    Bergweiler, W. (2001) Singularities in Baker domains, Comput. Methods and Funct. Theory, Vol. 1, pp. 41–49.MATHMathSciNetGoogle Scholar
  14. [14]
    Bergweiler, W. (2002) On the number of critical points in parabolic basins, Ergodic Theory Dynam. Systems, Vol. 22, pp. 655–669.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    Bergweiler, W. (2002) On proper analytic maps with one critical point, Value Distribution Theory and Complex Dynamics edited by W. Cherry & C.-C. Yang, Contemp. Math., Vol. 303, AMS, Providence, pp. 1–6.Google Scholar
  16. [16]
    Bergweiler, W., Haruta, M., Kriete, H., Meier, H. and Terglane, N. (1993) On the limit functions of iterates in wandering domains, Ann. Acad. Sci. Fenn., Ser. A, I. Math., Vol. 18, pp. 369–375.MathSciNetGoogle Scholar
  17. [17]
    Bergweiler, W. and Morosawa, S. (2002) Semihyperbolic entire functions, Nonlinearity, Vol. 15, pp. 1673–1684.CrossRefMathSciNetGoogle Scholar
  18. [18]
    Bergweiler, W. and Rohde, S. (1995) Omitted values in domains of normality, Proc. AMS, Vol. 123, pp. 1857–1858.MathSciNetGoogle Scholar
  19. [19]
    Bergweiler, W. and Terglane, N. (1996) Weakly repelling fixpoints and the connectivity of wandering domains, Trans. Amer. Math. Soc., Vol. 348, pp. 1–12.CrossRefMathSciNetGoogle Scholar
  20. [20]
    Bergweiler, W. and Terglane, N. (1998) On the zeros of solutions of linear differential equations of the second order, J. Lond. Math. Soc. (2), Vol. 58, pp. 311–330.MathSciNetGoogle Scholar
  21. [21]
    Berteloot, F. and Mayer, V. (2001) Rudiments de dynamique holomorphe, Cours Spécialisés 7, Soc. Math. France, Paris.Google Scholar
  22. [22]
    Bolsch, A. (1997) Iteration of meromorphic functions with countably many essential singularities, Dissertation, Berlin.Google Scholar
  23. [23]
    Bolsch, A. (1999) Periodic Fatou components of meromorphic functions, Bull. London Math. Soc., Vol. 31, pp. 543–555.CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    Bonfert, P. (1997) On iteration in planar domains, Mich. Math. J., Vol. 44, pp. 47–68.MATHMathSciNetGoogle Scholar
  25. [25]
    Bu, X. and Epstein, A.L. (2002) A parabolic Pommerenke-Levin-Yoccoz in-equality, Fund. Math., Vol. 172, pp. 249–289.MathSciNetGoogle Scholar
  26. [26]
    Carleson, L. and Gamelin, T.W. (1993) Complex dynamics, Springer-Verlag, New York.Google Scholar
  27. [27]
    Douady, A. (1987) Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, Vol. 1986/87, Astérisque, Vol. 152/153, pp. 151–172.MathSciNetGoogle Scholar
  28. [28]
    Eremenko, A.E. and Lyubich, M.Yu. (1987) Examples of entire functions with pathological dynamics, J. London Math. Soc. (2), Vol. 36, pp. 458–468.MathSciNetGoogle Scholar
  29. [29]
    Eremenko, A.E. and Lyubich, M.Yu. (1990) The dynamics of analytic transforms, Leningrad Math. J., Vol. 1, pp. 563–634.MathSciNetGoogle Scholar
  30. [30]
    Eremenko, A.E. and Lyubich, M.Yu. (1992) Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, Vol. 42, pp. 989–1020.MathSciNetGoogle Scholar
  31. [31]
    Fatou, P. (1919,1920) Sur les équations fonctionelles, Bull. Soc. Math. France, Vol. 47, pp. 161–271; Vol. 48, pp. 33–94, 208–314.MathSciNetGoogle Scholar
  32. [32]
    Fatou, P. (1926) Sur ľitération des fonctions transcendantes entières, Acta Math., Vol. 47, pp. 337–360.MATHGoogle Scholar
  33. [33]
    Geyer, L. (2001) Siegel discs, Herman rings and the Arnold family, Trans. Amer. Math. Soc., Vol. 353, pp. 3661–3683.CrossRefMATHMathSciNetGoogle Scholar
  34. [34]
    Goldberg, L.R. and Keen, L. (1986) A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems, Vol. 6, pp. 183–192.MathSciNetGoogle Scholar
  35. [35]
    Herman, M. (1979) Sur la conjugaison di érentiable des di éomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math., Vol. 49, pp. 5–233.MATHMathSciNetGoogle Scholar
  36. [36]
    Herman, M. (1985) Are there critical points on the boundary of singular domains?, Comm. Math. Phys., Vol. 99, pp. 593–612CrossRefMATHMathSciNetGoogle Scholar
  37. [37]
    Herring, M. (1998) Mapping properties of Fatou components, Ann. Acad. Sci. Fenn. Ser. A, I. Math., Vol. 23, pp. 263–274.MATHMathSciNetGoogle Scholar
  38. [38]
    Hinkkanen, A. (1992) Iteration and the zeros of the second derivative of a meromorphic function, Proc. London Math. Soc. (3), Vol. 65, pp. 629–650.MATHMathSciNetGoogle Scholar
  39. [39]
    Julia, G. (1918,1968) Sur ľitération des fonctions rationelles, J. Math. Pures Appl., (7), Vol. 4, pp. 47–245; Œuvres de Gaston Julia, Vol. I, Gauthier-Villars, Paris.Google Scholar
  40. [40]
    Kisaka, M. (1998) On the connectivity of Julia sets of transcendental entire functions, Ergodic Theory Dynam. Systems, Vol. 18, pp. 189–205.CrossRefMATHMathSciNetGoogle Scholar
  41. [41]
    König, H. (1999) Conformal conjugacies in Baker domains, J. London Math. Soc. (2), Vol. 59, pp. 153–170.MATHMathSciNetGoogle Scholar
  42. [42]
    Mañné, R. (1993) On a theorem of Fatou, Bol. Soc. Bras. Mat., Vol. 24, pp. 1–11.MathSciNetGoogle Scholar
  43. [43]
    Milnor, J. (1999) Dynamics in one complex variable, Vieweg, Braunschweig, Wiesbaden.Google Scholar
  44. [44]
    Morosawa, S. (1999) An example of cyclic Baker domains, Mem. Fac. Sci. Kochi Univ. Ser. A Math., Vol. 20, pp. 123–126.MATHMathSciNetGoogle Scholar
  45. [45]
    Morosawa, S., Nishimura, Y., Taniguchi, M. and Ueda, T. (2000) Holomorphic dynamics, Cambridge Stud. Adv. Math., Vol. 66, Cambridge Univ. Press, Cambridge.Google Scholar
  46. [46]
    Rempe, L. (2002) An answer to a question of Herman, Baker and Rippon concerning Siegel disks, Berichtsreihe Math. Sem. Kiel, Preprint 02–7.Google Scholar
  47. [47]
    Rippon, P.J. (1994) On the boundaries of certain Siegel discs, C. R. Acad. Sci. Paris Sér. I. Math., Vol. 319, pp. 821–826.MATHMathSciNetGoogle Scholar
  48. [48]
    Rippon, P.J. and Stallard, G.M. (1999) Families of Baker domains, I. Nonlinearity, Vol. 12, pp. 1005–1012.CrossRefMathSciNetGoogle Scholar
  49. [49]
    Rippon, O.J. (1999) Families of Baker domains, II. Conform. Geom. Dyn., Vol. 3, pp. 67–78.CrossRefMATHMathSciNetGoogle Scholar
  50. [50]
    Rogers, J.T. (1995) Critical points on the boundaries of Siegel disks, Bull. Amer. Math. Soc. (N.S.), Vol. 32, pp. 317–321.MATHMathSciNetGoogle Scholar
  51. [51]
    Rogers, J.T. (1998) Recent results on the boundaries of Siegel disks, Progress in holomorphic dynamics, Pitman Res. Notes Math. Ser., Vol. 387, Longman, Harlow, pp. 41–49.Google Scholar
  52. [52]
    Shishikura, M. (1991) On the parabolic bifurcation of holomorphic maps, Dynamical Systems and Related Topics (Nagoya, 1990), Adv. Ser. Dyn. Syst., Vol. 9, World Sci., River Edge, NJ, pp. 478–486.Google Scholar
  53. [53]
    Shishikura, M. and Tan, L. (2000) An alternative proof of R. Mañné’s theorem on non-expanding Julia sets. The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., Vol. 274, Cambridge Univ. Press, Cambridge, pp. 265–279.Google Scholar
  54. [54]
    Siegel, C.L. (1942) Iteration of analytic functions, Ann. Math., Vol. 43, pp. 607–612.MATHGoogle Scholar
  55. [55]
    Stallard, G.M. (1991) A class of meromorphic functions with no wandering domains, Ann. Acad. Sci. Fenn. Ser. A, I. Math., Vol. 16, pp. 211–226.MathSciNetGoogle Scholar
  56. [56]
    Steinmetz, N. (1993) Rational iteration, Walter de Gruyter, Berlin.Google Scholar
  57. [57]
    Sullivan, D. (1985) Quasiconformal homeomorphisms and dynamics I, Solution of the Fatou-Julia problem on wandering domains. Ann. Math., Vol. 122, pp. 401–418.MATHMathSciNetGoogle Scholar
  58. [58]
    Töpfer, H. (1939) Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sin z und cos z, Math. Ann., Vol. 117, pp. 65–84.MATHMathSciNetGoogle Scholar
  59. [59]
    Zakeri, S. (1999) Dynamics of cubic Siegel polynomials, Comm. Math. Phys., Vol. 206, pp. 185–233.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Walter Bergweiler
    • 1
  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany

Personalised recommendations