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On the distribution of Julia sets of holomorphic maps

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Abstract

In 1965 Baker first considered the distribution of Julia sets of transcendental entire maps and proved that the Julia set of an entire map cannot be contained in any finite set of straight lines. In this paper we shall consider the distribution problem of Julia sets of meromorphic maps. We shall show that the Julia set of a transcendental meromorphic map with at most finitely many poles cannot be contained in any finite set of straight lines. Meanwhile, examples show that the Julia sets of meromorphic maps with infinitely many poles may indeed be contained in straight lines. Moreover, we shall show that the Julia set of a transcendental analytic self-map of C* can neither contain a free Jordan arc nor be contained in any finite set of straight lines.

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References

  1. Beardon A F. Iteration of Rational Functions. New York, Berlin, Heidelberg: Springer, 1991

    Book  Google Scholar 

  2. Bergweiler W. Iteration of meromorphic functions. Bull Amer Math Soc (N S), 1993, 29: 151–188

    Article  MathSciNet  Google Scholar 

  3. McMullen C. Complex Dynamics and Renormalization. Annals of Mathematics Studies Vol 135. Princeton, NJ: Princeton University Press, 1994

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Baker I N. Sets of non-normality in iteration theory. J London Math Soc, 1965, 40: 499–502

    Article  MathSciNet  Google Scholar 

  6. Baker I N. Repulsive fixpoints of entire functions. Math Z, 1968, 104: 252–256

    Article  MathSciNet  Google Scholar 

  7. Baker I N. Completely invariant domains of entire functions//Shankar H. Mathematical Essays Dedicated to A J Macintyre. Ohio: Ohio University Press, 1970: 33–35

    Google Scholar 

  8. Baker I N. The domains of normality of an entire function. Ann Acad Sci Fenn (Ser A I Math), 1975, 1: 277–283

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Baker I N. Iteration of entire functions: an introductory survey//Lectures on Complex Analysis. Singapore, New Jersey, London, Hong Kong: World Scientific, 1987: 1–17

    MATH  Google Scholar 

  11. Rådström H. On the iteration of analytic functions. Math Scand, 1953, 1: 85–92

    Article  MathSciNet  Google Scholar 

  12. Baker I N. Wandering domains for maps of the punctured plane. Ann Acad Sci Fenn (Ser A I Math), 1987, 12: 191–198

    Article  MathSciNet  Google Scholar 

  13. Keen L. Dynamics of holomorphic self-maps of C*//Drasin D, Earle C J, Gehring F W, Kra I, Marden A, et al. Holomorphic functions and moduli I. New York, Berlin, Heidelberg: Springer, 1988: 9–30

    Chapter  Google Scholar 

  14. Kotus J. Iterated holomorphic maps of the punctured plane//Kurzhanski A B, Sigmund K. Dynamical systems. Lect Notes Econ & Math Syst 287. Berlin, Heidelberg, New York: Springer, 1987: 10–29

    Chapter  Google Scholar 

  15. Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions I. Ergod Th and Dynam Sys, 1991, 11: 241–248

    Article  MathSciNet  Google Scholar 

  16. Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions II: Examples of wandering domains. J London Math Soc, 1990, 42(2): 267–278

    Article  MathSciNet  Google Scholar 

  17. Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions III: Preperiodic domains. Ergod Th and Dynam Sys, 1991, 11: 603–618

    Article  MathSciNet  Google Scholar 

  18. Baker I N, Kotus J, Lü Y. Iterates of meromorphic functions IV: Critically finite functions. Results Math, 1992, 22: 651–656

    Article  MathSciNet  Google Scholar 

  19. Devaney R L, Keen L. Dynamics of meromorphic maps with polynomial Schwarzian derivative. Ann Sci École Norm Sup, 1989, 22: 55–81

    Article  MathSciNet  Google Scholar 

  20. Stallard G M. Meromorphic functions whose Julia sets contain a free Jordan arc. Ann Acad Sci Fenn (Ser A I Math), 1993, 18: 273–298

    MathSciNet  MATH  Google Scholar 

  21. Qiao J Y. On the distribution of Julia sets//Yang C C, Wen G C, Li K Y, Chiang Y M, et al. Complex Analysis and Its Applications. Pitman Res Notes Math Ser 305. Harlow: Longman Sci Tech, 1994: 246–257

    Google Scholar 

  22. Domínguez P. Dynamics of transcendental meromorphic functions. Ann Acad Sci Fenn (Ser A I Math), 1998, 28: 225–250

    MathSciNet  MATH  Google Scholar 

  23. Nevanlinna R. Analytic Functions. Berlin-Heidelberg-New York: Springer-Verlag, 1970

    Book  Google Scholar 

  24. Rippon P J, Stallard G M. Escaping points of meromorphic functions with a finite number of poles. J d’Anal Math, 2005, 96: 225–245

    Article  MathSciNet  Google Scholar 

  25. Bhattacharyya P. Iteration of Analytic Functions [D]. London: University of London, 1969

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Chunlei Cao

  2. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, China

    Yuefei Wang

  3. Academy of Mathemtics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Yuefei Wang

Authors
  1. Chunlei Cao
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  2. Yuefei Wang
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Corresponding author

Correspondence to Yuefei Wang.

Additional information

The work was supported by NSF of China (11688101).

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Cao, C., Wang, Y. On the distribution of Julia sets of holomorphic maps. Acta Math Sci 40, 903–909 (2020). https://doi.org/10.1007/s10473-020-0401-5

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  • DOI: https://doi.org/10.1007/s10473-020-0401-5

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