Skip to main content
SpringerLink
Log in

Escaping points of entire functions of small growth

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Baker I.N. (1958). Zusammensetzungen ganzer Funktionen. Math. Z. 69: 121–163

    Article  MATH  MathSciNet  Google Scholar 

  2. Baker I.N. (1981). The iteration of polynomials and transcendental entire functions. J. Aust. Math. Soc. Ser. A 30: 483–495

    MATH  Google Scholar 

  3. Baker I.N. (1984). Wandering domains in the iteration of entire functions. Proc. Lond. Math. Soc. 49(3): 563–576

    Article  MATH  Google Scholar 

  4. Baker I.N. (1985). Some entire functions with multiply-connected wandering domains. Ergod. Theory Dyn. Syst. 5: 163–169

    Article  MATH  Google Scholar 

  5. Baker I.N. (2001). Dynamics of slowly growing entire functions. Bull. Aust. Math. Soc. 63: 367–377

    MATH  Google Scholar 

  6. Barański K. (2007). Trees and hairs for some hyperbolic entire maps of finite order. Math. Z. 257: 33–59

    Article  MATH  MathSciNet  Google Scholar 

  7. Bergweiler W. (1993). Iteration of meromorphic functions. Bull. Am. Math. Soc. New Ser. 29: 151–188

    Article  MATH  MathSciNet  Google Scholar 

  8. Bergweiler W. and Eremenko A.E. (2000). Entire functions of slow growth whose Julia set coincides with the plane. Ergod. Theory Dyn. Syst. 20: 1577–1582

    Article  MATH  MathSciNet  Google Scholar 

  9. Bergweiler W. and Eremenko A.E. (2001). Correction to the paper: Entire functions of slow growth whose Julia set coincides with the plane. Ergodic Theory Dyn. Syst. 21: 637

    Article  MATH  MathSciNet  Google Scholar 

  10. Bergweiler W. and Hinkkanen A. (1999). On semiconjugation of entire functions. Math. Proc. Camb. Philos. Soc. 126: 565–574

    Article  MATH  MathSciNet  Google Scholar 

  11. Boyd D.A. (2002). An entire function with slow growth and simple dynamics. Ergod. Theory Dyn. Syst. 22: 317–322

    Article  MATH  Google Scholar 

  12. Cartwright M.L. (1962). Integral Functions. Cambridge Tracts in Mathematics and Mathematical Physics, vol 44. Cambridge University Press, London

    Google Scholar 

  13. Eremenko A.E. (1989). On the iteration of entire functions, Dynamical systems and ergodic theory, Banach Cent. Publ. 23. Polish Scientific Publishers, Warsaw, 339–345

    Google Scholar 

  14. Fatou P. (1926). Sur l’itération des fonctions transcendantes entières. Acta Math. 47: 337–360

    Article  MATH  MathSciNet  Google Scholar 

  15. Hayman W.K. (1974). The local growth of power series: a survey of the Wiman–Valiron method. Can. Math. Bull. 17: 317–358

    MATH  MathSciNet  Google Scholar 

  16. Hinkkanen, A.: Entire functions with bounded Fatou components. In: Transcendental Dynamics and Complex Analysis. LMS Lecture Note Series, no. 348, pp. 187–216. Cambridge University Press, London (2008, to appear)

  17. Hinkkanen, A., Miles, J.: Growth conditions for entire functions with only bounded Fatou components. J. Anal. Math. (to appear)

  18. Markushevich A.I. (1965). Theory of Functions of a Complex Variable, vol II. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  19. Rempe L. (2007). On a question of Eremenko concerning escaping components of entire functions. Bull. Lond. Math. Soc. 39: 661–666

    Article  MATH  MathSciNet  Google Scholar 

  20. Rippon P.J. and Stallard G.M. (2000). On sets where iterates of a meromorphic function zip towards infinity. Bull. Lond. Math. Soc. 32: 528–536

    Article  MATH  MathSciNet  Google Scholar 

  21. Rippon P.J. and Stallard G.M. (2005). On questions of Fatou and Eremenko. Proc. Am. Math. Soc. 133: 1119–1126

    Article  MATH  MathSciNet  Google Scholar 

  22. Rippon P.J. and Stallard G.M. (2005). Escaping points of meromorphic functions with a finite number of poles. J. Anal. Math. 96: 225–245

    Article  MATH  MathSciNet  Google Scholar 

  23. Rippon, P.J., Stallard, G.M.: Functions of small growth with no unbounded Fatou components. J. Anal. Math. arXiv:0801.3610 (to appear)

  24. Rottenfußer, G., Rückert, J., Rempe, L., Schleicher, D.: Dynamic rays of bounded-type entire functions (preprint). arXiv:0704.3213

  25. Stallard G.M. (1993). The iteration of entire functions of small growth. Math. Proc. Camb. Philos. Soc. 114: 43–55

    Article  MATH  MathSciNet  Google Scholar 

  26. Titchmarsh E.C. (1939). The Theory of Functions. Oxford University Press, New York

    MATH  Google Scholar 

  27. Wiman A. (1903). Über die angenäherte Darstellung von ganzen Funktionen. Ark. Mat. Astr. o. Fysik 1: 105–111

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK

    P. J. Rippon & G. M. Stallard

Authors
  1. P. J. Rippon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. M. Stallard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. M. Stallard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rippon, P.J., Stallard, G.M. Escaping points of entire functions of small growth. Math. Z. 261, 557–570 (2009). https://doi.org/10.1007/s00209-008-0339-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-008-0339-0

Keywords

Search

Navigation