Journal d’Analyse Mathématique

, Volume 96, Issue 1, pp 225–245

Escaping points of meromorphic functions with a finite number of poles

  • P. J. Rippon
  • G. M. Stallard
Article

Abstract

We establish several new properties of the escaping setI(f)={z∶fn(z)→∞ andfn(z)⇑∞ for eachn∈N} of a transcendental meromorphic functionf with a finite number of poles. By considering a subset ofI(f) where the iterates escape about as fast as possible, we show thatI(f) always contains at least one unbounded component. Also, iff has no Baker wandering domains, then the setI(f)J(f), whereJ(f) is the Julia set off, has at least one unbounded component. These results are false for transcendental meromorphic functions with infinitely many poles.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. N. Baker,Multiply connected domains of normality in iteration theory, Math. Z.81 (1963), 206–214.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    I. N. Baker,An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A22 (1976), 173–176.MATHMathSciNetGoogle Scholar
  3. [3]
    I. N. Baker,Wandering domains in the iteration of entire functions, Proc. London Math. Soc. (3)49 (1984), 563–576.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    I. N. Baker, J. Kotus and Lü Yinian,Iterates of meromorphic functions I, Ergodic Theory Dynam. Systems11 (1991), 241–248.MATHMathSciNetGoogle Scholar
  5. [5]
    W. Bergweiler,Iteration of meromorphic functions, Bull. Amer. Math. Soc.29 (1993), 151–188.MATHMathSciNetGoogle Scholar
  6. [6]
    W. Bergweiler,Invariant domains and singularities, Math. Proc. Camb. Phil. Soc.117 (1995), 525–532.MATHMathSciNetGoogle Scholar
  7. [7]
    W. Bergweiler,On the Julia set of analytic self-maps of the punctured plane, Analysis15 (1995), 251–256.MATHMathSciNetGoogle Scholar
  8. [8]
    W. Bergweiler and A. Hinkkanen,On semiconjugation of entire functions, Math. Proc. Camb. Phil. Soc.126 (1999), 565–574.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    E. F. Collingwood and A. J. Lohwater,The Theory of Cluster Sets, Cambridge University Press, 1966.Google Scholar
  10. [10]
    P. Domínguez,Connectedness properties of Julia sets of transcendental entire functions, Complex Variables32 (1997), 199–215.MATHGoogle Scholar
  11. [11]
    P. Domínguez,Dynamics of transcendental meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I Math.23 (1998), 225–250.Google Scholar
  12. [12]
    A. E. Eremenko,On the iteration of entire functions, inDynamical Systems and Ergodic Theory, Banach Center Publ., 23, Polish Scientific Publishers, Warsaw, 1989, pp. 339–345.Google Scholar
  13. [13]
    K. J. Falconer,The Geometry of Fractal Sets, Cambridge University Press, 1989.Google Scholar
  14. [14]
    M. Kisaka,On the connectivity of Julia sets of transcendental entire functions, Ergodic Theory Dynam. Systems18 (1998), 189–205.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    M. H. A. Newman,Elements of the Topology of Plane Sets of Points, Cambridge University Press, 1961.Google Scholar
  16. [16]
    P. J. Rippon,Asymptotic values of continuous functions in Euclidean space, Math. Proc. Camb. Phil. Soc.111 (1992), 309–318.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    P. J. Rippon and G. M. Stallard,On sets where iterates of a meromorphic function zip towards infinity, Bull. London Math. Soc.32 (2000), 528–536.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    P. J. Rippon and G. M. Stallard,On questions of Fatou and Eremenko, Proc. Amer. Math. Soc.133 (2005), 1109–1118.CrossRefMathSciNetGoogle Scholar
  19. [19]
    G. M. Stallard,Meromorphic functions whose Julia sets contain a free Jordan arc, Ann. Acad. Sci. Fenn. Ser. A I Math.18 (1993), 273–298.MATHMathSciNetGoogle Scholar
  20. [20]
    H. Töpfer,Über die Iteration der ganzen transzendenten Funktionen, insbesondere von sin z und cos z, Math. Ann.117 (1941), 65–84.CrossRefGoogle Scholar
  21. [21]
    G. Whyburn and E. Duda,Dynamic Topology, Springer-Verlag, Berlin, 1979.MATHGoogle Scholar

Copyright information

© The Hebrew University Magnes Press 2005

Authors and Affiliations

  • P. J. Rippon
    • 1
  • G. M. Stallard
    • 1
  1. 1.Department of Pure MathematicsThe Open UniversityMilton KeynesUK

Personalised recommendations