Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 551–572 | Cite as

Escape rate and Hausdorff measure for entire functions



The escaping set of an entire function is the set of points that tend to infinity under iteration. We consider subsets of the escaping set defined in terms of escape rates and obtain upper and lower bounds for the Hausdorff measure of these sets with respect to certain gauge functions.

Mathematics Subject Classification (2000)

Primary 37F10 Secondary 30D05 30D15 



We thank Lasse Rempe and the referee for a great number of very helpful comments and suggestions.


  1. 1.
    Ahlfors, L.: Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen. Acta Soc. Sci. Fenn. Nova Ser. A. 1(9) (1930)Google Scholar
  2. 2.
    Baker, I.N.: The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser A I Math. 1, 277–283 (1975)MathSciNetMATHGoogle Scholar
  3. 3.
    Barański, K.: Hausdorff dimension of hairs and ends for entire maps of finite order. Math. Proc. Camb. Philos. Soc. 145, 719–737 (2008)MATHCrossRefGoogle Scholar
  4. 4.
    Barański, K., Karpińska, B., Zdunik, A.: Hyperbolic dimension of Julia sets of meromorphic maps with logarithmic tracts. Int. Math. Res. Not. 615–624 (2009)Google Scholar
  5. 5.
    Bergweiler, W.: Iteration of meromorphic functions. Bull. Am. Math. Soc. (N.S.) 29, 151–188 (1993)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bergweiler, W.: Karpińska’s paradox in dimension 3. Duke Math. J. 154, 599–630 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bergweiler, W., Hinkkanen, A.: On semiconjugation of entire functions. Math. Proc. Camb. Philos. Soc. 126, 565–574 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bergweiler, W., Karpińska, B.: On the Hausdorff dimension of the Julia set of a regularly growing entire function. Math. Proc. Camb. Philos. Soc. 148, 531–551 (2010)MATHCrossRefGoogle Scholar
  9. 9.
    Bergweiler, W., Karpińska, B., Stallard, G.M.: The growth rate of an entire function and the Hausdorff dimension of its Julia set. J. Lond. Math. Soc. 80, 680–698 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bergweiler, W., Rippon, P.J., Stallard, G.M.: Dynamics of meromorphic functions with direct or logarithmic singularities. Proc. Lond. Math. Soc. 97, 368–400 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bishop, C.J.: A transcendental Julia set of dimension \(1\) (preprint)Google Scholar
  12. 12.
    Devaney, R.L., Krych, M.: Dynamics of exp(z). Ergodic Theory Dynam. Syst. 4, 35–52 (1984)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    de Guzmán, M.: Real variable methods in Fourier analysis. North-Holland Mathematics Studies, vol. 46, Amsterdam, New York (1981)Google Scholar
  14. 14.
    Eremenko, A.E.: On the iteration of entire functions. In: Dynamical systems and ergodic theory, vol. 23, pp. 339–345. Banach Center Publ., Warsaw (1989)Google Scholar
  15. 15.
    Eremenko, A.E., Lyubich, MYu.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier 42, 989–1020 (1992)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Falconer, K.: Fractal geometry—mathematical foundations and applications. Wiley, Chichester (1997)Google Scholar
  17. 17.
    Karpińska, B., Urbański, M.: How points escape to infinity under exponential maps. J. Lond Math. Soc. 73(2), 141–156 (2006)MATHCrossRefGoogle Scholar
  18. 18.
    Mayer, J.C.: An explosion point for the set of endpoints of the Julia set of \(\lambda \exp (z)\). Ergodic Theory Dynam. Syst. 10, 177–183 (1990)CrossRefGoogle Scholar
  19. 19.
    McMullen, C.: Area and Hausdorff dimension of Julia sets of entire functions. Trans. Am. Math. Soc. 300, 329–342 (1987)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Nevanlinna, R.: Eindeutige analytische Funktionen. Springer, Berlin, Heidelberg (1953)MATHCrossRefGoogle Scholar
  21. 21.
    Peter, J.: Hausdorff measure of Julia sets in the exponential family. J. Lond Math. Soc. 82(2), 229–255 (2010)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Peter, J.: Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order. Ergodic Theory Dynam. Systems. Available on CJO (2011). doi: 10.1017/S0143385711000745
  23. 23.
    Pólya, G., Szegö, G.: Aufgaben und Lehrsätze aus der Analysis I. Springer, Berlin (1925)MATHGoogle Scholar
  24. 24.
    Przytycki, F., Urbański, M.: Fractals in the plane: ergodic theory methods. London Mathematical Society Lecture Note Series 371. Cambridge University Press, Cambridge (2012)Google Scholar
  25. 25.
    Rempe, L.: Hyperbolic dimension and radial Julia sets of transcendental functions. Proc. Am. Math. Soc. 137, 1411–1420 (2009)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Rempe, L.: Hyperbolic entire functions with full hyperbolic dimension and approximation by Eremenko-Lyubich functions (preprint), arXiv:1106.3439v2 [math.CV]Google Scholar
  27. 27.
    Rempe, L., Rippon, P.J., Stallard, G.M.: Are Devaney hairs fast escaping? J. Differ. Equ. Appl. 16, 739–762 (2010)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Rempe, L., Stallard, G.M.: Hausdorff dimensions of escaping sets of transcendental entire functions. Proc. Am. Math. Soc. 138, 1657–1665 (2010)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Rippon, P.J., Stallard, G.M.: On questions of Fatou and Eremenko. Proc. Am. Math. Soc. 133, 1119–1126 (2005)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Rippon, P.J., Stallard, G.M.: Slow escaping points of meromorphic functions. Trans. Am. Math. Soc. 363, 4171–4201 (2011)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rippon, P.J., Stallard, G.M.: Fast escaping points of entire functions. Proc. Lond. Math. Soc. 105, 787–820 (2012)Google Scholar
  32. 32.
    Rottenfußer, G., Rückert, J., Rempe, L., Schleicher, D.: Dynamic rays of bounded type entire functions. Ann. Math. 173, 77–125 (2011)MATHCrossRefGoogle Scholar
  33. 33.
    Schubert, H.: Über die Hausdorff-Dimension der Juliamenge von Funktionen endlicher Ordnung. Dissertation. Christian-Albrechts-Universität zu Kiel (2007)Google Scholar
  34. 34.
    Stallard, G.M.: Entire functions with Julia sets of zero measure. Math. Proc. Camb. Philos. Soc. 108, 551–557 (1990)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions II. Math. Proc. Cambridge Philos. Soc. 119, 513–536 (1996)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Stallard, G.M.: The Hausdorff dimension of Julia sets of entire functions IV. J. Lond. Math. Soc. 61(2), 471–488 (2000)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Stallard, G.M.: Dimensions of Julia sets of transcendental meromorphic functions. In: Transcendental dynamics and complex analysis. London Mathematical Society Lecture Note Series 348, pp. 425–446. Cambridge University Press, Cambridge (2008)Google Scholar
  38. 38.
    Tyler, T.F.: Maximum curves and isolated points of entire functions. Proc. Am. Math. Soc. 128, 2561–2568 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Mathematisches Seminar, Christian-Albrechts-Universität zu KielKielGermany

Personalised recommendations