Chinese Annals of Mathematics, Series B

, Volume 40, Issue 4, pp 481–494 | Cite as

Fast Growth Entire Functions Whose Escaping Set Has Hausdorff Dimension Two

  • Jie Ding
  • Jun Wang
  • Zhuan YeEmail author


The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain the Hausdor. measure of their escaping set is infinity.


Dynamic systems Entire function Julia set Escaping set Hausdorff dimension 

2000 MR Subject Classification

37F10 37F35 30D05 30D15 


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The authors are grateful to the referees for their suggestions and comments which have improved the clarity of the paper.


  1. [1]
    Barański, K., Hausdorff dimension of hairs and ends for entire maps of finite order, Math. Proc. Camb. Phil. Soc., 145, 2008, 719–737.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Beardon, A. F., Iteration of Rational Functions, Spinger-Verlag, Berlin, 1991.CrossRefzbMATHGoogle Scholar
  3. [3]
    Bergweiler, W., Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N. S.), 29, 1993, 151–188.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Bergweiler, W., Lebesgue measure of Julia sets and escaping sets of certain entire functions, Fund. Math., 242, 2018, 281–301.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Bergweiler, W. and Chyzhykov, I, Lebesgue measure of escaping sets of entire functions of completely regular growth, J. London Math. Soc., 94(2), 2016, 639–661.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Bergweiler, W. and Hinkkanen, A., On semiconjugation of entire functions, Math. Pro. Camb. Phil. Soc., 126(3), 1999, 565–574.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Bergweiler, W. and Karpinska, B., On the Hausdorff dimension of the Julia set of a regularly growing entire function, Math. Pro. Camb. Phil. Soc., 148, 2010, 531–551.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Bergweiler, W., Karpinska, B. and Stallard, G. M., The growth rate of an entire function and the Hausdorff dimension of its Julia set, J. London Math. Soc., 80(2), 2009, 680–698.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Eremenko, A. E., On the Iteration of Entire Functions, Dynamical Systems and Ergodic Theory, 23, Banach Center Publ., Warsaw, 1989, 339–345.Google Scholar
  10. [10]
    Eremenko, A. E. and Lyubich, M. Yu., Dynamical properties of some classes of entire functions, Ann. Inst. Fourier, 42, 1992, 989–1020.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Goldberg, A. A. and Ostrovsikii, I. V., Value Distribution of Meromorphic Functions, Translation of Mathematical Monographs, 236, Amer. Math. Soc., Province, Rhode Island, 2008.CrossRefGoogle Scholar
  12. [12]
    Gundersen, G., Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc., 305, 1988, 415–429.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Hayman, W. K., Meromorphic Functions, Clarendon Press, Oxford, UK, 1964.zbMATHGoogle Scholar
  14. [14]
    McMullen, C., Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300, 1987, 329–342.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Milnor, J., Dynamics in One Complex Variable, Friedr. Vieweg & Sohn, Braunschweig, 1999.zbMATHGoogle Scholar
  16. [16]
    Peter, J., Hausdorff measure of Julia sets in the exponential family, J. London Math. Soc., 82(1), 2010, 229–255.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Peter, J., Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order, Ergod. Th. & Dynam. Sys., 33(1), 2013, 284–302.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Rippon, P. J. and Stallard, G. M., Fast escaping points of entire functions, Pro. London Math. Soc., 105(4), 2012, 787–820.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Schubert, H., Über die Hausdorff-Dimension der Juliamenge von Funktionen endlicher Ordnung, Dissertation, University of Kiel, 2007,
  20. [20]
    Sixsmith, D., Functions of genus zero for which the fast escaping set has Hausdorff dimension two, Pro. Amer. Math. Soc., 143(6), 2015, 2597–2612.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Stallard, G. M., Dimensions of Julia sets of transcendental meromorphic functions, Transcendental Dynamics and Complex Analysis, London Math. Soc. Lect. Note 348, Rippon, P. J. and Stallard, G. M.(eds.), Cambridge University Press, Cambridge, 2008, 425–446.Google Scholar
  22. [22]
    Taniguchi, M., Size of the Julia set of structurally finite transcendental entire function, Math. Proc. Camb. Phil. Soc., 135, 2003, 181–192.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Ye, Z., On Nevnalinna’s error terms, Duke Math. J., 64(2), 1991, 243–260.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Zheng, J. H., On multiply-connected Fatou components in iteration of meromorphic functions, J. Math. Anal. Appl., 313, 2006, 24–37.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.School of MathematicsTaiyuan University of TechnologyTaiyuanChina
  2. 2.School of MathematicsFudan UniversityShanghaiChina
  3. 3.Department of Mathematics and StatisticsUniveristy of North Carolina WilmingtonWilmingtonUSA

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