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Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 855–873 | Cite as

Lyapunov exponents and related concepts for entire functions

  • Walter Bergweiler
  • Xiao Yao
  • Jianhua Zheng
Article
  • 159 Downloads

Abstract

Let f be an entire function and denote by \(f^\#\) the spherical derivative of f and by \(f^n\) the n-th iterate of f. For an open set U intersecting the Julia set J(f), we consider how fast \(\sup _{z\in U} (f^n)^\#(z)\) and \(\int _U (f^n)^\#(z)^2 dx\,dy\) tend to \(\infty \). We also study the growth rate of the sequence \((f^n)^\#(z)\) for \(z\in J(f)\).

Mathematics Subject Classification

Primary 37F10 Secondary 30D05 

Notes

Acknowledgements

We thank Alexandre Eremenko, Dan Liu, Lasse Rempe-Gillen, Phil Rippon, Weixiao Shen and the referee for helpful comments.

References

  1. 1.
    Aspenberg, M., Bergweiler, W.: Entire functions with Julia sets of positive measure. Math. Ann. 352, 27–54 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baker, I.N.: Repulsive fixpoints of entire functions. Math. Z. 104, 252–256 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barrett, M., Eremenko, A.: On the spherical derivative of a rational function. Anal. Math. Phys. 4, 73–81 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bergweiler, W.: Iteration of meromorphic functions. Bull. Am. Math. Soc. (N.S.) 29, 151–188 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bergweiler, W.: A new proof of the Ahlfors five islands theorem. J. Anal. Math. 76, 337–347 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bergweiler, W.: Quasinormal families and periodic points. In: Complex Analysis and Dynamical Systems II. Contemp. Math., vol. 382. Amer. Math. Soc., Providence, pp. 55–63 (2005)Google Scholar
  7. 7.
    Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11, 355–373 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bergweiler, W., Eremenko, A.: On a property of the derivative of an entire function. Ann. Acad. Sci. Fenn. Math. 37, 301–307 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bergweiler, W., Hinkkanen, A.: On semiconjugation of entire functions. Math. Proc. Camb. Philos. Soc. 126, 565–574 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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. (2) 80, 680–698 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bergweiler, W., Rippon, P.J., Stallard, G.M.: Multiply connected wandering domains of entire functions. Proc. Lond. Math. Soc. (3) 107, 1261–1301 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Berteloot, F.: Lyapunov exponent of a rational map and multipliers of repelling cycles. Riv. Math. Univ. Parma (N.S.) 1, 263–269 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Clunie, J., Hayman, W.K.: On the spherical derivative of integral and meromorphic functions. Comment. Math. Helv. 40, 117–148 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dobbs, N.: Perturbing Misiurewicz parameters in the exponential family. Commun. Math. Phys. 335, 571–608 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eremenko, A.E.: On the iteration of entire functions. In: Dynamical Systems and Ergodic Theory. Banach Center Publications, vol. 23. Polish Scientific Publishers, Warsaw, pp. 339–345 (1989)Google Scholar
  16. 16.
    Eremenko, A.E., Levin, G.M.: Periodic points of polynomials. Ukr. Math. J. 41(1989), 1258–1262 (1990) [translation of Ukrain. Mat. Zh. 41, 1467–1471 (1989)]Google Scholar
  17. 17.
    Eremenko, A.E., Lyubich, M.Y.: Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble) 42, 989–1020 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gelfert, K., Przytycki, F., Rams, M.: On the Lyapunov spectrum for rational maps. Math. Ann. 348, 965–1004 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gelfert, K., Przytycki, F., Rams, M., Rivera-Letelier, J.: Lyapunov spectrum for exceptional rational maps. Ann. Acad. Sci. Fenn. Math. 38, 631–656 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Goldberg, A.A., Ostrovskii, I.V.: Value distribution of meromorphic functions. Amer. Math. Soc, Providence (2008)CrossRefzbMATHGoogle Scholar
  21. 21.
    Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)zbMATHGoogle Scholar
  22. 22.
    Langley, J.K.: On the multiple points of certain meromorphic functions. Proc. Am. Math. Soc. 123, 355–373 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Levin, G., Przytycki, F., Shen, W.: The Lyapunov exponent of holomorphic maps. Invent. Math. 205, 363–382 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Pommerenke, Ch.: Estimates for normal meromorphic functions. Ann. Acad. Sci. Fenn. Ser. A I 476, 10 (1970)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Przytycki, F.: Letter to Alexandre Eremenko. http://www.impan.pl/~feliksp/unpublished.html (1994)
  26. 26.
    Rippon, P.J., Stallard, G.M.: Functions of small growth with no unbounded wandering domains. J. Anal. Math. 108, 61–86 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rippon, P.J., Stallard, G.M.: Slow escaping points of meromorphic functions. Trans. Am. Math. Soc. 362, 4171–4201 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rippon, P.J., Stallard, G.M.: Fast escaping points of entire functions. Proc. Lond. Math. Soc. (3) 105, 787–820 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schleicher, D.: Dynamics of entire functions. In: Holomorphic dynamical systems. Lecture Notes Math., vol. 1998. Springer, Berlin, pp. 295–339 (2010)Google Scholar
  30. 30.
    Yao, X., Sun, D., Xuan, Z.: A new geometric characterization of the Julia set (preprint). arXiv: 1512.05144
  31. 31.
    Zdunik, A.: Characteristic exponents of rational functions. Bull. Polish Acad. Sci. Math. 62, 257–263 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Shanghai Center for Mathematical SciencesShanghaiPeople’s Republic of China
  3. 3.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

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