Arnold Mathematical Journal

, Volume 3, Issue 1, pp 83–95 | Cite as

On Postsingularly Finite Exponential Maps

  • Walter Bergweiler
Research Contribution


We consider parameters \(\lambda \) for which 0 is preperiodic under the map \(z\mapsto \lambda e^z\). Given k and l, let n(r) be the number of \(\lambda \) satisfying \(0<|\lambda |\le r\) such that 0 is mapped after k iterations to a periodic point of period l. We determine the asymptotic behavior of n(r) as r tends to \(\infty \).


Entire function Singular value Exponential function Periodic point Preperiodic point Postcritically finite Misiurewicz map Nevanlinna characteristic 



The results of this paper (except for Proposition 3) were presented in two talks in John H. Hubbard’s seminar at Cornell University in the fall of 1988. They were inspired by a talk by Ben Bielefeld in this seminar about the computation of Misiurewicz parameters using the spider algorithm (Bielefeld et al. 1992; Hubbard and Schleicher 1994). The proof of Proposition 2 given below is a simplified version of the one presented in the seminar. John Hubbard’s seminar was my first encounter with complex dynamics. (The purpose of my stay at Cornell University was to visit Wolfgang H. J. Fuchs, a leading figure in Nevanlinna theory; see Anderson et al. (1998) for Fuchs’ life and work.) I would like to take this opportunity—albeit very belatedly—to thank John Hubbard and the participants of his seminar for igniting my interest in complex dynamics and for helpful discussions. I thank Dierk Schleicher for encouraging me to make the results of my talks in this seminar available—and I also thank him, Saikat Batabyal and the referee for useful comments on this manuscript. Finally, I remain grateful to the Alexander von Humboldt Foundation for making my stay at Cornell University possible by granting me a Feodor Lynen research fellowship.


  1. Ahlfors, L.V.: Complex Analysis: An Introduction of the Theory of Analytic Functions of One Complex Variable, 2nd edn. McGraw-Hill, New York, Toronto, London (1966)Google Scholar
  2. Anderson, J.M., Drasin, D., Sons, L.R.: Wolfgang Heinrich Johannes Fuchs (1915-1997). Not. Am. Math. Soc. 45(11), 1472–1478 (1998)Google Scholar
  3. Baker, I.N.: The existence of fixpoints of entire functions. Math. Z. 73, 280–284 (1960)MathSciNetCrossRefMATHGoogle Scholar
  4. Benini, A.M.: Triviality of fibers for Misiurewicz parameters in the exponential family. Conform. Geom. Dyn. 15, 133–151 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. Bergweiler, W.: Periodic points of entire functions: proof of a conjecture of Baker. Complex Var. Theory Appl. 17(1–2), 57–72 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. Bielefeld, B., Fisher, Y., Hubbard, J.: The classification of critically preperiodic polynomials as dynamical systems. J. Am. Math. Soc. 5, 721–762 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. Clunie, J.: The composition of entire and meromorphic functions. In: Shankar, H. (ed.) Mathematical Essays Dedicated to A. J. Macintyre, pp. 75–92. Ohio University Press, Athens (1970)Google Scholar
  8. Devaney, R.L., Jarque, X.: Misiurewicz points for complex exponentials. Int. J. Bifur. Chaos Appl. Sci. Eng. 7(7), 1599–1615 (1997)MathSciNetCrossRefMATHGoogle Scholar
  9. Devaney, R.L., Jarque, X., Rocha, M.M.: Indecomposable continua and Misiurewicz points in exponential dynamics. Int. J. Bifur. Chaos Appl. Sci. Eng. 15(10), 3281–3293 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. Edrei, A., Fuchs, W.H.J.: On the zeros of \(f(g(z))\) where \(f\) and \(g\) are entire functions. J. Anal. Math. 12, 243–255 (1964)MathSciNetCrossRefMATHGoogle Scholar
  11. Goldberg, A.A., Ostrovskii, I.V.: Distribution of Values of Meromorphic Functions. Translations of Mathematical Monographs, vol. 236. American Mathematical Society, Providence (2008)Google Scholar
  12. Hayman, W.K.: A generalisation of Stirling’s formula. J. Reine Angew. Math. 196, 67–95 (1956)MathSciNetMATHGoogle Scholar
  13. Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)MATHGoogle Scholar
  14. Hubbard, J.H., Schleicher, D.: The spider algorithm. In: Complex Dynamical Systems (Cincinnati, OH, 1994). Proceedings of Symposia in Applied Mathematics, vol. 49, pp. 155–180. American Mathematical Society, Providence (1994)Google Scholar
  15. Hubbard, J., Schleicher, D., Shishikura, M.: Exponential Thurston maps and limits of quadratic differentials. J. Am. Math. Soc. 22(1), 77–117 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. Jarque, X.: On the connectivity of the escaping set for complex exponential Misiurewicz parameters. Proc. Am. Math. Soc. 139(6), 2057–2065 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. Laubner, B., Schleicher, D., Vicol, V.: A combinatorial classification of postsingularly finite complex exponential maps. Discrete Contin. Dyn. Syst. 22(3), 663–682 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. London, R.R.: The behaviour of certain entire functions near points of maximum modulus. J. Lond. Math. Soc. (2) 12(4), 485–504 (1975/1976)Google Scholar
  19. Schleicher, D., Zimmer, J.: Periodic points and dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 28, 327–354 (2003)MathSciNetMATHGoogle Scholar
  20. Yamanoi, K.: The second main theorem for small functions and related problems. Acta Math. 192(2), 225–294 (2004)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2016

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany

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