Abstract
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 \(\infty \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Anderson, J.M., Drasin, D., Sons, L.R.: Wolfgang Heinrich Johannes Fuchs (1915-1997). Not. Am. Math. Soc. 45(11), 1472–1478 (1998)
Baker, I.N.: The existence of fixpoints of entire functions. Math. Z. 73, 280–284 (1960)
Benini, A.M.: Triviality of fibers for Misiurewicz parameters in the exponential family. Conform. Geom. Dyn. 15, 133–151 (2011)
Bergweiler, W.: Periodic points of entire functions: proof of a conjecture of Baker. Complex Var. Theory Appl. 17(1–2), 57–72 (1991)
Bielefeld, B., Fisher, Y., Hubbard, J.: The classification of critically preperiodic polynomials as dynamical systems. J. Am. Math. Soc. 5, 721–762 (1992)
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)
Devaney, R.L., Jarque, X.: Misiurewicz points for complex exponentials. Int. J. Bifur. Chaos Appl. Sci. Eng. 7(7), 1599–1615 (1997)
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)
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)
Goldberg, A.A., Ostrovskii, I.V.: Distribution of Values of Meromorphic Functions. Translations of Mathematical Monographs, vol. 236. American Mathematical Society, Providence (2008)
Hayman, W.K.: A generalisation of Stirling’s formula. J. Reine Angew. Math. 196, 67–95 (1956)
Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)
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)
Hubbard, J., Schleicher, D., Shishikura, M.: Exponential Thurston maps and limits of quadratic differentials. J. Am. Math. Soc. 22(1), 77–117 (2009)
Jarque, X.: On the connectivity of the escaping set for complex exponential Misiurewicz parameters. Proc. Am. Math. Soc. 139(6), 2057–2065 (2011)
Laubner, B., Schleicher, D., Vicol, V.: A combinatorial classification of postsingularly finite complex exponential maps. Discrete Contin. Dyn. Syst. 22(3), 663–682 (2008)
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)
Schleicher, D., Zimmer, J.: Periodic points and dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. Math. 28, 327–354 (2003)
Yamanoi, K.: The second main theorem for small functions and related problems. Acta Math. 192(2), 225–294 (2004)
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bergweiler, W. On Postsingularly Finite Exponential Maps. Arnold Math J. 3, 83–95 (2017). https://doi.org/10.1007/s40598-016-0056-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40598-016-0056-4