On Postsingularly Finite Exponential Maps
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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 \(\infty \).
KeywordsEntire 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.
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