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A landing theorem for periodic rays of exponential maps

Author: Lasse Rempe
Journal: Proc. Amer. Math. Soc. 134 (2006), 2639-2648
MSC (2000): Primary 37F10; Secondary 30D05
Published electronically: March 22, 2006
MathSciNet review: 2213743
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Abstract: For the family of exponential maps $ z\mapsto \exp(z)+\kappa$, we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that $ g$ is a periodic dynamic ray of an exponential map with nonescaping singular value. Then $ g$ lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.

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  • [BR] I. Noël Baker and Philip J. Rippon, Iteration of exponential functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 49-77. MR 0752391 (86d:58065)
  • [BDD] Ranjit Bhattacharjee, Robert L. Devaney, R. E. Lee Deville, Krešimir Josić, and Monica Moreno-Rocha, Accessible points in the Julia sets of stable exponentials, Discrete Contin. Dyn. Syst. Ser. B 1 (2001), no. 3, 299–318. MR 1849820,
  • [BDG] Clara Bodelón, Robert L. Devaney, Michael Hayes, Gareth Roberts, Lisa R. Goldberg, and John H. Hubbard, Hairs for the complex exponential family, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 8, 1517–1534. MR 1721835,
  • [DGH] Robert L. Devaney, Lisa R. Goldberg, and John H. Hubbard, A dynamical approximation to the exponential map by polynomials, Preprint, MSRI Berkeley, 1986; compare [BDG].
  • [DH] Adrien Douady and John Hubbard, Etude dynamique des polynômes complexes, Prépublications mathémathiques d'Orsay (1984 / 1985), no. 2/4.
  • [EL] A. È. Erëmenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, 989–1020 (English, with English and French summaries). MR 1196102
  • [F] Markus Förster, Parameter rays for the exponential family, Diplomarbeit, Techn. Univ. München, 2003, Available as Thesis 2003-03 on the Stony Brook Thesis Server.
  • [FRS] Markus Förster, Lasse Rempe, and Dierk Schleicher, Classification of escaping exponential maps, Preprint, 2004, arXiv:math.DS/0311427, submitted for publication.
  • [FS] Markus Förster and Dierk Schleicher, Parameter rays for the exponential family, Preprint, 2005, arXiv:math.DS/0505097.
  • [MSS] Ricardo Mañé, Paulo Sad, and Dennis Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193-217. MR 0732343 (85j:58089)
  • [M1] John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
  • [M2] John Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account, Astérisque 261 (2000), xiii, 277–333 (English, with English and French summaries). Géométrie complexe et systèmes dynamiques (Orsay, 1995). MR 1755445
  • [R1] Lasse Rempe, Dynamics of exponential maps, doctoral thesis, Christian-Albrechts-Universität Kiel, 2003,
  • [R2] -, Topological dynamics of exponential maps on their escaping sets, Preprint, 2003, arXiv:math.DS/0309107, conditionally accepted for publication in Ergodic Theory Dynam. Systems.
  • [R3] -, Siegel disks and periodic rays of entire functions, Preprint, 2004, arXiv:math.DS/0408041, submitted for publication.
  • [R4] -, Nonlanding dynamic rays of exponential maps, Preprint, 2005, arXiv: math.DS/0511588, submitted for publication.
  • [R5] -, On entire functions with accessible singular value, in preparation.
  • [RS1] Lasse Rempe and Dierk Schleicher, Bifurcations in the space of exponential maps, Preprint #2004/03, Institute for Mathematical Sciences, SUNY Stony Brook, 2004, arXiv:math.DS/0311480, submitted for publication.
  • [RS2] -, Combinatorics of bifurcations in exponential parameter space, Preprint, 2004, arXiv:math.DS/0408011, submitted for publication.
  • [S1] Dierk Schleicher, On the dynamics of iterated exponential maps, Habilitation thesis, TU München, 1999.
  • [S2] Dierk Schleicher, Attracting dynamics of exponential maps, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 3–34. MR 1976827
  • [SZ1] Dierk Schleicher and Johannes Zimmer, Escaping points of exponential maps, J. London Math. Soc. (2) 67 (2003), no. 2, 380–400. MR 1956142,
  • [SZ2] Dierk Schleicher and Johannes Zimmer, Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 2, 327–354. MR 1996442

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Additional Information

Lasse Rempe
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Address at time of publication: Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, United Kingdom

Received by editor(s): July 30, 2003
Received by editor(s) in revised form: March 30, 2005
Published electronically: March 22, 2006
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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