Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Exponential Thurston maps and limits of quadratic differentials

Authors: John Hubbard, Dierk Schleicher and Mitsuhiro Shishikura
Journal: J. Amer. Math. Soc. 22 (2009), 77-117
MSC (2000): Primary 30F30; Secondary 30F60, 32G15, 37F20, 37F30
Published electronically: June 3, 2009
Previous version: Original version posted July 9, 2008
Corrected version: Current version corrects publisher's introduction of inconsistent spelling of "Teichmüller".
MathSciNet review: 2449055
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give a topological characterization of postsingularly finite topological exponential maps, i.e., universal covers $ g\colon\mathbb{C}\to\mathbb{C}\setminus\{0\}$ such that 0 has a finite orbit. Such a map either is Thurston equivalent to a unique holomorphic exponential map $ \lambda e^z$ or it has a topological obstruction called a degenerate Levy cycle. This is the first analog of Thurston's topological characterization theorem of rational maps, as published by Douady and Hubbard, for the case of infinite degree.

One main tool is a theorem about the distribution of mass of an integrable quadratic differential with a given number of poles, providing an almost compact space of models for the entire mass of quadratic differentials. This theorem is given for arbitrary Riemann surfaces of finite type in a uniform way.

References [Enhancements On Off] (What's this?)

  • [A1] Lars Ahlfors: Conformal invariants. McGraw-Hill (1973). MR 0357743 (50:10211)
  • [A2] Lars Ahlfors: Lectures on quasiconformal mappings. Van Nostrand (1966). MR 0200442 (34:336)
  • [BDGHHR] Clara Bodelon, Robert Devaney, Lisa Goldberg, Martin Hayes, John Hubbard, Gareth Roberts: Hairs for the complex exponential family. Internat. J. Bif. Chaos 9 No. 8 (1999), 1517-1534. MR 1721835 (2001a:37055)
  • [BFH] Ben Bielefeld, Yuval Fisher, John Hubbard: The classification of critically preperiodic polynomials as dynamical systems. Jour. AMS 5 4 (1992). MR 1149891 (93h:58128)
  • [BKS] Henk Bruin, Alexandra Kaffl, Dierk Schleicher: Symbolic dynamics of quadratic polynomials. Manuscript, in preparation. Earlier version circulated as [BS].
  • [BR] Noel Baker, Phil Rippon: Iteration of exponential functions. Ann. Acad. Sci. Fenn., Series A.I. Math. 9 (1984), 49-77. MR 752391 (86d:58065)
  • [BS] Henk Bruin, Dierk Schleicher: Symbolic dynamics of quadratic polynomials. Report, Institute Mittag-Leffler, Djursholm 7 (2001/02). (128 pp.)
  • [DGH] Robert Devaney, Lisa Goldberg, John Hubbard: A dynamical approximation to the exponential map of polynomials. Preprint, MSRI (1986).
  • [DH] Adrien Douady, John Hubbard: A proof of Thurston's topological characterization of rational functions. Acta Math. 171 (1993), 263-297. MR 1251582 (94j:58143)
  • [E] Leonhardt Euler: De formulis exponentialibus replicatis. Acta Acad. Petropolitanae 1 (1777), 38-60.
  • [EL] Alexandre Eremenko, Mikhail Lyubich: Dynamical properties of some classes of entire functions. Ann. Sci. Inst. Fourier, Grenoble 42 4 (1992), 989-1020. MR 1196102 (93k:30034)
  • [FRS] Markus Förster, Lasse Rempe, Dierk Schleicher: Classification of Escaping Exponential Maps. Proc. Am. Math. Soc. 136 2 (2008), 651-663. MR 2358507
  • [FS] Markus Förster, Dierk Schleicher: Parameter rays for the complex exponential family. Ergod. Thy and Dynam. Syst., to appear. ArXiv math.DS/050597.
  • [GL] Frederick Gardiner, Nikola Lakic: Quasiconformal Teichmüller theory. Mathematical Surveys and Monographs, 76. AMS (2000) MR 1730906 (2001d:32016)
  • [H] John Hubbard: Teichmüller theory and applications to geometry, topology, and dynamics, Volume I: Teichmüller theory. Matrix editions, Ithaca/NY (2006). MR 2245223
  • [HS] John Hubbard, Dierk Schleicher: The spider algorithm. In: Complex dynamical systems, Robert Devaney (ed.), AMS (1994), 155-180. MR 1315537
  • [IT] Yoichi Imayoshi, Masahiko Taniguchi: An introduction to Teichmüller spaces. Springer-Verlag, Tokyo (1992). MR 1215481 (94b:32031)
  • [K] Jan Kiwi: Rational laminations of complex polynomials. In: Laminations and foliations in dynamics, geometry and topology, Mikhail Lyubich, John Milnor, Yair Minsky (eds), Contemporary Mathematics 269, AMS (2000), 111-154. MR 1810538 (2002e:37063)
  • [L] Olli Lehto: Univalent functions and Teichmüller spaces. Graduate Texts in Mathematics 109, Springer-Verlag, New York (1987). MR 867407 (88f:30073)
  • [Le] Silvio Levy: Critically finite rational maps. Ph.D. thesis, Princeton (1985).
  • [LS] Eike Lau, Dierk Schleicher: Internal addresses in the Mandelbrot set and irreducibility of polynomials. Preprint, Inst. Math. Sci., Stony Brook 19 (1994). ArXiv math.DS/9411238.
  • [LSV] Bastian Laubner, Dierk Schleicher and Vlad Vicol: A dynamical classification of postsingularly finite exponential maps. Discr. and Cont. Dyn. Sys., to appear. ArXiv math.DS/0602602.
  • [M1] Curtis McMullen: Amenability, Poincaré series and quasiconformal maps. Invent. Math. 97 (1989), 95-127. MR 999314 (90e:30048)
  • [M2] Curtis McMullen: Complex dynamics and renormalization. Ann. Math. Studies 135, Princeton Univ. Press (1994). MR 1312365 (96b:58097)
  • [M3] Curtis McMullen: The moduli space of Riemann surfaces is Kähler hyperbolic. Annals of Mathematics 151 (2000), 327-357. MR 1745010 (2001m:32032)
  • [P] Alfredo Poirier: On postcritically finite polynomials I/II. Preprint, Inst. Math. Sci., Stony Brook 5/7 (1993).
  • [RS1] Lasse Rempe, Dierk Schleicher: Bifurcations in the space of exponential maps. Submitted. Preprint, Institute for Mathematical Sciences at Stony Brook 3 (2004). ArXiv math.DS/0311480.
  • [RS2] Lasse Rempe, Dierk Schleicher: Combinatorics of bifurcations in exponential parameter space. In: P. Rippon, G. Stallard (eds), Transcendental dynamics and complex analysis, volume in honour of Professor I. N. Baker, LMS Lecture Note Series. ArXiv math.DS/0408011.
  • [RS3] Lasse Rempe, Dierk Schleicher: Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity, with Lasse Rempe. In: M. Lyubich, M. Yampolsky (eds), Holomorphic Dynamics and Renormalization, in Honour of John Milnor's 75th birthday. Fields Institute Communications 53 (2008), to appear.
  • [S1] Dierk Schleicher: On the dynamics of iterated exponential maps. Habilitationsschrift, TU München (1999).
  • [S2] Dierk Schleicher: Attracting dynamics of exponential maps. Annales Academiae Scientiarum Fennicae 28 1 (2003), 3-34. MR 1976827 (2004k:37091)
  • [S3] Dierk Schleicher: Internal addresses in the Mandelbrot set and Galois groups of polynomials. Manuscript (2007), submitted. Earlier version circulated as [LS].
  • [S4] Dierk Schleicher: The dynamical fine structure of iterated cosine maps and a dimension paradox. Duke Mathematics Journal 136 2 (2007), 343-356. MR 2286634 (2008d:37078)
  • [S5] Dierk Schleicher: Hyperbolic Components in Exponential Parameter Space. Comptes Rendus -- Mathématiques 339 3 (2004), 223-228. ArXiv math.DS/0406256. MR 2078079 (2005e:37105)
  • [SZ1] Dierk Schleicher and Johannes Zimmer: Escaping points of exponential maps. Journal Lond. Math. Soc. (2) 67 (2003), 380-400. MR 1956142 (2003k:37067)
  • [SZ2] Dierk Schleicher and Johannes Zimmer: Periodic points and dynamic rays of exponential maps. Ann. Acad. Scient. Fenn. 28 2 (2003), 327-354. MR 1996442 (2004e:37068)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 30F30, 30F60, 32G15, 37F20, 37F30

Retrieve articles in all journals with MSC (2000): 30F30, 30F60, 32G15, 37F20, 37F30

Additional Information

John Hubbard
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853, and Centre de Mathématiques et d’Informatique, Université de Provence, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France

Dierk Schleicher
Affiliation: School of Engineering and Science, Jacobs University Bremen, Postfach 750 561, D-28725 Bremen, Germany

Mitsuhiro Shishikura
Affiliation: Department of Mathematics, Faculty of Sciences, Kyoto University, Kyoto 606-8502, Japan

Keywords: Quadratic differential, decomposition, limit model, iteration, exponential map, classification
Received by editor(s): March 28, 2006
Published electronically: June 3, 2009
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society