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Conformal Geometry and Dynamics

ISSN 1088-4173



Parametrized dynamics of the Weierstrass elliptic function

Authors: Jane Hawkins and Lorelei Koss
Translated by:
Journal: Conform. Geom. Dyn. 8 (2004), 1-35
MSC (2000): Primary 37F45
Published electronically: February 24, 2004
MathSciNet review: 2060376
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Abstract: We study parametrized dynamics of the Weierstrass elliptic $\wp$ function by looking at the underlying lattices; that is, we study parametrized families $\wp_{\Lambda}$ and let $\Lambda$ vary. Each lattice shape is represented by a point $\tau$ in a fundamental period in modular space; for a fixed lattice shape $\Lambda = [1, \tau]$ we study the parametrized space $k \Lambda$. We show that within each shape space there is a wide variety of dynamical behavior, and we conduct a deeper study into certain lattice shapes such as triangular and square. We also use the invariant pair $(g_2, g_3)$ to parametrize some lattices.

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  • 1. Baker, I. N., Kotus, J., Lü, Y. (1991), Iterates of meromorphic functions, I, ETDS 11: 241-248. MR 92m:58113
  • 2. -, (1990), Iterates of meromorphic functions, III, ETDS 11: 603-618. MR 92m:58115
  • 3. -, (1992), Iterates of meromorphic functions IV: Critically finite functions, Results Math. 22: 651-656. MR 94c:58166
  • 4. Bergweiler, W. (1993), Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29, no. 2: 151-188. MR 94c:30033
  • 5. Devaney, R., Keen, L. (1989), Dynamics of meromorphic maps with polynomial Schwarzian derivative, Ann. Sci. Ecole Norm. Sup. (4) 22: 55-81. MR 90e:58071
  • 6. Douady, A. and Hubbard, J. (1985), On the dynamics of polynomial-like mappings, Ann. Sci. Ec. Norm. Sup. 18, 287-344. MR 87f:58083
  • 7. DuVal, P. (1973), Elliptic Functions and Elliptic Curves, Cambridge University Press. MR 52:417
  • 8. Eremenko, A. E., Lyubich, M. Y. (1992), Dynamical properties of some classes of entire functions, A. Inst. Fourier (Grenoble) 42: 989-1020. MR 93k:30034
  • 9. Hawkins, J., Koss, L. (2002), Ergodic properties and Julia sets of Weierstrass elliptic functions, Monatsh. Math. 137 no. 4: 273-300. MR 2003j:37066
  • 10. -, (2003), Connectivity of Julia sets of elliptic functions, preprint.
  • 11. Jones, G., Singerman, D. (1997), Complex Functions: An algebraic and geometric viewpoint, Cambridge Univ. Press. MR 89b:30001
  • 12. Keen, L., Kotus, J. (1997), Dynamics of the family $\lambda \tan \,z$, Conform. Geom. Dyn. 1: 28-57. MR 98h:58159
  • 13. -, (1999), Ergodicity of some classes of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 24: 133-145. MR 2000a:30049
  • 14. Kotus, J., Urbanski, M. (2003), Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions, Bull. London Math. Soc. 35, 269-275. MR 2003j:37067
  • 15. Lyubich, M. (1987), The measurable dynamics of the exponential map, Siberian J. Math 28: 111-127. MR 89d:58071
  • 16. Mañé, R., Sad, P., Sullivan, D. (1983), On the dynamics of rational maps, Ann. Sci. Ecole Norm. Sup. 16 $4^e$série: 193-217. MR 85j:58089
  • 17. Mathematica, Wolfram Research, Inc. 1988-2002.
  • 18. McMullen, C. T. (1987), Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc. 300: 329-334. MR 88a:30057
  • 19. -, (2000), The Mandelbrot set is universal, The Mandelbrot Set, Theme and Variations, London Math Soc. Lecture Notes 274, Cambridge Univ. Press, 1-17. MR 2002f:37081
  • 20. McMullen, C. T., Sullivan, D. (1998), Quasiconformal homeomorphism and dynamics, III: The Teichmüller space of a holomorphic dynamical system. Adv. in Math. 135: 351-395. MR 99e:58145
  • 21. Milne-Thomson, L. (1950), Jacobian Elliptic Function Tables, Dover Publications, Inc.

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Additional Information

Jane Hawkins
Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, North Carolina 27599-3250

Lorelei Koss
Affiliation: Department of Mathematics and Computer Science, Dickinson College, P.O. Box 1773, Carlisle, Pennsylvania 17013

Keywords: Complex dynamics, meromorphic functions, Julia sets, holomorphic families, parameter space
Received by editor(s): May 21, 2003
Received by editor(s) in revised form: January 23, 2004
Published electronically: February 24, 2004
Additional Notes: The second author was supported in part by NSF Grant 9970575
Article copyright: © Copyright 2004 American Mathematical Society

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