Parametrized dynamics of the Weierstrass elliptic function
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- by Jane Hawkins and Lorelei Koss
- Conform. Geom. Dyn. 8 (2004), 1-35
- DOI: https://doi.org/10.1090/S1088-4173-04-00103-1
- Published electronically: February 24, 2004
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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.References
- I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. I, Ergodic Theory Dynam. Systems 11 (1991), no. 2, 241–248. MR 1116639, DOI 10.1017/S014338570000612X
- I. N. Baker, J. Kotus, and Yi Nian Lü, Iterates of meromorphic functions. III. Preperiodic domains, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 603–618. MR 1145612, DOI 10.1017/S0143385700006386
- I. N. Baker, J. Kotus, and Lü Yinian, Iterates of meromorphic functions. IV. Critically finite functions, Results Math. 22 (1992), no. 3-4, 651–656. MR 1189754, DOI 10.1007/BF03323112
- Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, DOI 10.1090/S0273-0979-1993-00432-4
- Robert L. Devaney and Linda Keen, Dynamics of meromorphic maps: maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 55–79. MR 985854, DOI 10.24033/asens.1575
- Adrien Douady and John Hamal Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 287–343. MR 816367, DOI 10.24033/asens.1491
- Patrick Du Val, Elliptic functions and elliptic curves, London Mathematical Society Lecture Note Series, No. 9, Cambridge University Press, London-New York, 1973. MR 0379512, DOI 10.1017/CBO9781107359901
- 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, DOI 10.5802/aif.1318
- Jane Hawkins and Lorelei Koss, Ergodic properties and Julia sets of Weierstrass elliptic functions, Monatsh. Math. 137 (2002), no. 4, 273–300. MR 1947915, DOI 10.1007/s00605-002-0504-1 hk3 —, (2003), Connectivity of Julia sets of elliptic functions, preprint.
- Gareth A. Jones and David Singerman, Complex functions, Cambridge University Press, Cambridge, 1987. An algebraic and geometric viewpoint. MR 890746, DOI 10.1017/CBO9781139171915
- Linda Keen and Janina Kotus, Dynamics of the family $\lambda \tan z$, Conform. Geom. Dyn. 1 (1997), 28–57. MR 1463839, DOI 10.1090/S1088-4173-97-00017-9
- Linda Keen and Janina Kotus, Ergodicity of some classes of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 24 (1999), no. 1, 133–145. MR 1670876
- Janina Kotus and Mariusz Urbański, Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions, Bull. London Math. Soc. 35 (2003), no. 2, 269–275. MR 1952406, DOI 10.1112/S0024609302001686
- M. Yu. Lyubich, The measurable dynamics of the exponential, Sibirsk. Mat. Zh. 28 (1987), no. 5, 111–127 (Russian). MR 924986
- R. Mañé, P. Sad, and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217. MR 732343, DOI 10.24033/asens.1446 mma Mathematica, Wolfram Research, Inc. 1988–2002.
- Curt McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc. 300 (1987), no. 1, 329–342. MR 871679, DOI 10.1090/S0002-9947-1987-0871679-3
- Curtis T. McMullen, The Mandelbrot set is universal, The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, pp. 1–17. MR 1765082
- Curtis T. McMullen and Dennis P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), no. 2, 351–395. MR 1620850, DOI 10.1006/aima.1998.1726 milne Milne-Thomson, L. (1950), Jacobian Elliptic Function Tables, Dover Publications, Inc.
Bibliographic Information
- Jane Hawkins
- Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250, Chapel Hill, North Carolina 27599-3250
- MR Author ID: 82840
- Email: jmh@math.unc.edu
- Lorelei Koss
- Affiliation: Department of Mathematics and Computer Science, Dickinson College, P.O. Box 1773, Carlisle, Pennsylvania 17013
- MR Author ID: 662937
- Email: koss@dickinson.edu
- 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
- © Copyright 2004 American Mathematical Society
- Journal: Conform. Geom. Dyn. 8 (2004), 1-35
- MSC (2000): Primary 37F45
- DOI: https://doi.org/10.1090/S1088-4173-04-00103-1
- MathSciNet review: 2060376