Parametrized dynamics of the Weierstrass elliptic function

Hawkins, Jane; Koss, Lorelei

doi:10.1090/S1088-4173-04-00103-1

Parametrized dynamics of the Weierstrass elliptic function
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by Jane Hawkins and Lorelei Koss PDF

Conform. Geom. Dyn. 8 (2004), 1-35 Request permission

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
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
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