A fundamental dichotomy for Julia sets of a family of elliptic functions

Koss, L.

doi:10.1090/S0002-9939-09-09967-5

A fundamental dichotomy for Julia sets of a family of elliptic functions

Author: L. Koss
Journal: Proc. Amer. Math. Soc. 137 (2009), 3927-3938
MSC (2000): Primary 54H20, 37F10; Secondary 37F20
DOI: https://doi.org/10.1090/S0002-9939-09-09967-5
Published electronically: June 29, 2009
MathSciNet review: 2529903
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate topological properties of Julia sets of iterated elliptic functions of the form $g = 1/\wp$ , where $\wp$ is the Weierstrass elliptic function, on triangular lattices. These functions can be parametrized by $\mathbb{C} - \{0\}$ , and they all have a superattracting fixed point at zero and three other distinct critical values. We prove that the Julia set of is either Cantor or connected, and we obtain examples of each type.

1. 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, https://doi.org/10.1007/BF03323112
2. Walter Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 151–188. MR 1216719, https://doi.org/10.1090/S0273-0979-1993-00432-4
3. Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229–325. MR 1194004, https://doi.org/10.1007/BF02392761
4. Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, https://doi.org/10.1007/BF02591353
5. Robert L. Devaney and Linda Keen, Dynamics of tangent, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 105–111. MR 970550, https://doi.org/10.1007/BFb0082826
6. Patrick Du Val, Elliptic functions and elliptic curves, Cambridge University Press, London-New York, 1973. London Mathematical Society Lecture Note Series, No. 9. MR 0379512
7. P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161–271 (French). MR 1504787
P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 33–94 (French). MR 1504792
8. J. Goldsmith and L. Koss, Dynamical properties of the derivative of the Weierstrass elliptic function, Involve. To appear.
9. G. Julia, Mémoire sur l'iteration des applications fonctionnelles, J. Math. Pures et Appl. 8 (1918), 47-245.
10. Jane Hawkins, Smooth Julia sets of elliptic functions for square rhombic lattices, Topology Proc. 30 (2006), no. 1, 265–278. Spring Topology and Dynamical Systems Conference. MR 2280672
11. J. Hawkins, A family of elliptic functions with Julia set the whole sphere (2008), J. Difference Equ. Appl. To appear.
12. Jane Hawkins and Lorelei Koss, Ergodic properties and Julia sets of Weierstrass elliptic functions, Monatsh. Math. 137 (2002), no. 4, 273–300. MR 1947915, https://doi.org/10.1007/s00605-002-0504-1
13. Jane Hawkins and Lorelei Koss, Parametrized dynamics of the Weierstrass elliptic function, Conform. Geom. Dyn. 8 (2004), 1–35. MR 2060376, https://doi.org/10.1090/S1088-4173-04-00103-1
14. Jane Hawkins and Lorelei Koss, Connectivity properties of Julia sets of Weierstrass elliptic functions, Topology Appl. 152 (2005), no. 1-2, 107–137. MR 2160809, https://doi.org/10.1016/j.topol.2004.08.018
15. J. Hawkins, L. Koss and J. Kotus, Elliptic functions with critical orbits approaching infinity (2008), J. Difference Equ. Appl. To appear.
16. Jane M. Hawkins and Daniel M. Look, Locally Sierpinski Julia sets of Weierstrass elliptic ℘ functions, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 5, 1505–1520. MR 2254870, https://doi.org/10.1142/S0218127406015453
17. Gareth A. Jones and David Singerman, Complex functions, Cambridge University Press, Cambridge, 1987. An algebraic and geometric viewpoint. MR 890746
18. Linda Keen and Janina Kotus, Dynamics of the family 𝜆tan𝑧, Conform. Geom. Dyn. 1 (1997), 28–57. MR 1463839, https://doi.org/10.1090/S1088-4173-97-00017-9
19. L. Koss, Cantor Julia sets in a family of even elliptic functions (2008), J. Difference Equ. Appl. To appear.
20. John Milnor, Dynamics in one complex variable, Friedr. Vieweg & Sohn, Braunschweig, 1999. Introductory lectures. MR 1721240
21. John Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), no. 1, 37–83. With an appendix by the author and Lei Tan. MR 1246482
22. P. J. Rippon and G. M. Stallard, Iteration of a class of hyperbolic meromorphic functions, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3251–3258. MR 1610785, https://doi.org/10.1090/S0002-9939-99-04942-4
23. Yong Cheng Yin, On the Julia sets of quadratic rational maps, Complex Variables Theory Appl. 18 (1992), no. 3-4, 141–147. MR 1157922