Connectivity of Julia sets for Weierstrass elliptic functions on square lattices
HTML articles powered by AMS MathViewer
- by Joshua J. Clemons PDF
- Proc. Amer. Math. Soc. 140 (2012), 1963-1972 Request permission
Abstract:
We show that the Julia set of a Weierstrass elliptic function on a square lattice is connected. The techniques used to prove this result are used to show a similar result for a related family of rational maps obtained from the Laurent series.References
- Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications, Inc., New York, 1992. Reprint of the 1972 edition. MR 1225604
- 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
- Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
- 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
- Joshua Clemons, Dynamical properties of Weierstrass elliptic functions on square lattices, University of North Carolina at Chapel Hill (2010), Ph.D. dissertation.
- H. Cremer, Über die schrödersche funktionalgleichung und das schwarzsche eckenabbildungsproblem, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math-Phys. Kl. 84 (1932), 291–324.
- Robert L. Devaney, Daniel M. Look, and David Uminsky, The escape trichotomy for singularly perturbed rational maps, Indiana Univ. Math. J. 54 (2005), no. 6, 1621–1634. MR 2189680, DOI 10.1512/iumj.2005.54.2615
- 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
- P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161–271 (French). MR 1504787, DOI 10.24033/bsmf.998
- P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 48 (1920), 208–314 (French). MR 1504797, DOI 10.24033/bsmf.1008
- 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
- 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
- Jane Hawkins and Lorelei Koss, Parametrized dynamics of the Weierstrass elliptic function, Conform. Geom. Dyn. 8 (2004), 1–35. MR 2060376, DOI 10.1090/S1088-4173-04-00103-1
- 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, DOI 10.1016/j.topol.2004.08.018
- Jane M. Hawkins and Daniel M. Look, Locally Sierpinski Julia sets of Weierstrass elliptic $\wp$ functions, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 16 (2006), no. 5, 1505–1520. MR 2254870, DOI 10.1142/S0218127406015453
- John Milnor, On rational maps with two critical points, Experiment. Math. 9 (2000), no. 4, 481–522. MR 1806289, DOI 10.1080/10586458.2000.10504657
Additional Information
- Joshua J. Clemons
- Affiliation: Department of Mathematics, Phillips Hall CB#3250, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250
- Email: jclemons@vt.edu
- Received by editor(s): September 8, 2010
- Received by editor(s) in revised form: January 24, 2011, and January 31, 2011
- Published electronically: September 27, 2011
- Additional Notes: This is part of completed Ph.D. work at the University of North Carolina under the supervision of Jane Hawkins
The author would like to thank the reviewer for the helpful suggestions that improved this paper. - Communicated by: Bryna Kra
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 1963-1972
- MSC (2010): Primary 37F10, 54H20; Secondary 37F20
- DOI: https://doi.org/10.1090/S0002-9939-2011-11079-7
- MathSciNet review: 2888184