A combination theorem for covering correspondences and an application to mating polynomial maps with Kleinian groups
HTML articles powered by AMS MathViewer
- by Shaun Bullett PDF
- Conform. Geom. Dyn. 4 (2000), 75-96 Request permission
Abstract:
The simplest version of the Maskit-Klein combination theorems concerns the action of a free product of two finite subgroups of $PSL(2,{\mathbb C})$ on the Riemann sphere $\hat {\mathbb C}$, when these subgroups have fundamental domains whose interiors together cover $\hat {\mathbb C}$. We prove an analogous combination theorem for covering correspondences of rational maps, making use of Douady and Hubbard’s Straightening Theorem for polynomial-like maps to describe the structure of the limit sets. We apply our theorem to construct holomorphic correspondences which are matings of polynomial maps with Hecke groups $C_p*C_q$, and we show how it may also be applied to the analysis of separable correspondences.References
- Shaun Bullett and Christopher Penrose, Mating quadratic maps with the modular group, Invent. Math. 115 (1994), no. 3, 483–511. MR 1262941, DOI 10.1007/BF01231770
- Shaun Bullett and Christopher Penrose, A gallery of iterated correspondences, Experiment. Math. 3 (1994), no. 2, 85–105. MR 1313875, DOI 10.1080/10586458.1994.10504282 bpreg S. Bullett and C. Penrose, Regular and limit sets for holomorphic correspondences, QMW preprint 1999.
- Shaun Bullett and Christopher Penrose, Perturbing circle-packing Kleinian groups as correspondences, Nonlinearity 12 (1999), no. 3, 635–672. MR 1690198, DOI 10.1088/0951-7715/12/3/313 bh S. Bullett and W. Harvey, Mating quadratic maps with Kleinian groups via quasiconformal surgery, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 21–30.
- 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 kl F. Klein, Neue Beiträge zur Riemann’schen Functiontheorie, Math. Ann. 21 (1883), 141–218.
- Bernard Maskit, On Klein’s combination theorem, Trans. Amer. Math. Soc. 120 (1965), 499–509. MR 192047, DOI 10.1090/S0002-9947-1965-0192047-1
- Bernard Maskit, On Klein’s combination theorem. II, Trans. Amer. Math. Soc. 131 (1968), 32–39. MR 223570, DOI 10.1090/S0002-9947-1968-0223570-1
- Bernard Maskit, On Klein’s combination theorem. III, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Studies, No. 66, Princeton Univ. Press, Princeton, N.J., 1971, pp. 297–316. MR 0289768
- Bernard Maskit, On Klein’s combination theorem. IV, Trans. Amer. Math. Soc. 336 (1993), no. 1, 265–294. MR 1137258, DOI 10.1090/S0002-9947-1993-1137258-0
Additional Information
- Shaun Bullett
- Affiliation: School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom
- Email: s.r.bullett@qmw.ac.uk
- Received by editor(s): September 30, 1999
- Received by editor(s) in revised form: January 20, 2000
- Published electronically: April 27, 2000
- Additional Notes: I would like to thank Christopher Penrose for many helpful discussions concerning this work.
- © Copyright 2000 American Mathematical Society
- Journal: Conform. Geom. Dyn. 4 (2000), 75-96
- MSC (2000): Primary 37F05; Secondary 30D05, 30F40
- DOI: https://doi.org/10.1090/S1088-4173-00-00056-4
- MathSciNet review: 1755900