Mating Kleinian groups isomorphic to with quadratic polynomials

Author:
Marianne Freiberger

Journal:
Conform. Geom. Dyn. **7** (2003), 11-33

MSC (2000):
Primary 37F45, 37F30, 37F05; Secondary 37F10

DOI:
https://doi.org/10.1090/S1088-4173-03-00087-0

Published electronically:
May 27, 2003

MathSciNet review:
1992035

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a quadratic polynomial and a representation of in satisfying certain conditions, we will construct a holomorphic correspondence on the sphere (given by a polynomial relation ) that *mates* the two actions: The sphere will be partitioned into two completely invariant sets and . The set consists of the disjoint union of two sets, and , each of which is conformally homeomorphic to the filled Julia set of a degree 4 polynomial . This filled Julia set contains infinitely many copies of the filled Julia set of . Suitable restrictions of the correspondence are conformally conjugate to on each of and . The set will not be connected, but it can be joined up using a family of completely invariant curves. The action of the correspondence on the complement of will then be conformally conjugate to the action of on a simply connected subset of its regular set.

**1.**L. Ahlfors,*Lectures on quasiconformal mappings*, Van Nostrand, 1966. MR**34:336****2.**Alan F. Beardon,*The Geometry of Discrete Groups*, volume 91 of*Graduate Texts in Mathematics*, Springer-Verlag, 1983. MR**85d:22026****3.**Alan F. Beardon,*Iteration of Rational Functions*, volume 132 of*Graduate Texts in Mathematics*, Springer-Verlag, 1991. MR**92j:30026****4.**S. Bullett and W. Harvey, Mating quadratic maps with Kleinian groups via quasiconformal surgery,*Electronic research announcements of the AMS*, 6:21-30, 2000. MR**2000m:37068****5.**S. Bullett and C. Penrose, Regular and limit sets for holomorphic correspondences,*Fundamenta Mathematica*, 167:111-171, 2001. MR**2002d:37068****6.**Shaun Bullett and Chris Penrose, Mating quadratic maps with the modular group,*Inventiones Mathematicae*, 115:483-511, 1994. MR**95c:58148****7.**R. Devaney, The complex dynamics of quadratic polynomials,*Complex Dynamical Systems: The mathematics behind the Mandelbrot and Julia sets, ed:R. Devaney, AMS, Proceedings of Symposia in applied mathematics*, 49:1-30, 1994. MR**95m:30036****8.**A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings,*Ann. Sci. Ec. Norm. Sup*, 18:263-297, 1985. MR**87f:58083****9.**Marianne Freiberger,*Matings between Hecke Groups and polynomials*, PhD thesis, Queen Mary, University of London, 2001.**10.**B. Maskit,*Kleinian Groups*, volume 287 of*Grundlagen der mathematischen Wissenschaften*, Springer-Verlag, 1987. MR**90a:30132****11.**S. Bullett and M. Freiberger,*Holomorphic correspondences mating quadratic maps with Hecke groups*, preprint, 2002.

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Additional Information

**Marianne Freiberger**

Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK

Email:
M.Freiberger@qmul.ac.uk

DOI:
https://doi.org/10.1090/S1088-4173-03-00087-0

Keywords:
Holomorphic correspondences,
rational maps,
Kleinian groups,
quasi-conformal surgery

Received by editor(s):
November 23, 2001

Received by editor(s) in revised form:
March 13, 2003

Published electronically:
May 27, 2003

Additional Notes:
The author was supported in part by the European Commission and EPSRC. The author would like to thank Shaun Bullett for many enlightening conversations. Also, thanks to the referee for useful comments

Article copyright:
© Copyright 2003
American Mathematical Society