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Conformal Geometry and Dynamics
Conformal Geometry and Dynamics
ISSN 1088-4173

 

A combination theorem for covering correspondences and an application to mating polynomial maps with Kleinian groups


Author: Shaun Bullett
Journal: Conform. Geom. Dyn. 4 (2000), 75-96
MSC (2000): Primary 37F05; Secondary 30D05, 30F40
Published electronically: April 27, 2000
MathSciNet review: 1755900
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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.


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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

DOI: http://dx.doi.org/10.1090/S1088-4173-00-00056-4
PII: S 1088-4173(00)00056-4
Keywords: Holomorphic dynamics, polynomial maps, Kleinian groups
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.
Article copyright: © Copyright 2000 American Mathematical Society