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Conformal Geometry and Dynamics

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Pinching holomorphic correspondences


Authors: Shaun Bullett and Peter Haïssinsky
Journal: Conform. Geom. Dyn. 11 (2007), 65-89
MSC (2000): Primary 37F05; Secondary 37F30
DOI: https://doi.org/10.1090/S1088-4173-07-00160-9
Published electronically: June 5, 2007
MathSciNet review: 2314243
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Abstract: For certain classes of holomorphic correspondences which are matings between Kleinian groups and polynomials, we prove the existence of pinching deformations, analogous to Maskit's deformations of Kleinian groups which pinch loxodromic elements to parabolic elements. We apply our results to establish the existence of matings between quadratic maps and the modular group, for a large class of quadratic maps, and of matings between the quadratic map $ z\to z^2$ and circle-packing representations of the free product $ C_2*C_3$ of cyclic groups of order $ 2$ and $ 3$.


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  • 1. A.F.Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics No. 91, Springer Verlag 1983. MR 698777 (85d:22026)
  • 2. L.Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94-97. MR 0111834 (22:2694)
  • 3. S.Bullett and M.Freiberger, Holomorphic correspondences mating Chebyshev-like maps and Hecke groups, Ergodic Theory and Dynamical Systems 25 (2005), 1057-1090. MR 2158397
  • 4. S.Bullett and W.J.Harvey, Mating quadratic maps with Kleinian groups via quasiconformal surgery, Electronic Research Announcements of the AMS 6 (2000), 21-30. MR 1751536 (2000m:37068)
  • 5. S.Bullett and C.Penrose, Mating quadratic maps with the modular group, Inventiones Mathematicae 115 (1994), 483-511. MR 1262941 (95c:58148)
  • 6. S.Bullett and P.Sentenac, Ordered orbits of the shift, square roots and the devil's staircase, Math. Proc. Cam. Phil. Soc. 115 (1994), 451-481. MR 1269932 (95j:58043)
  • 7. Cui, G.-Z., Dynamics of rational maps, topology, deformation and bifurcation, preprint, May 2002.
  • 8. A.Douady and J.H.Hubbard, Itération des polynômes quadratiques complexes, C.R. Acad. Sci. Paris 294 (1982), 123-126. MR 651802 (83m:58046)
  • 9. A.Douady and J.H.Hubbard, On the dynamics of polynomial-like mappings, An. de l'École Norm. Sup. 18 (1985), 287-343. MR 816367 (87f:58083)
  • 10. P.Haïssinsky, Chirurgie parabolique, C.R. Acad. Sci. Paris 327 (1998), 195-198. MR 1645124 (99i:58127)
  • 11. P.Haïssinsky, Rigidity and expansion for rational maps, J. London Math. Soc. (2) 63 (2001), no. 1, 128-140. MR 1802762 (2001m:37085)
  • 12. P.Haïssinsky, Pincement de polynômes, Comment. Math. Helv. 77 (2002), no. 1, 1-23. MR 1898391 (2003m:37061)
  • 13. P.Haïssinsky and Tan Lei, Matings of geometrically finite polynomials, Fund. Math. 181 (2004), 143-188. MR 2070668 (2005e:37106)
  • 14. B.Maskit, On Klein's combination theorem, Trans. AMS 120 (1965), 499-509. MR 0192047 (33:274)
  • 15. B.Maskit, Parabolic elements in Kleinian groups, Ann. Math. 117 (1983), 659-668. MR 701259 (85a:30073)
  • 16. C.T.McMullen, Automorphisms of rational maps, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 31-60, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988. MR 955807 (89m:58187)
  • 17. C.T.McMullen, Cusps are dense, Annals of Mathematics 133 (1991), 217-247. MR 1087348 (91m:30058)
  • 18. F.Przytycki and S.Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math. 155 (1998), no. 2, 189-199. MR 1606527 (2000b:37047)
  • 19. M.Rees, Realization of matings of polynomials as rational maps of degree two, manuscript, 1986.
  • 20. Tan Lei, Matings of quadratic polynomials, Erg. Th. and Dynam. Syst. 12 (1992), 589-620. MR 1182664 (93h:58129)

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Additional Information

Shaun Bullett
Affiliation: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom
Email: s.r.bullett@qmul.ac.uk

Peter Haïssinsky
Affiliation: LATP/CMI, Université de Provence, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France
Email: phaissin@cmi.univ-mrs.fr

DOI: https://doi.org/10.1090/S1088-4173-07-00160-9
Keywords: Holomorphic correspondences, matings, quasiconformal deformations, pinching, circle-packing
Received by editor(s): June 19, 2006
Published electronically: June 5, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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