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Components of degree two hyperbolic rational maps

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We examine the structure of hyperbolic components and some boundary points of these for degree two rational maps. Rather than using quasi-conformal deformation theory, we use a technique from a theorem of Thurston about the existence of rational maps. This involves regarding a rational map as acting on the Teichmüller space of punctured sphere.

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References

  • [A] Ahlfors, L.: Lectures on quasi-conformal mappings. D. van Nostrand Co., 1966

  • [B] Brolin, M.: Invariant sets under iteration of rational functions. Ark. Mat.6, 103–144 (1966)

    Google Scholar 

  • [D] Douady, A.: Systèmes dynamiques holomorphes. Séminaire Bourbaki 1982/3, Astérisque105/6, 39–63 (1983)

    Google Scholar 

  • [D-H1] Douady, A., Hubbard, J.H.: Etudes Dynamiques des Polynômes Complexes, avec la collaboration de P. Lavaurs, Tan Lei, P. Sentenac. Parts I and II, Publications Mathématiques d'Orsay, 1985

  • [D-H2] Douady, A., Hubbard, J.H.: Itération des polynômes quadratiques complexes. C.R. Acad. Sci. Paris, Série I, t.294, 123–126 (1982)

    Google Scholar 

  • [D-H3] Douady, A., Hubbard, J.H.: A proof of Thurston's Topological Classification of Rational Functions. Mitlag-Leffler Preprint, 1985

  • [Du] Duren, P. L.: Univalent Functions. New York: Springer 1983

    Google Scholar 

  • [F] Fatou, P.: Memoire sur les equations fonctionelles. Bull. Soc Math. France47, 161–271 (1919),47, 33–96; 208–314 (1920)

    Google Scholar 

  • [J] Julia, G.: Itération des applications fonctionelles. J. Math. Pures Appl.8, 47–245 (1918)

    Google Scholar 

  • [L] Levy, S.V.F.: Critically finite rational maps. Thesis. Princeton University, 1985

  • [M] Milnor, J.: Hyperbolic components in spaces of hyperbolic maps. Preprint, I.A.S. 1988

  • [McM] McMullen, C.: Automorphisms of Rational Maps. Preprint 1986

  • [M-S-S] Mane, R., Sad, P., Sullivan, D.: On the dynamics of rational maps. Ann. Sic. Ec. Norm. Super., IV. Ser.16, 193–217 (1983)

    Google Scholar 

  • [SI] Sullivan, D.: Quasi-conformal homeomorphisms, and dynamics I. Ann. Math.122, 401–418 (1985)

    Google Scholar 

  • [S2] Sullivan, D.: Quasi-conformal homeomorphisms and dynamics III: Topological conjugacy classes of analytic endomorphisms. Ann. Math. (to appear)

  • [T1] Thurston, W.P.: On the Combinatorics of Iterated Rational Maps. Preprint, Princeton University and I.A.S., 1985

  • [T2] Thurston, W.P., The Geometry and Topology of Three-manifolds. Notes, Princeton University

  • [TL] Tan Lei, Accouplements des polynômes complexes. Thèse, Université de Paris-Sud, Orsay, 1987

    Google Scholar 

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  1. Department of Pure Mathematics, University of Liverpool, P.O. Box 147, L69 3BX, Liverpool, UK

    M. Rees

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  1. M. Rees
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Rees, M. Components of degree two hyperbolic rational maps. Invent Math 100, 357–382 (1990). https://doi.org/10.1007/BF01231191

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