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The Carathéodory topology for multiply connected domains I

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We consider the convergence of pointed multiply connected domains in the Carathéodory topology. Behaviour in the limit is largely determined by the properties of the simple closed hyperbolic geodesics which separate components of the complement. Of particular importance are those whose hyperbolic length is as short as possible which we call meridians of the domain. We prove continuity results on convergence of such geodesics for sequences of pointed hyperbolic domains which converge in the Carathéodory topology to another pointed hyperbolic domain. Using these we describe an equivalent condition to Carathéodory convergence which is formulated in terms of Riemann mappings to standard slit domains.
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Bibliografia
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  • [5] Comerford M., Short separating geodesics for multiply connected domains, Cent. Eur. J. Math., 2011, 9(5), 984–996 http://dx.doi.org/10.2478/s11533-011-0065-4
  • [6] Comerford M., A straightening theorem for non-autonomous iteration, Comm. Appl. Nonlinear Anal., 2012, 19(2), 1–23
  • [7] Comerford M., The Carathéodory topology for multiply connected domains II, Cent. Eur. J. Math. (in press), preprint available at http://arxiv.org/abs/1103.2537
  • [8] Duren P.L., Univalent Functions, Grundlehren Math. Wiss., 259, Springer, New York, 1983
  • [9] Epstein A.L., Towers of Finite Type Complex Analytic Maps, PhD thesis, CUNY, New York, 1993
  • [10] Hejhal D.A., Universal covering maps for variable regions, Math. Z., 1974, 137, 7–20 http://dx.doi.org/10.1007/BF01213931
  • [11] Hubbard J.H., Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. I, Matrix Editions, Ithaca, 2006
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  • [13] McMullen C.T., Complex Dynamics and Renormalization, Ann. of Math. Stud., 135, Princeton University Press, Princeton, 1994
Typ dokumentu
Bibliografia
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bwmeta1.element.doi-10_2478_s11533-012-0136-1
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