The Carathéodory topology for multiply connected domains I


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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] Ahlfors L.V., Complex Analysis, 3rd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1978

  • [2] Beardon A.F., Iteration of Rational Functions, Grad. Texts in Math., 132, Springer, New York, 1991

  • [3] Carathéodory C., Conformal Representation, 2nd ed., Cambridge Tracts in Mathematics and Mathematical Physics, 28, Cambridge University Press, Cambridge, 1952

  • [4] Carleson L., Gamelin T.W., Complex Dynamics, Universitext Tracts Math., Springer, New York, 1993

  • [5] Comerford M., Short separating geodesics for multiply connected domains, Cent. Eur. J. Math., 2011, 9(5), 984–996

  • [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

  • [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

  • [11] Hubbard J.H., Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. I, Matrix Editions, Ithaca, 2006

  • [12] Keen L., Lakic N., Hyperbolic Geometry from a Local Viewpoint, London Math. Soc. Stud. Texts, 68, Cambridge University Press, Cambridge, 2007

  • [13] McMullen C.T., Complex Dynamics and Renormalization, Ann. of Math. Stud., 135, Princeton University Press, Princeton, 1994


Journal + Issues

Open Mathematics is a fully peer-reviewed, open access, electronic journal that publishes significant and original works in all areas of mathematics. The journal publishes both research papers and comprehensive and timely survey articles. Open Math aims at presenting high-impact and relevant research on topics across the full span of mathematics.