The Carathéodory topology for multiply connected domains I

Mark Comerford

Open Mathematics (2013)

  • Volume: 11, Issue: 2, page 322-340
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Mark Comerford. "The Carathéodory topology for multiply connected domains I." Open Mathematics 11.2 (2013): 322-340. <http://eudml.org/doc/269062>.

@article{MarkComerford2013,
abstract = {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.},
author = {Mark Comerford},
journal = {Open Mathematics},
keywords = {Carathéodory topology; Hyperbolic geodesics; Meridians; hyperbolic geodesics; meridians},
language = {eng},
number = {2},
pages = {322-340},
title = {The Carathéodory topology for multiply connected domains I},
url = {http://eudml.org/doc/269062},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Mark Comerford
TI - The Carathéodory topology for multiply connected domains I
JO - Open Mathematics
PY - 2013
VL - 11
IS - 2
SP - 322
EP - 340
AB - 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.
LA - eng
KW - Carathéodory topology; Hyperbolic geodesics; Meridians; hyperbolic geodesics; meridians
UR - http://eudml.org/doc/269062
ER -

References

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  6. [6] Comerford M., A straightening theorem for non-autonomous iteration, Comm. Appl. Nonlinear Anal., 2012, 19(2), 1–23 Zbl1259.37030
  7. [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. [8] Duren P.L., Univalent Functions, Grundlehren Math. Wiss., 259, Springer, New York, 1983 
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  12. [12] Keen L., Lakic N., Hyperbolic Geometry from a Local Viewpoint, London Math. Soc. Stud. Texts, 68, Cambridge University Press, Cambridge, 2007 http://dx.doi.org/10.1017/CBO9780511618789 Zbl1190.30001
  13. [13] McMullen C.T., Complex Dynamics and Renormalization, Ann. of Math. Stud., 135, Princeton University Press, Princeton, 1994 Zbl0822.30002

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