Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2018: 152.31

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 12, Issue 5

Issues

Volume 13 (2015)

The Carathéodory topology for multiply connected domains II

Mark Comerford
Published Online: 2014-02-15 | DOI: https://doi.org/10.2478/s11533-013-0365-y

Abstract

We continue our exposition concerning the Carathéodory topology for multiply connected domains which we began in [Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340] by introducing the notion of boundedness for a family of pointed domains of the same connectivity. The limit of a convergent sequence of n-connected domains which is bounded in this sense is again n-connected and will satisfy the same bounds. We prove a result which establishes several equivalent conditions for boundedness. This allows us to extend the notions of convergence and equicontinuity to families of functions defined on varying domains.

Keywords: Carathéodory Topology; Meridians; Bounded family of pointed domains

MSC: 30C75; 30C20; 30C45

  • [1] Ahlfors L.V., Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies, 10, Van Nostrand, Toronto, 1966 Google Scholar

  • [2] Beardon A.F., Pommerenke Ch., The Poincaré metric of plane domains, J. London Math. Soc., 1978, 18(3), 475–483 http://dx.doi.org/10.1112/jlms/s2-18.3.475CrossrefGoogle Scholar

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

  • [4] Comerford M., Short separating geodesics for multiply connected domains, Cent. Eur. J. Math., 2011, 9(5), 984–996 http://dx.doi.org/10.2478/s11533-011-0065-4CrossrefGoogle Scholar

  • [5] Comerford M., A straightening theorem for non-autonomous iteration, Comm. Appl. Nonlinear Anal., 2012, 19(2), 1–23 Google Scholar

  • [6] Comerford M., The Carathéodory topology for multiply connected domains I, Cent. Eur. J. Math., 2013, 11(2), 322–340 http://dx.doi.org/10.2478/s11533-012-0136-1CrossrefGoogle Scholar

  • [7] Conway J.B., Functions of One Complex Variable, Grad. Texts in Math., 11, Springer, New York-Heidelberg, 1972 Google Scholar

  • [8] Epstein A.L., Towers of Finite Type Complex Analytic Maps, PhD thesis, CUNY Graduate School, 1993 Google Scholar

  • [9] Herron D.A., Liu X.Y., Minda D., Ring domains with separating circles or separating annuli, J. Analyse Math., 1989, 53, 233–252 http://dx.doi.org/10.1007/BF02793416CrossrefGoogle Scholar

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

  • [11] Lang S., Complex Analysis, 3rd ed., Grad. Texts in Math., 103, Springer, New York, 1993 Google Scholar

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

  • [13] Newman M.H.A., Elements of the Topology of Plane Sets of Points, 2nd ed., Cambridge University Press, Cambridge, 1961 Google Scholar

  • [14] Pommerenke Ch., Uniformly perfect sets and the Poincaré metric, Arch. Math., 1979, 32(2), 192–199 http://dx.doi.org/10.1007/BF01238490CrossrefGoogle Scholar

About the article

Published Online: 2014-02-15

Published in Print: 2014-05-01


Citation Information: Open Mathematics, Volume 12, Issue 5, Pages 721–741, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-013-0365-y.

Export Citation

© 2014 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in