# The Carathéodory topology for multiply connected domains II

Open Mathematics (2014)

- Volume: 12, Issue: 5, page 721-741
- ISSN: 2391-5455

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topMark Comerford. "The Carathéodory topology for multiply connected domains II." Open Mathematics 12.5 (2014): 721-741. <http://eudml.org/doc/269107>.

@article{MarkComerford2014,

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.},

author = {Mark Comerford},

journal = {Open Mathematics},

keywords = {Carathéodory Topology; Meridians; Bounded family of pointed domains; Carathéodory topology; meridians; bounded family of pointed domains},

language = {eng},

number = {5},

pages = {721-741},

title = {The Carathéodory topology for multiply connected domains II},

url = {http://eudml.org/doc/269107},

volume = {12},

year = {2014},

}

TY - JOUR

AU - Mark Comerford

TI - The Carathéodory topology for multiply connected domains II

JO - Open Mathematics

PY - 2014

VL - 12

IS - 5

SP - 721

EP - 741

AB - 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.

LA - eng

KW - Carathéodory Topology; Meridians; Bounded family of pointed domains; Carathéodory topology; meridians; bounded family of pointed domains

UR - http://eudml.org/doc/269107

ER -

## References

top- [1] Ahlfors L.V., Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies, 10, Van Nostrand, Toronto, 1966
- [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/s2-18.3.475 Zbl0399.30008
- [3] Carleson L., Gamelin T.W., Complex Dynamics, Universitext Tracts Math., Springer, New York, 1993 Zbl0782.30022
- [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/s11533-011-0065-4 Zbl1277.30015
- [5] Comerford M., A straightening theorem for non-autonomous iteration, Comm. Appl. Nonlinear Anal., 2012, 19(2), 1–23 Zbl1259.37030
- [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/s11533-012-0136-1 Zbl1282.30017
- [7] Conway J.B., Functions of One Complex Variable, Grad. Texts in Math., 11, Springer, New York-Heidelberg, 1972
- [8] Epstein A.L., Towers of Finite Type Complex Analytic Maps, PhD thesis, CUNY Graduate School, 1993
- [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/BF02793416 Zbl0697.30021
- [10] Keen L., Lakic N., Hyperbolic Geometry from a Local Viewpoint, London Math. Soc. Stud. Texts, 68, Cambridge University Press, Cambridge, 2007 Zbl1190.30001
- [11] Lang S., Complex Analysis, 3rd ed., Grad. Texts in Math., 103, Springer, New York, 1993
- [12] McMullen C.T., Complex Dynamics and Renormalization, Ann. of Math. Stud., 135, Princeton University Press, Princeton, 1994 Zbl0822.30002
- [13] Newman M.H.A., Elements of the Topology of Plane Sets of Points, 2nd ed., Cambridge University Press, Cambridge, 1961
- [14] Pommerenke Ch., Uniformly perfect sets and the Poincaré metric, Arch. Math., 1979, 32(2), 192–199 http://dx.doi.org/10.1007/BF01238490 Zbl0393.30005