Hostname: page-component-5b777bbd6c-rbv74 Total loading time: 0 Render date: 2025-06-22T23:28:34.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity of condenser capacity under holomorphic motions

Part of: Two-dimensional theory Complex dynamical systems Geometric function theory

Published online by Cambridge University Press:  29 June 2020

Stamatis Pouliasis
Stamatis Pouliasis*
Affiliation:
Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas79409, USA
*
e-mail: stamatis.pouliasis@ttu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that condenser capacity varies continuously under holomorphic motions, and the corresponding family of the equilibrium measures of the condensers is continuous with respect to the weak-star convergence. We also study the behavior of uniformly perfect sets under holomorphic motions.

Keywords

Condenser capacityequilibrium measureuniformly perfect setsholomorphic motions

MSC classification

Primary: 31A15: Potentials and capacity, harmonic measure, extremal length 31A05: Harmonic, subharmonic, superharmonic functions 30C85: Capacity and harmonic measure in the complex plane 37F45: Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 2 , June 2021 , pp. 340 - 348
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Astala, K., Iwaniec, T., and Martin, G., Elliptic partial differential equations and quasiconformal mappings in the plane . Princeton Mathematical Series, 48, Princeton University Press, Princeton, NJ, 2009.Google Scholar
Astala, K. and Martin, G. J., Holomorphic motions . In: Papers on analysis, Rep. Univ. Jyväskylä Dep. Math. Stat., 83, Univ. Jyväskylä, 2001, pp. 27–40.Google Scholar
Bagby, T., The modulus of a plane condenser . J. Math. Mech. 17(1967), 315–329. https://doi.org/10.1512/iumj.1968.17.17017 Google Scholar
Dubinin, V. N., Condenser capacities and symmetrization in geometric function theory . Translated from Russian by Nikolai G. Kruzhilin, Springer, Basel, 2014. https://doi.org/10.1007/978-3-0348-0843-9 CrossRefGoogle Scholar
Earle, C. J. and Mitra, S., Variation of moduli under holomorphic motions . In: In the tradition of Ahlfors and Bers, Contemp. Math., 256, Amer. Math. Soc., Providence, RI, 2000, pp. 39–67. https://doi.org/10.1090/conm/256/03996 CrossRefGoogle Scholar
Folland, G. B., Real analysis. Modern techniques and their applications . 2nd ed., Pure and Applied Mathematics (New York), John Wiley and Sons, New York, 1999.Google Scholar
Goluzin, G. M., Geometric theory of functions of a complex variable . Translations of Mathematical Monographs, 26, Amer. Math. Soc., Providence, RI, 1969.Google Scholar
Helms, L. L., Introduction to potential theory . Pure and Applied Mathematics, 22, Wiley-Interscience, New York-London-Sydney, 1969.Google Scholar
Järvi, P. and Vuorinen, M., Uniformly perfect sets and quasiregular mappings . J. Lond. Math. Soc. (2) 54(1996), no. 3, 515–529. https://doi.org/10.1112/jlms/54.3.515 CrossRefGoogle Scholar
Jiang, Y. and Mitra, S., Douady-Earle section, holomorphic motions, and some applications . In: Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces, Contemp. Math., 575, Amer. Math. Soc., Providence, RI, 2012, pp. 219–251. https://doi.org/10.1090/conm/575/11400 CrossRefGoogle Scholar
Landkof, N. S., Foundations of modern potential theory . Springer-Verlag, New York-Heidelberg, 1972.CrossRefGoogle Scholar
Mañé, R., Sad, P., and Sullivan, D., On the dynamics of rational maps . Ann. Sci. École Norm. Sup. (4) 16(1983), 193–217.CrossRefGoogle Scholar
Martin, G. J., Holomorphic motions and quasicircles . Proc. Amer. Math. Soc. 141(2013), no. 11, 3911–3918. https://doi.org/10.1090/S0002-9939-2013-11684-9 CrossRefGoogle Scholar
Pommerenke, C., Uniformly perfect sets and the Poincarémetric . Arch. Math. 32(1979), no. 2, 192–199. https://doi.org/10.1007/BF1238490 CrossRefGoogle Scholar
Pouliasis, S., Condenser energy under holomorphic motions . Illinois J. Math. 55(2011), 1119–1134.CrossRefGoogle Scholar
Pouliasis, S., Ransford, T. J., and Younsi, M., Analytic capacity and holomorphic motions . Conform. Geom. Dyn. 23(2019), 130–134. https://doi.org/10.1090/ecgd/336 CrossRefGoogle Scholar
Ransford, T. J., Potential theory in the complex plane . Cambridge University Press, Cambridge, UK, 1995. https://doi.org/10.1017/CB09780511623776 CrossRefGoogle Scholar
Ransford, T. J., Variation of Hausdorff dimension of Julia sets . Ergod. Theory Dyn. Syst. 13(1993), 167–174. https://doi.org/10.1017/S0143385700007276 CrossRefGoogle Scholar
Ransford, T. J., Younsi, M., and Ai, W.-H., Continuity of capacity of a holomorphic motion . Preprint, 2020. arXiv:2004.05266.CrossRefGoogle Scholar
Ruelle, D., Repellers for real analytic maps . Ergod. Theory Dyn. Syst. 2(1982), 99–107. https://doi.org/10.1017/s0143385700009603 CrossRefGoogle Scholar
Słodkowski, Z., Holomorphic motions and polynomial hulls . Proc. Amer. Math. Soc. 111(1991), 347–355. https://doi.org/10.2307/2048323 CrossRefGoogle Scholar