We study explicit examples of Loewner chains generated by absolutely continuous driving measures, and discuss how properties of driving measures are reflected in the shapes of the growing Loewner hulls.
@article{bwmeta1.element.ojs-doi-10_17951_a_2013_67_1_83-101,
author = {Alan Sola},
title = {Elementary examples of Loewner chains generated by densities},
journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
volume = {67},
year = {2013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2013_67_1_83-101}
}
Alan Sola. Elementary examples of Loewner chains generated by densities. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 67 (2013) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2013_67_1_83-101/
Berkson, E., Porta, H., Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), 101–115.
Bracci, F., Contreras, M. D., Dıaz-Madrigal, S., Evolution families and the Loewner equation I: the unit disk, J. Reine Angew. Math., to appear.
Bracci, F., Contreras, M. D., Dıaz-Madrigal, S., Regular poles and -numbers in the theory of holomorphic semigroups, arxiv.org/abs/1201.4705.
Carleson, L., Makarov, N., Aggregation in the plane and Loewner’s equation, Comm. Math. Phys. 216 (2001), 583–607.
Carleson, L., Makarov, N., Laplacian path models. Dedicated to the memory of Thomas H. Wolff, J. Anal. Math. 87 (2002), 103–150.
Contreras, M. D., Dıaz-Madrigal, S., Gumenyuk, P., Geometry behind Loewner chains, Complex Anal. Oper. Theory 4 (2010), 541–587.
Contreras, M. D., Dıaz-Madrigal, S., Gumenyuk, P., Local duality in Loewner equations, arxiv.org/abs/1202.2334.
Contreras, M. D., Dıaz-Madrigal, S., Pommerenke, Ch., On boundary critical points for semigroups of analytic functions, Math. Scand. 98 (2006), 125–142.
Duran, M. A., Vasconcelos, G. L., Interface growth in two dimensions: A Loewner equation approach, Phys. Rev. E 82 (2010), 031601.
Elin, M., Shoikhet, D., Linearization Models for Complex Dynamical Systems, Birkhauser Verlag, Basel, 2010.
Gubiec, T., Szymczak, P., Fingered growth in channel geometry: A Loewner equation approach, Phys. Rev. E 77 (2008), 041602.
Hastings, M., Levitov, L., Laplacian growth as one-dimensional turbulence, Phys. D: Nonlinear Phenomena 116 (1998), 244–252.
Ivanov, G., Prokhorov, D., Vasil’ev, A., Singular solutions to the Loewner equation, Bull. Sci. Math. 136 (2012), 328–341.
Johansson Viklund, F., Sola, A., Turner, A., Scaling limits of anisotropic Hastings-Levitov clusters, Ann. Inst. H. Poincar´e Probab. Stat. 48 (2012), 235–257.
Kager, W., Nienhuis, B., Kadanoff, L. P., Exact solutions for Loewner evolutions, J. Statist. Phys. 115 (2004), 805–822.
Kuznetsov, A., Boundary behaviour of Loewner chains, arxiv.org/abs/0705.4564.
Lawler, G., Conformally Invariant Processes in the Plane, American Mathematical Society, Providence, 2005.
Lind, J., A sharp condition for the Loewner equation to generate slits, Ann. Acad. Sci. Fenn. Math. 30 (2005), 143–158.
Lind, J., Marshall, D. E., Rohde, S., Collisions and spirals of Loewner traces, Duke Math. J. 154 (2010), 527–573.
Marshall, D. E., Rohde, S., The Loewner differential equation and slit mappings, J. Amer. Math. Soc. 18 (2005), 763–778.
Pommerenke, Ch., Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975.
Pommerenke, Ch., Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin–Heidelberg, 1992.
Popescu, M. N., Hentschel, H. G. E., Family, F., Anisotropic diffusion-limited aggregation, Phys. Rev. E 69 (2004), 061403.
Rohde, S., Zinsmeister, M., Some remarks on Laplacian growth, Topology Appl. 152 (2005), 26–43.
Selander, G., Two deterministic growth models related to diffusion-limited aggregation. Doctoral dissertation, Thesis (Dr. Tech.), Kungliga Tekniska Hogskolan, 1999, pp. 101.
Siskakis, A. G., Semi-groups of composition operators on spaces of analytic functions, a review, Studies on composition operators (Laramie, WY, 1996), 229–252, Contemp. Math., 213, Amer. Math. Soc., Providence, R.I., 1998.
Vasil’ev, A., Evolution of conformal maps with quasiconformal extensions, Bull. Sci. Math. 129 (2005), 831–859.