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.doi-10_2478_v10062-012-0024-y,
author = {Alan Sola},
title = {Elementary examples of Loewner chains generated by densities},
journal = {Annales UMCS, Mathematica},
volume = {67},
year = {2013},
pages = {83-101},
zbl = {1291.30114},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0024-y}
}
Alan Sola. Elementary examples of Loewner chains generated by densities. Annales UMCS, Mathematica, Tome 67 (2013) pp. 83-101. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_v10062-012-0024-y/
[1] Berkson, E., Porta, H., Semigroups of analytic functions and composition operators, Michigan Math. J. 25 (1978), 101-115. | Zbl 0382.47017
[2] Bracci, F., Contreras, M. D., D´ıaz-Madrigal, S., Evolution families and the Loewnerequation I: the unit disk, J. Reine Angew. Math., to appear.
[3] Bracci, F., Contreras, M. D., D´ıaz-Madrigal, S., Regular poles and β-numbers in thetheory of holomorphic semigroups, arxiv.org/abs/1201.4705.
[4] Carleson, L., Makarov, N., Aggregation in the plane and Loewner’s equation, Comm. Math. Phys. 216 (2001), 583-607. | Zbl 1042.82039
[5] Carleson, L., Makarov, N., Laplacian path models. Dedicated to the memory ofThomas H. Wolff, J. Anal. Math. 87 (2002), 103-150. | Zbl 1040.30011
[6] Contreras, M. D., D´ıaz-Madrigal, S., Gumenyuk, P., Geometry behind Loewnerchains, Complex Anal. Oper. Theory 4 (2010), 541-587. | Zbl 1209.30010
[7] Contreras, M. D., D´ıaz-Madrigal, S., Gumenyuk, P., Local duality in Loewner equations, arxiv.org/abs/1202.2334. | Zbl 1316.30025
[8] Contreras, M. D., D´ıaz-Madrigal, S., Pommerenke, Ch., On boundary critical pointsfor semigroups of analytic functions, Math. Scand. 98 (2006), 125-142. | Zbl 1143.30013
[9] Dur´an, M. A., Vasconcelos, G. L., Interface growth in two dimensions: A Loewnerequation approach, Phys. Rev. E 82 (2010), 031601.
[10] Elin, M., Shoikhet, D., Linearization Models for Complex Dynamical Systems, Birkh¨auser Verlag, Basel, 2010. | Zbl 1198.37001
[11] Gubiec, T., Szymczak, P., Fingered growth in channel geometry: A Loewner equationapproach, Phys. Rev. E 77 (2008), 041602.
[12] Hastings, M., Levitov, L., Laplacian growth as one-dimensional turbulence, Phys. D: Nonlinear Phenomena 116 (1998), 244-252. | Zbl 0962.76542
[13] Ivanov, G., Prokhorov, D., Vasil’ev, A., Singular solutions to the Loewner equation, Bull. Sci. Math. 136 (2012), 328-341.
[14] 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. | Zbl 1251.82025
[15] Kager, W., Nienhuis, B., Kadanoff, L. P., Exact solutions for Loewner evolutions, J. Statist. Phys. 115 (2004), 805-822. | Zbl 1056.30005
[16] Kuznetsov, A., Boundary behaviour of Loewner chains, arxiv.org/abs/0705.4564.
[17] Lawler, G., Conformally Invariant Processes in the Plane, American Mathematical Society, Providence, 2005. | Zbl 1074.60002
[18] Lind, J., A sharp condition for the Loewner equation to generate slits, Ann. Acad. Sci. Fenn. Math. 30 (2005), 143-158. | Zbl 1069.30012
[19] Lind, J., Marshall, D. E., Rohde, S., Collisions and spirals of Loewner traces, Duke Math. J. 154 (2010), 527-573.[WoS] | Zbl 1206.30024
[20] Marshall, D. E., Rohde, S., The Loewner differential equation and slit mappings, J. Amer. Math. Soc. 18 (2005), 763-778. | Zbl 1078.30005
[21] Pommerenke, Ch., Univalent Functions, Vandenhoeck & Ruprecht, G¨ottingen, 1975.
[22] Pommerenke, Ch., Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin- Heidelberg, 1992. | Zbl 0534.30008
[23] Popescu, M. N., Hentschel, H. G. E., Family, F., Anisotropic diffusion-limited aggregation, Phys. Rev. E 69 (2004), 061403.
[24] Rohde, S., Zinsmeister, M., Some remarks on Laplacian growth, Topology Appl. 152 (2005), 26-43. | Zbl 1077.60040
[25] Selander, G., Two deterministic growth models related to diffusion-limited aggregation. Doctoral dissertation, Thesis (Dr. Tech.), Kungliga Tekniska Hogskolan, 1999, pp. 101.
[26] 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. | Zbl 0904.47030
[27] Vasil’ev, A., Evolution of conformal maps with quasiconformal extensions, Bull. Sci. Math. 129 (2005), 831-859.