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Random conformal snowflakes. (English) Zbl 1204.30015

The authors report on their search for extremal fractals. They argue that one should study random fractals instead of deterministic ones, introduce a new class of random fractals, called random conformal snowflakes, and investigate their properties. As a consequence, the authors significantly improve known estimates for the multifractal spectra of the harmonic measure. It is shown that, for this class, the average integral means spectrum, \(\overline\beta(t)\), is related to the main eigenvalue of a particular integral operator. Also, fractal approximation for this class is proved.
An example of a snowflake is given. For this snowflake, it is proved that \(\overline\beta(t)> 0.2308\). This significantly improves the previously known estimate for the universal spectrum, \(B(1) > 0.17\), due to Ch. Pommerenke [Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbücher. Band XXV. Göttingen: Vandenhoeck & Ruprecht (1975; Zbl 0298.30014)].

MSC:

30C75 Extremal problems for conformal and quasiconformal mappings, other methods
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
30C85 Capacity and harmonic measure in the complex plane
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions

Citations:

Zbl 0298.30014

References:

[1] D. Beliaev, ”Integral means spectrum of random conformal snowflakes,” Nonlinearity, vol. 21, iss. 7, pp. 1435-1442, 2008. · Zbl 1154.30018 · doi:10.1088/0951-7715/21/7/003
[2] D. Beliaev, ”Harmonic measure on random fractals,” PhD Thesis , Royal Institute of Technology, 2005. · Zbl 1079.30026
[3] D. Beliaev and S. Smirnov, ”Harmonic measure on fractal sets,” in European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 41-59. · Zbl 1079.30026
[4] L. Carleson and P. W. Jones, ”On coefficient problems for univalent functions and conformal dimension,” Duke Math. J., vol. 66, iss. 2, pp. 169-206, 1992. · Zbl 0765.30005 · doi:10.1215/S0012-7094-92-06605-1
[5] L. de Branges, ”A proof of the Bieberbach conjecture,” Acta Math., vol. 154, iss. 1-2, pp. 137-152, 1985. · Zbl 0573.30014 · doi:10.1007/BF02392821
[6] T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman, ”Fractal measures and their singularities: the characterization of strange sets,” Phys. Rev. A, vol. 33, iss. 2, pp. 1141-1151, 1986. · Zbl 1184.37028 · doi:10.1103/PhysRevA.34.1601
[7] H. Hedenmalm and S. Shimorin, ”Weighted Bergman spaces and the integral means spectrum of conformal mappings,” Duke Math. J., vol. 127, iss. 2, pp. 341-393, 2005. · Zbl 1075.30005 · doi:10.1215/S0012-7094-04-12725-3
[8] J. E. Littlewood, ”On inequalities in the theory of functions,” Proc. L.M.S., vol. 23, pp. 481-519, 1925. · JFM 51.0247.03
[9] N. G. Makarov, ”Fine structure of harmonic measure,” Algebra i Analiz, vol. 10, iss. 2, pp. 1-62, 1998. · Zbl 0909.30016
[10] N. G. Makarov and C. Pommerenke, ”On coefficients, boundary size and Hölder domains,” Ann. Acad. Sci. Fenn. Math., vol. 22, iss. 2, pp. 305-312, 1997. · Zbl 0890.30010
[11] B. B. Mandelbrot, ”Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence,” in Statistical Models Turbulence, Proc. Sympos. Univ. California, San Diego, 1972, pp. 333-351. · Zbl 0227.76081
[12] B. B. Mandelbrot, ”Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier,” J. Fluid Mech., vol. 62, pp. 331-358, 1974. · Zbl 0289.76031 · doi:10.1017/S0022112074000711
[13] C. Pommerenke, ”On the coefficients of univalent functions,” J. London Math. Soc., vol. 42, pp. 471-474, 1967. · Zbl 0177.33602 · doi:10.1112/jlms/s1-42.1.471
[14] C. Pommerenke, Univalent Functions, Göttingen: Vandenhoeck & Ruprecht, 1975. · Zbl 0298.30014
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