Integral Means Spectrum of Random Conformal Snowflakes

It is known that extremal configurations in many important problems in classical complex analysis exhibit complicated fractal structure. This makes such problems extremely difficult. The classical example is the coefficient problem for the class of bounded univalent functions. Let φ(z) = z + a2z 2 + a3z 3 + . . . be a bounded univalent map in the unit disc. One can ask what are the maximal possible values of coefficients an, especially when n tends to infinity. We define γ as the best possible constant such that |an| decays as n. In [4] Carleson and Jones showed that this problem is related to another classical problem about the growth rate of the length of Greens’ lines. In particular, they showed that the extremal configurations for both of this problems should be of a fractal nature. During the last decade it became clear that the right language for these problems, as well as many other classical problems, is the maltifractal analysis. It turned out that all these problems could be reduced to the problem of finding the maximal value of the integral means spectrum. In the recent paper [2] S. Smirnov and the author introduced and studied a new class of random fractals, the so-called random conformal snowflakes. In particular, they proved the fractal approximation for this class, which means that one can find conformal snowflakes with spectra arbitrary close to the maximal possible spectrum. In this paper we report on our search for snowflakes with large spectrum. The paper organized as follows: in the introduction we give some basic information about integral means spectrum, define random conformal snowflakes, and state the main facts about spectrum of the snowflakes. In the Section 2 we give numerical estimates of the spectra of several snowflakes for different values of parameters. In the last Section we give rigorous lower bound for the spectrum at t = 1.