Random conformal snowflakes

• 成果类型: Article
• 署名作者: Beliaev, Dmitri; Smirnov, Stanislav
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2010.172.597
• 发表日期: 2010
• 页码: 597-615
• 关键词: integral means spectrum; univalent-functions; coefficients; dimension

摘要

In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structures, making it hard to describe them and to attack such problems. This is particularly true for questions related to the multifractal analysis of harmonic measure. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals: random conformal snowflakes, and investigate their properties, developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behavior of harmonic measure, showing how to construct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.