2007/01/16 by D. Beliaev, Stanislav Smirnov, Beliaev, D. +2 · 1 citation
Mathematics · #30Cxx #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Probability (math.PR) #math.CV #math.PR #msc:30Cxx
paper · pdf · doi:10.48550/arxiv.math/0701463
arxiv created 2007/01/16 · openalex publication_date 2007/01/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structure. This makes it much harder to describe extremals and to attack such problems. Many of these problems are 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 its 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 behaviour of harmonic measure, showing how to constuct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.