Random conformal snowflakes
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...
| Main Authors: | , |
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| Format: | Journal article |
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2007
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| _version_ | 1797063163006418944 |
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| author | Beliaev, D Smirnov, S |
| author_facet | Beliaev, D Smirnov, S |
| author_sort | Beliaev, D |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T20:55:54Z |
| format | Journal article |
| id | oxford-uuid:393d29d3-3d13-4334-a680-b0d4cdcd8ef5 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T20:55:54Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:393d29d3-3d13-4334-a680-b0d4cdcd8ef52022-03-26T13:54:25ZRandom conformal snowflakesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:393d29d3-3d13-4334-a680-b0d4cdcd8ef5Symplectic Elements at Oxford2007Beliaev, DSmirnov, SIn 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. |
| spellingShingle | Beliaev, D Smirnov, S Random conformal snowflakes |
| title | Random conformal snowflakes |
| title_full | Random conformal snowflakes |
| title_fullStr | Random conformal snowflakes |
| title_full_unstemmed | Random conformal snowflakes |
| title_short | Random conformal snowflakes |
| title_sort | random conformal snowflakes |
| work_keys_str_mv | AT beliaevd randomconformalsnowflakes AT smirnovs randomconformalsnowflakes |