Geodesic
Random conformal snowflakes
Dmitri Beliaev1 ; Stanislav Smirnov2
1 Department of Mathematics, Princeton University, Fine Hall - Washington Rd., Princeton, NJ 08544, United States
2 Université de Genève, Section de Mathématiques, 2-4, rue du Lièvre, c.p. 64, CH-1211 Genève 4, Switzerland
Annals of mathematics, Tome 172 (2010) no. 1, pp. 597-615.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2010.172.597

Dmitri Beliaev 1 ; Stanislav Smirnov 2

1 Department of Mathematics, Princeton University, Fine Hall - Washington Rd., Princeton, NJ 08544, United States
2 Université de Genève, Section de Mathématiques, 2-4, rue du Lièvre, c.p. 64, CH-1211 Genève 4, Switzerland 
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Dmitri Beliaev; Stanislav Smirnov. Random conformal snowflakes. Annals of mathematics, Tome 172 (2010) no. 1, pp. 597-615. doi : 10.4007/annals.2010.172.597. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.597/

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