By L. Olsen (auth.), Christoph Bandt, Siegfried Graf, Martina Zähle (eds.)

The moment convention on Fractal Geometry and Stochastics used to be held at Greifs wald/Koserow, Germany from August 28 to September 2, 1998. 4 years had handed after the 1st convention with this subject matter and through this era the curiosity within the topic had speedily elevated. a couple of hundred mathematicians from twenty-two nations attended the second one convention and such a lot of them provided their most modern effects. because it is very unlikely to gather some of these contributions in a booklet of average dimension we made up our minds to invite the thirteen major audio system to put in writing an account in their topic of curiosity. The corresponding articles are accrued during this quantity. lots of them mix a comic strip of the historic improvement with an intensive dialogue of the newest result of the fields thought of. We think that those surveys are of profit to the readers who are looking to be brought to the topic in addition to to the experts. We additionally imagine that this booklet displays the most instructions of analysis during this thriving region of arithmetic. We exhibit our gratitude to the Deutsche Forschungsgemeinschaft whose monetary help enabled us to prepare the convention. The Editors creation Fractal geometry bargains with geometric items that exhibit a excessive measure of irregu larity on all degrees of importance and, for this reason, can't be investigated by means of equipment of classical geometry yet, however, are fascinating types for phenomena in physics, chemistry, biology, astronomy and different sciences.

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**Sample text**

F log E[3~] l' . t)-a. a. x E lRn. t) < 00. 1. The Non-Trivial case. t) x ')'n,m-a. a. £(q) + COb1{mL(II+x)(q) 25 Multifractal Geometry :<::: cob fLn (1t=L(II+x))(q) :<::: cobfL(q) + COb1t =L(II+x)(q) + max (0, q,(q), ~m(l ~ q)) . If n~ m < dimj1, ,(q) = 0 and qrnin < q :<::: 1, then (Hz,tLsupp j1) x ,n,m -a. a. (x, II) E lFt n x G(n,m) satisfy CObfLn(1t=L(II+x))(q) = CObfL(q) + COb1t=L(II+x)(q). 2. The Trivial case. If dim SUpp j1 < n~ m, then (Hz,tLsupp j1) x ,n,m -a. a. (x, II) E lFt n x G(n, m) satisfy j1 n (Hm L (II + x)) = 0 , where 0 denotes the zero measure.

Ii) v).. (x) = v).. (x) = 2"1[F).. (X-I) -,X- + F).. (X+l)] -,X- . 1991 Mathematics Subject Classification. Primary 42A85; Secondary HR06, 26A46, 26A30, 28A 78, 28A80 Key words and phrases. self-similar measures, Hausdorff dimension, Salem numbers Research of Peres was partially supported by NSF grant #DMS-9803597. Research of Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright foundation, and the Institute of Mathematics at the Hebrew University of Jerusalem. 40 Yuval Peres, Wilhelm Schlag, and Boris Solomyak In other words, 1/)..

We do not analyse the critical case in this paper. The trivial case: This case is defined by dim supp fJ < n - m. In the trivial case the measure fJ is "smaller" than the Hausdorff measre Hn-mLII-L restricted to II -L for II E G (n, m). Since the natural slices fJ n (Hm L (II + x)) are obtained by disintegrating fJ with respect to Hn-m L II -L, we therefore expect that most slices fJ n (Hm L (II + x)) are "small" and have a trivial multifractal structure in the trivial case. Indeed, it can be proved (cf.