# Quelques Interactions Entre Analyse, Probabilités et Fractals

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*J. Barral; J. Berestycki; J. Bertoin; A. H. Fan; B. Haas; S. Jaffard; G. Miermont; J. Peyrière*

A publication of the Société Mathématique de France

Following the seminal contributions of Benoît
Mandelbrot in the 1970s, concepts derived from fractal geometry gave a
new impulse to several areas of mathematics. The goal of this volume
is to present syntheses on two subjects where important advances
occurred in the last 15 years: multiplicative processes and
fragmentation. One arose from harmonic analysis (Riesz products) and
the other from a probabilistic model proposed by N. Kolmogorov
to explain experimental observations on rock fragmentation.
However, they share analogies and use common mathematical tools issued
from the study of random fractals.

The first paper introduces basic concepts in fractal analysis. It
starts with the description of the historical developments that led to their
introduction and interactions. The definitions of fractional dimensions are
introduced, and pertinent tools in geometric measure theory are recalled.
Examples of multifractal functions and measures are studied. Finally, ubiquity
systems, which play an increasing role in multifractal analysis, are
introduced.

The second paper deals with fine geometric properties of measures
obtained as limits of multiplicative processes. It starts by showing in which
contexts they appear and by describing their key properties. The notions of
dimension of a measure and of multifractal analysis are introduced in a general
setting and illustrated on the aforementioned examples. Finally, the
efficiency of these measures for the description of percolation on trees, and
for dynamical or random coverings, is shown.

The third paper describes the time evolution of objects that
disaggregate in a random way, and the fragments of which evolve
independently. A statistical self-similarity assumption endows them
with a structure of random fractal. The foundations of fragmentation
theory are given, and the laws of these processes are shown to be
characterized by a self-similarity index, a dislocation measure, and an
erosion coefficient. Then, a random tree endowed with a
distance is considered, which leads to a description of the genealogy of the
process. Finally, the speed with which the fragment
containing a given point decays is studied. This leads to the introduction of a
multifractal spectrum of speeds of fragmentation.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

#### Readership

Graduate students and research mathematicians.