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Quelques Interactions Entre Analyse, Probabilités et Fractals
J. Barral, Université Paris 13, Villetaneuse, France, J. Berestycki and J. Bertoin, Université Pierre et Marie Curie, Paris, France, A. H. Fan, Université de Picardie, Amiens, France, B. Haas, Université Paris-Dauphine, France, S. Jaffard, Université Paris Est Créteil, France, G. Miermont, Université Paris-Sud Bâtiment, Orsay, France, and J. Peyrière, Université Paris-Sud 11, Orsay, France
A publication of the Société Mathématique de France.
Panoramas et Synthèses
2010; 243 pp; softcover
Number: 32
ISBN-10: 2-85629-313-1
ISBN-13: 978-2-85629-313-3
List Price: US$60
Member Price: US$48
Order Code: PASY/32
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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.


Graduate students and research mathematicians.

Table of Contents

  • S. Jaffard -- Introduction
  • J. Barral, A. H. Fan, and J. Peyrière -- Mesures engendrées par multiplications
  • J. Berestycki, J. Bertoin, B. Haas, and G. Miermont -- Quelques aspects fractals des fragmentations aléatoires
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