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Fractal geometries: broken space-time. (Les géométries fractales: l’espace-temps brisé.) (French) Zbl 0834.58002

Traité des Nouvelles Technologies. Série Images. Paris: Hermes. 198 p. (1990).
As the author says, the aim of the book is to give new insights in the physics of irreversible phenomena regarded from the point of view of fractal geometry. It is based on the notes of the lectures on the transfer of energy, delivered by the author at Orsay University.
In the first part of the book (Chapters 1-4) theoretical ideas that are necessary to approach the concept of time in the physics of a fractal medium are presented. The author describes the classical constructions of Koch curves, Cantor sets and defines the notion of fractal dimension. As a novelty, it is defined the mass fractal dimension of “an object”, a notion useful when there is interest not only in the fractal boundary of a set, but also in the distribution of mass. This kind of dimension leads to the codimension of a fractal set embedded in a \(d\)-dimensional space. The codimension of a fractal set is the difference \(d- d_M\), where \(d_M\) is the mass fractal dimension.
The central idea of Chapter 2 is that irreversibility is the core of fractality. Descriptive definitions of Hausdorff dimension, Minkowski- Bouligand and packing dimension are given. The method of boxes is presented as a method for estimation of the fractal and Minkowski- Bouligand dimension. For the evaluation of the Minkows-Bouligand dimension of the graph of a function it is described Claude Tricot’s method of variation. In fact, the theoretical part of the book follows the works by Claude Tricot (École Polytechnique de Montreal). Finally the parametrization of a fractal curve, and the concept of irreversibility as it appears in this context are discussed.
Experimental studies led to the notion of non-integer derivative. Chapter 3 of the book presents the properties of this kind of derivative and its applications to the analysis of a nowhere differentiable function. Chapter 4 examines in an elementary fashion the multifractality in the Mandelbrot sense and raises the question of the existence of other scaling laws based upon the non-integer derivative. The last chapter of the book presents some applications of the developed theory: fractal morphogenesis, wavelet transformation of a fractal set, diffraction on a fractal set, generation of a fractal set through diffusion, distribution of energy and non-integer derivative. The author places emphasis on the applications in the theory of electro-chemistry – his field of interest.
The book is recommended for specialists working in the author’s field of research and also for students.

MSC:

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
28A80 Fractals
28A78 Hausdorff and packing measures
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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