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Book review

Author(s) Bélair, Jacques
Dubuc, Serge
Title Fractal Geometry and Analysis
Publisher Kluwer Academic Publishers
Year of publication 1991
Reviewed by ?

Fractal Geometry and Analysis, edited by Jacques Bélair and Serge Dubuc, is a collection of articles devoted to the geometry of fractal sets and to the analytical tools used to investigate them. Fractal applications of measure theory, dynamical systems, iteration theory, branching processes are reviewed by authoritative authors, a number of new results are also being reported. This book should be of interest both for mathematicians and for scientists working on applications of fractals, for those who already have some background in the field and would like to have, along with the latest review, an idea in which direction the mathematical theory of fractals is going to develop in the nearest future.

The main body of the book consists of 10 articles, most of them with illustrations in black and white (which are sometimes not of the best graphical quality). The articles are largely unrelated to each other, some are written in mathematically rigorous manner, while others are more like a general discussion of the latest ideas.

Different definitions of fractal dimensions and methods of their estimation are discussed in articles by Mendès-France, Bedford, Falconer, Tricot and Hata. Both deterministic and random recurrent methods of construction of fractal sets are described by Dekking, Bedford, Hata and Vrscay. Iterated function systems are covered by Falconer, Hata and Dubuc. Questions related to iteration of complex mappings are discussed by Blanchard. An interpretation of multifractal spectrum as a probability distribution is presented by Kahane. Relation between fractal geometry and mathematical analysis is touched in articles by Kahane and by Dubuc.

It is not unimportant to note that articles by Dubuc and by Kahane are in French (but, I believe, a dedicated reader can find them useful even if the knowledge of the language is lacking).

There is a preface, which, along with the table of contents, is useful for navigation purposes, a list of the participants of the 28th session of the Séminaire de mathématiques supérieures of the Université de Montréal, on occasion of which the collection has been published, a list of contributors (with postal addresses), and, at the end of the book, a short index of the main terms.

Although at the time of submission (July 1989) the articles reflected the latest developments in the field quite comprehensively, since some of the subjects covered, like, for example, the theory of iterated function systems, develop very rapidly, at present in certain cases it should be recommended to combine reading the book with an independent study of recent literature.