This is a wonderful book, introducing the reader into the modern theory of fractals. It uses tools from Analysis and Probability very elegantly, and starting from the basics ends with a selection of deep and important results. The authors worked hard to achieve clarity; the book contains many original proofs which are expository gems. The book would serve very well for a graduate course; it is highly recommended both for students and for experts. A notable feature is a wide selection of exercises, some quite challenging, but made more accessible with an appendix containing selected hints.
Boris Solomyak, Bar-Ilan University, Israel
This is a very valuable contribution to the field of geometric measure theory and its interactions with other branches of mathematical analysis and probability. The notions of Hausdorff measure, Hausdorff dimension, and Minkowski dimension are central objects in this text, as in other books on geometric measure theory. What is special in this text, written by two major experts in geometric analysis and probability, is the emphasis on problems lying in the intersection of probability and analysis. In particular, the book studies a variety questions in connection with self-similar sets, Frostman's theory, Weierstrass functions, Brownian motion and its relationship with the Dirichlet problem for harmonic functions, Besicovitch-Kakeya sets, and Jones' traveling salesman theorem. Many of the problems considered in the book are difficult to find in the literature. Further, very often their proofs are based on new and illuminating arguments. All in all, I think that this is a great book.
Xavier Tolsa, Universitat Autonoma de Barcelona
Fractal sets are now a key ingredient of much of mathematics, ranging from dynamical systems, transformation groups, stochastic processes, to modern analysis. This delightful book gives a correspondingly broad view of fractal sets. The presentation is original, clear and thoughtful, often with new and interesting approaches. It is suited both to graduate students and researchers, discussing reasonably easily accessible questions as well as research topics that are being actively investigated today. For example, in addition to learning about fractals, students will get new insights into some core topics, such as Brownian motion, while researchers will find new ideas for up-to-date research, for example related to analyst's travelling salesman problems. The book is splendid for a variety of graduate courses, most sections being essentially independent of each other, and is supported by a very large number of exercises of varying level with hints and solutions.
Pertti Mattila, University of Helsinki
This book, written by two of the best specialists in the world, is centered on the probabilistic aspects of geometric measure theory and fractals, but also contains beautiful pure analysis arguments. The point of view is very concrete, often based on many interesting examples or methods rather than a general theory. The most impressive aspect of the book is the huge collection of exercises of all levels, which will make a serious reading of the book both a pleasure and, if the reader wants to do them all, a performance.
Guy David, Universite Paris Sud