About the Book
This is a mathematically rigorous introduction to fractals which emphasizes examples and fundamental ideas. Building up from basic techniques of geometric measure theory and probability, central topics such as Hausdorff dimension, self-similar sets and Brownian motion are introduced, as are more specialized topics, including Kakeya sets, capacity, percolation on trees and the traveling salesman theorem. The broad range of techniques presented enables key ideas to be highlighted, without the distraction of excessive technicalities. The authors incorporate some novel proofs which are simpler than those available elsewhere. Where possible, chapters are designed to be read independently so the book can be used to teach a variety of courses, with the clear structure offering students an accessible route into the topic.
Table of Contents:
1. Minkowski and Hausdorff dimensions; 2. Self-similarity and packing dimension; 3. Frostman's theory and capacity; 4. Self-affine sets; 5. Graphs of continuous functions; 6. Brownian motion, part I; 7. Brownian motion, part II; 8. Random walks, Markov chains and capacity; 9. Besicovitch–Kakeya sets; 10. The traveling salesman theorem; Appendix A. Banach's fixed-point theorem; Appendix B. Frostman's lemma for analytic sets; Appendix C. Hints and solutions to selected exercises; References; Index.
About the Author :
Christopher J. Bishop is a professor in the Department of Mathematics at Stony Brook University, New York. He has made contributions to the theory of function algebras, Kleinian groups, harmonic measure, conformal and quasiconformal mapping, holomorphic dynamics and computational geometry. Yuval Peres is a Principal Researcher at Microsoft Research in Redmond, Washington. He is particularly known for his research in topics such as fractals and Hausdorff measures, random walks, Brownian motion, percolation and Markov chain mixing times.
Review :
'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 analysts' traveling 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 levels with hints and solutions.' Pertti Mattila, University of Helsinki
'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 of 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, ICREA, Catalan Institution for Research and Advanced Studies, and Universitat Autònoma de Barcelona
'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, Université de Paris Sud
'There are at least two outstanding features of Bishop-Peres's new textbook that will help it stand with self-assurance … The first feature is the remarkable clarity of exposition. The proofs are beautifully presented, with a stress on communicating ideas and methods (over technicalities). This leads the authors to study the simplest cases of problems/results that already contain the most important ideas. The second feature, which moves this text into its own class among existing graduate texts on the subject, is an exceptional list of 378 exercises.' Tushar Das, MAA Reviews
'This is a technical monograph suited to practioners of geometric measure theory and analysis written by two of the world's leaders in the field. It would make a serious study for graduate students, containing a large number of helpful examples.' Chris Athorne, Contemporary Physics
