Fractals in Probability and Analysis

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.

• 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.

'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