Fractals in Probability and Analysis

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.

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.

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.