Fractals in Probability and Analysis

A mathematically rigorous introduction to fractals, emphasizing examples and fundamental ideas while minimizing technicalities.

Christopher J. Bishop (Author), Yuval Peres (Author)

9781107134119, Cambridge University Press

Hardback, published 22 December 2016

412 pages, 75 b/w illus. 380 exercises
23.5 x 15.9 x 2.6 cm, 0.68 kg

'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

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.

Subject Areas: Stochastics [PBWL], Probability & statistics [PBT], Complex analysis, complex variables [PBKD], Real analysis, real variables [PBKB]