Fractured Fractals and Broken Dreams: Self-Similar Geometry Through Metric and Measure

by Guy R. David, Stephen W. Semmes

Overview

Fractal patterns have emerged in many contexts, but what exactly is a pattern? How can one make precise the structures lying within objects and the relationships between them? This book proposes new notions of coherent geometric structure to provide a fresh approach to this familiar field. It develops a new concept of self-similarity called "BPI" or "big pieces of itself," which makes the field much easier for people to enter. This new framework is quite broad, however, and has the potential to lead to significant discoveries. The text covers a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. Although fractal geometries arise in many different ways mathematically, comparing them has been difficult. This new approach combines accessibility with powerful tools for comparing fractal geometries, making it an ideal source for researchers in different areas to find both common ground and basic information.

Product Details

ISBN-13:	9780198501664
Publisher:	Oxford University Press, USA
Publication date:	02/19/1998
Series:	Oxford Lecture Series in Mathematics and Its Applications Series
Pages:	224
Product dimensions:	6.40(w) x 9.10(h) x 0.80(d)

About the Author

Institut Universitaire de France

Rice University

Table of Contents

1. Basic definitions
2. Examples
3. Comparison with rectifiability
4. The Heisenberg group
5. Background information
6. Stronger self-similarity for BPI spaces
7. BPI equivalence
8. Convergence of metric spaces
9. Weak tangents
10. Rest stop
11. Spaces looking down on other spaces
12. Regular mappings
13. Sets made from nested cubes
14. Big pieces of bilipschitz mappings
15. Uniformly disconnected spaces
16. Doubling measures and geometry
17. Deformations of BPI spaces
18. Snapshots
19. Some sets that are far from BPI
20. A few more questions