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

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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.

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Product Details

Meet the Author

Institut Universitaire de France

Rice University

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Table of Contents

1 Basic definitions 1
2 Examples 5
3 Comparison with rectifiability 13
4 The Heisenberg group 16
5 Background information 19
6 Stronger self-similarity for BPI spaces 26
7 BPI equivalence 35
8 Convergence of metric spaces 52
9 Weak tangents 71
10 Rest stop 84
11 Spaces looking down on other spaces 85
12 Regular mappings 102
13 Sets made from nested cubes 122
14 Big pieces of bilipschitz mappings 136
15 Uniformly disconnected spaces 156
16 Doubling measures and geometry 172
17 Deformations of BPI spaces 189
18 Snapshots 194
19 Some sets that are far from BPI 197
20 A few more questions 205
References 207
Index 211
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