Fractured Fractals and Broken Dreams: Self-similar Geometry through Metric and Measure

Hardcover | August 1, 1997

byGuy David, Stephen Semmes

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This book proposes new notions of coherent geometric structure. Fractal patterns have emerged in many contexts, but what exactly is a "pattern" and what is not? How can one make precise the structures lying within objects and the relationships between them? The foundations laid herein providea fresh approach to a familiar field. From this emerges a wide range of open problems, large and small, and a variety of examples with diverse connections to other parts of mathematics. One of the main features of the present text is that the basic framework is completely new. This makes it easier for people to get into the field. There are many open problems, with plenty of opportunities that are likely to be close at hand, particularly as concerns the exploration of examples. Onthe other hand the general framework is quite broad and provides the possibility for future discoveries of some magnitude. Fractual geometries can arise in many different ways mathematically, but there is not so much general language for making comparisons. This book provides some tools for doing this, and a place where researchers in different areas can find common ground and basic information.

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This book proposes new notions of coherent geometric structure. Fractal patterns have emerged in many contexts, but what exactly is a "pattern" and what is not? How can one make precise the structures lying within objects and the relationships between them? The foundations laid herein providea fresh approach to a familiar field. From t...

Guy David is a Professor of Mathematics at University Paris XI and Institut Universitaire de France. Stephen Semmes is a Professor of Mathematics at Rice University, Texas.

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Format:HardcoverDimensions:222 pages, 9.21 × 6.14 × 0.67 inPublished:August 1, 1997Publisher:Oxford University Press

The following ISBNs are associated with this title:

ISBN - 10:0198501668

ISBN - 13:9780198501664

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Table of Contents

1. Basic definitions2. Examples3. Comparison4. The Heisenberg group5. Background information6. Stronger self-similarity for BPI spaces7. BPI equivalence8. Convergence of metric spaces9. Weak tangents10. Rest stop11. Spaces looking down on other spaces12. Regular mappings13. Sets made from nested cubes14. Big pieces of bilipschitz mappings15. Uniformly disconnected spaces16. Doubling measures and geometry17. Deformations of BPI spaces18. Snapshots19. Some sets that are far from BPI20. A few more questionsReferencesIndex

Editorial Reviews

Most of the material in this book is completely new and the style, though unusual, is a refreshing change from convetional texts. The authors have taken a natural but not too stront notion relating to sets of fine structure, and follwed through its properties, relationships and applications.They freely admit that their framework is not theonly possible one, but by the end of the book they have more than justified theri claim that their approach is both rich and flexible. The book is recommended not only for those interested in the broad subject of he geometry of fractal sets andmeasures but also as a fine insight into how two eminent mathematicians explore and develop a new area.