Fractured fractals and broken dreams : self-similar geometry through metric and measure
著者
書誌事項
Fractured fractals and broken dreams : self-similar geometry through metric and measure
（Oxford lecture series in mathematics and its applications, 7）
Clarendon Press , Oxford University Press, 1997
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注記
Bibliography: p. [207]-210
Includes index
内容説明・目次
内容説明
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 provide a 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. On the 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.
目次
- 1. Basic definitions
- 2. Examples
- 3. Comparison
- 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
- References
- Index
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