Assouad Dimension and Fractal Geometry
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Part of Cambridge Tracts in Mathematics
- Author: Jonathan M. Fraser, University of St Andrews, Scotland
- Date Published: December 2020
- availability: Available
- format: Hardback
- isbn: 9781108478656
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The Assouad dimension is a notion of dimension in fractal geometry that has been the subject of much interest in recent years. This book, written by a world expert on the topic, is the first thorough account of the Assouad dimension and its many variants and applications in fractal geometry and beyond. It places the theory of the Assouad dimension in context among up-to-date treatments of many key advances in fractal geometry, while also emphasising its diverse connections with areas of mathematics including number theory, dynamical systems, harmonic analysis, and probability theory. A final chapter detailing open problems and future directions for research brings readers to the cutting edge of this exciting field. This book will be an indispensable part of the modern fractal geometer's library and a valuable resource for pure mathematicians interested in the beauty and many applications of the Assouad dimension.
Read more- The first thorough treatment of the Assouad dimension in the context of fractal geometry
- Provides a systematic study using consistent notation of the many variants of the Assouad dimension
- Discusses several key applications to other fields: number theory, dynamical systems, harmonic analysis, and probability theory
Reviews & endorsements
'Assouad dimension has recently emerged as a key concept in fractal geometry and its applications. Starting with the basic notions of Assouad dimension and its variants, the book goes on to present elegant applications across mathematics, including to Kleinian groups, spirals and number-theoretic questions. Clearly written and full of recent research, much of which is due to the author, the book will undoubtedly become the definitive work on the topic.' Kenneth Falconer, University of St Andrews
See more reviews'Assouad dimension is a relative newcomer to the family of fractal dimensions and this clear, concise and conversational text is the first to thoroughly treat its basic properties, independent of a particular application. Especially appealing is the extensive list of detailed examples and the summaries of various (often surprising) problems where Assouad dimension arises naturally. This accessible introduction by a leading expert is sure to become a standard reference on the topic.' Christopher J. Bishop, Stony Brook University
'This monograph provides a comprehensive treatment of the Assouad dimension from the perspective of classical fractal geometry. Concrete applications of the Assouad dimension in fields as diverse as probability, functional analysis, logic, and number theory enrich the text. The author’s passion for, and deep knowledge of, the field are manifestly evident, and the book rewards the dedicated reader with a wealth of information on a branch of fractal geometry of increasing importance.' Jeremy Tyson, University of Illinois
‘The book is very well written and illustrated. The reader gets to know an almost complete spectrum of resent results and historical developments concerning Assouad dimension.’ Jörg Neunhäuserer, European Mathematical Society
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×Product details
- Date Published: December 2020
- format: Hardback
- isbn: 9781108478656
- length: 284 pages
- dimensions: 235 x 156 x 20 mm
- weight: 0.53kg
- availability: Available
Table of Contents
Part I. Theory:
1. Fractal geometry and dimension theory
2. The Assouad dimension
3. Some variations on the Assouad dimension
4. Dimensions of measures
5. Weak tangents and microsets
Part II. Examples:
6. Iterated function systems
7. Self-similar sets
8. Self-affine sets
9. Further examples: attractors and limit sets
10. Geometric constructions
11. Two famous problems in geometric measure theory
12. Conformal dimension
Part III. Applications:
13. Applications in embedding theory
14. Applications in number theory
15. Applications in probability theory
16. Applications in functional analysis
17. Future directions
References
List of notation
Index.
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