Dimensions, Embeddings, and Attractors
Part of Cambridge Tracts in Mathematics
- Author: James C. Robinson, University of Warwick
- Date Published: December 2010
- availability: Available
- format: Hardback
- isbn: 9780521898058
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This accessible research monograph investigates how 'finite-dimensional' sets can be embedded into finite-dimensional Euclidean spaces. The first part brings together a number of abstract embedding results, and provides a unified treatment of four definitions of dimension that arise in disparate fields: Lebesgue covering dimension (from classical 'dimension theory'), Hausdorff dimension (from geometric measure theory), upper box-counting dimension (from dynamical systems), and Assouad dimension (from the theory of metric spaces). These abstract embedding results are applied in the second part of the book to the finite-dimensional global attractors that arise in certain infinite-dimensional dynamical systems, deducing practical consequences from the existence of such attractors: a version of the Takens time-delay embedding theorem valid in spatially extended systems, and a result on parametrisation by point values. This book will appeal to all researchers with an interest in dimension theory, particularly those working in dynamical systems.
Read more- Introduces alternative definitions to researchers who traditionally use only one
- An authoritative summary which assembles results scattered through the literature
- Provides up-to-date results and abstract background for researchers in dynamical systems
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×Product details
- Date Published: December 2010
- format: Hardback
- isbn: 9780521898058
- length: 218 pages
- dimensions: 235 x 160 x 20 mm
- weight: 0.45kg
- contains: 10 b/w illus. 60 exercises
- availability: Available
Table of Contents
Preface
Introduction
Part I. Finite-Dimensional Sets:
1. Lebesgue covering dimension
2. Hausdorff measure and Hausdorff dimension
3. Box-counting dimension
4. An embedding theorem for subsets of RN
5. Prevalence, probe spaces, and a crucial inequality
6. Embedding sets with dH(X-X) finite
7. Thickness exponents
8. Embedding sets of finite box-counting dimension
9. Assouad dimension
Part II. Finite-Dimensional Attractors:
10. Partial differential equations and nonlinear semigroups
11. Attracting sets in infinite-dimensional systems
12. Bounding the box-counting dimension of attractors
13. Thickness exponents of attractors
14. The Takens time-delay embedding theorem
15. Parametrisation of attractors via point values
Solutions to exercises
References
Index.
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