CiNii 図書 - Dimensions, embeddings, and attractors

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.

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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Dimensions, embeddings, and attractors

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