Dimensions, embeddings, and attractors
Author(s)
Bibliographic Information
Dimensions, embeddings, and attractors
（Cambridge tracts in mathematics, 186）
Cambridge University Press, 2011
- : hardback
Available at 35 libraries
-
Library, Research Institute for Mathematical Sciences, Kyoto University数研
ROB||37||2200019998322
Note
Includes bibliographical references (p. 196-201) and index
Description and Table of Contents
Description
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.
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.
by "Nielsen BookData"