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Dimensions, Embeddings, and Attractors

Details

  • 10 b/w illus. 60 exercises
  • Page extent: 218 pages
  • Size: 228 x 152 mm
  • Weight: 0.45 kg

Library of Congress

  • Dewey number: 515/.39
  • Dewey version: 22
  • LC Classification: QA611.3 .R63 2011
  • LC Subject headings:
    • Dimension theory (Topology)
    • Attractors (Mathematics)
    • Topological imbeddings

Library of Congress Record

Hardback

 (ISBN-13: 9780521898058)

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.

• 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

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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