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Dimensions, Embeddings, and Attractors

$83.99 (C)

Part of Cambridge Tracts in Mathematics

  • Date Published: January 2011
  • availability: Available
  • format: Hardback
  • isbn: 9780521898058

$ 83.99 (C)
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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.

    • 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: January 2011
    • 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.

  • Author

    James C. Robinson, University of Warwick
    James C. Robinson is Professor of Mathematics at Warwick University.

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