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  3. Dimensions, Embeddings, and Attractors
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    • Publisher:
      Cambridge University Press
      Publication date:
      10 January 2011
      16 December 2010
      ISBN:
      9780511933912
      9780521898058
      DOI:
      https://doi.org/10.1017/CBO9780511933912
      Dimensions:
      (228 x 152 mm)
      Weight & Pages:
      0.45kg, 218 Pages
      Dimensions:
      Weight & Pages:
      • Subjects:
      Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
      Series:
      Cambridge Tracts in Mathematics (186)
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    Subjects:
    Mathematics (general), Mathematics, Differential and Integral Equations, Dynamical Systems and Control Theory, Abstract Analysis
    Series:
    Cambridge Tracts in Mathematics (186)
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    Book 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.

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