Dimensions, embeddings, and attractors / James C. Robinson.Material type: TextSeries: Cambridge tracts in mathematics: 186.Publisher: Cambridge : Cambridge University Press, 2011, ³2011Description: xii, 205 pages : UKP 51.00 illustrations ; 24 cm.Content type: text Media type: unmediated Carrier type: volumeISBN: 9780521898058 (hardback); 0521898056 (hardback).Subject(s): Dimension theory (Topology) | Attractors (Mathematics) | Topological imbeddingsDDC classification: 515/.39 Online resources: Cover image
|Item type||Current location||Call number||Status||Date due||Barcode||Item holds|
|Book||Chennai Mathematical Institute General Stacks||515.39 ROB (Browse shelf)||Available||10059|
Includes bibliographical references (p. 196-201) and index.
Finite-dimensional sets. Lebesgue covering dimension -- Hausdorff measure and Hausdorff dimension -- Box-counting dimension -- An embedding theorem for subsets of RN -- Prevalence, probe spaces, and a crucial inequality -- Embedding sets with dH(X-X) finite -- Thickness exponents -- Embedding sets of finite box-counting dimension -- Assouad dimension -- Finite-dimensional attractors. Partial differential equations and nonlinear semigroups -- Attracting sets in infinite-dimensional systems -- Bounding the box-counting dimension of attractors -- Thickness exponents of attractors -- The Takens time-delay embedding theorem -- Parametrisation of attractors via point values.
"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"-- Provided by publisher.