TY  - BOOK
AU  - Robinson,James C.
TI  - Dimensions, embeddings, and attractors
T2  - Cambridge tracts in mathematics
SN  - 9780511933912 (ebook)
AV  - QA611.3 .R63 2011
U1  - 515/.39 22
PY  - 2011///
CY  - Cambridge
PB  - Cambridge University Press
KW  - Dimension theory (Topology)
KW  - Attractors (Mathematics)
KW  - Topological imbeddings
N1  - Title from publisher's bibliographic system (viewed on 05 Oct 2015); 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
N2  - 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
UR  - https://doi.org/10.1017/CBO9780511933912
ER  -