Dimensions, embeddings, and attractors

Cover; Half-title; Title; Copyright; Dedication; 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 in terms of the upper box-counting dimension; 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.
"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"--