Dimensions, Embeddings, and Attractors (Cambridge Tracts in Mathematics) Book

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.