Dimensions, embeddings, and attractors / James C. Robinson.

Format:

Book

Author/Creator:

Robinson, James C. (James Cooper), 1969- author.

Series:

Cambridge tracts in mathematics ; 186.

Cambridge tracts in mathematics ; 186

Language:

English

Subjects (All):

Dimension theory (Topology).

Attractors (Mathematics).

Topological imbeddings.

Physical Description:

1 online resource (xii, 205 pages) : digital, PDF file(s).

Other Title:

Dimensions, Embeddings, & Attractors

Place of Publication:

Cambridge : Cambridge University Press, 2011.

Language Note:

English

Summary:

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.

Contents:

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.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Includes bibliographical references and index.

ISBN:

1-107-21957-4

1-282-93179-2

9786612931796

0-511-93217-0

0-511-93083-6

0-511-92830-0

0-511-93353-3

0-511-93391-6

0-511-92580-8

OCLC:

702372870