Dimensions, Embeddings, and Attractors

Abstract

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.

An estimate on the fractal dimension of attractors of gradient-like dynamical systems

“…Note that our argument implies that if we approach a bifurcation point λ = n 2 , n ∈ N, our estimate on the fractal dimension of the attractor A λ explodes, since the rate of exponential attraction ω approaches to zero (see [20] where it is proved that this attraction is in fact polynomial). However, we know that the fractal dimension of the above Chafee-Infante equation is finite and of order √ λ for all values of λ ≥ λ 1 (the first eigenvalue of the Laplacian operator) (see, for instance, [36,33] the exponential attraction to hyperbolic equilibria. Moreover, for any sequence {λ n : n ∈ N} such that λ n is away from the endpoints of the interval (n 2 , (n + 1) 2 ), uniformly for n ∈ N, our estimate is of order √ λ n .…”

“…In this case, the topological dimension dim T (K) of K is the minimum d with this property. With this notion, a subset of R n with non-empty interior has topological dimension n and, if K is a compact metric space with topological dimension dim T (K) < ∞, then it is homeomorphic to a subset of R n with n = 2dim T (K) + 1 (see [27], [33]). …”

“…Since µ It is known (see [33]) that dim T (K) dim H (K). Now we turn our attention to the attractors of gradient semigroups in Banach spaces.…”

Structure of attractors and estimates of their fractal dimension

“…Before we proceed, let us briefly recall the definitions of with n = 2dim T (K ) + 1 (see [33], [41]). For a more detailed discussion, the reader can see [6].…”

“…Then one might ask why to study the fractal dimension, since it is the worst estimate for the dimensions of the same compact set K , and one particular result that makes the fractal dimension a very interesting object of research is the following result (see [31] The inverse of the projection restricted to K is continuous. In fact, in some situations, this inverse is Hölder continuous (see [20,41]). Another aspect is that the fractal dimension computation is fairly easy when we compare it with the computation of the Hausdorff and topological dimensions.…”