The term ‘prevalence’ was coined by Hunt et al. (1992), for a generalisation of the notion of ‘almost every’ that is appropriate for infinite-dimensional spaces. Essentially the same definition was used earlier by Christensen (1973), although for him a set was prevalent if its complement was a Haar null set; we adopt here the more recent and more descriptive terminology. A nice review of the theory of prevalence is given by Ott & Yorke (2005). We only develop the theory here as far as we will need it in what follows; more details can be found in the above papers and in Benyamini & Lindenstrauss (2000, Chapter 6).

Once we have introduced prevalence, we show how the idea can be adapted to treat certain classes of linear maps from infinite-dimensional spaces into finite-dimensional Euclidean spaces (Section 5.2), and then prove a generalisation of the inequality (4.1) that is a key element of the subsequent embedding proofs.

Prevalence

Let V be a normed linear space. First we define what it means for a subset of V to be ‘shy’, the equivalent in this setting of ‘having measure zero’; the complement of a shy set is said to be ‘prevalent’.

There are a number of definitions of dimension that are invariant under homeomorphisms, i.e. that are topological invariants – in particular, the large and small inductive dimensions, and the Lebesgue covering dimension. Although different a priori, the large inductive dimension and the Lebesgue covering dimension are equal in any metric space (Katětov, 1952; Morita, 1954; Chapter 4 of Engelking, 1978), and all three definitions coincide for separable metric spaces (Proposition III.5 A and Theorem V.8 in Hurewicz & Wallman (1941)). A beautiful exposition of the theory of ‘topological dimension’ is given in the classic text by Hurewicz & Wallman (1941), which treats separable spaces throughout and makes much capital out of the equivalence of these definitions. Chapter 1 of Engelking (1978) recapitulates these results, while the rest of his book discusses dimension theory in more general spaces in some detail.

This chapter concentrates on one of these definitions, the Lebesgue covering dimension, which we will denote by dim(X), and refer to simply as the covering dimension. Among the three definitions mentioned above, it is the covering dimension that is most suitable for proving an embedding result: we will show in Theorem 1.12, the central result of this chapter, that if dim(X) ≤ n then a generic set of continuous maps from X into ℝ2n+1 are homeomorphisms, i.e. provide an embedding of X into ℝ2n+1.

The main purpose of this book is to bring together a number of results concerning the embedding of ‘finite-dimensional’ compact sets into Euclidean spaces, where an ‘embedding’ of a metric space (X, ϱ) into ℝn is to be understood as a homeomorphism from X onto its image. A secondary aim is to present, alongside such ‘abstract’ embedding theorems, more concrete embedding results for the finite-dimensional attractors that have been shown to exist in many infinite-dimensional dynamical systems.

In addition to its summary of embedding results, the book also gives a unified survey of four major definitions of dimension (Lebesgue covering dimension, Hausdorff dimension, upper box-counting dimension, and Assouad dimension). In particular, it provides a more sustained exposition of the properties of the boxcounting dimension than can be found elsewhere; indeed, the abstract results for sets with finite box-counting dimension are those that are taken further in the second part of the book, which treats finite-dimensional attractors.

While the various measures of dimension discussed here find a natural application in the theory of fractals, this is not a book about fractals. An example to which we will return continually is an orthogonal sequence in an infinite-dimensional Hilbert space, which is very far from being a ‘fractal’. In particular, this class of examples can be used to show the sharpness of three of the embedding theorems that are proved here.

We now give the first application of the constructions of the previous chapter to prove a ‘prevalent’ version of a result first due to Mañé (1981). He showed that if X is a subset of a Banach space ℬ and dH(X − X) < k, then a residual subset of the space of projections onto any subspace of dimension at least k are injective on X.

We show here that in general no linear embedding into any ℝk is possible if we only assume that dH(X) is finite (Section 6.1). If we want an embedding theorem for such sets, we must fall back on Theorem 1.12 which guarantees the existence of generic embeddings of sets with finite covering dimension (we can apply this result since dim(X) ≤ dH(X) by Theorem 2.11).

While we prove in Theorem 6.2 the existence of a prevalent set of linear embeddings into ℝk when dH(X − X) < k, we will see that even with this assumption one cannot guarantee any particular degree of continuity for the inverse of the linear mapping that provides the embedding (Section 6.3).

In this chapter and those that follow, we will often wish to show that certain embedding results are sharp, in the sense that the information we obtain on the modulus of continuity for the inverse of the embedding map cannot be improved.

Part I of this book treats four different definitions of dimension, and investigates what being ‘finite dimensional’ implies in terms of embeddings into Euclidean spaces for each of these definitions.

Whitney (1936) showed that any abstract n-dimensional Cr manifold is Cr-homeomorphic to an analytic submanifold in ℝ2n+1. This book treats embeddings for much more general sets that need not have such a smooth structure; one might say ‘fractals’, but we will not be concerned with the fractal nature of these sets (whatever one takes that to mean).

We will consider four major definitions of dimension:

1. (i) The (Lebesgue) covering dimension dim(X), based on the maximum number of simultaneously intersecting sets in refinements of open covers of X (Chapter 1). This definition is topologically invariant, and is primarily used in the classical and abstract ‘Dimension Theory’, elegantly developed in Hurewicz & Wallman's 1941 text, and subsequently by Engelking (1978), who updates and extends their treatment.

2. (ii) The Hausdorff dimension dH(X), the value of d where the ‘d-dimensional Hausdorff measure’ of X switches from ∞ to zero (Chapter 2). Hausdorff measures (and hence the Hausdorff dimension) play a large role in geometric measure theory (Federer, 1969), and in the theory of dynamical systems (see Pesin (1997)); the standard reference is Falconer's 1985 tract, and subsequent volumes (Falconer, 1990, 1997).

3. […]

Powerful techniques are available for bounding the box-counting dimension of attractors in Hilbert spaces, the case most often encountered in applications. The most widely-used method was developed for finite-dimensional dynamical systems by Douady & Oesterlé (1980), and was extended to treat subsets of infinite-dimensional Hilbert spaces by Constantin & Foias (1985). Much effort has also been expended in refining the resulting estimates for particular models, in particular for the two-dimensional Navier–Stokes equations (for a nice overview see Doering & Gibbon (1995)).

However, general results providing bounds on the dimension of compact invariant sets go back to Mallet-Paret (1976), who showed that if K is a compact subset of a Hilbert space H, f : H → H is continuously differentiable, f(K) ⊇ K (‘K is negatively invariant’), and the derivative of f is everywhere equal to the sum of a compact map and a contraction, then the upper boxcounting dimension of K is finite. Mañé (1981) generalised this argument to treat subsets of Banach spaces (this was in the same paper in which he proved a ‘generic’ embedding theorem for sets with dH(X − X) finite, cf. our Theorem 6.2).

The Hilbert space method is already cleanly and clearly presented in a number of texts that concentrate more specifically on estimating the dimension of attractors (e.g. Chepyzhov & Vishik, 2002; Robinson, 2001; Temam, 1988), and a general technique that covers the Banach space case seems more in keeping with the rest of this book.