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.