Smooth attractors have zero "thickness"

A finite-dimensional global attractor A can be embedded, using some linear map L, into a Euclidean space R-k of sufficiently high dimension. The Holder exponent of L-1 depends upon k and upon tau(A), the "thickness exponent" of A. We show that global attractors which are uniformly bounded in the Sobolev spaces H-s for all s > 0 have tau(A = 0. It follows, using a result of B. R. Hunt and V. Y. Kaloshin, that the Holder constant of the inverse of a typical linear embedding into R-k (or rank k orthogonal projection) can be chosen arbitrarily close to 1 if k is large enough. (C) 1999 Academic Press