Smooth Attractors Have Zero “Thickness”

Abstract

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