Smooth Attractors Have Zero ‘‘Thickness’’

A finite-dimensional global attractor A can be embedded, using some linear map L, into a Euclidean space Rk of sufficiently high dimension. The H¨older exponent of Ly1 depends upon k and upon t 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 t A.s0. It follows, using a result of B. R. Hunt and V. Y. Kaloshin, that the H¨older 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