Kaloshin (1999 Nonlinearity 12 1263-75) proved that it is possible to embed a compact subset X of a Hilbert space with upper box-counting dimension d < k into R N for any N > 2k + 1, using a linear map L whose inverse is Hölder continuous with exponent α < (N − 2d)/N (1 + τ (X)/2), where τ (X) is the 'thickness exponent' of X. More recently, Ott et al (2006 Ergod. Theory Dyn. Syst. 26 869-91) studied the effect of such embeddings on the Hausdorff dimension of X, and showed that for 'most' linear mapsThey also conjectured that 'many of the attractors associated with the evolution equations of mathematical physics have thickness exponent zero'. In this paper we introduce a variant of the thickness exponent, the Lipschitz deviation dev(X): we show that in both of the above results this can be used in place of the thickness exponent, and-appealing to results from the theory of approximate inertial manifolds-we prove that dev(X) = 0 for the attractors of a wide class of semilinear parabolic equations, thus providing a partial answer to the conjecture of Ott, Hunt and Kaloshin. In particular, dev(X) = 0 for the attractor of the 2D Navier-Stokes equations with forcing f ∈ L 2 , while current results only guarantee that τ (X) = 0, when f ∈ C ∞ .

Abstract. Suppose that A is the global attractor associated with a dissipative dynamical system on a Hilbert space H.If the set A − A has finite Assouad dimension d, then for any m > d there are linear homeomorphisms L : A → R m+1 such that LA is a cellular subset of R m+1 and L −1 is log-Lipschitz (i.e. Lipschitz to within logarithmic corrections). We give a relatively simple proof that a compact subset X of R k is the global attractor of some smooth ordinary differential equation on R k if and only if it is cellular, and hence we obtain a dynamical system on R k for which LA is the global attractor. However, LA consists entirely of stationary points.In order for the dynamics on LA to reproduce those on LA we need to make an additional assumption, namely that the dynamics restricted to A are generated by a log-Lipschitz continuous vector field (this appears overly restrictive when H is infinite-dimensional, but is clearly satisfied when the initial dynamical system is generated by a Lipschitz ordinary differential equation on R N ). Given this we can construct an ordinary differential equation in some R k (where k is determined by d and α) that has unique solutions and reproduces the dynamics on A. Moreover, the dynamical system generated by this new ordinary differential equation has a global attractor X arbitrarily close to LA.

This paper focuses on the regularity of linear embeddings of finite-dimensional subsets of Hilbert and Banach spaces into Euclidean spaces. We study orthogonal sequences in a Hilbert space H, whose elements tend to zero, and similar sequences in the space c 0 of null sequences. The examples studied show that the results due to Hunt and Kaloshin (Regularity of embeddings of infinite-dimensional fractal sets into finitedimensional spaces, Nonlinearity 12 (1999) 1263-1275) and Robinson (Linear embeddings of finite-dimensional subsets of Banach spaces into Euclidean spaces, Nonlinearity 22 (2009) 711-728) for subsets of Hilbert and Banach spaces with finite box-counting dimension are asymptotically sharp. An analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by Olson and Robinson (Almost bi-Lipschitz embeddings and almost homogeneous sets, Trans. Amer. Math. Soc. 362 (1) (2010) 145-168) for subsets X of Hilbert spaces when X − X has finite Assouad dimension.

We discuss various issues related to the finite-dimensionality of the asymptotic dynamics of solutions of parabolic equations. In particular, we study the regularity of the vector field on the global attractor associated with these equations. We show that certain dissipative partial differential equations possess a linear term that is log-Lipschitz continuous on the attractor. We then prove that this property implies that the associated global attractor A lies within a small neighbourhood of a smooth manifold, given as a Lipschitz graph over a finite number of Fourier modes. Consequently, the global attractor A has zero Lipschitz deviation and, therefore, there are linear maps L into finite-dimensional spaces, whose inverses restricted to LA are Hölder continuous with an exponent arbitrarily close to one.

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