  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 $\mathcal A$ 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 $\mathcal A$ has zero Lipschitz deviation and, therefore,
there are linear maps $L$ into finite-dimensional spaces, whose inverses
restricted to $L\mathcal A$ are H\"older continuous with an exponent
arbitrarily close to one.
