  We study the thermodynamic formalism for generalized Gibbs measures, such as
renormalization group transformations of Gibbs measures or joint measures of
disordered spin systems. We first show existence of the relative entropy
density and obtain a familiar expression in terms of entropy and relative
energy for "almost Gibbsian measures" (almost sure continuity of conditional
probabilities). We also describe these measures as equilibrium states and
establish an extension of the usual variational principle. As a corollary, we
obtain a full variational principle for quasilocal measures. For the joint
measures of the random field Ising model, we show that the weak Gibbs property
holds, with an almost surely rapidly decaying translation invariant potential.
For these measures we show that the variational principle fails as soon as the
measures loses the almost Gibbs property. These examples suggest that the class
of weakly Gibbsian measures is too broad from the perspective of a reasonable
thermodynamic formalism.
