We examine the interplay between the thermodynamic formalism and the multifractal formalism on the so-called self-affine symbolic spaces, under the specification property assumption. We investigate the properties of a weighted variational principle to derive a new result concerning the approximation of any invariant probability measure $\mu$ by sequences of weighted equilibrium states whose weighted entropies converge to the weighted entropy of $\mu$. This is a key property in the estimation of the Hausdorff dimension of sets of generic points, and then in the multifractal analysis of non homogeneous Birkhoff averages.
"Weighted thermodynamic formalism on subshifts and applications." Asian J. Math. 16 (2) 319 - 352, June 2012.