Geometrical constructions of equilibrium states

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying in symbolic dynamics) and even provide new information in the classical case. For such systems, we give in particular a constructive proof of the existence of the SRB measure and of the entropy maximizing measure. It is also established very fine statistical properties (Bernoulliness), and it is given a characterization of equilibrium states in terms of their conditional measures in the stable/unstable lamination, similar to the SRB case. The construction is applied to obtain the uniqueness of quasi-invariant measures associated to H\"older Jacobian for the horocyclic flow.