Let f be an Anosov diffeomorphism of T^3 which is partially hyperbolic with expanding center. One can associate to the fast unstable direction of f a dynamical invariant: the unstable topological entropy.

I this talk, I will explain how one can compute this invariant and construct a measure of maximal unstable entropy using systems of leaf measures supported on the strong unstable and center stable bundles, similar to the constructions of the measure of maximal entropy due to Sinai and Margulis.

These systems of measures are constructed combining ideas coming from dynamics and more analytical ideas. In particular, we show that these systems of measures can be interpreted as eigenfunctions (the Ruelle (co)-resonant states) for the spectral theory of f (the so called Ruelle resonances) acting on a certain bundle of forms.