u-Gibbs measures of partially hyperbolic Anosov diffeomorphisms of T^3

It is well known that every transitive Anosov diffeomorphism admits a unique physical measure: this measure is SRB, i.e. it is smooth along the unstable leaves. Considering Anosov systems which also preserve a sub-foliation by strong unstable leaves, one can similarly consider u-Gibbs measures, i.e. measures whose conditional measures along strong u-leaves are smooth. In this talk, we will discuss the following question for Anosov diffeomorphisms of T^3: do u-Gibbs measures necessarily coincide with the SRB measure? Under a generic transversality assumption, this question was answered affirmatively by Katz and independently by S. Alvarez, M. Leguil, D. Obata and B. Santiago (ALOS). In a joint work in progress with A., L., O. and S., we get a similar conclusion in the remaining case, that is when the stable and the strong unstable foliations jointly integrate. Our approach involves the construction of a ``horocyclic flow’’ on a suspended space.