Generalized u-Gibbs Measures for $C^\infty$ Diffeomorphisms

We show that for every $C^\infty$ diffeomorphism of a closed Riemannian manifold, if there exists a positive volume set of points with a positive Lyapunov exponent (in a weak sense) then there exists an invariant probability measure with a disintegration by absolutely continuous conditionals on disks subordinated to unstable leaves. As an application, we prove a strong version of the strong Viana conjecture in any dimension. This is joint work with D. Burguet.