In dimension three, we present three conservative partially hyperbolic settings (derived from Anosov systems, time one-maps of geodesic flows on negatively curved surfaces, and skew-product with circle fibers) where the following dichotomy holds: the diffeomorphism is either Anosov or it supports and (ergodic) nonhyperbolic measure.
We also see that the second possibility is quite frequent.
Join work with J, Yang (UFF, Niterói, Brazil) J. Zhang (Beihaan, Beijing, China).