  In this work we obtain some metric and ergodic properties of $C^{1+}$
partially hyperbolic diffeomorphisms with one-dimensional topological neutral
center, mainly regarding the behavior of its center foliation. Based on a
trichotomy for the center conditional measures of any invariant ergodic
measure, we show that if these conditionals have full support, then the center
foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in
the $C^{1+}$ case, and an invariance principle occurs in the sense that M may
be covered by a finite number of open sets where the system of center
conditionals is continuous and su-invariant. Using this invariance principle we
show that if a local accessibility hypothesis occurs then the center foliation
must be as regular as the partially hyperbolic dynamics.
