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

Ergodic properties of partially hyperbolic diffeomorphisms with topological neutral center