  In this paper, we study the limit measures of the empirical measures of
Lebesgue almost every point in the basin of a partially hyperbolic attractor.
They are strongly related to a notion named Gibbs u-state, which can be defined
in a large class of diffeomorphisms with less regularity and which is the same
as Pesin-Sinai's notion for partially hyperbolic attractors of $C^{1+\alpha}$
diffeomorphisms.
  In particular, we prove that for partially hyperbolic $C^{1+\alpha}$
diffeomorphisms with one-dimensional center, and for Lebesgue almost every
point: (1) the center Lyapunov exponent is well defined, but (2) the sequence
of empirical measures may not converge.
  We also give some consequences on SRB measures and large deviations.
