  In this paper, we provide a technique result on the existence of Gibbs
cu-states for diffeomorphisms with dominated splittings. More precisely, for
given $C^2$ diffeomorphim $f$ with dominated splitting $T_{\Lambda}M = E\oplus
F$ on an attractor $\Lambda$, by considering some suitable random perturbation
of $f$, we show that for any zero noise limit of ergodic stationary measures,
if it has positive integrable Lyapunov exponents along invariant sub-bundle
$E$, then its ergodic components contain Gibbs cu-states associated to $E$.
With this technique, we show the existence of SRB measures and physical
measures for some systems exhibitting dominated splittings and weak
hyperbolicity.
