Physical measures for partially hyperbolic systems with $u$-definite subbundles

In this paper, we study $C^{1+\alpha }$ diffeomorphisms of a compact Riemannian manifold $\mathbf{M}$ with an attracting set $\mathrm{\Lambda }$ admitting a partially hyperbolic splitting $$T_{\mathrm{\Lambda }}\mathbf{M}=E^u\oplus C_1\oplus \cdots \oplus C_k\oplus E^s\mathrm{.}$$ We assume that, for every ergodic Gibbs $u$-state supported on $\mathrm{\Lambda }$, the Lyapunov exponents along each $C_i$ are either all positive or all negative. We prove that $\mathrm{\Lambda }$ supports finitely many ergodic hyperbolic SRB measures whose basins cover Lebesgue almost every point of its topological basin. Our approach relies on an empirical basin-covering theorem for attracting sets with dominated splittings. Its proof hinges on a new notion of sliding suffix times, which allows us to synchronize expansion and contraction along a common empirical subsequence.