Physical measures and absolute continuity for one-dimensional center direction

For a class of partially hyperbolic C^k$C^k$, k>1$k>1$ diffeomorphisms with circle center leaves we prove the existence and finiteness of physical (or Sinai–Ruelle–Bowen) measures, whose basins cover a full Lebesgue measure subset of the ambient manifold. Our conditions hold for an open and dense subset of all C^k$C^k$ partially hyperbolic skew-products on compact circle bundles.