Physical measures for partially hyperbolic flows with mostly contracting centre

In this paper, we study the physical measures for a class of partially hyperbolic flows with mostly contracting centre. Let X be a vector field on a compact Riemannian manifold M with partially hyperbolic splitting . We prove that if the centre direction exhibits the asmptotically sectionally contracting behaviour with respect to Gibbs u-states, then X admits finitely many physical measures, and their basins cover Lebesgue almost all points of the ambient manifold. Moreover, when the unstable manifolds are dense, we prove that X admits only one physical measure whose basin covers a full Lebesgue measure subset. By tracing the physical measures via typical hyperbolic periodic orbits, we study the statistical stability of this kind of partially hyperbolic flows.