SRB Measures for Partially Hyperbolic Flows with Mostly Expanding Center

Abstract

We prove that a partially hyperbolic attractor for a $$C^1$$ vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is $$C^2$$ , and the center bundle admits the sectionally expanding condition w.r.t. Gibbs u-states, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.