Existence and smoothness of the stable foliation for sectional hyperbolic attractors

We prove the existence of a contracting invariant topological foliation in a full neighbourhood for partially hyperbolic attractors. Under certain bunching conditions, it can then be shown that this stable foliation is smooth. Specialising to sectional hyperbolic attractors, we give a verifiable condition for bunching. In particular, we show that the stable foliation for the classical Lorenz equation (and nearby vector fields) is better than C1, which is crucial for recent results on exponential decay of correlations. In fact, the foliation is at least C1.278