Despite some groundbreaking advances on the exponential mixing rate of the SRB measure for contact Anosov flows, Anosov flows with smooth invariant foliations and codimension one Anosov flows, the general picture concerning the decay of correlations for Anosov flows is still far from complete. There are two main conjectures in this area:
1. every topologically mixing Axiom A attractor mixes exponentially fast
2. an Axiom A attractor is topologically mixing if and only if the strong stable and strong unstable subbundles are jointly nonintegrable
Partial answers have been obtained by Tsujii, O. Butterley-War and Araújo in lower (co)dimension in the recent years.