Exponential decay of correlations for Gibbs measures on attractors of Axiom A flows

In this paper we study the decay of correlations for Gibbs measures associated to codimension one Axiom A attractors for flows. We prove that a codimension one Axiom A attractors whose strong stable foliation is $C^{1+\alpha }$ either have exponential decay of correlations with respect to all Gibbs measures associated to H"older continuous potentials or their stable and unstable bundles are jointly integrable. As a consequence, there exist $C^1$-open sets of $C^3$-vector fields generating Axiom A flows having attractors so that: (i) mix exponentially with respect to equilibrium states associated with H"older continuous potentials, (ii) their time-1 maps satisfy an almost sure invariance principle, and (iii) the growth of the number of closed orbits of length $T$ is described by the topological entropy of the attractor.