Gibbs measures for suspension semiflows over C1+α piecewise expanding maps: the absence of the Federer property and their exponential decay of correlations

Decay of correlations, Gibbs measure, equilibrium states, uniform hyperbolicity.

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 C1+α either have exponential decay of correlations with respect to all Gibbs measures associated to Holder continuous potentials or their stable and unstable bundles are jointly integrable. As a consequence, there exist C1-open sets of C3-vector fields generating Axiom A flows having attractors so that it mix exponentially with respect to equilibrium states associated with Holder continuous potentials.