. We consider piecewise smooth, uniformly hyperbolic systems on a Riemannian manifold, where we allow the angle between the unstable direction and the singularity manifolds to vanish. Under natural assumptions we prove that such systems exhibit Exponential Decay of Correlations and satisfy a Central Limit Theorem with respect to a mixing srb-measure. These results have been shown previously for systems in which the angle between the singularity manifold and unstable direction is uniformly bounded away from zero. 1. Introduction Anosov and Axiom A systems, together with their srb (Sinai-Ruelle-Bowen) measures, are frequently used to model many particle statistical mechanical systems, both at equilibrium and at steady states not too far from equilibrium. In such applications, physicists are generally interested in strong statistical properties, like exponential decay of correlations (EDC) and central limit theorems (CLT). In the seventies already, exponential decay was shown for smooth...