Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations

Given any smooth Anosov map, we construct a Banach space on which the associated transfer operator is quasi-compact. The peculiarity of such a space is that, in the case of expanding maps, it reduces exactly to the usual space of functions of bounded variation which has proved to be particularly successful in studying the statistical properties of piecewise expanding maps. Our approach is based on a new method of studying the absolute continuity of foliations, which provides new information that could prove useful in treating hyperbolic systems with singularities.