  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 proven 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.
