  On a smooth compact Riemannian manifold without boundary, we construct a
finite dimensional cohomological complex of currents that are invariant by an
Axiom A flow verifying Smale's transversality assumptions. The cohomology of
that complex is isomorphic to the De Rham cohomology via certain spectral
projectors. This construction is achieved by defining anisotropic Sobolev
spaces adapted to the global dynamics of Axiom A flows. In the particular case
of Morse-Smale gradient flows, this complex coincides with the classical Morse
complex.
