  On a smooth, compact and oriented manifold without boundary, we give a
complete description of the correlation function of a Morse-Smale gradient flow
satisfying a certain nonresonance assumption. This is done by analyzing
precisely the spectrum of the generator of such a flow acting on certain
anisotropic spaces of currents. In particular, we prove that this dynamical
spectrum is given by linear combinations with integer coefficients of the
Lyapunov exponents at the critical points of the Morse function. Via this
spectral analysis and in analogy with Hodge-de Rham theory, we give an
interpretation of the Morse complex as the image of the de Rham complex under
the spectral projector on the kernel of the generator of the flow. This allows
us to recover classical results from differential topology such as the Morse
inequalities and Poincar{\'e} duality.
