  We prove that the twisted De Rham cohomology of a flat vector bundleover some
smooth manifold is isomorphic to the cohomology of invariant
Pollicott--Ruelleresonant states associated with Anosov and Morse--Smale flows.
As a consequence, weobtain generalized Morse inequalities for such flows. In
the case of Morse--Smale flows,we relate the resonances lying on the imaginary
axis with the twisted Fuller measuresused by Fried in his work on Reidemeister
torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that
the Reidemeister torsion can be recovered from the onlyknowledge of dynamical
resonances on the imaginary axis by expressing the torsion as azeta regularized
infinite product of these resonances.
