  We investigate Ruelle-Pollicott resonances of Anosov diffeomorphisms with
respect to the measure of maximal entropy. We highlight a profound connection
between resonances and eigenvalues of the action induced by the dynamics on de
Rham cohomology. We finally exploit this relation to get information about the
Ruelle-Pollicott asymptotics of the correlation function and establish a
cohomological bound for the speed of mixing of Anosov diffeomorphisms.
