We study the asymptotic behaviour of open quantum maps and relate the decays to the eigenvalues of a coarse grained completely positive superoperator. In specific ranges of coarse graining, and for chaotic maps, these decay rates are given by the Ruelle-Pollicott resonances of the classical map. Application to the entropy growth and the decay of the Loschnidt echo will be presented. ( nlin.CD/0301025)