Abstract: We analyze spectral properties of quantum stochastic maps and formulate a quantum analogue of the Frobenius-Perron theorem concerning spectral properties of stochastic matrices. Under assumption of strong chaos in the corresponding classical system and a strong decoherence  (i.e. strong coupling with an M-dimensional environment) the spectrum of a quantum map  Phi displays a universal behaviour: it  contains the leading eigenvalue \lambda_1 = 1, while all other eigenvalues are restricted to the disk of radius R = M^{-1/2}. Sequential action of the map Phi brings all pure states  exponentially fast to the invariant state, while the convergence rate is determined by modulus of the subleading eigenvalue, R = |\lambda_2|.