**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|.