  We compare the quantum and classical properties of the (Quantum) Isoperiodic
Stable Structures -- (Q)ISSs -- which organize the parameter space of a
paradigmatic dissipative ratchet model, i.e. the dissipative modified kicked
rotator. We study the spectral behavior of the corresponding classical
Perron-Frobenius operators with thermal noise and the quantum superoperators
without it for small $\hbar_{\rm eff}$ values. We find a remarkable similarity
between the classical and quantum spectra. This finding significantly extends
previous results -- obtained for the mean currents and asymptotic distributions
only -- and on the other hand unveils a classical to quantum correspondence
mechanism where the classical noise is qualitatively different from the quantum
one. This is crucial not only for simple attractors but also for chaotic ones,
where just analyzing the asymptotic distribution reveals insufficient.
Moreover, we provide with a detailed characterization of relevant eigenvectors
by means of the corresponding Weyl-Wigner distributions, in order to better
identify similarities and differences. Finally, this model being generic, it
allows us to conjecture that this classical to quantum correspondence mechanism
is a universal feature of dissipative systems.
