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By analyzing a paradigmatic example of the theory of dissipative systems --the classical and quantum dissipative standard map -- we are able to explainthe main features of the decay to the quantum equilibrium state. The classicalisoperiodic stable structures typically present in the parameter space of thesekind of systems play a fundamental role. In fact, we have found that the periodof stable structures that are near in this space determines the phase of theleading eigenstates of the corresponding quantum superoperator. Moreover, theeigenvectors show a strong localization on the corresponding periodic orbits(limit cycles). We show that this sort of scarring phenomenon (an establishedproperty of Hamiltonian and projectively open systems) is present in thedissipative case and it is of extreme simplicity.

Classical properties of the leading eigenstates of quantum dissipative systems