  We analyze a model quantum dynamical system subjected to periodic interaction
with an environment, which can describe quantum measurements. Under the
condition of strong classical chaos and strong decoherence due to large
coupling with the measurement device, the spectra of the evolution operator
exhibit an universal behavior. A generic spectrum consists of a single
eigenvalue equal to unity, which corresponds to the invariant state of the
system, while all other eigenvalues are contained in a disk in the complex
plane. Its radius depends on the number of the Kraus measurement operators, and
determines the speed with which an arbitrary initial state converges to the
unique invariant state. These spectral properties are characteristic of an
ensemble of random quantum maps, which in turn can be described by an ensemble
of real random Ginibre matrices. This will be proven in the limit of large
dimension.
