  Spectral properties of evolution operators corresponding to random maps and
quantized chaotic systems strongly interacting with an environment can be
described by the ensemble of non-hermitian random matrices from the real
Ginibre ensemble. We analyze evolution operators Psi=Psi_s...Psi_1 representing
the composition of s random maps and demonstrate that their complex eigenvalues
are asymptotically described by the law of Burda et al. obtained for a product
of s independent random complex Ginibre matrices. Numerical data support the
conjecture that the same results are applicable to characterize the
distribution of eigenvalues of the s-th power of a random Ginibre matrix.
Squared singular values of Psi are shown to be described by the Fuss-Catalan
distribution of order s. Results obtained for products of random Ginibre
matrices are also capable to describe the s-step evolution operator for a model
deterministic dynamical system - a generalized quantum baker map subjected to
strong interaction with an environment.
