  For a piecewise linear version of the periodic map with anomalous diffusion,
the evolution of statistical averages of a class of observables with respect to
piecewise constant initial densities is investigated and generalized
eigenfunctions of the Frobenius-Perron operator are explicitly derived. The
evolution of the averages is controlled by real eigenvalues as well as
continuous spectra terminating at the unit circle. Appropriate scaling limits
are shown to give a normal diffusion if the reduced map is in the stationary
regime with normal fluctuations, a L\'evy flight if the reduced map is in the
stationary regime with L\'evy-type fluctuations and a transport of ballistic
type if the reduced map is in the non-stationary regime.
