  The main objective of this paper is understanding the propagation laws obeyed
by high-frequency limits of Wigner distributions associated to solutions to the
Schroedinger equation on the standard d-dimensional torus T^{d}. From the point
of view of semiclassical analysis, our setting corresponds to performing the
semiclassical limit at times of order 1/h, as the characteristic wave-length h
of the initial data tends to zero. It turns out that, in spite that for fixed h
every Wigner distribution satisfies a Liouville equation, their limits are no
longer uniquely determined by those of the Wigner distributions of the initial
data. We characterize them in terms of a new object, the resonant Wigner
distribution, which describes high-frequency effects associated to the fraction
of the energy of the sequence of initial data that concentrates around the set
of resonant frequencies in phase-space T^{*}T^{d}. This construction is related
to that of the so-called two-microlocal semiclassical measures. We prove that
any limit \mu of the Wigner distributions corresponding to solutions to the
Schroedinger equation on the torus is completely determined by the limits of
both the Wigner distribution and the resonant Wigner distribution of the
initial data; moreover, \mu follows a propagation law described by a family of
density-matrix Schroedinger equations on the periodic geodesics of T^{d}.
Finally, we present some connections with the study of the dispersive behavior
of the Schroedinger flow (in particular, with Strichartz estimates). Among
these, we show that the limits of sequences of position densities of solutions
to the Schroedinger equation on T^2 are absolutely continuous with respect to
the Lebesgue measure.
