  We show that the time-dependence of electromagnetic field in a parametrically
modulated cavity can be effectively analyzed using a $Floquet$ $map$. The map
relates the field states separated by one period of the drive; iterative
application of the map allows to determine field configuration after arbitrary
number of drive periods. For resonant and near-resonant drives, the map has
stable and unstable fixed points, which are the loci of infinite energy
concentration in the long time limit. The Floquet map method can be applied
both to classical and quantum massless field problems, including the dynamical
Casimir effect. The stroboscopic time evolution implemented by the map can be
interpreted in terms of the wave propagation in a curved space, with the fixed
points of the map corresponding to the black hole and white hole horizons. More
practically, the map can be used to design protocols for signal
compression/decompression, cooling, and sensing.
