  We study the problem of the behavior of a quantum massless scalar field in
the space between two parallel infinite perfectly conducting plates, one of
them stationary, the other moving periodically. We reformulate the physical
problem into a problem about the asymptotic behavior of the iterates of a map
of the circle, and then apply results from theory of dynamical systems to study
the properties of the map. Many of the general mathematical properties of maps
of the circle translate into properties of the field in the cavity. For
example, we give a complete classification of the possible resonances in the
system, and show that small enough perturbations do not destroy the resonances.
We use some mathematical identities to give transparent physical interpretation
of the processes of creation and amplification of the quantum field due to the
motion of the boundary and to elucidate the similarities and the differences
between the classical and quantum fields in domains with moving boundaries.
