  A scattering process can be described by suitably closing the system and
considering the first return map from the entrance onto itself. This scattering
map may be singular and discontinuous, but it will be measure preserving as a
consequence of the recurrence theorem applied to any region of a simpler map.
In the case of a billiard this is the Birkhoff map. The semiclassical
quantization of the Birkhoff map can be subdivided into an entrance and a
repeller. The construction of a scattering operator then follows in exact
analogy to the classical process. Generically, the approximate unitarity of the
semiclassical Birkhoff map is inherited by the S-matrix, even for highly
resonant scattering where direct quantization of the scattering map breaks
down.
