  We find a normal form which describes the high energy dynamics of a class of
piecewise smooth Fermi-Ulam ping pong models; depending on the value of a
single real parameter, the dynamics can be either hyperbolic or elliptic. In
the first case we prove that the set of orbits undergoing Fermi acceleration
has zero measure but full Hausdorff dimension. We also show that for almost
every orbit the energy eventually falls below a fixed threshold. In the second
case we prove that, generically, we have stable periodic orbits for arbitrarily
high energies, and that the set of Fermi accelerating orbits may have infinite
measure.
