  We consider the static wall approximation to the dynamics of a particle
bouncing on a periodically oscillating infinitely heavy plate while subject to
a potential force. We assume the case of a potential given by a power of the
particle's height and sinusoidal motions of the plate. We find that for powers
smaller than 1 the set of escaping orbits has full Hausdorff dimension for all
motions and obtain existence of elliptic island of period 2 for arbitrarily
high energies for a full-measure set of motions. Moreover we obtain conditions
on the potential to ensure that the total (Lebesgue) measure of elliptic
islands of period 2 is either finite or infinite.
