We analyze the dynamics of a valence electron of the buckminsterfullerene
molecule (C60) subjected to a circularly polarized laser field by modeling it
with the motion of a classical particle in an annular billiard. We show that
the phase space of the billiard model gives rise to three distinct
trajectories: "Whispering gallery orbits", which only hit the outer billiard
wall, "daisy orbits" which hit both billiard walls (while rotating solely
clockwise or counterclockwise for all time), and orbits which only visit the
downfield part of the billiard, as measured relative to the laser term. These
trajectories, in general, maintain their distinct features, even as intensity
is increased from 10^10 to 10^14 W*cm^-2. We attribute this robust separation
of phase space to the existence of twistless tori
