  We study the motion of a particle on the two-dimensional honeycomb lattice,
whose sites are occupied by either flipping rotators or flipping mirrors, which
scatter the particle according to a deterministic rule. For both types of
scatterers we find a new type of motion that has not been observed in a Lorentz
Lattice gas, where the particle's trajectory is a self-avoiding walk between
returns to its initial position. We show that this behavior is a consequence of
the deterministic scattering rule and the particular class of initial scatterer
configurations we consider. Since self-avoiding walks are one of the main tools
used to model the growth of crystals and polymers, the particle's motion in
this class of systems is potentially important for the study of these
processes.
