  A single spin flip stochastic energy conserving dynamics for the XY model is
considered. We study the ergodicity properties of the dynamics. It is shown
that phase space trajectories densely fill the geometrically connected parts of
the energy surface. We also show that while the dynamics is discrete and the
phase point jumps around, it cannot make transitions between closed
disconnected parts of the energy surface. Thus the number of distinct sectors
depends on the number of geometrically disconnected parts of the energy
surface. Information on the connectivity of the surfaces is obtained by
studying the critical points of the energy function. We study in detail the
case of two spins and find that the number of sectors can be either one or two,
depending on the external fields and the energy. For a periodic lattice in $d$
dimensions, we find regions in phase space where the dynamics is non-ergodic
and obtain a lower bound on the number of disconnected sectors. We provide some
numerical evidence which suggests that such regions might be of small measure
so that the dynamics is effectively ergodic.
