  We describe the theory of the dynamics of atoms in two-dimensional
quasicrystalline optical lattices. We focus on a regime of shallow lattice
depths under which the applied force can cause Landau-Zener tunneling past a
dense hierarchy of gaps in the quasiperiodic energy spectrum. We derive
conditions on the external force that allow for a "semiadiabatic" regime in
which semiclassical equations of motion can apply, leading to Bloch
oscillations between the edges of a pseudo-Brillouin-zone. We verify this
semiclassical theory by comparing to the results of an exact numerical
solution. Interesting features appear in the semiclassical dynamics for the
quasicrystal for a particle driven in a cyclic trajectory around the corner of
the pseudo-Brillouin-zone: The particle fails to return to its initial state,
providing a realization of a "spiral holonomy" in the dynamics. We show that
there can appear anomalous velocity contibutions, associated with nonzero Berry
curvature. We relate these to the Berry phase associated with the spiral
holonomy, and show how the Berry curvature can be accessed from the
semiclassical dynamics. Finally, by identifying the pseudo-Brillouin-zone as a
higher genus surface, we show that the Chern number classification for periodic
systems can be extended to a quasicrystal, thereby determining a topological
index for the system.
