  Dynamical systems on an infinite translation surface with the lattice
property are studied. The geodesic flow on this surface is found to be
recurrent in all but countably many rational directions. Hyperbolic elements of
the affine automorphism group are found to be nonrecurrent, and a precise
formula regarding their action on cylinders is proven. A deformation of the
surface in the space of translation surfaces is found, which "behaves nicely"
with the geodesic flow and action of the affine automorphism group.
