  An open question in the study of dilation surfaces is to determine the
typical dynamical behavior of the directional flow on a fixed dilation surface.
We show that on any one-holed dilation torus, in all but a measure zero Cantor
set of directions, the directional flow has an attracting periodic orbit, is
minimal, or is completely periodic. We further show that for directions in this
Cantor set, the directional flow is attracted to either a saddle connection or
a lamination on the surface that is locally the product of a measure zero
Cantor set and an interval.
